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hi everybody  um  glad to have you all back  for our next to last installment of  theory of computation  today um we are going to embark on the  very last  big topic for the semester  and that is in some ways going to be  following on what we  started  a couple of lectures back when we looked  at probabilistic  turing machines  and probabilistic computation  and its associated class bpp  now what we're going to  discuss  is in some sense  a probabilistic version of np  and that's going to be a complexity  class called ip  which stands for interactive proof  systems  um and so we're going to present that  model  and look at a couple of examples  uh  i would  just like to say at the beginning that  the  this model is a very important one it's  really uh has been the starting point  for a great deal of  research  in complexity theory so we're just  really going to be touching on it but  there's a lot more that people have  pursued with this model and it's also a  connection in to the cryptography field  which also makes use of the interactive  proof system model in fact  some of the  genesis of that model comes out of  cryptography  where you're having uh  multiple parties either communicating or  in some ways  interacting  uh to achieve certain uh goals of  communication or  signing or passwords or or or what have  you so uh this is a both a an applied  area and also one that has a lot of very  interesting theory uh um associated to  it  so with that one we we're going to jump  in um and uh  start out by uh  making myself smaller  and  just do a an introduction i'm going to  introduce  the model or the the concept of an  interactive proof  with an example  and that example involves the  graph isomorphism problem  that's the problem of testing whether  two graphs are isomorphic  what we mean by two graphs being  isomorphic is that they're really just  the same graph  with one of them perhaps being  re-labeled or permuted  so that  they may look superficially different  they may appear with a different  sequence of labels or the nodes are  appearing in a different order  but  except for that it's really just the  same graph so i'm kind of illustrating  that here if you can see those two  graphs here which look different from  each other both on eight notes they are  in fact the same graph  as i can illustrate by a little  animation which will convert this one  into that one  so uh  the  um  [Music]  so the  two graphs um  these graphs being the same um they're  we call that isomorphic  so these are graphs g and h and they're  really the same graph so we we um call  them isomorphic graphs  and we  are  have an associated computational problem  called iso  which is  given a pair of graphs we'd like to know  are they isomorphic or not so the iso is  the collection of pairs of graphs which  are isomorphic  and  it's easy  to see that this problem is an np  problem  because  all you need to do in order to see or to  give a certificate that the two graphs  are isomorphic to each other is tell you  it's just to say which nodes in the one  graph correspond to which other nodes in  the other graph  um and then you all you need to check is  that the edge relationships are  consistent with that  mapping or that isomorphism as it's  called  um  so it's easy to see that the iso problem  is an np  and if you're not getting that make sure  you uh understand because this is the  whole first part of the lecture will be  lost if you don't understand this iso  problem  now the question of whether  you can test two graphs  being isomorphic in polynomial time  is not clear and in fact that's an  unsolved problem to this day  uh  and it's a problem that has generated an  enormous literature uh  there are hundreds of papers  on the graph isomorphism problem as it's  called um to try to uh resolve  um you know uh to try to see if one can  find a polynomial time algorithm and in  fact it was a very big result  just in the last 10 years where there  was a sub-exponential algorithm given so  that was more  than faster than the brute force search  approach but didn't get it all the way  down to polynomial now why is there so  much attention just to this one  particular np problem it's because  it's not known whether the graph  isomorphism problem is np-complete iso  is not known to be an np-complete  problem and that puts it into a very  very small class of problems in np  which are no not known to be either in p  or np-complete  it's kind of a curiosity that for np  problems  almost all of them have ended up being  in one side or the other  and in fact  um it's a  uh  um  in fact the  the um  i i think it's the only problem that  just involves graphs  that's not known to be either in p or an  np  um  so so i got a question here  what would be in between exponential and  polynomial for example i don't remember  what the the bound is but it's it's  something in the uh  in the range of n to the log n  uh  time complexity for the graphite some  more i might be getting that wrong um i  don't remember exactly what the bound is  but um  that's significantly better than two to  the n or some  some exponential amount of time but it  it's more than enter any constant so  it's more than any polynomial time  so um  another question uh of the same sort is  whether the complementary problem is in  np or whether whether iso is in co-np or  let's let's talk about it in terms of  the complement whether the  complement of iso which i'll refer to as  the non-iso problem whether that's known  to be in np so that's also not known um  in other words if i give you two graphs  and i ask you to show that they're not  isomorphic suppose they aren't  isomorphic and you go through the effort  of you know determining that by a brute  force search and now you want to  prove  that they're not isomorphic well that's  not  it's not known to be an np either so  there's no known  short certificate certificate of  two graphs not being isomorphic  we don't know how to do that either  but there's something that's very  interesting nevertheless  um  and  it has to do with the ability to for one  party to prove to another  that graphs are either isomorphic or not  isomorphic  so if you're just having like a prover  we haven't really been necessarily  formulating that this way so much in  this class but this is a completely  equivalent way of formulating the notion  of np whether you have a polynomial time  verifier an approver who can produce  certificates say it's a powerful prover  so um if you have a problem that's in np  approver can convince a polynomial time  verifier  that uh strings are in the language if  in fact they are so in the case of the  iso problem a prover can convince a  polynomial time verifier the graphs are  isomorphic just by exhibiting the  isomorphism  now  for the non-isomorphism case  we don't know that that problem's in np  but it's still possible  for approver to convince a verifier that  graphs are not isomorphic  if you change the rules of the game  slightly  um  so even though the non-iso problem  is not known to be an np  approver can still convince a polynomial  time verifier that graphs are not  isomorphic assuming they they are in  fact not isomorphic um provided  that  um  the prover and the verifier can interact  with one another so the verifier can ask  questions of the prover  and the verifier gets to be  probabilistic  so that's in this  in that's in sense in which i mean  that this notion is a kind of a  probabilistic version of np  okay so um  let me  show you how that's done  so before we jump in to the  uh to the method for  approver to um show a verifier that  graphs are not isomorphic  um let me let let's try to get a little  clearer on the model so i'm going to  first show it to you informally and then  we'll look at it formally  okay so  in in interactive proofs there are two  parties  um  and i'm going to think about them as one  of them is going to be the professor  okay so the professor is going to play  the role of the verifier in a sense but  it's like that the one who checks  um  and uh the professor being kind of old  and tired he's been teaching too long  maybe  can only operate in  probabilistic polynomial time  so the professor if wants to tell  whether two graphs are isomorphic or not  probabilistic polynomial time doesn't  seem to be enough  to tell whether two graphs are  isomorphic or not because it seems to be  a more than polynomial problem  however  the professor  has  um  help  it has  an army  of  graduate students  and the graduate students  they're not limited uh in the same way  the professor is the graduate students  are young  they  are energetic they can stay up all night  they know how to code  so  the graduate students  have unlimited computational ability  so that we're going to think of the  graduate students playing the role of  the approver because they're not  they're not limited in their  capabilities we'll assume  the professor on the other hand is  limited so the professor wants to know  if the two graphs are isomorphic  let's say whatever they are  um  can't do it by himself so he's going to  ask his students to figure out the  answer  and report back  now there's only one problem  the professor knows that students  uh  well in the old days they'd like to  party i guess these days they like to  play on uh  play computer games a lot  and so they're not really that eager to  spend all their time  figuring out whether graphs are  isomorphic  so he's worried that the  the students will just come up with some  answer and figure that he won't be able  to tell the difference  so the professor does not trust the  students it's not enough to he for the  professor to give the problem to the  students and just take any answer that  they're going to give the professor  wants  wants to be convinced  okay so  um  now  how could the students convince the  professor of the answer that they've  really done the work and figured out  whether the graphs are isomorphic or not  well if the graphs are isomorphic if it  turns out that the graphs were  isomorphic and the students figure that  out  then life is good  because what are they going to do to  convince the professor  they're going to hand over the  isomorphism and show yeah i mean they  are  you know those graphs really are  isomorphic and here's how the  correspondence works professor can check  oh yeah i i now not now i'm convinced  but suppose the graphs were not  isomorphic  what are we going to do then  um the students have figured out where  after night else the professor wants  wants to be convinced oh no what are we  going to do  well in fact we're going to engage the  the professor and the students are going  to engage in the following protocol  dialogue  what's going to happen is  now you have to make sure you're you're  this is critical to follow to understand  this little  part of the story here because it's  really going to set the pattern for  everything in today's and tomorrow and  to today's lecture and the and the next  lecture  okay so  we're going to engage in a following  interaction between the students and the  professor which is going to enable the  students to convince the professor that  the two graphs really are not isomorphic  so how is that going to work  this is a beautiful little uh thing by  the way  so the professor is going to take the  two graphs  and pick one of them  at random  because the two graphs g and h  um let's say they're not they really are  not isomorphic the professor doesn't  know that for sure that's what the  students claim the professor really  wants to not be convinced that the  students are right  um so the professor's gonna pick one of  the two at random  randomly permute that  uh that choice the one that he picked  and hand it over to the students  say okay  here is one of those two graphs randomly  scrambled  then i'm going to ask the students  which one did i pick  okay now  if the graphs were really not isomorphic  the students can check  whether that randomly scrambled graph  is isomorphic to either g or to h it's  going to be isomorphic to one or the  other  and then they students can figure it out  and they say oh you picked g or no you  picked h  as the case may be the students can  figure that out  but if the graphs were isomorphic  then that  scrambled version of g or h could  equally well have come from either of  them  and the students would have no way of  knowing  which one the professor picked  so that there's nothing they could do  which would be better than guessing  so if we do that a bunch of times the  professor picks at random sometimes  secretly of course the picks the grip  picks either g or picks h  and the students get it right every time  either the students are really doing the  work  and the graphs are really not isomorphic  or  the students are just incredibly lucky  they're managing to guess right let's  say a hundred times  so how the  the the professor randomly and secretly  picks grh uses this uses its probabilism  flips a coin  just a two-sided coin and says okay  sometimes if we're going to do g  sometimes they're going to do h just  completely at random  picks one or the other  and then with some more randomness  gets finds a random permutation of the  one that he picked and then sends that  over to the students and say which one  did it come from  um  i'm not sure okay so let's pause here  let's let's make sure we all understand  this because this is really important um  so i'm getting a question here how do we  i'm not sure what your question is um  okay so let me just say yeah the  professor's going to play the role the  verifier the graduate students play  they're all approver that's coming but i  really want to understand this protocol  here okay so how is the professor  picking the graph skin if you're  okay i don't you know picking the graphs  at random you have just two graphs  they're in part of the input uh the both  the students and the professor can see  the graphs  and the professors are just picking one  of them at random using a coin so i'm  not sure i understand the question there  could p and v engage in a protocol where  the secretary is on the prover side  instead  the question of revealing the  isomorphism i there is no why so i'm not  sure i understand this question either  um maybe we'll make this clear  you know if  for  for this little illustration  the professor  doesn't know the graphs could be  isomorphic or they could be not  isomorphic  and so uh the professor wants to be  convinced either way whatever the  students whatever answer the students  come up with we're going to shift this  into a problem about a um deciding a  language  next but right now i'm just trying to  give a sense of the how the model works  i want to move from this informal model  and now i'm going to formalize that in  terms of  model which will be deciding a language  okay so  so the interactive proof  system  model  we have two interacting parties a  verifier which is probabilistic  polynomial time playing played by the  professor in the previous slide  and the prover which is unlimited  computational power played by the  students in the previous slide  both of them get to see the input  which in the previous case  well  it could be for example the pair of  graphs  they exchange a polynomial number of  polynomial size messages so the whole  exchange  including the verifier's own computation  is going to be polynomial the only thing  that's not not not included within the  computational cost is the prover's work  which is unlimited  um  after that the verifier after the  interaction the verifier will accept or  reject  and we're going to define the  probability that the verifier together  with a particular approver ends up  accepting  as  you  look over the different possible coin  tosses  of the verifier  which could lead to different  behavior on the part of the verifier and  therefore different behavior on the part  of the approver  so over all the different possibility  possibilities for the verifiers  computation  we're going to look at the probability  that the verifier with this particular  approver ends up accepting and i've  written it this way this is the  probability of the verifier interacting  with the prover accepts the input  is just simply that  um and so we're going to work through an  example  we're going to  work through the previous example  more  precisely in a second  the class ip for interactive proofs  stands for it's the class of languages  such that for some verifier and approver  um  for strings in the language  the prover makes the verifier accept  with high probability  and here's the interesting part  for strings not in the language the  prover makes it except with low  probability but every there's no prover  which can make it except with high  probability  so there's no way to cheat  if you think about it in the case of the  graphic non-isomorphism  there's nothing you know  if  if the graphs were really isomorphic  and the students were trying to in a  devious way  prove  through that protocol that they're not  isomorphic they would fail because  there's nothing they can do  if the graphs were isomorphic  then um  when the  verifier the the professor picks one or  the other at random  um and scrambles it the students would  have no way of telling which one the  professor did so no matter what kind of  scheme they try to come up with  they're going to be out of luck  so  it's no mat for any strategy  for strings that are not in the language  for any s any prover calling that p with  a tilde to stand for a devious or  crooked  prover  for any uh possibly crooked prover  even that with working with the verifier  is still going to end up accepting with  low probability  so strings in the language there's going  to be an honest prover who just follows  the protocol  in the correct way which makes the  verifier accept with high probability  for strings not in the language every  prover  is going to fail  to make it accept with high probability  um  okay so that i mean the way i like to  think about it is that p tilde is  a possibly crooked proverb which is  trying to make the verifier accept when  it shouldn't because the string is not  in the language  it's like you know  it's like even you can think of this in  the case of um satisfiability  um  you know  you  a crooked prover might try to convince  of the verifier that the formula is  satisfiable when it isn't by by somehow  trying to produce a satisfying  assignment but that's going to be  impossible there's nothing any strategy  can possibly work when the formula is  not satisfiable if that's what the  verifier is going to check it's going to  be looking for that satisfying  assignment  okay  and by the way this is we're not going  to prove this but it's really going to  be proved in the same way you can make  that one third error that could that  occurs here  something very tiny  by  the same kind of repetition argument  okay  so let's see um  so why can't the prover in the first  case be crooked um  the prover in the first case would could  be crooked but that's not going to serve  the purposes um  you know what what we want to show  um  you think about it like we think about  np  for strings in the language there exists  a certificate there is a proof  that you're in the language  so if somebody is going to  not produce the proof  that's irrelevant  the question is if you look at the best  possible case the best possible prover  um  you know who's going to be able  we're asking does there exist a way to  convince the verifier that the um  string is in the language  so it doesn't matter that there might be  some other  uh  silly way that doesn't work we're just  looking at the best possible way  so the best possible way when you're in  the language is going to end up with a  verifier having high probability  when you're not in the language the best  possible way is still going to end up  with low probability  when when i talk about best possible i'm  trying to maximize the probability that  the verifier is going to end up  accepting let's continue um not sure as  clear as i would like but  um  maybe again we're going to we're going  to stick with that example because this  is a very uh  helpful example and to try to  understand the setup  and uh so we're gonna i'm gonna revisit  that previous example about  non-isomorphism but now in the context  of this  thinking about as a language  so we're going to take this  non-isomorphism  um  uh yeah  we're going to take the non-isomorphism  problem  and show that it's an ip so there's  going to be a verifier together with  approver which are going to make the  verifier accept with high probability  for strings in the language namely  graphs not ice being isomorphic  and  nothing there's going to be no way to  make the verifier except with high  probability for strings out of the  language therefore that's when the  graphs are isomorphic  okay um so the protocol is just gonna  we're gonna repeat the following thing  twice  you know i said in the previous case do  it a hundred times just to  help us think about it but actually  twice is gonna be enough to get the  bound we need  so the verifier is going to operate like  this in terms of this is the verifiers  in first  communicating  sending messages to the approver it's  going to randomly choose grh  just like what the professor did last  time  randomly permute the result to get a new  graph k  which was going to be which was  which is isomorphic either to grh  depending upon the choice the verifier  made  and then send that graph k  now the provers turn is going to respond  by the proof is going to compare k with  the two one of the both of the original  graphs it's got to be isomorphic to one  or the other  and it's going to report back which one  just going to say well you pick g no or  you picked h  because the prover with its unlimited  capabilities can determine that  um  and then v accepts if the approval was  right both times  um  and if the approval was ever not right  the verify says oh something's fishy  here because we know that the prover has  unlimited capability so could get it  right if you know um if there was if  this was an honest approver um and so um  if uh  if it's not getting it right then the  verify is going to reject  so if the graphs are not isomorphic the  proofer can tell which one it picked  randomly so therefore  if the graphs are not isomorphic the  verifier with that that honest prover  will accept with probability one  because that honest proof is always  going to get the right answer  which is at least two-thirds is what the  bound we need  uh we don't care about the space used in  answer to a question  um if  we were not in the language so g and i h  are not isomorphic  then there's nothing any crooked pervert  could possibly do  because it gets a graph  can't tell there's no way to tell  whether it came from g or came from h  um  so that crooked proofer would have all  it could the best thing you could do is  guess sort of a 50 chance advancing  correctly each time and only a 25 chance  for doing it twice  and that's why i did it twice in order  to get that error um  uh  to be small so there's only a 25 chance  of the approver getting lucky so that  would be an  error case  if the proof of just by chance picked  the right answer twice even though the  graphs were isomorphic so therefore  for the isomorphic case  the verifier interacting with  any prover is going to accept that input  with at most one quarter 25 of the time  which is less than a third so those are  that's just to achieve that bound  okay  so  let's let's answer some questions first  and then i'll try to um  uh  i'll ask you  you understand this so this it's i think  it's worth trying to understand this  model um of this interactive proof  system  it's a little little slippery i i  realize but um  if you just hold your hold on to your  intuition of the prover trying to  convince  you know of uh  you know a powerful prover trying to  convince a limited verifier  um  of  some  string being in a language  uh  you want the proven to be able to  succeed when the string is in the  language but fail when the string is not  in the language  yes we are going to somebody's asking if  the prover is identifying grh by brute  force yes the prover is going to use its  unlimited capabilities to determine  given k  whether it came from g or h  the um  the computational  cost of the approver is irrelevant for  this it's just like when we think about  a certificate  um you know for satisfiability we don't  talk about the cost of finding that  certificate  uh for np  for ip again we don't talk about the  cost of the prover running  so somebody's asking does the crooked  per prover answer just randomly or does  uh can the cr quick approver has have a  strategy the crooked perimeter can have  a strategy  now we're not we're assuming the crooked  approver is devious  but it's still going to fail  okay um let's do the check-in  suppose we change the model  so that the prover can watch the  verifier picking its random choices  so uh the verifier cannot act in secret  anymore but the the prover can watch the  verifier  now let's suppose we had the same  protocol that i just described  what language do we end up with is it  the same language different language and  what is that language  okay so  i want to hopefully  um  let's let's  it'll  give me some sense of how well you're  following me by how well this uh this  goes  yeah someone's asking about how this  connects up for example with np so we're  going to look at that  also in a second  okay so this is reassuring that most of  you i think are on the right track at  least for this check-in uh do we assume  p uses this access to guess right  what access p is not really guessing the  p is tr is actually i don't think a p is  non-deterministic or anything like that  p is actually trying to get the right  answer  and using its  computational ability to do that if it's  possible  may not be possible then there's nothing  you can do  okay so let's uh end this are you all in  two seconds left  please vote  vote now or never  okay ending  um  yeah so c is the correct answer here  if the prover can watch what the verify  is doing the prover can see  what graph the verifier picked right  from the beginning  and so the prover without having to do  any work  can say you know it proves looks over  the verifier's shoulder and says oh you  picked g  and now you're randomly permitting it  but i don't care about that i just i  know you pick g  uh so uh the  proof is going to respond back uh g even  if the graphs were isomorphic the proof  is going to be able to get the right  answer  kind of interestingly um  uh  you can make a you can change the  protocol somewhat um to make it uh that  even if the  prover  has access to the verifier's randomness  you can still achieve this but not with  the same protocol  um  so that's uh that's a separate question  um okay so let's move on here i want to  get too bogged down  okay here's another check-in um  uh  okay so  you have to tell me which of the  following  statements are true  as far as you know  now you have to think about a little bit  how this  these uh  relate  to um how np and ip or bpp  and ip relate to one another  okay  how are we doing on this  okay so we're gonna have to close this  pretty soon too  do the best you can  interesting okay  closing up shop  last last vote  okay one two three  there's one more person out there who  hasn't voted who voted last time oh well  uh  all right  in fact they're all true  um  let's see why is np contained with ip  contained in ip uh well many of you have  seen this already so let's just quickly  go through it um  [Music]  if we just had a deterministic v  um  you know the uh  you can  uh  maybe it's just  that that can be enough of deterministic  v  i think it's just going to be equivalent  but actually just to be doubly sure the  deterministic v and the approver just  sends a message to the verifier and then  checks it  that's the way we normally think about a  certificate for np  uh i don't think it's going to change  anything but should double check that if  the verifier can still ask questions but  i think as long as the verifier is  deterministic you're going to get  exactly np  here  um  and um  now how about bpp  well  there you don't even need approver  because the verifier is already  probabilistic  so  verifier can ignore the approver  and this one is a little tricky uh i p  containing p space because we haven't  covered that so there's no way for you  to know that unless you happen to read  ahead in the book but it's in fact true  um in some ways it's a little bit like  the proof that  uh  np  is contained in p space ip is sort of an  enhanced version of np  and you and there's just a basically a  uh a piece based brute force algorithm  that goes through the entire tree of  possibilities um of the verifier  exchange  verifier with exchanges with approver  and can uh determine that um the  verifier is either going to accept for  some approver or is going to end up  projecting for every approver um  so we're not going to prove this  statement but something good for you to  know anyway they're just a fact  uh but we're going to do what the  surprising thing  in reference to part c  is that the containment also goes the  other way this is the amazing  um uh  was is an amazing result  that  everything in p space  you can do with in ip  so this is ip is actually turns out to  be incredibly powerful gives you  everything up in p space you get i p  equals p space  so that says that any problem that you  can solve in p space like any of the a  game for example  um if you can imagine you know  formulating you know checkers or  chess  as a piece based problem which you know  depending upon some details of the rules  you can do because you know you have to  generalize it to an end by end board but  okay  let's not quibble um  uh then  uh  um  [Music]  we don't know which side has a forced  win in chess  um and even if somebody goes through the  effort of going through the game tree  um and determines that let's say white  has a forced win  uh there's no way for them to there's no  short certificate we don't know that  that problem is not an np  but  by  going through an interactive proof  an all-powerful prover  could still convince somebody that white  had a force  you know convince somebody in polynomial  time that a  white has a forced wind let's say in um  in chess  again little  uh stretching things because this is  you know you really need to talk about  this as an n by n not an eight by eight  but i think in this the spirit is is uh  fair  so um  okay uh so let's continue  so wha  when we're not going to quite prove that  that p base  is contained in ip we're going to prove  a somewhat weaker statement but very  similar  um  is that uh and historically came first  that co-np is contained in ip  so not only is np contained in ip but  we're going to prove that co np is  contained in ip and this actually has  most of the i most of the idea for the p  space being contained in ip and itself  it's just an amazing proof  a little easier  um  okay  and th this was done  if i'm remembering somebody's asking me  when how old is this it's something in  the 19 i think late 90s but i'm not i  don't remember but maybe early 90s i  think it's late 90s  when this was shown  so it's been a while now  um  okay  so  um  yeah so in terms of the relationship  with cryptography there were two  parallel threads  um that both independently came up with  the notion of an interactive purge  system  uh i was a little bit personally  involved with this  in in a way as well but but mainly that  the the there was one group in  cryptography  um working on this and there was another  group who was actually coming out of the  graph isomorphism world  working on it and they they came up with  two separate models one involving the  private randomness and one involving the  public randomness um and it was turned  out that was  that they're actually equivalent  um  and uh  it's an interesting story but  unfortunately we don't have time for it  uh so  why don't we move on and i'm gonna start  showing you how the proof that  co-np  is contained in ip  goes and what we're going to do is work  with a problem that's  almost like co p complete but um  going to be uh well it's going to be  this number sat problem we'll see the  connection with co-np in a second  uh  so cohen p  so it's supposed to be exactly k  satisfying assignments  fee comma k  is the set of  pairs where the formula fee has exactly  k satisfying assignments so really  that's a problem of counting how many  satisfying assignments you have in a  formula  um  so you know for  np you have at least one  um but i'm i want to know exactly how  many  uh so the numbers that problem is the  pairs formula and the count  so  um  and uh so if we define the count  number fee is the number of satisfying  assignments of a fee  then another way of writing this uh  number stat problem  is  uh  the pairs phi k where k is the number of  satisfying assignments of fee  so we're going to be using this notation  number fee a lot so just make sure you  get you got that notation this is the  number of satisfying assignments of that  formula  okay  and here's a definition i probably  should have given you earlier in the  term but better late than never um  uh so the notion that a language is np  hard  it's like  np complete  except without being  necessarily being in np  so this is just the reduction part a  language is np-hard or co-np-hard or  p-space hard or any of those other  classes that we looked at  if every problem in the class is  reducible to that language  but you don't know whether this that  that language is in the class  so we just call it np hard instead of mp  complete  so you could say  the language is np complete if it's hard  and it's in np  okay and so we're going to show that  this number set problem is co np hard  so everything in cohen p is polynomial  time reducible to number set that's easy  because what we're going to do is take a  co np complete problem  which is the  unsatisfiability problem the complement  of satisfiability and sure that reduces  to the numbers that problem  and that's easy because  a formula is unsatisfiable exactly when  it has zero satisfying assignments  so if you can tell  how many satisfying assignments  something has exactly  or you can answer the question you know  does a formula have exactly  a thousand  satisfying assignments if you can do  that in general then you can solve co  and p uh you can solve the  unsatisfiability problem by asking with  zero satisfying assignments  and that allows you to solve anything in  cohen p  okay  so we're gonna just work with this one  problem the number set problem and show  that that problem's in ip  okay  let's take a quick break  um  okay feel free to send me let me see if  i can catch up with some of the  questions that have been  cropping up here so if the approver  knows the random choices of the verifier  can it can flip the answer to make the  verifier reject  not sure what that you mean in the  context just of the graph isomorphism  problem or  um something in general i'm not sure  you'll have to explain sorry i'll  respond with a question mark what else  can i answer for you guys so i got a  question  might  if i p equals p space does that mean  that iso or the or non-iso might be um  in might be p space complete but no  that's not known  so we're about out of time  okay  let's continue here  [Music]  okay so this this is where we're kind of  getting  going to start to get into the meat of  things um  [Music]  and if you didn't quite understand  everything up till now maybe just try to  keep your intuition  about how do i  you know how does a powerful party  convince a  um a polynomial probabilistic polynomial  time party  of the number of satisfying assignments  exact number not  not at least but you won't know exactly  the number um of satisfying assignments  um so it could be zero for example how  do you convince that how do you convince  someone that there were zero assignments  and and and you know you can have an  interaction which does that and that's  not obvious at all  how you're gonna do that  um  uh all right  so um  okay so we're gonna have to introduce  some notation which i  don't hope this doesn't  cause heartburn here  uh  so  let's say  again here is the the  computational the language we're working  with number set  and we have  a fee where that has m variables  x1 to xm  now  here's our notation i'm going to if i  write phi with this  free phi of 0  that just means the formula that i get  by  plugging in 0 for x one  and leaving all the rest of the  variables uh alone  okay so i substitute zero for x one  where zero means false and one means  true as usual  and but that's it's going it's still  going to be some other formula but just  with that substitution  if i write fee 0 1 that means i preset  the first two variables to 0 and 1.  um  if i write fee with  a bunch of  preset values i'm just setting the first  i variables x1 to xi to some values  and leaving the other variables  as unset  so i'm calling the ones that i'm  nailing in there as i'm already saying  these are the presets  okay so this is just converting some  formulas into other formulas that have  somewhat fewer variables  all right  now let's recall that number notation  the number sign notation number fee is  the number of satisfying assignments  now if i say number fee of 0  that's the number of satisfying  assignments  when i've preset x1 to 0.  similarly if i preset the first i  variables  to some  values  and then i take i want to think how many  how many satisfying assignments subject  to those prefix presets i write it this  way  so i'm going to use this notation a lot  you have to understand this notation ask  if you don't none if you don't get it so  another way of writing it i don't know  if this is helpful but another way of  writing number fee of a1 to a i  remember we have m variables all  together  that means i take the  variables which i have not yet preset  um  and i allow them to range over all  possible zeros and ones and i add up  the  the formulas values for all of those so  there's a one every time i satisfy and a  zero every time i don't satisfy so i'm  adding up all the satisfying assignments  subject to these i presets  okay  so here are two critical facts about  this  number sign notation  first of all  if i preset the first i values to  something  now i can  in addition  set the next variable either to zero or  to one  and i get this  relationship  which is just simply a generalization of  the fact that  the total number of satisfying  assignments of the formula  is equal to the number of satisfying  assignments when x1 is 0 plus the number  of satisfying assignments when x1 is 1.  right they together have to add up to  the total number because x1 is going to  be the either zero or one  so that's fact number one  fact number two is that if i preset  everything  all of the variables so there are no  variables left then the number of  satisfying assignments  subject to that preset  of everything is just whether or not  i've satisfied the formula which is the  value of the formula on that those  presets  okay  both two simple facts but it's going to  be critical in the protocol i'm about to  describe  questions on this  i think actually i do have a question  for you  so let's just see  what do you think  just to check your understanding  okay  got about 80  getting this  um  i'm not sure that's good but uh  all right almost done  closing  okay  okay so uh yes a is the correct answer  you know if if there are nine satisfying  assignments all together and there are  six satisfying assignments where the  first variable is set to zero  then there's only three satisfying  assignments with the first variable set  to one because  nine has got to be equal to six plus  three  that's actually this fact number one  it's not going to be 15  this is not true either  so  it's just a  good  okay  okay so let's try to  with with that knowledge let's try to  see how we can put  number sat in ip  so this is not going to quite work  but it's really going to set us up to do  this to finish this next time  um  so  you might immediately see where this is  going wrong but you'll  you'll have to put up with it  because um  the setup is what's important  um  okay  so understand now here's the  here's the the setup we have  um  the input is a formula  and  a number  where that number is supposed to be the  number of satisfying assignments  you know it could be wrong and in which  case we're not in the language  um but if it's right you're in the  language  so the approver is supposed to convince  the verifier  that it's correct if it is correct  and it's not gonna it's gonna fail no  matter what it tries to do  if  it's uh not  correct so this is the approver is going  to send first of all  um  so  so the proof is going to send a claim  about the number of satisfying  assignments  going to send when i say this value here  this is what the prover  if it's honest is going to send the  right value of course the verifier does  not know if the approver is honest  but i'm describing how the honest  approver is going to operate and we'll  have to understand what happens if the  prover tries to cheat  so the proof is going to send the honest  proof is going to send the number of  satisfying assignments all together and  the prover verifier just makes sure that  that matches up with the input  if it doesn't match up with the input uh  the verifier is just gonna  you know you know the the verifier is  gonna not be convinced that the input is  um in the language  so it's gonna just uh reject at that  point  um  okay uh  then now the verifier says okay that was  very good that you sent me this how do i  know that's right  so what the verif prover is going to do  to try to convince the verifier that  this value was correct  is  uh  unravel that by one level by say well  you know there were nine satisfying  assignments all together uh six of them  were when the when x one is zero and  three of them were when x one is one  to verify what does the verify have to  check  that  these add up correctly when i preset x1  to zero and to one  it had better add up to the total number  of satisfying assignments  if that works out the verifier is happy  is still being it's still consistent  with being convinced that this k was the  right value  um  so um  the next step is  well the verifier says well how do i  know those two values are correct the  prover says okay well i'm going to prove  send  un unravel them one level further  then here's a number of satisfying  assignments when the next variable is  set to both possibilities for each of  the possibilities of the first variable  now if you're  understanding me  about what the prover is sending you  should start to be getting a little  nervous  because  something is i mean this is going to be  correct  but it's going to start it looks like  it's starting to blow up in terms of the  number of  amount of work that's involved and  that's actually a problem  but let's bear with that for the moment  let's just worry about correctness not  about complexity for the moment  so  the proof is going to now send the  number of satisfying assignments for  each of those four possible ways of  presetting the first two variables  and the verifier is going to check that  that was consistent with the information  the prover sent in the previous round  right by again checking this identity  here so then the proof is going to  continue on doing that until it's got  it's done that  through m rounds where m is the number  of variables  so at this point  the proof is going to send  all possible ways of presetting all of  the variables so that now there's two to  the m possibilities here  again this is hopelessly  not allowed but okay ignoring that  the proofer's got to use this at the nth  round to check what happens at the at  the previous round so that's when they  were m minus one values sent because  each one has one more  uh  uh you're extending the presets by one  so which using this to check that the  previous round was the values were  correct so it's looking for  um  you know the m minus one  presets  have to add up correctly um you know in  terms of the m the presets of m uh  values  uh for each of those ways of uh doing  those uh m minus one presets  and so now the provers send all of those  two to the m  counts which are by the way ones and  zeros because at this point we have  preset all of the  values of the variables and so there's  only one possible assignment at most  that there can be  and now  the verifier  the approver is done the verifier is  going to check  by itself  that these values make sense that these  values are correct  so it's going to do that by looking back  at the formula  so far up at this point the verifier has  not been looking at the formula it's  just been checking the internal  consistency of the provers messages with  each other  but now at the end the verifier is going  to take the messages these these values  that the approver sent for each of the  two to the m presets and see if it  matches up with what the formula would  do  remember that was the other sort of the  base case  of the the end the fact number two  um from the uh slider to ago  make sure that these agree  okay and now the verifier says well okay  if everything has checked out and all of  these are  are in agreement  then the verifier is  going to be convinced  that um  that uh fee had k satisfying assignments  but if anywhere along the way one of  these checks fails  the approver is not the verifier is not  going to be convinced and is going to  reject  okay so in a sense this is kind of dopey  you know you know we've just  i'm just kind of giving you a  complicated way of just counting up one  by one each of the satisfying  assignments of the formula and seeing if  that matches k  but nevertheless  this way of looking at it is gonna be  uh  help us to understand  um the way to fix this so  so bear with me for another minute on  this one  so  another way of looking at this which i  think is is particularly useful  is to think of  what happens  well okay  we'll get there in a second i want to  look at what happens if k was wrong but  before i do that let's look at the  i'm going to give a kind of a graphic a  graphical  view  of um  the information that the prover sends  and the and the verifiers actions in  this protocol so the the values that the  prover sending are going to be in yellow  so the  and the information that the verifier  has or checks is going to be in white  so the the verifier has the k the the  input value which is supposed to be the  number of satisfying assignments  and the prover  sends some value and the verifier checks  that this value which is supposed to be  the number of satisfying assignments  corresponds with k  so that's one of the checks it does  then the approver is going to send kind  of take to justify this value  it  um sends  the number of satisfying assignments  when you have  x1 set to zero  or set to one  the verify adds those up to give you  and it's supposed to equal the total  number of satisfying assignments and so  this is if you understood this protocol  this is just i'm writing it out in a  sort of a simplified way perhaps  okay  and so um  keeps checking that these things add up  correctly until you get down to  setting all m values in all two to the  impossible ways and now the verifier is  going to then check to make sure that  that equals  the what the formula would say  um  okay  okay so now  let's what happens  if  k was the wrong value  it did not agree with the number of  satisfying assignments  um  and  what does what happens now um  could the prover  what happens if the approver tries to  make the verifier accept anyway  so  um  so the only thing the prover can do at  the very first step would be to lie  um about  you know if the approver sends the if k  is wrong  and the proof approver sends the correct  value  for  the the the total count the verifier is  going to reject so i'm trying to see  is it could the approver try to make the  verifier accept what what happens  so the prover has to lie here and i'm  going to indicate that by  saying the approver is sending in  the wrong value  for um  uh the the total count  well if the proof is going to lie here  um  then  just like you know if you  you know you have a child  who um  tells a lie  and then you start you know as the  parent you start asking questions to try  to  see if the story is consistent  one lie is going to lead to another lie  um and that's what happens here  if the uh  in order to justify this lie  um the proof is going to have to a lie  in one or perhaps both but at least one  of these two values because you can't  have the two correct values adding up to  the incorrect value  because you have to think about what's  going on here  so if this is a lie that's going to  force a lie  at one of one side of the other one  level down which is then going to force  a lie  to propagate down and so there's a lie  at every stage is going to force a lie  at least in one  one place or another to propagate all  the way down to the bottom  and then at the bottom the verifier will  see  that the check doesn't work  as when it turns when it tries to  connect it up with the formula itself  and the form verifier will reject  okay so just a way of looking at this  um if the form if the value  if the input was not in the language  um so uh  but the problem is  that as i said that this is exponential  so how are we going to fix that so  just looking ahead to what we're going  to do on tuesday  okay let's see if there's any questions  here first of all  uh  okay yes i got a question should this be  uh  should this be a minus i i purposely  made this bracket only go  not include the very last zero yeah  there's a total of m zeroes here all  together but the  i left out the last zero that's why i  said n minus one maybe it would have  been better to say m  okay so and you got another interesting  question here why can't we reject right  away if k is wrong  um  uh  well  the verifier is probabilistic polynomial  time how does the verifier know if k is  wrong  um  so it means or or or right  so  what we're trying to do is something  like you know like np where we have a  certificate but now we have this kind of  interactive certificate in the form of  disprover maybe that's another way to  look at it  um  where if you're in the language there  should be some way for the prover to  convince to get make you accept but if  you're not not in the language there  should be no way  for the proverb to make you accept  um  uh  so the verifier just can't  reject right away because there's no way  to tell how does the verifier know it's  going to start rejecting things when it  shouldn't if it's just going to be  rejecting willy-nilly here  um  okay how does the verifier need to  determine if the prover is  internally consistent instead of just  asking  so why does the verifier need to  determine if the approver is internally  consistent instead of just asking the  questions in step n plus one  yeah so maybe that's  because it looks like all of the work is  happening at the very end  um  but i'm really presenting this to you  as a preparation  for what we're going to do on tuesday  um so it's important to think about the  connection from each step to the next  each step is going to be justified by  what happens at the next step  until we get to the very end  so you have to just understand it for  what it is don't try to make it more  efficient yeah i realize this is kind of  dumb  um  good point we're not using the  probabilism here  um  and  moreover we're not really even using the  interaction here the prover is doing all  the sending the verifier is just  accepting at the end  yeah  this is we're not using the power and  we're getting a weaker result so let's  move on before we run out of time here  so how do we gonna fix this  so the problem is is blowing up  each to justify each stage where  each value  we're  um  needing to present two values which add  up to it  um  and that's uh good leading to a blow up  now it would be nice if we can do  something where each value was  supported by just a single value  at the next level  so we know here's an idea well you know  in order to understand to see that  uh that that this total count is correct  why don't we just pick at random either  zero or one and only follow that one  down  well the problem with doing that  is because  the the  the sequence of lies is could be just a  single path through this uh tree  and the chances you're going to find  that path  down to a contradiction at the bottom is  very low if you're just doing it at  random  um so just randomly picking zeros and  ones as as the as the one you're going  to justify used to justify the previous  value is not going to be good enough  so what  but this is what we're going to do  however the values that we're going to  pick  for um  for these random  um  inputs  are not going to be boolean values  we're going to pick  non-boolean assignments to the variables  which again just as with the branching  program case didn't make any sense  on the surface of it we're going to have  to make it make sense  and  we'll have to see how to do that on tues  in tuesday's lecture  okay so that's kind of the setup  okay  um  yeah so in a similar question why is  this any different from just  non-deterministically guessing the  assignments it's because of this we're  really setting the stage  okay so what we did today was we  introduced them the the model and  defined the complexity class  we can we did show this one in its  full glory we showed that non-iso is an  ip  really worth understanding this  uh protocol here making sure you  you're comfortable with that  and and also the the model itself  and so for tuesday's lecture we're going  to finish this up uh well we started  showing that number set as an ip which  is what we need to to do to prove co-np  is an ip  and we'll finish that  next time  which will be our last time  okay so that's it for today  i'll stick around for questions  so  a good question here why can't um  v just  reject if some of the checks are  incorrect yes v could as soon as  there's a check that fails we can just  reject at that stage  i'm just trying to argue that at some  point along the way if the input is not  in the language  there's going to be a check that fails  so i mean i said reject at the end but  yeah i mean  you could have rejected at any point  along the way  um  okay  um someone's asking for what what role  did i play uh so i did  my my own personal role in this i were  to which was twofold first of all i i  came up with the idea  uh of  well not the idea i came up with the  name  interactive proof uh  i remember when sylvia mccauley was  explaining this to me in my apartment  many many years ago he had a kind of a  little bit complicated and i don't even  remember what the protocol was for it  was not for something simple was  something involving uh prime numbers  and i said oh  that's a kind of an interactive proof  and it's and it it stuck from that point  on so  um that was one thing but the other  thing in terms of more mathematically  i my role was so uh  shafi goldwater and i approved the  equivalence of the two models the public  coin and the private coin version  um so that that was uh  my uh  my role in this back and when this was  all first coming out  approved it approved it on an airplane  on the way to a conference somewhere  so i think we're going to  notice any other questions  i think we'll head out take care  everybody  see one  see you on tuesday  bye bye  you

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hi everybody  um  glad to have you all back  for our next to last installment of  theory of computation  today um we are going to embark on the  very last  big topic for the semester  and that is in some ways going to be  following on what we  started  a couple of lectures back when we looked  at probabilistic  turing machines  and probabilistic computation  and its associated class bpp  now what we're going to  discuss  is in some sense  a probabilistic version of np  and that's going to be a complexity  class called ip  which stands for interactive proof  systems  um and so we're going to present that  model  and look at a couple of examples  uh  i would  just like to say at the beginning that  the  this model is a very important one it's  really uh has been the starting point  for a great deal of  research  in complexity theory so we're just  really going to be touching on it but  there's a lot more that people have  pursued with this model and it's also a  connection in to the cryptography field  which also makes use of the interactive  proof system model in fact  some of the  genesis of that model comes out of  cryptography  where you're having uh  multiple parties either communicating or  in some ways  interacting  uh to achieve certain uh goals of  communication or  signing or passwords or or or what have  you so uh this is a both a an applied  area and also one that has a lot of very  interesting theory uh um associated to  it  so with that one we we're going to jump  in um and uh  start out by uh  making myself smaller  and  just do a an introduction i'm going to  introduce  the model or the the concept of an  interactive proof  with an example  and that example involves the  graph isomorphism problem  that's the problem of testing whether  two graphs are isomorphic  what we mean by two graphs being  isomorphic is that they're really just  the same graph  with one of them perhaps being  re-labeled or permuted  so that  they may look superficially different  they may appear with a different  sequence of labels or the nodes are  appearing in a different order  but  except for that it's really just the  same graph so i'm kind of illustrating  that here if you can see those two  graphs here which look different from  each other both on eight notes they are  in fact the same graph  as i can illustrate by a little  animation which will convert this one  into that one  so uh  the  um  [Music]  so the  two graphs um  these graphs being the same um they're  we call that isomorphic  so these are graphs g and h and they're  really the same graph so we we um call  them isomorphic graphs  and we  are  have an associated computational problem  called iso  which is  given a pair of graphs we'd like to know  are they isomorphic or not so the iso is  the collection of pairs of graphs which  are isomorphic  and  it's easy  to see that this problem is an np  problem  because  all you need to do in order to see or to  give a certificate that the two graphs  are isomorphic to each other is tell you  it's just to say which nodes in the one  graph correspond to which other nodes in  the other graph  um and then you all you need to check is  that the edge relationships are  consistent with that  mapping or that isomorphism as it's  called  um  so it's easy to see that the iso problem  is an np  and if you're not getting that make sure  you uh understand because this is the  whole first part of the lecture will be  lost if you don't understand this iso  problem  now the question of whether  you can test two graphs  being isomorphic in polynomial time  is not clear and in fact that's an  unsolved problem to this day  uh  and it's a problem that has generated an  enormous literature uh  there are hundreds of papers  on the graph isomorphism problem as it's  called um to try to uh resolve  um you know uh to try to see if one can  find a polynomial time algorithm and in  fact it was a very big result  just in the last 10 years where there  was a sub-exponential algorithm given so  that was more  than faster than the brute force search  approach but didn't get it all the way  down to polynomial now why is there so  much attention just to this one  particular np problem it's because  it's not known whether the graph  isomorphism problem is np-complete iso  is not known to be an np-complete  problem and that puts it into a very  very small class of problems in np  which are no not known to be either in p  or np-complete  it's kind of a curiosity that for np  problems  almost all of them have ended up being  in one side or the other  and in fact  um it's a  uh  um  in fact the  the um  i i think it's the only problem that  just involves graphs  that's not known to be either in p or an  np  um  so so i got a question here  what would be in between exponential and  polynomial for example i don't remember  what the the bound is but it's it's  something in the uh  in the range of n to the log n  uh  time complexity for the graphite some  more i might be getting that wrong um i  don't remember exactly what the bound is  but um  that's significantly better than two to  the n or some  some exponential amount of time but it  it's more than enter any constant so  it's more than any polynomial time  so um  another question uh of the same sort is  whether the complementary problem is in  np or whether whether iso is in co-np or  let's let's talk about it in terms of  the complement whether the  complement of iso which i'll refer to as  the non-iso problem whether that's known  to be in np so that's also not known um  in other words if i give you two graphs  and i ask you to show that they're not  isomorphic suppose they aren't  isomorphic and you go through the effort  of you know determining that by a brute  force search and now you want to  prove  that they're not isomorphic well that's  not  it's not known to be an np either so  there's no known  short certificate certificate of  two graphs not being isomorphic  we don't know how to do that either  but there's something that's very  interesting nevertheless  um  and  it has to do with the ability to for one  party to prove to another  that graphs are either isomorphic or not  isomorphic  so if you're just having like a prover  we haven't really been necessarily  formulating that this way so much in  this class but this is a completely  equivalent way of formulating the notion  of np whether you have a polynomial time  verifier an approver who can produce  certificates say it's a powerful prover  so um if you have a problem that's in np  approver can convince a polynomial time  verifier  that uh strings are in the language if  in fact they are so in the case of the  iso problem a prover can convince a  polynomial time verifier the graphs are  isomorphic just by exhibiting the  isomorphism  now  for the non-isomorphism case  we don't know that that problem's in np  but it's still possible  for approver to convince a verifier that  graphs are not isomorphic  if you change the rules of the game  slightly  um  so even though the non-iso problem  is not known to be an np  approver can still convince a polynomial  time verifier that graphs are not  isomorphic assuming they they are in  fact not isomorphic um provided  that  um  the prover and the verifier can interact  with one another so the verifier can ask  questions of the prover  and the verifier gets to be  probabilistic  so that's in this  in that's in sense in which i mean  that this notion is a kind of a  probabilistic version of np  okay so um  let me  show you how that's done  so before we jump in to the  uh to the method for  approver to um show a verifier that  graphs are not isomorphic  um let me let let's try to get a little  clearer on the model so i'm going to  first show it to you informally and then  we'll look at it formally  okay so  in in interactive proofs there are two  parties  um  and i'm going to think about them as one  of them is going to be the professor  okay so the professor is going to play  the role of the verifier in a sense but  it's like that the one who checks  um  and uh the professor being kind of old  and tired he's been teaching too long  maybe  can only operate in  probabilistic polynomial time  so the professor if wants to tell  whether two graphs are isomorphic or not  probabilistic polynomial time doesn't  seem to be enough  to tell whether two graphs are  isomorphic or not because it seems to be  a more than polynomial problem  however  the professor  has  um  help  it has  an army  of  graduate students  and the graduate students  they're not limited uh in the same way  the professor is the graduate students  are young  they  are energetic they can stay up all night  they know how to code  so  the graduate students  have unlimited computational ability  so that we're going to think of the  graduate students playing the role of  the approver because they're not  they're not limited in their  capabilities we'll assume  the professor on the other hand is  limited so the professor wants to know  if the two graphs are isomorphic  let's say whatever they are  um  can't do it by himself so he's going to  ask his students to figure out the  answer  and report back  now there's only one problem  the professor knows that students  uh  well in the old days they'd like to  party i guess these days they like to  play on uh  play computer games a lot  and so they're not really that eager to  spend all their time  figuring out whether graphs are  isomorphic  so he's worried that the  the students will just come up with some  answer and figure that he won't be able  to tell the difference  so the professor does not trust the  students it's not enough to he for the  professor to give the problem to the  students and just take any answer that  they're going to give the professor  wants  wants to be convinced  okay so  um  now  how could the students convince the  professor of the answer that they've  really done the work and figured out  whether the graphs are isomorphic or not  well if the graphs are isomorphic if it  turns out that the graphs were  isomorphic and the students figure that  out  then life is good  because what are they going to do to  convince the professor  they're going to hand over the  isomorphism and show yeah i mean they  are  you know those graphs really are  isomorphic and here's how the  correspondence works professor can check  oh yeah i i now not now i'm convinced  but suppose the graphs were not  isomorphic  what are we going to do then  um the students have figured out where  after night else the professor wants  wants to be convinced oh no what are we  going to do  well in fact we're going to engage the  the professor and the students are going  to engage in the following protocol  dialogue  what's going to happen is  now you have to make sure you're you're  this is critical to follow to understand  this little  part of the story here because it's  really going to set the pattern for  everything in today's and tomorrow and  to today's lecture and the and the next  lecture  okay so  we're going to engage in a following  interaction between the students and the  professor which is going to enable the  students to convince the professor that  the two graphs really are not isomorphic  so how is that going to work  this is a beautiful little uh thing by  the way  so the professor is going to take the  two graphs  and pick one of them  at random  because the two graphs g and h  um let's say they're not they really are  not isomorphic the professor doesn't  know that for sure that's what the  students claim the professor really  wants to not be convinced that the  students are right  um so the professor's gonna pick one of  the two at random  randomly permute that  uh that choice the one that he picked  and hand it over to the students  say okay  here is one of those two graphs randomly  scrambled  then i'm going to ask the students  which one did i pick  okay now  if the graphs were really not isomorphic  the students can check  whether that randomly scrambled graph  is isomorphic to either g or to h it's  going to be isomorphic to one or the  other  and then they students can figure it out  and they say oh you picked g or no you  picked h  as the case may be the students can  figure that out  but if the graphs were isomorphic  then that  scrambled version of g or h could  equally well have come from either of  them  and the students would have no way of  knowing  which one the professor picked  so that there's nothing they could do  which would be better than guessing  so if we do that a bunch of times the  professor picks at random sometimes  secretly of course the picks the grip  picks either g or picks h  and the students get it right every time  either the students are really doing the  work  and the graphs are really not isomorphic  or  the students are just incredibly lucky  they're managing to guess right let's  say a hundred times  so how the  the the professor randomly and secretly  picks grh uses this uses its probabilism  flips a coin  just a two-sided coin and says okay  sometimes if we're going to do g  sometimes they're going to do h just  completely at random  picks one or the other  and then with some more randomness  gets finds a random permutation of the  one that he picked and then sends that  over to the students and say which one  did it come from  um  i'm not sure okay so let's pause here  let's let's make sure we all understand  this because this is really important um  so i'm getting a question here how do we  i'm not sure what your question is um  okay so let me just say yeah the  professor's going to play the role the  verifier the graduate students play  they're all approver that's coming but i  really want to understand this protocol  here okay so how is the professor  picking the graph skin if you're  okay i don't you know picking the graphs  at random you have just two graphs  they're in part of the input uh the both  the students and the professor can see  the graphs  and the professors are just picking one  of them at random using a coin so i'm  not sure i understand the question there  could p and v engage in a protocol where  the secretary is on the prover side  instead  the question of revealing the  isomorphism i there is no why so i'm not  sure i understand this question either  um maybe we'll make this clear  you know if  for  for this little illustration  the professor  doesn't know the graphs could be  isomorphic or they could be not  isomorphic  and so uh the professor wants to be  convinced either way whatever the  students whatever answer the students  come up with we're going to shift this  into a problem about a um deciding a  language  next but right now i'm just trying to  give a sense of the how the model works  i want to move from this informal model  and now i'm going to formalize that in  terms of  model which will be deciding a language  okay so  so the interactive proof  system  model  we have two interacting parties a  verifier which is probabilistic  polynomial time playing played by the  professor in the previous slide  and the prover which is unlimited  computational power played by the  students in the previous slide  both of them get to see the input  which in the previous case  well  it could be for example the pair of  graphs  they exchange a polynomial number of  polynomial size messages so the whole  exchange  including the verifier's own computation  is going to be polynomial the only thing  that's not not not included within the  computational cost is the prover's work  which is unlimited  um  after that the verifier after the  interaction the verifier will accept or  reject  and we're going to define the  probability that the verifier together  with a particular approver ends up  accepting  as  you  look over the different possible coin  tosses  of the verifier  which could lead to different  behavior on the part of the verifier and  therefore different behavior on the part  of the approver  so over all the different possibility  possibilities for the verifiers  computation  we're going to look at the probability  that the verifier with this particular  approver ends up accepting and i've  written it this way this is the  probability of the verifier interacting  with the prover accepts the input  is just simply that  um and so we're going to work through an  example  we're going to  work through the previous example  more  precisely in a second  the class ip for interactive proofs  stands for it's the class of languages  such that for some verifier and approver  um  for strings in the language  the prover makes the verifier accept  with high probability  and here's the interesting part  for strings not in the language the  prover makes it except with low  probability but every there's no prover  which can make it except with high  probability  so there's no way to cheat  if you think about it in the case of the  graphic non-isomorphism  there's nothing you know  if  if the graphs were really isomorphic  and the students were trying to in a  devious way  prove  through that protocol that they're not  isomorphic they would fail because  there's nothing they can do  if the graphs were isomorphic  then um  when the  verifier the the professor picks one or  the other at random  um and scrambles it the students would  have no way of telling which one the  professor did so no matter what kind of  scheme they try to come up with  they're going to be out of luck  so  it's no mat for any strategy  for strings that are not in the language  for any s any prover calling that p with  a tilde to stand for a devious or  crooked  prover  for any uh possibly crooked prover  even that with working with the verifier  is still going to end up accepting with  low probability  so strings in the language there's going  to be an honest prover who just follows  the protocol  in the correct way which makes the  verifier accept with high probability  for strings not in the language every  prover  is going to fail  to make it accept with high probability  um  okay so that i mean the way i like to  think about it is that p tilde is  a possibly crooked proverb which is  trying to make the verifier accept when  it shouldn't because the string is not  in the language  it's like you know  it's like even you can think of this in  the case of um satisfiability  um  you know  you  a crooked prover might try to convince  of the verifier that the formula is  satisfiable when it isn't by by somehow  trying to produce a satisfying  assignment but that's going to be  impossible there's nothing any strategy  can possibly work when the formula is  not satisfiable if that's what the  verifier is going to check it's going to  be looking for that satisfying  assignment  okay  and by the way this is we're not going  to prove this but it's really going to  be proved in the same way you can make  that one third error that could that  occurs here  something very tiny  by  the same kind of repetition argument  okay  so let's see um  so why can't the prover in the first  case be crooked um  the prover in the first case would could  be crooked but that's not going to serve  the purposes um  you know what what we want to show  um  you think about it like we think about  np  for strings in the language there exists  a certificate there is a proof  that you're in the language  so if somebody is going to  not produce the proof  that's irrelevant  the question is if you look at the best  possible case the best possible prover  um  you know who's going to be able  we're asking does there exist a way to  convince the verifier that the um  string is in the language  so it doesn't matter that there might be  some other  uh  silly way that doesn't work we're just  looking at the best possible way  so the best possible way when you're in  the language is going to end up with a  verifier having high probability  when you're not in the language the best  possible way is still going to end up  with low probability  when when i talk about best possible i'm  trying to maximize the probability that  the verifier is going to end up  accepting let's continue um not sure as  clear as i would like but  um  maybe again we're going to we're going  to stick with that example because this  is a very uh  helpful example and to try to  understand the setup  and uh so we're gonna i'm gonna revisit  that previous example about  non-isomorphism but now in the context  of this  thinking about as a language  so we're going to take this  non-isomorphism  um  uh yeah  we're going to take the non-isomorphism  problem  and show that it's an ip so there's  going to be a verifier together with  approver which are going to make the  verifier accept with high probability  for strings in the language namely  graphs not ice being isomorphic  and  nothing there's going to be no way to  make the verifier except with high  probability for strings out of the  language therefore that's when the  graphs are isomorphic  okay um so the protocol is just gonna  we're gonna repeat the following thing  twice  you know i said in the previous case do  it a hundred times just to  help us think about it but actually  twice is gonna be enough to get the  bound we need  so the verifier is going to operate like  this in terms of this is the verifiers  in first  communicating  sending messages to the approver it's  going to randomly choose grh  just like what the professor did last  time  randomly permute the result to get a new  graph k  which was going to be which was  which is isomorphic either to grh  depending upon the choice the verifier  made  and then send that graph k  now the provers turn is going to respond  by the proof is going to compare k with  the two one of the both of the original  graphs it's got to be isomorphic to one  or the other  and it's going to report back which one  just going to say well you pick g no or  you picked h  because the prover with its unlimited  capabilities can determine that  um  and then v accepts if the approval was  right both times  um  and if the approval was ever not right  the verify says oh something's fishy  here because we know that the prover has  unlimited capability so could get it  right if you know um if there was if  this was an honest approver um and so um  if uh  if it's not getting it right then the  verify is going to reject  so if the graphs are not isomorphic the  proofer can tell which one it picked  randomly so therefore  if the graphs are not isomorphic the  verifier with that that honest prover  will accept with probability one  because that honest proof is always  going to get the right answer  which is at least two-thirds is what the  bound we need  uh we don't care about the space used in  answer to a question  um if  we were not in the language so g and i h  are not isomorphic  then there's nothing any crooked pervert  could possibly do  because it gets a graph  can't tell there's no way to tell  whether it came from g or came from h  um  so that crooked proofer would have all  it could the best thing you could do is  guess sort of a 50 chance advancing  correctly each time and only a 25 chance  for doing it twice  and that's why i did it twice in order  to get that error um  uh  to be small so there's only a 25 chance  of the approver getting lucky so that  would be an  error case  if the proof of just by chance picked  the right answer twice even though the  graphs were isomorphic so therefore  for the isomorphic case  the verifier interacting with  any prover is going to accept that input  with at most one quarter 25 of the time  which is less than a third so those are  that's just to achieve that bound  okay  so  let's let's answer some questions first  and then i'll try to um  uh  i'll ask you  you understand this so this it's i think  it's worth trying to understand this  model um of this interactive proof  system  it's a little little slippery i i  realize but um  if you just hold your hold on to your  intuition of the prover trying to  convince  you know of uh  you know a powerful prover trying to  convince a limited verifier  um  of  some  string being in a language  uh  you want the proven to be able to  succeed when the string is in the  language but fail when the string is not  in the language  yes we are going to somebody's asking if  the prover is identifying grh by brute  force yes the prover is going to use its  unlimited capabilities to determine  given k  whether it came from g or h  the um  the computational  cost of the approver is irrelevant for  this it's just like when we think about  a certificate  um you know for satisfiability we don't  talk about the cost of finding that  certificate  uh for np  for ip again we don't talk about the  cost of the prover running  so somebody's asking does the crooked  per prover answer just randomly or does  uh can the cr quick approver has have a  strategy the crooked perimeter can have  a strategy  now we're not we're assuming the crooked  approver is devious  but it's still going to fail  okay um let's do the check-in  suppose we change the model  so that the prover can watch the  verifier picking its random choices  so uh the verifier cannot act in secret  anymore but the the prover can watch the  verifier  now let's suppose we had the same  protocol that i just described  what language do we end up with is it  the same language different language and  what is that language  okay so  i want to hopefully  um  let's let's  it'll  give me some sense of how well you're  following me by how well this uh this  goes  yeah someone's asking about how this  connects up for example with np so we're  going to look at that  also in a second  okay so this is reassuring that most of  you i think are on the right track at  least for this check-in uh do we assume  p uses this access to guess right  what access p is not really guessing the  p is tr is actually i don't think a p is  non-deterministic or anything like that  p is actually trying to get the right  answer  and using its  computational ability to do that if it's  possible  may not be possible then there's nothing  you can do  okay so let's uh end this are you all in  two seconds left  please vote  vote now or never  okay ending  um  yeah so c is the correct answer here  if the prover can watch what the verify  is doing the prover can see  what graph the verifier picked right  from the beginning  and so the prover without having to do  any work  can say you know it proves looks over  the verifier's shoulder and says oh you  picked g  and now you're randomly permitting it  but i don't care about that i just i  know you pick g  uh so uh the  proof is going to respond back uh g even  if the graphs were isomorphic the proof  is going to be able to get the right  answer  kind of interestingly um  uh  you can make a you can change the  protocol somewhat um to make it uh that  even if the  prover  has access to the verifier's randomness  you can still achieve this but not with  the same protocol  um  so that's uh that's a separate question  um okay so let's move on here i want to  get too bogged down  okay here's another check-in um  uh  okay so  you have to tell me which of the  following  statements are true  as far as you know  now you have to think about a little bit  how this  these uh  relate  to um how np and ip or bpp  and ip relate to one another  okay  how are we doing on this  okay so we're gonna have to close this  pretty soon too  do the best you can  interesting okay  closing up shop  last last vote  okay one two three  there's one more person out there who  hasn't voted who voted last time oh well  uh  all right  in fact they're all true  um  let's see why is np contained with ip  contained in ip uh well many of you have  seen this already so let's just quickly  go through it um  [Music]  if we just had a deterministic v  um  you know the uh  you can  uh  maybe it's just  that that can be enough of deterministic  v  i think it's just going to be equivalent  but actually just to be doubly sure the  deterministic v and the approver just  sends a message to the verifier and then  checks it  that's the way we normally think about a  certificate for np  uh i don't think it's going to change  anything but should double check that if  the verifier can still ask questions but  i think as long as the verifier is  deterministic you're going to get  exactly np  here  um  and um  now how about bpp  well  there you don't even need approver  because the verifier is already  probabilistic  so  verifier can ignore the approver  and this one is a little tricky uh i p  containing p space because we haven't  covered that so there's no way for you  to know that unless you happen to read  ahead in the book but it's in fact true  um in some ways it's a little bit like  the proof that  uh  np  is contained in p space ip is sort of an  enhanced version of np  and you and there's just a basically a  uh a piece based brute force algorithm  that goes through the entire tree of  possibilities um of the verifier  exchange  verifier with exchanges with approver  and can uh determine that um the  verifier is either going to accept for  some approver or is going to end up  projecting for every approver um  so we're not going to prove this  statement but something good for you to  know anyway they're just a fact  uh but we're going to do what the  surprising thing  in reference to part c  is that the containment also goes the  other way this is the amazing  um uh  was is an amazing result  that  everything in p space  you can do with in ip  so this is ip is actually turns out to  be incredibly powerful gives you  everything up in p space you get i p  equals p space  so that says that any problem that you  can solve in p space like any of the a  game for example  um if you can imagine you know  formulating you know checkers or  chess  as a piece based problem which you know  depending upon some details of the rules  you can do because you know you have to  generalize it to an end by end board but  okay  let's not quibble um  uh then  uh  um  [Music]  we don't know which side has a forced  win in chess  um and even if somebody goes through the  effort of going through the game tree  um and determines that let's say white  has a forced win  uh there's no way for them to there's no  short certificate we don't know that  that problem is not an np  but  by  going through an interactive proof  an all-powerful prover  could still convince somebody that white  had a force  you know convince somebody in polynomial  time that a  white has a forced wind let's say in um  in chess  again little  uh stretching things because this is  you know you really need to talk about  this as an n by n not an eight by eight  but i think in this the spirit is is uh  fair  so um  okay uh so let's continue  so wha  when we're not going to quite prove that  that p base  is contained in ip we're going to prove  a somewhat weaker statement but very  similar  um  is that uh and historically came first  that co-np is contained in ip  so not only is np contained in ip but  we're going to prove that co np is  contained in ip and this actually has  most of the i most of the idea for the p  space being contained in ip and itself  it's just an amazing proof  a little easier  um  okay  and th this was done  if i'm remembering somebody's asking me  when how old is this it's something in  the 19 i think late 90s but i'm not i  don't remember but maybe early 90s i  think it's late 90s  when this was shown  so it's been a while now  um  okay  so  um  yeah so in terms of the relationship  with cryptography there were two  parallel threads  um that both independently came up with  the notion of an interactive purge  system  uh i was a little bit personally  involved with this  in in a way as well but but mainly that  the the there was one group in  cryptography  um working on this and there was another  group who was actually coming out of the  graph isomorphism world  working on it and they they came up with  two separate models one involving the  private randomness and one involving the  public randomness um and it was turned  out that was  that they're actually equivalent  um  and uh  it's an interesting story but  unfortunately we don't have time for it  uh so  why don't we move on and i'm gonna start  showing you how the proof that  co-np  is contained in ip  goes and what we're going to do is work  with a problem that's  almost like co p complete but um  going to be uh well it's going to be  this number sat problem we'll see the  connection with co-np in a second  uh  so cohen p  so it's supposed to be exactly k  satisfying assignments  fee comma k  is the set of  pairs where the formula fee has exactly  k satisfying assignments so really  that's a problem of counting how many  satisfying assignments you have in a  formula  um  so you know for  np you have at least one  um but i'm i want to know exactly how  many  uh so the numbers that problem is the  pairs formula and the count  so  um  and uh so if we define the count  number fee is the number of satisfying  assignments of a fee  then another way of writing this uh  number stat problem  is  uh  the pairs phi k where k is the number of  satisfying assignments of fee  so we're going to be using this notation  number fee a lot so just make sure you  get you got that notation this is the  number of satisfying assignments of that  formula  okay  and here's a definition i probably  should have given you earlier in the  term but better late than never um  uh so the notion that a language is np  hard  it's like  np complete  except without being  necessarily being in np  so this is just the reduction part a  language is np-hard or co-np-hard or  p-space hard or any of those other  classes that we looked at  if every problem in the class is  reducible to that language  but you don't know whether this that  that language is in the class  so we just call it np hard instead of mp  complete  so you could say  the language is np complete if it's hard  and it's in np  okay and so we're going to show that  this number set problem is co np hard  so everything in cohen p is polynomial  time reducible to number set that's easy  because what we're going to do is take a  co np complete problem  which is the  unsatisfiability problem the complement  of satisfiability and sure that reduces  to the numbers that problem  and that's easy because  a formula is unsatisfiable exactly when  it has zero satisfying assignments  so if you can tell  how many satisfying assignments  something has exactly  or you can answer the question you know  does a formula have exactly  a thousand  satisfying assignments if you can do  that in general then you can solve co  and p uh you can solve the  unsatisfiability problem by asking with  zero satisfying assignments  and that allows you to solve anything in  cohen p  okay  so we're gonna just work with this one  problem the number set problem and show  that that problem's in ip  okay  let's take a quick break  um  okay feel free to send me let me see if  i can catch up with some of the  questions that have been  cropping up here so if the approver  knows the random choices of the verifier  can it can flip the answer to make the  verifier reject  not sure what that you mean in the  context just of the graph isomorphism  problem or  um something in general i'm not sure  you'll have to explain sorry i'll  respond with a question mark what else  can i answer for you guys so i got a  question  might  if i p equals p space does that mean  that iso or the or non-iso might be um  in might be p space complete but no  that's not known  so we're about out of time  okay  let's continue here  [Music]  okay so this this is where we're kind of  getting  going to start to get into the meat of  things um  [Music]  and if you didn't quite understand  everything up till now maybe just try to  keep your intuition  about how do i  you know how does a powerful party  convince a  um a polynomial probabilistic polynomial  time party  of the number of satisfying assignments  exact number not  not at least but you won't know exactly  the number um of satisfying assignments  um so it could be zero for example how  do you convince that how do you convince  someone that there were zero assignments  and and and you know you can have an  interaction which does that and that's  not obvious at all  how you're gonna do that  um  uh all right  so um  okay so we're gonna have to introduce  some notation which i  don't hope this doesn't  cause heartburn here  uh  so  let's say  again here is the the  computational the language we're working  with number set  and we have  a fee where that has m variables  x1 to xm  now  here's our notation i'm going to if i  write phi with this  free phi of 0  that just means the formula that i get  by  plugging in 0 for x one  and leaving all the rest of the  variables uh alone  okay so i substitute zero for x one  where zero means false and one means  true as usual  and but that's it's going it's still  going to be some other formula but just  with that substitution  if i write fee 0 1 that means i preset  the first two variables to 0 and 1.  um  if i write fee with  a bunch of  preset values i'm just setting the first  i variables x1 to xi to some values  and leaving the other variables  as unset  so i'm calling the ones that i'm  nailing in there as i'm already saying  these are the presets  okay so this is just converting some  formulas into other formulas that have  somewhat fewer variables  all right  now let's recall that number notation  the number sign notation number fee is  the number of satisfying assignments  now if i say number fee of 0  that's the number of satisfying  assignments  when i've preset x1 to 0.  similarly if i preset the first i  variables  to some  values  and then i take i want to think how many  how many satisfying assignments subject  to those prefix presets i write it this  way  so i'm going to use this notation a lot  you have to understand this notation ask  if you don't none if you don't get it so  another way of writing it i don't know  if this is helpful but another way of  writing number fee of a1 to a i  remember we have m variables all  together  that means i take the  variables which i have not yet preset  um  and i allow them to range over all  possible zeros and ones and i add up  the  the formulas values for all of those so  there's a one every time i satisfy and a  zero every time i don't satisfy so i'm  adding up all the satisfying assignments  subject to these i presets  okay  so here are two critical facts about  this  number sign notation  first of all  if i preset the first i values to  something  now i can  in addition  set the next variable either to zero or  to one  and i get this  relationship  which is just simply a generalization of  the fact that  the total number of satisfying  assignments of the formula  is equal to the number of satisfying  assignments when x1 is 0 plus the number  of satisfying assignments when x1 is 1.  right they together have to add up to  the total number because x1 is going to  be the either zero or one  so that's fact number one  fact number two is that if i preset  everything  all of the variables so there are no  variables left then the number of  satisfying assignments  subject to that preset  of everything is just whether or not  i've satisfied the formula which is the  value of the formula on that those  presets  okay  both two simple facts but it's going to  be critical in the protocol i'm about to  describe  questions on this  i think actually i do have a question  for you  so let's just see  what do you think  just to check your understanding  okay  got about 80  getting this  um  i'm not sure that's good but uh  all right almost done  closing  okay  okay so uh yes a is the correct answer  you know if if there are nine satisfying  assignments all together and there are  six satisfying assignments where the  first variable is set to zero  then there's only three satisfying  assignments with the first variable set  to one because  nine has got to be equal to six plus  three  that's actually this fact number one  it's not going to be 15  this is not true either  so  it's just a  good  okay  okay so let's try to  with with that knowledge let's try to  see how we can put  number sat in ip  so this is not going to quite work  but it's really going to set us up to do  this to finish this next time  um  so  you might immediately see where this is  going wrong but you'll  you'll have to put up with it  because um  the setup is what's important  um  okay  so understand now here's the  here's the the setup we have  um  the input is a formula  and  a number  where that number is supposed to be the  number of satisfying assignments  you know it could be wrong and in which  case we're not in the language  um but if it's right you're in the  language  so the approver is supposed to convince  the verifier  that it's correct if it is correct  and it's not gonna it's gonna fail no  matter what it tries to do  if  it's uh not  correct so this is the approver is going  to send first of all  um  so  so the proof is going to send a claim  about the number of satisfying  assignments  going to send when i say this value here  this is what the prover  if it's honest is going to send the  right value of course the verifier does  not know if the approver is honest  but i'm describing how the honest  approver is going to operate and we'll  have to understand what happens if the  prover tries to cheat  so the proof is going to send the honest  proof is going to send the number of  satisfying assignments all together and  the prover verifier just makes sure that  that matches up with the input  if it doesn't match up with the input uh  the verifier is just gonna  you know you know the the verifier is  gonna not be convinced that the input is  um in the language  so it's gonna just uh reject at that  point  um  okay uh  then now the verifier says okay that was  very good that you sent me this how do i  know that's right  so what the verif prover is going to do  to try to convince the verifier that  this value was correct  is  uh  unravel that by one level by say well  you know there were nine satisfying  assignments all together uh six of them  were when the when x one is zero and  three of them were when x one is one  to verify what does the verify have to  check  that  these add up correctly when i preset x1  to zero and to one  it had better add up to the total number  of satisfying assignments  if that works out the verifier is happy  is still being it's still consistent  with being convinced that this k was the  right value  um  so um  the next step is  well the verifier says well how do i  know those two values are correct the  prover says okay well i'm going to prove  send  un unravel them one level further  then here's a number of satisfying  assignments when the next variable is  set to both possibilities for each of  the possibilities of the first variable  now if you're  understanding me  about what the prover is sending you  should start to be getting a little  nervous  because  something is i mean this is going to be  correct  but it's going to start it looks like  it's starting to blow up in terms of the  number of  amount of work that's involved and  that's actually a problem  but let's bear with that for the moment  let's just worry about correctness not  about complexity for the moment  so  the proof is going to now send the  number of satisfying assignments for  each of those four possible ways of  presetting the first two variables  and the verifier is going to check that  that was consistent with the information  the prover sent in the previous round  right by again checking this identity  here so then the proof is going to  continue on doing that until it's got  it's done that  through m rounds where m is the number  of variables  so at this point  the proof is going to send  all possible ways of presetting all of  the variables so that now there's two to  the m possibilities here  again this is hopelessly  not allowed but okay ignoring that  the proofer's got to use this at the nth  round to check what happens at the at  the previous round so that's when they  were m minus one values sent because  each one has one more  uh  uh you're extending the presets by one  so which using this to check that the  previous round was the values were  correct so it's looking for  um  you know the m minus one  presets  have to add up correctly um you know in  terms of the m the presets of m uh  values  uh for each of those ways of uh doing  those uh m minus one presets  and so now the provers send all of those  two to the m  counts which are by the way ones and  zeros because at this point we have  preset all of the  values of the variables and so there's  only one possible assignment at most  that there can be  and now  the verifier  the approver is done the verifier is  going to check  by itself  that these values make sense that these  values are correct  so it's going to do that by looking back  at the formula  so far up at this point the verifier has  not been looking at the formula it's  just been checking the internal  consistency of the provers messages with  each other  but now at the end the verifier is going  to take the messages these these values  that the approver sent for each of the  two to the m presets and see if it  matches up with what the formula would  do  remember that was the other sort of the  base case  of the the end the fact number two  um from the uh slider to ago  make sure that these agree  okay and now the verifier says well okay  if everything has checked out and all of  these are  are in agreement  then the verifier is  going to be convinced  that um  that uh fee had k satisfying assignments  but if anywhere along the way one of  these checks fails  the approver is not the verifier is not  going to be convinced and is going to  reject  okay so in a sense this is kind of dopey  you know you know we've just  i'm just kind of giving you a  complicated way of just counting up one  by one each of the satisfying  assignments of the formula and seeing if  that matches k  but nevertheless  this way of looking at it is gonna be  uh  help us to understand  um the way to fix this so  so bear with me for another minute on  this one  so  another way of looking at this which i  think is is particularly useful  is to think of  what happens  well okay  we'll get there in a second i want to  look at what happens if k was wrong but  before i do that let's look at the  i'm going to give a kind of a graphic a  graphical  view  of um  the information that the prover sends  and the and the verifiers actions in  this protocol so the the values that the  prover sending are going to be in yellow  so the  and the information that the verifier  has or checks is going to be in white  so the the verifier has the k the the  input value which is supposed to be the  number of satisfying assignments  and the prover  sends some value and the verifier checks  that this value which is supposed to be  the number of satisfying assignments  corresponds with k  so that's one of the checks it does  then the approver is going to send kind  of take to justify this value  it  um sends  the number of satisfying assignments  when you have  x1 set to zero  or set to one  the verify adds those up to give you  and it's supposed to equal the total  number of satisfying assignments and so  this is if you understood this protocol  this is just i'm writing it out in a  sort of a simplified way perhaps  okay  and so um  keeps checking that these things add up  correctly until you get down to  setting all m values in all two to the  impossible ways and now the verifier is  going to then check to make sure that  that equals  the what the formula would say  um  okay  okay so now  let's what happens  if  k was the wrong value  it did not agree with the number of  satisfying assignments  um  and  what does what happens now um  could the prover  what happens if the approver tries to  make the verifier accept anyway  so  um  so the only thing the prover can do at  the very first step would be to lie  um about  you know if the approver sends the if k  is wrong  and the proof approver sends the correct  value  for  the the the total count the verifier is  going to reject so i'm trying to see  is it could the approver try to make the  verifier accept what what happens  so the prover has to lie here and i'm  going to indicate that by  saying the approver is sending in  the wrong value  for um  uh the the total count  well if the proof is going to lie here  um  then  just like you know if you  you know you have a child  who um  tells a lie  and then you start you know as the  parent you start asking questions to try  to  see if the story is consistent  one lie is going to lead to another lie  um and that's what happens here  if the uh  in order to justify this lie  um the proof is going to have to a lie  in one or perhaps both but at least one  of these two values because you can't  have the two correct values adding up to  the incorrect value  because you have to think about what's  going on here  so if this is a lie that's going to  force a lie  at one of one side of the other one  level down which is then going to force  a lie  to propagate down and so there's a lie  at every stage is going to force a lie  at least in one  one place or another to propagate all  the way down to the bottom  and then at the bottom the verifier will  see  that the check doesn't work  as when it turns when it tries to  connect it up with the formula itself  and the form verifier will reject  okay so just a way of looking at this  um if the form if the value  if the input was not in the language  um so uh  but the problem is  that as i said that this is exponential  so how are we going to fix that so  just looking ahead to what we're going  to do on tuesday  okay let's see if there's any questions  here first of all  uh  okay yes i got a question should this be  uh  should this be a minus i i purposely  made this bracket only go  not include the very last zero yeah  there's a total of m zeroes here all  together but the  i left out the last zero that's why i  said n minus one maybe it would have  been better to say m  okay so and you got another interesting  question here why can't we reject right  away if k is wrong  um  uh  well  the verifier is probabilistic polynomial  time how does the verifier know if k is  wrong  um  so it means or or or right  so  what we're trying to do is something  like you know like np where we have a  certificate but now we have this kind of  interactive certificate in the form of  disprover maybe that's another way to  look at it  um  where if you're in the language there  should be some way for the prover to  convince to get make you accept but if  you're not not in the language there  should be no way  for the proverb to make you accept  um  uh  so the verifier just can't  reject right away because there's no way  to tell how does the verifier know it's  going to start rejecting things when it  shouldn't if it's just going to be  rejecting willy-nilly here  um  okay how does the verifier need to  determine if the prover is  internally consistent instead of just  asking  so why does the verifier need to  determine if the approver is internally  consistent instead of just asking the  questions in step n plus one  yeah so maybe that's  because it looks like all of the work is  happening at the very end  um  but i'm really presenting this to you  as a preparation  for what we're going to do on tuesday  um so it's important to think about the  connection from each step to the next  each step is going to be justified by  what happens at the next step  until we get to the very end  so you have to just understand it for  what it is don't try to make it more  efficient yeah i realize this is kind of  dumb  um  good point we're not using the  probabilism here  um  and  moreover we're not really even using the  interaction here the prover is doing all  the sending the verifier is just  accepting at the end  yeah  this is we're not using the power and  we're getting a weaker result so let's  move on before we run out of time here  so how do we gonna fix this  so the problem is is blowing up  each to justify each stage where  each value  we're  um  needing to present two values which add  up to it  um  and that's uh good leading to a blow up  now it would be nice if we can do  something where each value was  supported by just a single value  at the next level  so we know here's an idea well you know  in order to understand to see that  uh that that this total count is correct  why don't we just pick at random either  zero or one and only follow that one  down  well the problem with doing that  is because  the the  the sequence of lies is could be just a  single path through this uh tree  and the chances you're going to find  that path  down to a contradiction at the bottom is  very low if you're just doing it at  random  um so just randomly picking zeros and  ones as as the as the one you're going  to justify used to justify the previous  value is not going to be good enough  so what  but this is what we're going to do  however the values that we're going to  pick  for um  for these random  um  inputs  are not going to be boolean values  we're going to pick  non-boolean assignments to the variables  which again just as with the branching  program case didn't make any sense  on the surface of it we're going to have  to make it make sense  and  we'll have to see how to do that on tues  in tuesday's lecture  okay so that's kind of the setup  okay  um  yeah so in a similar question why is  this any different from just  non-deterministically guessing the  assignments it's because of this we're  really setting the stage  okay so what we did today was we  introduced them the the model and  defined the complexity class  we can we did show this one in its  full glory we showed that non-iso is an  ip  really worth understanding this  uh protocol here making sure you  you're comfortable with that  and and also the the model itself  and so for tuesday's lecture we're going  to finish this up uh well we started  showing that number set as an ip which  is what we need to to do to prove co-np  is an ip  and we'll finish that  next time  which will be our last time  okay so that's it for today  i'll stick around for questions  so  a good question here why can't um  v just  reject if some of the checks are  incorrect yes v could as soon as  there's a check that fails we can just  reject at that stage  i'm just trying to argue that at some  point along the way if the input is not  in the language  there's going to be a check that fails  so i mean i said reject at the end but  yeah i mean  you could have rejected at any point  along the way  um  okay  um someone's asking for what what role  did i play uh so i did  my my own personal role in this i were  to which was twofold first of all i i  came up with the idea  uh of  well not the idea i came up with the  name  interactive proof uh  i remember when sylvia mccauley was  explaining this to me in my apartment  many many years ago he had a kind of a  little bit complicated and i don't even  remember what the protocol was for it  was not for something simple was  something involving uh prime numbers  and i said oh  that's a kind of an interactive proof  and it's and it it stuck from that point  on so  um that was one thing but the other  thing in terms of more mathematically  i my role was so uh  shafi goldwater and i approved the  equivalence of the two models the public  coin and the private coin version  um so that that was uh  my uh  my role in this back and when this was  all first coming out  approved it approved it on an airplane  on the way to a conference somewhere  so i think we're going to  notice any other questions  i think we'll head out take care  everybody  see one  see you on tuesday  bye bye  you
 

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hi everybody  um  glad to have you all back  for our next to last installment of  theory of computation  today um we are going to embark on the  very last  big topic for the semester  and that is in some ways going to be  following on what we  started  a couple of lectures back when we looked  at probabilistic  turing machines  and probabilistic computation  and its associated class bpp  now what we're going to  discuss  is in some sense  a probabilistic version of np  and that's going to be a complexity  class called ip  which stands for interactive proof  systems  um and so we're going to present that  model  and look at a couple of examples  uh  i would  just like to say at the beginning that  the  this model is a very important one it's  really uh has been the starting point  for a great deal of  research  in complexity theory so we're just  really going to be touching on it but  there's a lot more that people have  pursued with this model and it's also a  connection in to the cryptography field  which also makes use of the interactive  proof system model in fact  some of the  genesis of that model comes out of  cryptography  where you're having uh  multiple parties either communicating or  in some ways  interacting  uh to achieve certain uh goals of  communication or  signing or passwords or or or what have  you so uh this is a both a an applied  area and also one that has a lot of very  interesting theory uh um associated to  it  so with that one we we're going to jump  in um and uh  start out by uh  making myself smaller  and  just do a an introduction i'm going to  introduce  the model or the the concept of an  interactive proof  with an example  and that example involves the  graph isomorphism problem  that's the problem of testing whether  two graphs are isomorphic  what we mean by two graphs being  isomorphic is that they're really just  the same graph  with one of them perhaps being  re-labeled or permuted  so that  they may look superficially different  they may appear with a different  sequence of labels or the nodes are  appearing in a different order  but  except for that it's really just the  same graph so i'm kind of illustrating  that here if you can see those two  graphs here which look different from  each other both on eight notes they are  in fact the same graph  as i can illustrate by a little  animation which will convert this one  into that one  so uh  the  um  [Music]  so the  two graphs um  these graphs being the same um they're  we call that isomorphic  so these are graphs g and h and they're  really the same graph so we we um call  them isomorphic graphs  and we  are  have an associated computational problem  called iso  which is  given a pair of graphs we'd like to know  are they isomorphic or not so the iso is  the collection of pairs of graphs which  are isomorphic  and  it's easy  to see that this problem is an np  problem  because  all you need to do in order to see or to  give a certificate that the two graphs  are isomorphic to each other is tell you  it's just to say which nodes in the one  graph correspond to which other nodes in  the other graph  um and then you all you need to check is  that the edge relationships are  consistent with that  mapping or that isomorphism as it's  called  um  so it's easy to see that the iso problem  is an np  and if you're not getting that make sure  you uh understand because this is the  whole first part of the lecture will be  lost if you don't understand this iso  problem  now the question of whether  you can test two graphs  being isomorphic in polynomial time  is not clear and in fact that's an  unsolved problem to this day  uh  and it's a problem that has generated an  enormous literature uh  there are hundreds of papers  on the graph isomorphism problem as it's  called um to try to uh resolve  um you know uh to try to see if one can  find a polynomial time algorithm and in  fact it was a very big result  just in the last 10 years where there  was a sub-exponential algorithm given so  that was more  than faster than the brute force search  approach but didn't get it all the way  down to polynomial now why is there so  much attention just to this one  particular np problem it's because  it's not known whether the graph  isomorphism problem is np-complete iso  is not known to be an np-complete  problem and that puts it into a very  very small class of problems in np  which are no not known to be either in p  or np-complete  it's kind of a curiosity that for np  problems  almost all of them have ended up being  in one side or the other  and in fact  um it's a  uh  um  in fact the  the um  i i think it's the only problem that  just involves graphs  that's not known to be either in p or an  np  um  so so i got a question here  what would be in between exponential and  polynomial for example i don't remember  what the the bound is but it's it's  something in the uh  in the range of n to the log n  uh  time complexity for the graphite some  more i might be getting that wrong um i  don't remember exactly what the bound is  but um  that's significantly better than two to  the n or some  some exponential amount of time but it  it's more than enter any constant so  it's more than any polynomial time  so um  another question uh of the same sort is  whether the complementary problem is in  np or whether whether iso is in co-np or  let's let's talk about it in terms of  the complement whether the  complement of iso which i'll refer to as  the non-iso problem whether that's known  to be in np so that's also not known um  in other words if i give you two graphs  and i ask you to show that they're not  isomorphic suppose they aren't  isomorphic and you go through the effort  of you know determining that by a brute  force search and now you want to  prove  that they're not isomorphic well that's  not  it's not known to be an np either so  there's no known  short certificate certificate of  two graphs not being isomorphic  we don't know how to do that either  but there's something that's very  interesting nevertheless  um  and  it has to do with the ability to for one  party to prove to another  that graphs are either isomorphic or not  isomorphic  so if you're just having like a prover  we haven't really been necessarily  formulating that this way so much in  this class but this is a completely  equivalent way of formulating the notion  of np whether you have a polynomial time  verifier an approver who can produce  certificates say it's a powerful prover  so um if you have a problem that's in np  approver can convince a polynomial time  verifier  that uh strings are in the language if  in fact they are so in the case of the  iso problem a prover can convince a  polynomial time verifier the graphs are  isomorphic just by exhibiting the  isomorphism  now  for the non-isomorphism case  we don't know that that problem's in np  but it's still possible  for approver to convince a verifier that  graphs are not isomorphic  if you change the rules of the game  slightly  um  so even though the non-iso problem  is not known to be an np  approver can still convince a polynomial  time verifier that graphs are not  isomorphic assuming they they are in  fact not isomorphic um provided  that  um  the prover and the verifier can interact  with one another so the verifier can ask  questions of the prover  and the verifier gets to be  probabilistic  so that's in this  in that's in sense in which i mean  that this notion is a kind of a  probabilistic version of np  okay so um  let me  show you how that's done  so before we jump in to the  uh to the method for  approver to um show a verifier that  graphs are not isomorphic  um let me let let's try to get a little  clearer on the model so i'm going to  first show it to you informally and then  we'll look at it formally  okay so  in in interactive proofs there are two  parties  um  and i'm going to think about them as one  of them is going to be the professor  okay so the professor is going to play  the role of the verifier in a sense but  it's like that the one who checks  um  and uh the professor being kind of old  and tired he's been teaching too long  maybe  can only operate in  probabilistic polynomial time  so the professor if wants to tell  whether two graphs are isomorphic or not  probabilistic polynomial time doesn't  seem to be enough  to tell whether two graphs are  isomorphic or not because it seems to be  a more than polynomial problem  however  the professor  has  um  help  it has  an army  of  graduate students  and the graduate students  they're not limited uh in the same way  the professor is the graduate students  are young  they  are energetic they can stay up all night  they know how to code  so  the graduate students  have unlimited computational ability  so that we're going to think of the  graduate students playing the role of  the approver because they're not  they're not limited in their  capabilities we'll assume  the professor on the other hand is  limited so the professor wants to know  if the two graphs are isomorphic  let's say whatever they are  um  can't do it by himself so he's going to  ask his students to figure out the  answer  and report back  now there's only one problem  the professor knows that students  uh  well in the old days they'd like to  party i guess these days they like to  play on uh  play computer games a lot  and so they're not really that eager to  spend all their time  figuring out whether graphs are  isomorphic  so he's worried that the  the students will just come up with some  answer and figure that he won't be able  to tell the difference  so the professor does not trust the  students it's not enough to he for the  professor to give the problem to the  students and just take any answer that  they're going to give the professor  wants  wants to be convinced  okay so  um  now  how could the students convince the  professor of the answer that they've  really done the work and figured out  whether the graphs are isomorphic or not  well if the graphs are isomorphic if it  turns out that the graphs were  isomorphic and the students figure that  out  then life is good  because what are they going to do to  convince the professor  they're going to hand over the  isomorphism and show yeah i mean they  are  you know those graphs really are  isomorphic and here's how the  correspondence works professor can check  oh yeah i i now not now i'm convinced  but suppose the graphs were not  isomorphic  what are we going to do then  um the students have figured out where  after night else the professor wants  wants to be convinced oh no what are we  going to do  well in fact we're going to engage the  the professor and the students are going  to engage in the following protocol  dialogue  what's going to happen is  now you have to make sure you're you're  this is critical to follow to understand  this little  part of the story here because it's  really going to set the pattern for  everything in today's and tomorrow and  to today's lecture and the and the next  lecture  okay so  we're going to engage in a following  interaction between the students and the  professor which is going to enable the  students to convince the professor that  the two graphs really are not isomorphic  so how is that going to work  this is a beautiful little uh thing by  the way  so the professor is going to take the  two graphs  and pick one of them  at random  because the two graphs g and h  um let's say they're not they really are  not isomorphic the professor doesn't  know that for sure that's what the  students claim the professor really  wants to not be convinced that the  students are right  um so the professor's gonna pick one of  the two at random  randomly permute that  uh that choice the one that he picked  and hand it over to the students  say okay  here is one of those two graphs randomly  scrambled  then i'm going to ask the students  which one did i pick  okay now  if the graphs were really not isomorphic  the students can check  whether that randomly scrambled graph  is isomorphic to either g or to h it's  going to be isomorphic to one or the  other  and then they students can figure it out  and they say oh you picked g or no you  picked h  as the case may be the students can  figure that out  but if the graphs were isomorphic  then that  scrambled version of g or h could  equally well have come from either of  them  and the students would have no way of  knowing  which one the professor picked  so that there's nothing they could do  which would be better than guessing  so if we do that a bunch of times the  professor picks at random sometimes  secretly of course the picks the grip  picks either g or picks h  and the students get it right every time  either the students are really doing the  work  and the graphs are really not isomorphic  or  the students are just incredibly lucky  they're managing to guess right let's  say a hundred times  so how the  the the professor randomly and secretly  picks grh uses this uses its probabilism  flips a coin  just a two-sided coin and says okay  sometimes if we're going to do g  sometimes they're going to do h just  completely at random  picks one or the other  and then with some more randomness  gets finds a random permutation of the  one that he picked and then sends that  over to the students and say which one  did it come from  um  i'm not sure okay so let's pause here  let's let's make sure we all understand  this because this is really important um  so i'm getting a question here how do we  i'm not sure what your question is um  okay so let me just say yeah the  professor's going to play the role the  verifier the graduate students play  they're all approver that's coming but i  really want to understand this protocol  here okay so how is the professor  picking the graph skin if you're  okay i don't you know picking the graphs  at random you have just two graphs  they're in part of the input uh the both  the students and the professor can see  the graphs  and the professors are just picking one  of them at random using a coin so i'm  not sure i understand the question there  could p and v engage in a protocol where  the secretary is on the prover side  instead  the question of revealing the  isomorphism i there is no why so i'm not  sure i understand this question either  um maybe we'll make this clear  you know if  for  for this little illustration  the professor  doesn't know the graphs could be  isomorphic or they could be not  isomorphic  and so uh the professor wants to be  convinced either way whatever the  students whatever answer the students  come up with we're going to shift this  into a problem about a um deciding a  language  next but right now i'm just trying to  give a sense of the how the model works  i want to move from this informal model  and now i'm going to formalize that in  terms of  model which will be deciding a language  okay so  so the interactive proof  system  model  we have two interacting parties a  verifier which is probabilistic  polynomial time playing played by the  professor in the previous slide  and the prover which is unlimited  computational power played by the  students in the previous slide  both of them get to see the input  which in the previous case  well  it could be for example the pair of  graphs  they exchange a polynomial number of  polynomial size messages so the whole  exchange  including the verifier's own computation  is going to be polynomial the only thing  that's not not not included within the  computational cost is the prover's work  which is unlimited  um  after that the verifier after the  interaction the verifier will accept or  reject  and we're going to define the  probability that the verifier together  with a particular approver ends up  accepting  as  you  look over the different possible coin  tosses  of the verifier  which could lead to different  behavior on the part of the verifier and  therefore different behavior on the part  of the approver  so over all the different possibility  possibilities for the verifiers  computation  we're going to look at the probability  that the verifier with this particular  approver ends up accepting and i've  written it this way this is the  probability of the verifier interacting  with the prover accepts the input  is just simply that  um and so we're going to work through an  example  we're going to  work through the previous example  more  precisely in a second  the class ip for interactive proofs  stands for it's the class of languages  such that for some verifier and approver  um  for strings in the language  the prover makes the verifier accept  with high probability  and here's the interesting part  for strings not in the language the  prover makes it except with low  probability but every there's no prover  which can make it except with high  probability  so there's no way to cheat  if you think about it in the case of the  graphic non-isomorphism  there's nothing you know  if  if the graphs were really isomorphic  and the students were trying to in a  devious way  prove  through that protocol that they're not  isomorphic they would fail because  there's nothing they can do  if the graphs were isomorphic  then um  when the  verifier the the professor picks one or  the other at random  um and scrambles it the students would  have no way of telling which one the  professor did so no matter what kind of  scheme they try to come up with  they're going to be out of luck  so  it's no mat for any strategy  for strings that are not in the language  for any s any prover calling that p with  a tilde to stand for a devious or  crooked  prover  for any uh possibly crooked prover  even that with working with the verifier  is still going to end up accepting with  low probability  so strings in the language there's going  to be an honest prover who just follows  the protocol  in the correct way which makes the  verifier accept with high probability  for strings not in the language every  prover  is going to fail  to make it accept with high probability  um  okay so that i mean the way i like to  think about it is that p tilde is  a possibly crooked proverb which is  trying to make the verifier accept when  it shouldn't because the string is not  in the language  it's like you know  it's like even you can think of this in  the case of um satisfiability  um  you know  you  a crooked prover might try to convince  of the verifier that the formula is  satisfiable when it isn't by by somehow  trying to produce a satisfying  assignment but that's going to be  impossible there's nothing any strategy  can possibly work when the formula is  not satisfiable if that's what the  verifier is going to check it's going to  be looking for that satisfying  assignment  okay  and by the way this is we're not going  to prove this but it's really going to  be proved in the same way you can make  that one third error that could that  occurs here  something very tiny  by  the same kind of repetition argument  okay  so let's see um  so why can't the prover in the first  case be crooked um  the prover in the first case would could  be crooked but that's not going to serve  the purposes um  you know what what we want to show  um  you think about it like we think about  np  for strings in the language there exists  a certificate there is a proof  that you're in the language  so if somebody is going to  not produce the proof  that's irrelevant  the question is if you look at the best  possible case the best possible prover  um  you know who's going to be able  we're asking does there exist a way to  convince the verifier that the um  string is in the language  so it doesn't matter that there might be  some other  uh  silly way that doesn't work we're just  looking at the best possible way  so the best possible way when you're in  the language is going to end up with a  verifier having high probability  when you're not in the language the best  possible way is still going to end up  with low probability  when when i talk about best possible i'm  trying to maximize the probability that  the verifier is going to end up  accepting let's continue um not sure as  clear as i would like but  um  maybe again we're going to we're going  to stick with that example because this  is a very uh  helpful example and to try to  understand the setup  and uh so we're gonna i'm gonna revisit  that previous example about  non-isomorphism but now in the context  of this  thinking about as a language  so we're going to take this  non-isomorphism  um  uh yeah  we're going to take the non-isomorphism  problem  and show that it's an ip so there's  going to be a verifier together with  approver which are going to make the  verifier accept with high probability  for strings in the language namely  graphs not ice being isomorphic  and  nothing there's going to be no way to  make the verifier except with high  probability for strings out of the  language therefore that's when the  graphs are isomorphic  okay um so the protocol is just gonna  we're gonna repeat the following thing  twice  you know i said in the previous case do  it a hundred times just to  help us think about it but actually  twice is gonna be enough to get the  bound we need  so the verifier is going to operate like  this in terms of this is the verifiers  in first  communicating  sending messages to the approver it's  going to randomly choose grh  just like what the professor did last  time  randomly permute the result to get a new  graph k  which was going to be which was  which is isomorphic either to grh  depending upon the choice the verifier  made  and then send that graph k  now the provers turn is going to respond  by the proof is going to compare k with  the two one of the both of the original  graphs it's got to be isomorphic to one  or the other  and it's going to report back which one  just going to say well you pick g no or  you picked h  because the prover with its unlimited  capabilities can determine that  um  and then v accepts if the approval was  right both times  um  and if the approval was ever not right  the verify says oh something's fishy  here because we know that the prover has  unlimited capability so could get it  right if you know um if there was if  this was an honest approver um and so um  if uh  if it's not getting it right then the  verify is going to reject  so if the graphs are not isomorphic the  proofer can tell which one it picked  randomly so therefore  if the graphs are not isomorphic the  verifier with that that honest prover  will accept with probability one  because that honest proof is always  going to get the right answer  which is at least two-thirds is what the  bound we need  uh we don't care about the space used in  answer to a question  um if  we were not in the language so g and i h  are not isomorphic  then there's nothing any crooked pervert  could possibly do  because it gets a graph  can't tell there's no way to tell  whether it came from g or came from h  um  so that crooked proofer would have all  it could the best thing you could do is  guess sort of a 50 chance advancing  correctly each time and only a 25 chance  for doing it twice  and that's why i did it twice in order  to get that error um  uh  to be small so there's only a 25 chance  of the approver getting lucky so that  would be an  error case  if the proof of just by chance picked  the right answer twice even though the  graphs were isomorphic so therefore  for the isomorphic case  the verifier interacting with  any prover is going to accept that input  with at most one quarter 25 of the time  which is less than a third so those are  that's just to achieve that bound  okay  so  let's let's answer some questions first  and then i'll try to um  uh  i'll ask you  you understand this so this it's i think  it's worth trying to understand this  model um of this interactive proof  system  it's a little little slippery i i  realize but um  if you just hold your hold on to your  intuition of the prover trying to  convince  you know of uh  you know a powerful prover trying to  convince a limited verifier  um  of  some  string being in a language  uh  you want the proven to be able to  succeed when the string is in the  language but fail when the string is not  in the language  yes we are going to somebody's asking if  the prover is identifying grh by brute  force yes the prover is going to use its  unlimited capabilities to determine  given k  whether it came from g or h  the um  the computational  cost of the approver is irrelevant for  this it's just like when we think about  a certificate  um you know for satisfiability we don't  talk about the cost of finding that  certificate  uh for np  for ip again we don't talk about the  cost of the prover running  so somebody's asking does the crooked  per prover answer just randomly or does  uh can the cr quick approver has have a  strategy the crooked perimeter can have  a strategy  now we're not we're assuming the crooked  approver is devious  but it's still going to fail  okay um let's do the check-in  suppose we change the model  so that the prover can watch the  verifier picking its random choices  so uh the verifier cannot act in secret  anymore but the the prover can watch the  verifier  now let's suppose we had the same  protocol that i just described  what language do we end up with is it  the same language different language and  what is that language  okay so  i want to hopefully  um  let's let's  it'll  give me some sense of how well you're  following me by how well this uh this  goes  yeah someone's asking about how this  connects up for example with np so we're  going to look at that  also in a second  okay so this is reassuring that most of  you i think are on the right track at  least for this check-in uh do we assume  p uses this access to guess right  what access p is not really guessing the  p is tr is actually i don't think a p is  non-deterministic or anything like that  p is actually trying to get the right  answer  and using its  computational ability to do that if it's  possible  may not be possible then there's nothing  you can do  okay so let's uh end this are you all in  two seconds left  please vote  vote now or never  okay ending  um  yeah so c is the correct answer here  if the prover can watch what the verify  is doing the prover can see  what graph the verifier picked right  from the beginning  and so the prover without having to do  any work  can say you know it proves looks over  the verifier's shoulder and says oh you  picked g  and now you're randomly permitting it  but i don't care about that i just i  know you pick g  uh so uh the  proof is going to respond back uh g even  if the graphs were isomorphic the proof  is going to be able to get the right  answer  kind of interestingly um  uh  you can make a you can change the  protocol somewhat um to make it uh that  even if the  prover  has access to the verifier's randomness  you can still achieve this but not with  the same protocol  um  so that's uh that's a separate question  um okay so let's move on here i want to  get too bogged down  okay here's another check-in um  uh  okay so  you have to tell me which of the  following  statements are true  as far as you know  now you have to think about a little bit  how this  these uh  relate  to um how np and ip or bpp  and ip relate to one another  okay  how are we doing on this  okay so we're gonna have to close this  pretty soon too  do the best you can  interesting okay  closing up shop  last last vote  okay one two three  there's one more person out there who  hasn't voted who voted last time oh well  uh  all right  in fact they're all true  um  let's see why is np contained with ip  contained in ip uh well many of you have  seen this already so let's just quickly  go through it um  [Music]  if we just had a deterministic v  um  you know the uh  you can  uh  maybe it's just  that that can be enough of deterministic  v  i think it's just going to be equivalent  but actually just to be doubly sure the  deterministic v and the approver just  sends a message to the verifier and then  checks it  that's the way we normally think about a  certificate for np  uh i don't think it's going to change  anything but should double check that if  the verifier can still ask questions but  i think as long as the verifier is  deterministic you're going to get  exactly np  here  um  and um  now how about bpp  well  there you don't even need approver  because the verifier is already  probabilistic  so  verifier can ignore the approver  and this one is a little tricky uh i p  containing p space because we haven't  covered that so there's no way for you  to know that unless you happen to read  ahead in the book but it's in fact true  um in some ways it's a little bit like  the proof that  uh  np  is contained in p space ip is sort of an  enhanced version of np  and you and there's just a basically a  uh a piece based brute force algorithm  that goes through the entire tree of  possibilities um of the verifier  exchange  verifier with exchanges with approver  and can uh determine that um the  verifier is either going to accept for  some approver or is going to end up  projecting for every approver um  so we're not going to prove this  statement but something good for you to  know anyway they're just a fact  uh but we're going to do what the  surprising thing  in reference to part c  is that the containment also goes the  other way this is the amazing  um uh  was is an amazing result  that  everything in p space  you can do with in ip  so this is ip is actually turns out to  be incredibly powerful gives you  everything up in p space you get i p  equals p space  so that says that any problem that you  can solve in p space like any of the a  game for example  um if you can imagine you know  formulating you know checkers or  chess  as a piece based problem which you know  depending upon some details of the rules  you can do because you know you have to  generalize it to an end by end board but  okay  let's not quibble um  uh then  uh  um  [Music]  we don't know which side has a forced  win in chess  um and even if somebody goes through the  effort of going through the game tree  um and determines that let's say white  has a forced win  uh there's no way for them to there's no  short certificate we don't know that  that problem is not an np  but  by  going through an interactive proof  an all-powerful prover  could still convince somebody that white  had a force  you know convince somebody in polynomial  time that a  white has a forced wind let's say in um  in chess  again little  uh stretching things because this is  you know you really need to talk about  this as an n by n not an eight by eight  but i think in this the spirit is is uh  fair  so um  okay uh so let's continue  so wha  when we're not going to quite prove that  that p base  is contained in ip we're going to prove  a somewhat weaker statement but very  similar  um  is that uh and historically came first  that co-np is contained in ip  so not only is np contained in ip but  we're going to prove that co np is  contained in ip and this actually has  most of the i most of the idea for the p  space being contained in ip and itself  it's just an amazing proof  a little easier  um  okay  and th this was done  if i'm remembering somebody's asking me  when how old is this it's something in  the 19 i think late 90s but i'm not i  don't remember but maybe early 90s i  think it's late 90s  when this was shown  so it's been a while now  um  okay  so  um  yeah so in terms of the relationship  with cryptography there were two  parallel threads  um that both independently came up with  the notion of an interactive purge  system  uh i was a little bit personally  involved with this  in in a way as well but but mainly that  the the there was one group in  cryptography  um working on this and there was another  group who was actually coming out of the  graph isomorphism world  working on it and they they came up with  two separate models one involving the  private randomness and one involving the  public randomness um and it was turned  out that was  that they're actually equivalent  um  and uh  it's an interesting story but  unfortunately we don't have time for it  uh so  why don't we move on and i'm gonna start  showing you how the proof that  co-np  is contained in ip  goes and what we're going to do is work  with a problem that's  almost like co p complete but um  going to be uh well it's going to be  this number sat problem we'll see the  connection with co-np in a second  uh  so cohen p  so it's supposed to be exactly k  satisfying assignments  fee comma k  is the set of  pairs where the formula fee has exactly  k satisfying assignments so really  that's a problem of counting how many  satisfying assignments you have in a  formula  um  so you know for  np you have at least one  um but i'm i want to know exactly how  many  uh so the numbers that problem is the  pairs formula and the count  so  um  and uh so if we define the count  number fee is the number of satisfying  assignments of a fee  then another way of writing this uh  number stat problem  is  uh  the pairs phi k where k is the number of  satisfying assignments of fee  so we're going to be using this notation  number fee a lot so just make sure you  get you got that notation this is the  number of satisfying assignments of that  formula  okay  and here's a definition i probably  should have given you earlier in the  term but better late than never um  uh so the notion that a language is np  hard  it's like  np complete  except without being  necessarily being in np  so this is just the reduction part a  language is np-hard or co-np-hard or  p-space hard or any of those other  classes that we looked at  if every problem in the class is  reducible to that language  but you don't know whether this that  that language is in the class  so we just call it np hard instead of mp  complete  so you could say  the language is np complete if it's hard  and it's in np  okay and so we're going to show that  this number set problem is co np hard  so everything in cohen p is polynomial  time reducible to number set that's easy  because what we're going to do is take a  co np complete problem  which is the  unsatisfiability problem the complement  of satisfiability and sure that reduces  to the numbers that problem  and that's easy because  a formula is unsatisfiable exactly when  it has zero satisfying assignments  so if you can tell  how many satisfying assignments  something has exactly  or you can answer the question you know  does a formula have exactly  a thousand  satisfying assignments if you can do  that in general then you can solve co  and p uh you can solve the  unsatisfiability problem by asking with  zero satisfying assignments  and that allows you to solve anything in  cohen p  okay  so we're gonna just work with this one  problem the number set problem and show  that that problem's in ip  okay  let's take a quick break  um  okay feel free to send me let me see if  i can catch up with some of the  questions that have been  cropping up here so if the approver  knows the random choices of the verifier  can it can flip the answer to make the  verifier reject  not sure what that you mean in the  context just of the graph isomorphism  problem or  um something in general i'm not sure  you'll have to explain sorry i'll  respond with a question mark what else  can i answer for you guys so i got a  question  might  if i p equals p space does that mean  that iso or the or non-iso might be um  in might be p space complete but no  that's not known  so we're about out of time  okay  let's continue here  [Music]  okay so this this is where we're kind of  getting  going to start to get into the meat of  things um  [Music]  and if you didn't quite understand  everything up till now maybe just try to  keep your intuition  about how do i  you know how does a powerful party  convince a  um a polynomial probabilistic polynomial  time party  of the number of satisfying assignments  exact number not  not at least but you won't know exactly  the number um of satisfying assignments  um so it could be zero for example how  do you convince that how do you convince  someone that there were zero assignments  and and and you know you can have an  interaction which does that and that's  not obvious at all  how you're gonna do that  um  uh all right  so um  okay so we're gonna have to introduce  some notation which i  don't hope this doesn't  cause heartburn here  uh  so  let's say  again here is the the  computational the language we're working  with number set  and we have  a fee where that has m variables  x1 to xm  now  here's our notation i'm going to if i  write phi with this  free phi of 0  that just means the formula that i get  by  plugging in 0 for x one  and leaving all the rest of the  variables uh alone  okay so i substitute zero for x one  where zero means false and one means  true as usual  and but that's it's going it's still  going to be some other formula but just  with that substitution  if i write fee 0 1 that means i preset  the first two variables to 0 and 1.  um  if i write fee with  a bunch of  preset values i'm just setting the first  i variables x1 to xi to some values  and leaving the other variables  as unset  so i'm calling the ones that i'm  nailing in there as i'm already saying  these are the presets  okay so this is just converting some  formulas into other formulas that have  somewhat fewer variables  all right  now let's recall that number notation  the number sign notation number fee is  the number of satisfying assignments  now if i say number fee of 0  that's the number of satisfying  assignments  when i've preset x1 to 0.  similarly if i preset the first i  variables  to some  values  and then i take i want to think how many  how many satisfying assignments subject  to those prefix presets i write it this  way  so i'm going to use this notation a lot  you have to understand this notation ask  if you don't none if you don't get it so  another way of writing it i don't know  if this is helpful but another way of  writing number fee of a1 to a i  remember we have m variables all  together  that means i take the  variables which i have not yet preset  um  and i allow them to range over all  possible zeros and ones and i add up  the  the formulas values for all of those so  there's a one every time i satisfy and a  zero every time i don't satisfy so i'm  adding up all the satisfying assignments  subject to these i presets  okay  so here are two critical facts about  this  number sign notation  first of all  if i preset the first i values to  something  now i can  in addition  set the next variable either to zero or  to one  and i get this  relationship  which is just simply a generalization of  the fact that  the total number of satisfying  assignments of the formula  is equal to the number of satisfying  assignments when x1 is 0 plus the number  of satisfying assignments when x1 is 1.  right they together have to add up to  the total number because x1 is going to  be the either zero or one  so that's fact number one  fact number two is that if i preset  everything  all of the variables so there are no  variables left then the number of  satisfying assignments  subject to that preset  of everything is just whether or not  i've satisfied the formula which is the  value of the formula on that those  presets  okay  both two simple facts but it's going to  be critical in the protocol i'm about to  describe  questions on this  i think actually i do have a question  for you  so let's just see  what do you think  just to check your understanding  okay  got about 80  getting this  um  i'm not sure that's good but uh  all right almost done  closing  okay  okay so uh yes a is the correct answer  you know if if there are nine satisfying  assignments all together and there are  six satisfying assignments where the  first variable is set to zero  then there's only three satisfying  assignments with the first variable set  to one because  nine has got to be equal to six plus  three  that's actually this fact number one  it's not going to be 15  this is not true either  so  it's just a  good  okay  okay so let's try to  with with that knowledge let's try to  see how we can put  number sat in ip  so this is not going to quite work  but it's really going to set us up to do  this to finish this next time  um  so  you might immediately see where this is  going wrong but you'll  you'll have to put up with it  because um  the setup is what's important  um  okay  so understand now here's the  here's the the setup we have  um  the input is a formula  and  a number  where that number is supposed to be the  number of satisfying assignments  you know it could be wrong and in which  case we're not in the language  um but if it's right you're in the  language  so the approver is supposed to convince  the verifier  that it's correct if it is correct  and it's not gonna it's gonna fail no  matter what it tries to do  if  it's uh not  correct so this is the approver is going  to send first of all  um  so  so the proof is going to send a claim  about the number of satisfying  assignments  going to send when i say this value here  this is what the prover  if it's honest is going to send the  right value of course the verifier does  not know if the approver is honest  but i'm describing how the honest  approver is going to operate and we'll  have to understand what happens if the  prover tries to cheat  so the proof is going to send the honest  proof is going to send the number of  satisfying assignments all together and  the prover verifier just makes sure that  that matches up with the input  if it doesn't match up with the input uh  the verifier is just gonna  you know you know the the verifier is  gonna not be convinced that the input is  um in the language  so it's gonna just uh reject at that  point  um  okay uh  then now the verifier says okay that was  very good that you sent me this how do i  know that's right  so what the verif prover is going to do  to try to convince the verifier that  this value was correct  is  uh  unravel that by one level by say well  you know there were nine satisfying  assignments all together uh six of them  were when the when x one is zero and  three of them were when x one is one  to verify what does the verify have to  check  that  these add up correctly when i preset x1  to zero and to one  it had better add up to the total number  of satisfying assignments  if that works out the verifier is happy  is still being it's still consistent  with being convinced that this k was the  right value  um  so um  the next step is  well the verifier says well how do i  know those two values are correct the  prover says okay well i'm going to prove  send  un unravel them one level further  then here's a number of satisfying  assignments when the next variable is  set to both possibilities for each of  the possibilities of the first variable  now if you're  understanding me  about what the prover is sending you  should start to be getting a little  nervous  because  something is i mean this is going to be  correct  but it's going to start it looks like  it's starting to blow up in terms of the  number of  amount of work that's involved and  that's actually a problem  but let's bear with that for the moment  let's just worry about correctness not  about complexity for the moment  so  the proof is going to now send the  number of satisfying assignments for  each of those four possible ways of  presetting the first two variables  and the verifier is going to check that  that was consistent with the information  the prover sent in the previous round  right by again checking this identity  here so then the proof is going to  continue on doing that until it's got  it's done that  through m rounds where m is the number  of variables  so at this point  the proof is going to send  all possible ways of presetting all of  the variables so that now there's two to  the m possibilities here  again this is hopelessly  not allowed but okay ignoring that  the proofer's got to use this at the nth  round to check what happens at the at  the previous round so that's when they  were m minus one values sent because  each one has one more  uh  uh you're extending the presets by one  so which using this to check that the  previous round was the values were  correct so it's looking for  um  you know the m minus one  presets  have to add up correctly um you know in  terms of the m the presets of m uh  values  uh for each of those ways of uh doing  those uh m minus one presets  and so now the provers send all of those  two to the m  counts which are by the way ones and  zeros because at this point we have  preset all of the  values of the variables and so there's  only one possible assignment at most  that there can be  and now  the verifier  the approver is done the verifier is  going to check  by itself  that these values make sense that these  values are correct  so it's going to do that by looking back  at the formula  so far up at this point the verifier has  not been looking at the formula it's  just been checking the internal  consistency of the provers messages with  each other  but now at the end the verifier is going  to take the messages these these values  that the approver sent for each of the  two to the m presets and see if it  matches up with what the formula would  do  remember that was the other sort of the  base case  of the the end the fact number two  um from the uh slider to ago  make sure that these agree  okay and now the verifier says well okay  if everything has checked out and all of  these are  are in agreement  then the verifier is  going to be convinced  that um  that uh fee had k satisfying assignments  but if anywhere along the way one of  these checks fails  the approver is not the verifier is not  going to be convinced and is going to  reject  okay so in a sense this is kind of dopey  you know you know we've just  i'm just kind of giving you a  complicated way of just counting up one  by one each of the satisfying  assignments of the formula and seeing if  that matches k  but nevertheless  this way of looking at it is gonna be  uh  help us to understand  um the way to fix this so  so bear with me for another minute on  this one  so  another way of looking at this which i  think is is particularly useful  is to think of  what happens  well okay  we'll get there in a second i want to  look at what happens if k was wrong but  before i do that let's look at the  i'm going to give a kind of a graphic a  graphical  view  of um  the information that the prover sends  and the and the verifiers actions in  this protocol so the the values that the  prover sending are going to be in yellow  so the  and the information that the verifier  has or checks is going to be in white  so the the verifier has the k the the  input value which is supposed to be the  number of satisfying assignments  and the prover  sends some value and the verifier checks  that this value which is supposed to be  the number of satisfying assignments  corresponds with k  so that's one of the checks it does  then the approver is going to send kind  of take to justify this value  it  um sends  the number of satisfying assignments  when you have  x1 set to zero  or set to one  the verify adds those up to give you  and it's supposed to equal the total  number of satisfying assignments and so  this is if you understood this protocol  this is just i'm writing it out in a  sort of a simplified way perhaps  okay  and so um  keeps checking that these things add up  correctly until you get down to  setting all m values in all two to the  impossible ways and now the verifier is  going to then check to make sure that  that equals  the what the formula would say  um  okay  okay so now  let's what happens  if  k was the wrong value  it did not agree with the number of  satisfying assignments  um  and  what does what happens now um  could the prover  what happens if the approver tries to  make the verifier accept anyway  so  um  so the only thing the prover can do at  the very first step would be to lie  um about  you know if the approver sends the if k  is wrong  and the proof approver sends the correct  value  for  the the the total count the verifier is  going to reject so i'm trying to see  is it could the approver try to make the  verifier accept what what happens  so the prover has to lie here and i'm  going to indicate that by  saying the approver is sending in  the wrong value  for um  uh the the total count  well if the proof is going to lie here  um  then  just like you know if you  you know you have a child  who um  tells a lie  and then you start you know as the  parent you start asking questions to try  to  see if the story is consistent  one lie is going to lead to another lie  um and that's what happens here  if the uh  in order to justify this lie  um the proof is going to have to a lie  in one or perhaps both but at least one  of these two values because you can't  have the two correct values adding up to  the incorrect value  because you have to think about what's  going on here  so if this is a lie that's going to  force a lie  at one of one side of the other one  level down which is then going to force  a lie  to propagate down and so there's a lie  at every stage is going to force a lie  at least in one  one place or another to propagate all  the way down to the bottom  and then at the bottom the verifier will  see  that the check doesn't work  as when it turns when it tries to  connect it up with the formula itself  and the form verifier will reject  okay so just a way of looking at this  um if the form if the value  if the input was not in the language  um so uh  but the problem is  that as i said that this is exponential  so how are we going to fix that so  just looking ahead to what we're going  to do on tuesday  okay let's see if there's any questions  here first of all  uh  okay yes i got a question should this be  uh  should this be a minus i i purposely  made this bracket only go  not include the very last zero yeah  there's a total of m zeroes here all  together but the  i left out the last zero that's why i  said n minus one maybe it would have  been better to say m  okay so and you got another interesting  question here why can't we reject right  away if k is wrong  um  uh  well  the verifier is probabilistic polynomial  time how does the verifier know if k is  wrong  um  so it means or or or right  so  what we're trying to do is something  like you know like np where we have a  certificate but now we have this kind of  interactive certificate in the form of  disprover maybe that's another way to  look at it  um  where if you're in the language there  should be some way for the prover to  convince to get make you accept but if  you're not not in the language there  should be no way  for the proverb to make you accept  um  uh  so the verifier just can't  reject right away because there's no way  to tell how does the verifier know it's  going to start rejecting things when it  shouldn't if it's just going to be  rejecting willy-nilly here  um  okay how does the verifier need to  determine if the prover is  internally consistent instead of just  asking  so why does the verifier need to  determine if the approver is internally  consistent instead of just asking the  questions in step n plus one  yeah so maybe that's  because it looks like all of the work is  happening at the very end  um  but i'm really presenting this to you  as a preparation  for what we're going to do on tuesday  um so it's important to think about the  connection from each step to the next  each step is going to be justified by  what happens at the next step  until we get to the very end  so you have to just understand it for  what it is don't try to make it more  efficient yeah i realize this is kind of  dumb  um  good point we're not using the  probabilism here  um  and  moreover we're not really even using the  interaction here the prover is doing all  the sending the verifier is just  accepting at the end  yeah  this is we're not using the power and  we're getting a weaker result so let's  move on before we run out of time here  so how do we gonna fix this  so the problem is is blowing up  each to justify each stage where  each value  we're  um  needing to present two values which add  up to it  um  and that's uh good leading to a blow up  now it would be nice if we can do  something where each value was  supported by just a single value  at the next level  so we know here's an idea well you know  in order to understand to see that  uh that that this total count is correct  why don't we just pick at random either  zero or one and only follow that one  down  well the problem with doing that  is because  the the  the sequence of lies is could be just a  single path through this uh tree  and the chances you're going to find  that path  down to a contradiction at the bottom is  very low if you're just doing it at  random  um so just randomly picking zeros and  ones as as the as the one you're going  to justify used to justify the previous  value is not going to be good enough  so what  but this is what we're going to do  however the values that we're going to  pick  for um  for these random  um  inputs  are not going to be boolean values  we're going to pick  non-boolean assignments to the variables  which again just as with the branching  program case didn't make any sense  on the surface of it we're going to have  to make it make sense  and  we'll have to see how to do that on tues  in tuesday's lecture  okay so that's kind of the setup  okay  um  yeah so in a similar question why is  this any different from just  non-deterministically guessing the  assignments it's because of this we're  really setting the stage  okay so what we did today was we  introduced them the the model and  defined the complexity class  we can we did show this one in its  full glory we showed that non-iso is an  ip  really worth understanding this  uh protocol here making sure you  you're comfortable with that  and and also the the model itself  and so for tuesday's lecture we're going  to finish this up uh well we started  showing that number set as an ip which  is what we need to to do to prove co-np  is an ip  and we'll finish that  next time  which will be our last time  okay so that's it for today  i'll stick around for questions  so  a good question here why can't um  v just  reject if some of the checks are  incorrect yes v could as soon as  there's a check that fails we can just  reject at that stage  i'm just trying to argue that at some  point along the way if the input is not  in the language  there's going to be a check that fails  so i mean i said reject at the end but  yeah i mean  you could have rejected at any point  along the way  um  okay  um someone's asking for what what role  did i play uh so i did  my my own personal role in this i were  to which was twofold first of all i i  came up with the idea  uh of  well not the idea i came up with the  name  interactive proof uh  i remember when sylvia mccauley was  explaining this to me in my apartment  many many years ago he had a kind of a  little bit complicated and i don't even  remember what the protocol was for it  was not for something simple was  something involving uh prime numbers  and i said oh  that's a kind of an interactive proof  and it's and it it stuck from that point  on so  um that was one thing but the other  thing in terms of more mathematically  i my role was so uh  shafi goldwater and i approved the  equivalence of the two models the public  coin and the private coin version  um so that that was uh  my uh  my role in this back and when this was  all first coming out  approved it approved it on an airplane  on the way to a conference somewhere  so i think we're going to  notice any other questions  i think we'll head out take care  everybody  see one  see you on tuesday  bye bye  you

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00:07:09,830 --> 00:07:11,430

 

211
00:07:11,430 --> 00:07:13,909

 

212
00:07:13,909 --> 00:07:15,830

 

213
00:07:15,830 --> 00:07:17,189

 

214
00:07:17,189 --> 00:07:20,710

 

215
00:07:20,710 --> 00:07:20,720

 

216
00:07:20,720 --> 00:07:22,710


217
00:07:22,710 --> 00:07:24,469

 

218
00:07:24,469 --> 00:07:26,950

 

219
00:07:26,950 --> 00:07:29,029

 

220
00:07:29,029 --> 00:07:31,510

 

221
00:07:31,510 --> 00:07:33,270

 

222
00:07:33,270 --> 00:07:35,270

 

223
00:07:35,270 --> 00:07:37,830

 

224
00:07:37,830 --> 00:07:40,150

 

225
00:07:40,150 --> 00:07:44,950

 

226
00:07:44,950 --> 00:07:47,270

 

227
00:07:47,270 --> 00:07:50,150

 

228
00:07:50,150 --> 00:07:52,070

 

229
00:07:52,070 --> 00:07:55,670

 

230
00:07:55,670 --> 00:07:57,110

 

231
00:07:57,110 --> 00:07:59,830

 

232
00:07:59,830 --> 00:08:01,990

 

233
00:08:01,990 --> 00:08:04,950

 

234
00:08:04,950 --> 00:08:10,550

 

235
00:08:10,550 --> 00:08:13,430

 

236
00:08:13,430 --> 00:08:15,589

 

237
00:08:15,589 --> 00:08:17,350

 

238
00:08:17,350 --> 00:08:18,869

 

239
00:08:18,869 --> 00:08:20,869

 

240
00:08:20,869 --> 00:08:22,950

 

241
00:08:22,950 --> 00:08:22,960

 

242
00:08:22,960 --> 00:08:23,909


243
00:08:23,909 --> 00:08:26,710

 

244
00:08:26,710 --> 00:08:26,720

 

245
00:08:26,720 --> 00:08:27,430


246
00:08:27,430 --> 00:08:29,189

 

247
00:08:29,189 --> 00:08:30,710

 

248
00:08:30,710 --> 00:08:34,070

 

249
00:08:34,070 --> 00:08:36,389

 

250
00:08:36,389 --> 00:08:38,230

 

251
00:08:38,230 --> 00:08:39,430

 

252
00:08:39,430 --> 00:08:41,430

 

253
00:08:41,430 --> 00:08:41,440

 

254
00:08:41,440 --> 00:08:42,550


255
00:08:42,550 --> 00:08:42,560

 

256
00:08:42,560 --> 00:08:44,070


257
00:08:44,070 --> 00:08:47,110

 

258
00:08:47,110 --> 00:08:49,670

 

259
00:08:49,670 --> 00:08:51,590

 

260
00:08:51,590 --> 00:08:51,600

 

261
00:08:51,600 --> 00:08:52,790


262
00:08:52,790 --> 00:08:55,190

 

263
00:08:55,190 --> 00:08:57,110

 

264
00:08:57,110 --> 00:08:59,110

 

265
00:08:59,110 --> 00:09:00,870

 

266
00:09:00,870 --> 00:09:02,710

 

267
00:09:02,710 --> 00:09:05,269

 

268
00:09:05,269 --> 00:09:07,670

 

269
00:09:07,670 --> 00:09:10,870

 

270
00:09:10,870 --> 00:09:14,790

 

271
00:09:14,790 --> 00:09:17,110

 

272
00:09:17,110 --> 00:09:17,120

 

273
00:09:17,120 --> 00:09:18,310


274
00:09:18,310 --> 00:09:20,389

 

275
00:09:20,389 --> 00:09:23,670

 

276
00:09:23,670 --> 00:09:26,070

 

277
00:09:26,070 --> 00:09:27,990

 

278
00:09:27,990 --> 00:09:29,829

 

279
00:09:29,829 --> 00:09:29,839

 

280
00:09:29,839 --> 00:09:32,790


281
00:09:32,790 --> 00:09:32,800

 

282
00:09:32,800 --> 00:09:34,710


283
00:09:34,710 --> 00:09:37,990

 

284
00:09:37,990 --> 00:09:40,150

 

285
00:09:40,150 --> 00:09:42,470

 

286
00:09:42,470 --> 00:09:45,509

 

287
00:09:45,509 --> 00:09:49,750

 

288
00:09:49,750 --> 00:09:51,269

 

289
00:09:51,269 --> 00:09:51,279

 

290
00:09:51,279 --> 00:09:53,110


291
00:09:53,110 --> 00:09:53,120

 

292
00:09:53,120 --> 00:09:53,990


293
00:09:53,990 --> 00:09:57,269

 

294
00:09:57,269 --> 00:09:59,269

 

295
00:09:59,269 --> 00:10:01,350

 

296
00:10:01,350 --> 00:10:03,269

 

297
00:10:03,269 --> 00:10:05,190

 

298
00:10:05,190 --> 00:10:09,350

 

299
00:10:09,350 --> 00:10:09,360

 

300
00:10:09,360 --> 00:10:10,389


301
00:10:10,389 --> 00:10:10,399

 

302
00:10:10,399 --> 00:10:11,350


303
00:10:11,350 --> 00:10:13,910

 

304
00:10:13,910 --> 00:10:16,470

 

305
00:10:16,470 --> 00:10:18,230

 

306
00:10:18,230 --> 00:10:19,509

 

307
00:10:19,509 --> 00:10:19,519

 

308
00:10:19,519 --> 00:10:21,829


309
00:10:21,829 --> 00:10:23,910

 

310
00:10:23,910 --> 00:10:26,389

 

311
00:10:26,389 --> 00:10:28,630

 

312
00:10:28,630 --> 00:10:33,910

 

313
00:10:33,910 --> 00:10:37,509

 

314
00:10:37,509 --> 00:10:38,550

 

315
00:10:38,550 --> 00:10:40,710

 

316
00:10:40,710 --> 00:10:42,949

 

317
00:10:42,949 --> 00:10:45,030

 

318
00:10:45,030 --> 00:10:48,389

 

319
00:10:48,389 --> 00:10:50,790

 

320
00:10:50,790 --> 00:10:53,030

 

321
00:10:53,030 --> 00:10:54,630

 

322
00:10:54,630 --> 00:10:56,550

 

323
00:10:56,550 --> 00:10:59,990

 

324
00:10:59,990 --> 00:11:01,030

 

325
00:11:01,030 --> 00:11:03,750

 

326
00:11:03,750 --> 00:11:03,760

 

327
00:11:03,760 --> 00:11:05,110


328
00:11:05,110 --> 00:11:05,120

 

329
00:11:05,120 --> 00:11:06,310


330
00:11:06,310 --> 00:11:08,870

 

331
00:11:08,870 --> 00:11:12,069

 

332
00:11:12,069 --> 00:11:14,949

 

333
00:11:14,949 --> 00:11:17,670

 

334
00:11:17,670 --> 00:11:20,069

 

335
00:11:20,069 --> 00:11:20,079

 

336
00:11:20,079 --> 00:11:21,269


337
00:11:21,269 --> 00:11:24,630

 

338
00:11:24,630 --> 00:11:26,949

 

339
00:11:26,949 --> 00:11:26,959

 

340
00:11:26,959 --> 00:11:28,630


341
00:11:28,630 --> 00:11:31,670

 

342
00:11:31,670 --> 00:11:34,710

 

343
00:11:34,710 --> 00:11:36,870

 

344
00:11:36,870 --> 00:11:39,670

 

345
00:11:39,670 --> 00:11:41,269

 

346
00:11:41,269 --> 00:11:42,870

 

347
00:11:42,870 --> 00:11:44,790

 

348
00:11:44,790 --> 00:11:47,269

 

349
00:11:47,269 --> 00:11:50,310

 

350
00:11:50,310 --> 00:11:50,320

 

351
00:11:50,320 --> 00:11:51,590


352
00:11:51,590 --> 00:11:53,269

 

353
00:11:53,269 --> 00:11:53,279

 

354
00:11:53,279 --> 00:11:54,829


355
00:11:54,829 --> 00:11:54,839

 

356
00:11:54,839 --> 00:11:56,710


357
00:11:56,710 --> 00:11:56,720

 

358
00:11:56,720 --> 00:11:58,310


359
00:11:58,310 --> 00:11:59,350

 

360
00:11:59,350 --> 00:12:01,430

 

361
00:12:01,430 --> 00:12:01,440

 

362
00:12:01,440 --> 00:12:02,389


363
00:12:02,389 --> 00:12:05,190

 

364
00:12:05,190 --> 00:12:07,190

 

365
00:12:07,190 --> 00:12:09,269

 

366
00:12:09,269 --> 00:12:11,269

 

367
00:12:11,269 --> 00:12:12,550

 

368
00:12:12,550 --> 00:12:12,560

 

369
00:12:12,560 --> 00:12:14,790


370
00:12:14,790 --> 00:12:17,350

 

371
00:12:17,350 --> 00:12:19,350

 

372
00:12:19,350 --> 00:12:19,360

 

373
00:12:19,360 --> 00:12:20,069


374
00:12:20,069 --> 00:12:21,590

 

375
00:12:21,590 --> 00:12:26,389

 

376
00:12:26,389 --> 00:12:27,350

 

377
00:12:27,350 --> 00:12:28,550

 

378
00:12:28,550 --> 00:12:31,269

 

379
00:12:31,269 --> 00:12:32,230

 

380
00:12:32,230 --> 00:12:33,990

 

381
00:12:33,990 --> 00:12:35,829

 

382
00:12:35,829 --> 00:12:38,470

 

383
00:12:38,470 --> 00:12:42,150

 

384
00:12:42,150 --> 00:12:44,389

 

385
00:12:44,389 --> 00:12:44,399

 

386
00:12:44,399 --> 00:12:45,590


387
00:12:45,590 --> 00:12:48,389

 

388
00:12:48,389 --> 00:12:50,069

 

389
00:12:50,069 --> 00:12:50,079

 

390
00:12:50,079 --> 00:12:52,710


391
00:12:52,710 --> 00:12:54,470

 

392
00:12:54,470 --> 00:12:56,310

 

393
00:12:56,310 --> 00:12:59,030

 

394
00:12:59,030 --> 00:12:59,040

 

395
00:12:59,040 --> 00:13:00,150


396
00:13:00,150 --> 00:13:01,350

 

397
00:13:01,350 --> 00:13:03,110

 

398
00:13:03,110 --> 00:13:04,870

 

399
00:13:04,870 --> 00:13:06,870

 

400
00:13:06,870 --> 00:13:08,470

 

401
00:13:08,470 --> 00:13:09,990

 

402
00:13:09,990 --> 00:13:11,269

 

403
00:13:11,269 --> 00:13:11,279

 

404
00:13:11,279 --> 00:13:12,790


405
00:13:12,790 --> 00:13:14,790

 

406
00:13:14,790 --> 00:13:16,230

 

407
00:13:16,230 --> 00:13:18,150

 

408
00:13:18,150 --> 00:13:21,430

 

409
00:13:21,430 --> 00:13:24,389

 

410
00:13:24,389 --> 00:13:27,110

 

411
00:13:27,110 --> 00:13:28,629

 

412
00:13:28,629 --> 00:13:30,470

 

413
00:13:30,470 --> 00:13:31,990

 

414
00:13:31,990 --> 00:13:32,000

 

415
00:13:32,000 --> 00:13:33,269


416
00:13:33,269 --> 00:13:37,430

 

417
00:13:37,430 --> 00:13:39,350

 

418
00:13:39,350 --> 00:13:39,360

 

419
00:13:39,360 --> 00:13:42,550


420
00:13:42,550 --> 00:13:42,560

 

421
00:13:42,560 --> 00:13:45,350


422
00:13:45,350 --> 00:13:46,870

 

423
00:13:46,870 --> 00:13:48,550

 

424
00:13:48,550 --> 00:13:49,910

 

425
00:13:49,910 --> 00:13:52,230

 

426
00:13:52,230 --> 00:13:54,790

 

427
00:13:54,790 --> 00:13:55,910

 

428
00:13:55,910 --> 00:13:57,430

 

429
00:13:57,430 --> 00:13:57,440

 

430
00:13:57,440 --> 00:13:59,030


431
00:13:59,030 --> 00:14:00,829

 

432
00:14:00,829 --> 00:14:03,269

 

433
00:14:03,269 --> 00:14:05,509

 

434
00:14:05,509 --> 00:14:06,470

 

435
00:14:06,470 --> 00:14:09,030

 

436
00:14:09,030 --> 00:14:09,040

 

437
00:14:09,040 --> 00:14:11,430


438
00:14:11,430 --> 00:14:12,870

 

439
00:14:12,870 --> 00:14:14,150

 

440
00:14:14,150 --> 00:14:16,550

 

441
00:14:16,550 --> 00:14:20,550

 

442
00:14:20,550 --> 00:14:22,389

 

443
00:14:22,389 --> 00:14:22,399

 

444
00:14:22,399 --> 00:14:24,870


445
00:14:24,870 --> 00:14:27,670

 

446
00:14:27,670 --> 00:14:30,550

 

447
00:14:30,550 --> 00:14:32,710

 

448
00:14:32,710 --> 00:14:34,550

 

449
00:14:34,550 --> 00:14:35,509

 

450
00:14:35,509 --> 00:14:38,150

 

451
00:14:38,150 --> 00:14:39,670

 

452
00:14:39,670 --> 00:14:43,030

 

453
00:14:43,030 --> 00:14:43,040

 

454
00:14:43,040 --> 00:14:45,269


455
00:14:45,269 --> 00:14:46,870

 

456
00:14:46,870 --> 00:14:48,949

 

457
00:14:48,949 --> 00:14:50,949

 

458
00:14:50,949 --> 00:14:52,310

 

459
00:14:52,310 --> 00:14:53,750

 

460
00:14:53,750 --> 00:14:55,350

 

461
00:14:55,350 --> 00:14:57,110

 

462
00:14:57,110 --> 00:14:58,949

 

463
00:14:58,949 --> 00:14:58,959

 

464
00:14:58,959 --> 00:15:00,230


465
00:15:00,230 --> 00:15:02,470

 

466
00:15:02,470 --> 00:15:04,949

 

467
00:15:04,949 --> 00:15:07,590

 

468
00:15:07,590 --> 00:15:10,230

 

469
00:15:10,230 --> 00:15:13,430

 

470
00:15:13,430 --> 00:15:17,910

 

471
00:15:17,910 --> 00:15:20,949

 

472
00:15:20,949 --> 00:15:23,430

 

473
00:15:23,430 --> 00:15:24,710

 

474
00:15:24,710 --> 00:15:27,030

 

475
00:15:27,030 --> 00:15:28,310

 

476
00:15:28,310 --> 00:15:30,069

 

477
00:15:30,069 --> 00:15:31,670

 

478
00:15:31,670 --> 00:15:34,949

 

479
00:15:34,949 --> 00:15:37,590

 

480
00:15:37,590 --> 00:15:39,110

 

481
00:15:39,110 --> 00:15:40,310

 

482
00:15:40,310 --> 00:15:42,069

 

483
00:15:42,069 --> 00:15:44,069

 

484
00:15:44,069 --> 00:15:45,430

 

485
00:15:45,430 --> 00:15:47,110

 

486
00:15:47,110 --> 00:15:48,629

 

487
00:15:48,629 --> 00:15:50,870

 

488
00:15:50,870 --> 00:15:54,310

 

489
00:15:54,310 --> 00:15:56,629

 

490
00:15:56,629 --> 00:15:58,150

 

491
00:15:58,150 --> 00:16:00,470

 

492
00:16:00,470 --> 00:16:00,480

 

493
00:16:00,480 --> 00:16:03,269


494
00:16:03,269 --> 00:16:04,949

 

495
00:16:04,949 --> 00:16:12,870

 

496
00:16:12,870 --> 00:16:14,069

 

497
00:16:14,069 --> 00:16:18,710

 

498
00:16:18,710 --> 00:16:21,110

 

499
00:16:21,110 --> 00:16:24,310

 

500
00:16:24,310 --> 00:16:26,710

 

501
00:16:26,710 --> 00:16:28,310

 

502
00:16:28,310 --> 00:16:28,320

 

503
00:16:28,320 --> 00:16:29,910


504
00:16:29,910 --> 00:16:31,430

 

505
00:16:31,430 --> 00:16:33,910

 

506
00:16:33,910 --> 00:16:35,110

 

507
00:16:35,110 --> 00:16:36,949

 

508
00:16:36,949 --> 00:16:38,550

 

509
00:16:38,550 --> 00:16:42,629

 

510
00:16:42,629 --> 00:16:43,670

 

511
00:16:43,670 --> 00:16:46,870

 

512
00:16:46,870 --> 00:16:48,870

 

513
00:16:48,870 --> 00:16:48,880

 

514
00:16:48,880 --> 00:16:51,910


515
00:16:51,910 --> 00:16:53,269

 

516
00:16:53,269 --> 00:16:53,279

 

517
00:16:53,279 --> 00:16:56,389


518
00:16:56,389 --> 00:16:58,949

 

519
00:16:58,949 --> 00:17:01,269

 

520
00:17:01,269 --> 00:17:04,789

 

521
00:17:04,789 --> 00:17:06,470

 

522
00:17:06,470 --> 00:17:08,949

 

523
00:17:08,949 --> 00:17:12,390

 

524
00:17:12,390 --> 00:17:17,350

 

525
00:17:17,350 --> 00:17:21,270

 

526
00:17:21,270 --> 00:17:23,110

 

527
00:17:23,110 --> 00:17:23,120

 

528
00:17:23,120 --> 00:17:23,990


529
00:17:23,990 --> 00:17:26,390

 

530
00:17:26,390 --> 00:17:26,400

 

531
00:17:26,400 --> 00:17:27,429


532
00:17:27,429 --> 00:17:30,310

 

533
00:17:30,310 --> 00:17:32,310

 

534
00:17:32,310 --> 00:17:38,150

 

535
00:17:38,150 --> 00:17:39,750

 

536
00:17:39,750 --> 00:17:42,310

 

537
00:17:42,310 --> 00:17:46,150

 

538
00:17:46,150 --> 00:17:47,750

 

539
00:17:47,750 --> 00:17:50,070

 

540
00:17:50,070 --> 00:17:51,510

 

541
00:17:51,510 --> 00:17:53,270

 

542
00:17:53,270 --> 00:17:55,029

 

543
00:17:55,029 --> 00:17:56,710

 

544
00:17:56,710 --> 00:17:59,590

 

545
00:17:59,590 --> 00:18:01,909

 

546
00:18:01,909 --> 00:18:04,390

 

547
00:18:04,390 --> 00:18:06,549

 

548
00:18:06,549 --> 00:18:11,990

 

549
00:18:11,990 --> 00:18:12,000

 

550
00:18:12,000 --> 00:18:15,669


551
00:18:15,669 --> 00:18:18,390

 

552
00:18:18,390 --> 00:18:20,310

 

553
00:18:20,310 --> 00:18:23,350

 

554
00:18:23,350 --> 00:18:25,350

 

555
00:18:25,350 --> 00:18:27,909

 

556
00:18:27,909 --> 00:18:29,750

 

557
00:18:29,750 --> 00:18:31,110

 

558
00:18:31,110 --> 00:18:32,630

 

559
00:18:32,630 --> 00:18:34,070

 

560
00:18:34,070 --> 00:18:35,590

 

561
00:18:35,590 --> 00:18:38,070

 

562
00:18:38,070 --> 00:18:40,470

 

563
00:18:40,470 --> 00:18:42,070

 

564
00:18:42,070 --> 00:18:43,669

 

565
00:18:43,669 --> 00:18:46,230

 

566
00:18:46,230 --> 00:18:48,390

 

567
00:18:48,390 --> 00:18:49,669

 

568
00:18:49,669 --> 00:18:50,950

 

569
00:18:50,950 --> 00:18:52,710

 

570
00:18:52,710 --> 00:18:54,230

 

571
00:18:54,230 --> 00:18:56,070

 

572
00:18:56,070 --> 00:18:57,590

 

573
00:18:57,590 --> 00:18:57,600

 

574
00:18:57,600 --> 00:18:58,789


575
00:18:58,789 --> 00:18:59,990

 

576
00:18:59,990 --> 00:19:02,070

 

577
00:19:02,070 --> 00:19:03,909

 

578
00:19:03,909 --> 00:19:06,630

 

579
00:19:06,630 --> 00:19:07,750

 

580
00:19:07,750 --> 00:19:07,760

 

581
00:19:07,760 --> 00:19:08,870


582
00:19:08,870 --> 00:19:11,110

 

583
00:19:11,110 --> 00:19:13,190

 

584
00:19:13,190 --> 00:19:14,630

 

585
00:19:14,630 --> 00:19:16,070

 

586
00:19:16,070 --> 00:19:16,080

 

587
00:19:16,080 --> 00:19:18,789


588
00:19:18,789 --> 00:19:20,950

 

589
00:19:20,950 --> 00:19:22,470

 

590
00:19:22,470 --> 00:19:24,150

 

591
00:19:24,150 --> 00:19:26,630

 

592
00:19:26,630 --> 00:19:29,990

 

593
00:19:29,990 --> 00:19:30,000

 

594
00:19:30,000 --> 00:19:31,029


595
00:19:31,029 --> 00:19:32,549

 

596
00:19:32,549 --> 00:19:35,029

 

597
00:19:35,029 --> 00:19:36,950

 

598
00:19:36,950 --> 00:19:38,470

 

599
00:19:38,470 --> 00:19:40,150

 

600
00:19:40,150 --> 00:19:43,750

 

601
00:19:43,750 --> 00:19:45,990

 

602
00:19:45,990 --> 00:19:47,830

 

603
00:19:47,830 --> 00:19:47,840

 

604
00:19:47,840 --> 00:19:49,430


605
00:19:49,430 --> 00:19:49,440

 

606
00:19:49,440 --> 00:19:50,470


607
00:19:50,470 --> 00:19:52,950

 

608
00:19:52,950 --> 00:19:54,630

 

609
00:19:54,630 --> 00:19:56,630

 

610
00:19:56,630 --> 00:19:58,710

 

611
00:19:58,710 --> 00:20:00,710

 

612
00:20:00,710 --> 00:20:02,789

 

613
00:20:02,789 --> 00:20:08,149

 

614
00:20:08,149 --> 00:20:10,549

 

615
00:20:10,549 --> 00:20:12,390

 

616
00:20:12,390 --> 00:20:12,400

 

617
00:20:12,400 --> 00:20:13,110


618
00:20:13,110 --> 00:20:15,110

 

619
00:20:15,110 --> 00:20:15,120

 

620
00:20:15,120 --> 00:20:17,270


621
00:20:17,270 --> 00:20:19,110

 

622
00:20:19,110 --> 00:20:21,830

 

623
00:20:21,830 --> 00:20:21,840

 

624
00:20:21,840 --> 00:20:24,070


625
00:20:24,070 --> 00:20:27,110

 

626
00:20:27,110 --> 00:20:29,190

 

627
00:20:29,190 --> 00:20:32,070

 

628
00:20:32,070 --> 00:20:34,630

 

629
00:20:34,630 --> 00:20:37,990

 

630
00:20:37,990 --> 00:20:38,000

 

631
00:20:38,000 --> 00:20:39,990


632
00:20:39,990 --> 00:20:41,750

 

633
00:20:41,750 --> 00:20:43,590

 

634
00:20:43,590 --> 00:20:43,600

 

635
00:20:43,600 --> 00:20:45,190


636
00:20:45,190 --> 00:20:46,830

 

637
00:20:46,830 --> 00:20:49,430

 

638
00:20:49,430 --> 00:20:51,430

 

639
00:20:51,430 --> 00:20:51,440

 

640
00:20:51,440 --> 00:20:52,950


641
00:20:52,950 --> 00:20:52,960

 

642
00:20:52,960 --> 00:20:54,390


643
00:20:54,390 --> 00:20:54,400

 

644
00:20:54,400 --> 00:20:55,590


645
00:20:55,590 --> 00:20:57,750

 

646
00:20:57,750 --> 00:20:57,760

 

647
00:20:57,760 --> 00:20:59,270


648
00:20:59,270 --> 00:21:01,350

 

649
00:21:01,350 --> 00:21:04,549

 

650
00:21:04,549 --> 00:21:06,310

 

651
00:21:06,310 --> 00:21:07,909

 

652
00:21:07,909 --> 00:21:10,390

 

653
00:21:10,390 --> 00:21:12,230

 

654
00:21:12,230 --> 00:21:14,149

 

655
00:21:14,149 --> 00:21:14,159

 

656
00:21:14,159 --> 00:21:15,430


657
00:21:15,430 --> 00:21:17,029

 

658
00:21:17,029 --> 00:21:18,710

 

659
00:21:18,710 --> 00:21:20,870

 

660
00:21:20,870 --> 00:21:22,149

 

661
00:21:22,149 --> 00:21:24,710

 

662
00:21:24,710 --> 00:21:27,590

 

663
00:21:27,590 --> 00:21:31,590

 

664
00:21:31,590 --> 00:21:33,830

 

665
00:21:33,830 --> 00:21:33,840

 

666
00:21:33,840 --> 00:21:34,870


667
00:21:34,870 --> 00:21:35,909

 

668
00:21:35,909 --> 00:21:38,230

 

669
00:21:38,230 --> 00:21:38,240

 

670
00:21:38,240 --> 00:21:39,029


671
00:21:39,029 --> 00:21:42,950

 

672
00:21:42,950 --> 00:21:45,830

 

673
00:21:45,830 --> 00:21:49,190

 

674
00:21:49,190 --> 00:21:53,590

 

675
00:21:53,590 --> 00:21:53,600

 

676
00:21:53,600 --> 00:21:54,390


677
00:21:54,390 --> 00:21:57,110

 

678
00:21:57,110 --> 00:21:59,270

 

679
00:21:59,270 --> 00:22:01,669

 

680
00:22:01,669 --> 00:22:03,590

 

681
00:22:03,590 --> 00:22:05,990

 

682
00:22:05,990 --> 00:22:07,750

 

683
00:22:07,750 --> 00:22:10,149

 

684
00:22:10,149 --> 00:22:11,350

 

685
00:22:11,350 --> 00:22:11,360

 

686
00:22:11,360 --> 00:22:12,390


687
00:22:12,390 --> 00:22:15,990

 

688
00:22:15,990 --> 00:22:17,350

 

689
00:22:17,350 --> 00:22:20,149

 

690
00:22:20,149 --> 00:22:21,909

 

691
00:22:21,909 --> 00:22:21,919

 

692
00:22:21,919 --> 00:22:23,510


693
00:22:23,510 --> 00:22:26,950

 

694
00:22:26,950 --> 00:22:28,710

 

695
00:22:28,710 --> 00:22:30,789

 

696
00:22:30,789 --> 00:22:30,799

 

697
00:22:30,799 --> 00:22:32,230


698
00:22:32,230 --> 00:22:34,310

 

699
00:22:34,310 --> 00:22:36,230

 

700
00:22:36,230 --> 00:22:38,470

 

701
00:22:38,470 --> 00:22:40,470

 

702
00:22:40,470 --> 00:22:42,310

 

703
00:22:42,310 --> 00:22:43,270

 

704
00:22:43,270 --> 00:22:45,909

 

705
00:22:45,909 --> 00:22:48,149

 

706
00:22:48,149 --> 00:22:50,470

 

707
00:22:50,470 --> 00:22:51,909

 

708
00:22:51,909 --> 00:22:54,710

 

709
00:22:54,710 --> 00:22:57,029

 

710
00:22:57,029 --> 00:22:59,029

 

711
00:22:59,029 --> 00:22:59,039

 

712
00:22:59,039 --> 00:22:59,990


713
00:22:59,990 --> 00:23:03,029

 

714
00:23:03,029 --> 00:23:04,870

 

715
00:23:04,870 --> 00:23:08,390

 

716
00:23:08,390 --> 00:23:11,110

 

717
00:23:11,110 --> 00:23:11,120

 

718
00:23:11,120 --> 00:23:12,149


719
00:23:12,149 --> 00:23:12,159

 

720
00:23:12,159 --> 00:23:13,350


721
00:23:13,350 --> 00:23:16,310

 

722
00:23:16,310 --> 00:23:18,549

 

723
00:23:18,549 --> 00:23:20,390

 

724
00:23:20,390 --> 00:23:23,510

 

725
00:23:23,510 --> 00:23:25,430

 

726
00:23:25,430 --> 00:23:27,270

 

727
00:23:27,270 --> 00:23:28,549

 

728
00:23:28,549 --> 00:23:29,990

 

729
00:23:29,990 --> 00:23:32,070

 

730
00:23:32,070 --> 00:23:34,310

 

731
00:23:34,310 --> 00:23:34,320

 

732
00:23:34,320 --> 00:23:35,430


733
00:23:35,430 --> 00:23:36,950

 

734
00:23:36,950 --> 00:23:39,190

 

735
00:23:39,190 --> 00:23:39,200

 

736
00:23:39,200 --> 00:23:42,230


737
00:23:42,230 --> 00:23:43,750

 

738
00:23:43,750 --> 00:23:46,710

 

739
00:23:46,710 --> 00:23:49,269

 

740
00:23:49,269 --> 00:23:50,870

 

741
00:23:50,870 --> 00:23:52,549

 

742
00:23:52,549 --> 00:23:54,310

 

743
00:23:54,310 --> 00:23:55,590

 

744
00:23:55,590 --> 00:23:57,590

 

745
00:23:57,590 --> 00:24:00,630

 

746
00:24:00,630 --> 00:24:00,640

 

747
00:24:00,640 --> 00:24:02,070


748
00:24:02,070 --> 00:24:03,669

 

749
00:24:03,669 --> 00:24:03,679

 

750
00:24:03,679 --> 00:24:04,549


751
00:24:04,549 --> 00:24:06,549

 

752
00:24:06,549 --> 00:24:08,789

 

753
00:24:08,789 --> 00:24:11,590

 

754
00:24:11,590 --> 00:24:12,870

 

755
00:24:12,870 --> 00:24:13,990

 

756
00:24:13,990 --> 00:24:16,310

 

757
00:24:16,310 --> 00:24:18,549

 

758
00:24:18,549 --> 00:24:20,390

 

759
00:24:20,390 --> 00:24:21,830

 

760
00:24:21,830 --> 00:24:23,190

 

761
00:24:23,190 --> 00:24:23,200

 

762
00:24:23,200 --> 00:24:24,470


763
00:24:24,470 --> 00:24:24,480

 

764
00:24:24,480 --> 00:24:25,430


765
00:24:25,430 --> 00:24:27,190

 

766
00:24:27,190 --> 00:24:28,470

 

767
00:24:28,470 --> 00:24:30,789

 

768
00:24:30,789 --> 00:24:32,789

 

769
00:24:32,789 --> 00:24:33,990

 

770
00:24:33,990 --> 00:24:35,669

 

771
00:24:35,669 --> 00:24:35,679

 

772
00:24:35,679 --> 00:24:36,470


773
00:24:36,470 --> 00:24:39,990

 

774
00:24:39,990 --> 00:24:40,000

 

775
00:24:40,000 --> 00:24:40,870


776
00:24:40,870 --> 00:24:44,870

 

777
00:24:44,870 --> 00:24:46,470

 

778
00:24:46,470 --> 00:24:49,190

 

779
00:24:49,190 --> 00:24:50,950

 

780
00:24:50,950 --> 00:24:53,269

 

781
00:24:53,269 --> 00:24:55,909

 

782
00:24:55,909 --> 00:24:58,710

 

783
00:24:58,710 --> 00:24:58,720

 

784
00:24:58,720 --> 00:24:59,590


785
00:24:59,590 --> 00:25:02,070

 

786
00:25:02,070 --> 00:25:02,080

 

787
00:25:02,080 --> 00:25:03,510


788
00:25:03,510 --> 00:25:05,350

 

789
00:25:05,350 --> 00:25:07,510

 

790
00:25:07,510 --> 00:25:09,430

 

791
00:25:09,430 --> 00:25:12,470

 

792
00:25:12,470 --> 00:25:17,029

 

793
00:25:17,029 --> 00:25:18,470

 

794
00:25:18,470 --> 00:25:20,710

 

795
00:25:20,710 --> 00:25:23,430

 

796
00:25:23,430 --> 00:25:23,440

 

797
00:25:23,440 --> 00:25:24,470


798
00:25:24,470 --> 00:25:26,549

 

799
00:25:26,549 --> 00:25:28,789

 

800
00:25:28,789 --> 00:25:34,070

 

801
00:25:34,070 --> 00:25:35,830

 

802
00:25:35,830 --> 00:25:37,269

 

803
00:25:37,269 --> 00:25:39,110

 

804
00:25:39,110 --> 00:25:39,120

 

805
00:25:39,120 --> 00:25:39,990


806
00:25:39,990 --> 00:25:41,590

 

807
00:25:41,590 --> 00:25:43,669

 

808
00:25:43,669 --> 00:25:45,110

 

809
00:25:45,110 --> 00:25:46,549

 

810
00:25:46,549 --> 00:25:49,029

 

811
00:25:49,029 --> 00:25:50,390

 

812
00:25:50,390 --> 00:25:51,909

 

813
00:25:51,909 --> 00:25:53,510

 

814
00:25:53,510 --> 00:25:55,590

 

815
00:25:55,590 --> 00:25:57,750

 

816
00:25:57,750 --> 00:25:59,029

 

817
00:25:59,029 --> 00:26:02,070

 

818
00:26:02,070 --> 00:26:03,909

 

819
00:26:03,909 --> 00:26:03,919

 

820
00:26:03,919 --> 00:26:05,669


821
00:26:05,669 --> 00:26:07,190

 

822
00:26:07,190 --> 00:26:08,870

 

823
00:26:08,870 --> 00:26:12,149

 

824
00:26:12,149 --> 00:26:14,789

 

825
00:26:14,789 --> 00:26:17,830

 

826
00:26:17,830 --> 00:26:21,029

 

827
00:26:21,029 --> 00:26:22,870

 

828
00:26:22,870 --> 00:26:25,430

 

829
00:26:25,430 --> 00:26:26,470

 

830
00:26:26,470 --> 00:26:28,549

 

831
00:26:28,549 --> 00:26:30,310

 

832
00:26:30,310 --> 00:26:30,320

 

833
00:26:30,320 --> 00:26:32,630


834
00:26:32,630 --> 00:26:32,640

 

835
00:26:32,640 --> 00:26:37,110


836
00:26:37,110 --> 00:26:38,710

 

837
00:26:38,710 --> 00:26:40,470

 

838
00:26:40,470 --> 00:26:40,480

 

839
00:26:40,480 --> 00:26:41,669


840
00:26:41,669 --> 00:26:43,510

 

841
00:26:43,510 --> 00:26:45,350

 

842
00:26:45,350 --> 00:26:47,029

 

843
00:26:47,029 --> 00:26:49,909

 

844
00:26:49,909 --> 00:26:51,750

 

845
00:26:51,750 --> 00:26:53,990

 

846
00:26:53,990 --> 00:26:54,000

 

847
00:26:54,000 --> 00:26:55,190


848
00:26:55,190 --> 00:26:56,630

 

849
00:26:56,630 --> 00:26:58,310

 

850
00:26:58,310 --> 00:27:00,149

 

851
00:27:00,149 --> 00:27:01,750

 

852
00:27:01,750 --> 00:27:03,990

 

853
00:27:03,990 --> 00:27:07,190

 

854
00:27:07,190 --> 00:27:08,470

 

855
00:27:08,470 --> 00:27:08,480

 

856
00:27:08,480 --> 00:27:09,350


857
00:27:09,350 --> 00:27:11,350

 

858
00:27:11,350 --> 00:27:13,350

 

859
00:27:13,350 --> 00:27:14,870

 

860
00:27:14,870 --> 00:27:16,230

 

861
00:27:16,230 --> 00:27:17,669

 

862
00:27:17,669 --> 00:27:19,830

 

863
00:27:19,830 --> 00:27:22,470

 

864
00:27:22,470 --> 00:27:23,750

 

865
00:27:23,750 --> 00:27:23,760

 

866
00:27:23,760 --> 00:27:25,510


867
00:27:25,510 --> 00:27:27,269

 

868
00:27:27,269 --> 00:27:30,470

 

869
00:27:30,470 --> 00:27:31,990

 

870
00:27:31,990 --> 00:27:32,000

 

871
00:27:32,000 --> 00:27:33,269


872
00:27:33,269 --> 00:27:35,590

 

873
00:27:35,590 --> 00:27:37,110

 

874
00:27:37,110 --> 00:27:39,029

 

875
00:27:39,029 --> 00:27:41,430

 

876
00:27:41,430 --> 00:27:43,029

 

877
00:27:43,029 --> 00:27:43,039

 

878
00:27:43,039 --> 00:27:44,230


879
00:27:44,230 --> 00:27:48,149

 

880
00:27:48,149 --> 00:27:51,190

 

881
00:27:51,190 --> 00:27:54,230

 

882
00:27:54,230 --> 00:27:56,230

 

883
00:27:56,230 --> 00:27:57,909

 

884
00:27:57,909 --> 00:27:59,350

 

885
00:27:59,350 --> 00:28:02,789

 

886
00:28:02,789 --> 00:28:05,029

 

887
00:28:05,029 --> 00:28:06,630

 

888
00:28:06,630 --> 00:28:08,149

 

889
00:28:08,149 --> 00:28:11,510

 

890
00:28:11,510 --> 00:28:11,520

 

891
00:28:11,520 --> 00:28:13,269


892
00:28:13,269 --> 00:28:15,269

 

893
00:28:15,269 --> 00:28:17,590

 

894
00:28:17,590 --> 00:28:17,600

 

895
00:28:17,600 --> 00:28:19,269


896
00:28:19,269 --> 00:28:21,669

 

897
00:28:21,669 --> 00:28:23,430

 

898
00:28:23,430 --> 00:28:26,149

 

899
00:28:26,149 --> 00:28:28,389

 

900
00:28:28,389 --> 00:28:31,430

 

901
00:28:31,430 --> 00:28:35,110

 

902
00:28:35,110 --> 00:28:36,789

 

903
00:28:36,789 --> 00:28:38,950

 

904
00:28:38,950 --> 00:28:43,190

 

905
00:28:43,190 --> 00:28:45,430

 

906
00:28:45,430 --> 00:28:47,269

 

907
00:28:47,269 --> 00:28:50,630

 

908
00:28:50,630 --> 00:28:52,630

 

909
00:28:52,630 --> 00:28:55,110

 

910
00:28:55,110 --> 00:28:58,710

 

911
00:28:58,710 --> 00:29:00,389

 

912
00:29:00,389 --> 00:29:02,230

 

913
00:29:02,230 --> 00:29:03,669

 

914
00:29:03,669 --> 00:29:06,950

 

915
00:29:06,950 --> 00:29:09,110

 

916
00:29:09,110 --> 00:29:10,830

 

917
00:29:10,830 --> 00:29:13,909

 

918
00:29:13,909 --> 00:29:16,470

 

919
00:29:16,470 --> 00:29:18,149

 

920
00:29:18,149 --> 00:29:20,149

 

921
00:29:20,149 --> 00:29:21,590

 

922
00:29:21,590 --> 00:29:23,750

 

923
00:29:23,750 --> 00:29:25,190

 

924
00:29:25,190 --> 00:29:27,909

 

925
00:29:27,909 --> 00:29:27,919

 

926
00:29:27,919 --> 00:29:29,830


927
00:29:29,830 --> 00:29:32,070

 

928
00:29:32,070 --> 00:29:33,590

 

929
00:29:33,590 --> 00:29:36,149

 

930
00:29:36,149 --> 00:29:39,190

 

931
00:29:39,190 --> 00:29:41,750

 

932
00:29:41,750 --> 00:29:43,669

 

933
00:29:43,669 --> 00:29:46,549

 

934
00:29:46,549 --> 00:29:46,559

 

935
00:29:46,559 --> 00:29:47,669


936
00:29:47,669 --> 00:29:50,789

 

937
00:29:50,789 --> 00:29:52,310

 

938
00:29:52,310 --> 00:29:53,269

 

939
00:29:53,269 --> 00:29:54,470

 

940
00:29:54,470 --> 00:29:56,710

 

941
00:29:56,710 --> 00:29:59,110

 

942
00:29:59,110 --> 00:30:02,389

 

943
00:30:02,389 --> 00:30:04,549

 

944
00:30:04,549 --> 00:30:06,549

 

945
00:30:06,549 --> 00:30:09,269

 

946
00:30:09,269 --> 00:30:12,310

 

947
00:30:12,310 --> 00:30:14,310

 

948
00:30:14,310 --> 00:30:16,710

 

949
00:30:16,710 --> 00:30:16,720

 

950
00:30:16,720 --> 00:30:17,669


951
00:30:17,669 --> 00:30:17,679

 

952
00:30:17,679 --> 00:30:18,630


953
00:30:18,630 --> 00:30:20,870

 

954
00:30:20,870 --> 00:30:24,070

 

955
00:30:24,070 --> 00:30:24,080

 

956
00:30:24,080 --> 00:30:27,909


957
00:30:27,909 --> 00:30:30,710

 

958
00:30:30,710 --> 00:30:33,029

 

959
00:30:33,029 --> 00:30:34,789

 

960
00:30:34,789 --> 00:30:37,029

 

961
00:30:37,029 --> 00:30:37,039

 

962
00:30:37,039 --> 00:30:39,590


963
00:30:39,590 --> 00:30:41,269

 

964
00:30:41,269 --> 00:30:43,510

 

965
00:30:43,510 --> 00:30:45,110

 

966
00:30:45,110 --> 00:30:47,350

 

967
00:30:47,350 --> 00:30:47,360

 

968
00:30:47,360 --> 00:30:48,470


969
00:30:48,470 --> 00:30:50,389

 

970
00:30:50,389 --> 00:30:51,909

 

971
00:30:51,909 --> 00:30:55,510

 

972
00:30:55,510 --> 00:30:55,520

 

973
00:30:55,520 --> 00:30:56,710


974
00:30:56,710 --> 00:30:56,720

 

975
00:30:56,720 --> 00:30:57,909


976
00:30:57,909 --> 00:30:57,919

 

977
00:30:57,919 --> 00:30:58,870


978
00:30:58,870 --> 00:31:01,509

 

979
00:31:01,509 --> 00:31:01,519

 

980
00:31:01,519 --> 00:31:03,190


981
00:31:03,190 --> 00:31:04,549

 

982
00:31:04,549 --> 00:31:06,389

 

983
00:31:06,389 --> 00:31:08,230

 

984
00:31:08,230 --> 00:31:11,590

 

985
00:31:11,590 --> 00:31:13,990

 

986
00:31:13,990 --> 00:31:16,389

 

987
00:31:16,389 --> 00:31:18,789

 

988
00:31:18,789 --> 00:31:21,669

 

989
00:31:21,669 --> 00:31:22,789

 

990
00:31:22,789 --> 00:31:28,389

 

991
00:31:28,389 --> 00:31:30,230

 

992
00:31:30,230 --> 00:31:31,990

 

993
00:31:31,990 --> 00:31:33,669

 

994
00:31:33,669 --> 00:31:35,590

 

995
00:31:35,590 --> 00:31:36,870

 

996
00:31:36,870 --> 00:31:39,269

 

997
00:31:39,269 --> 00:31:40,710

 

998
00:31:40,710 --> 00:31:40,720

 

999
00:31:40,720 --> 00:31:41,830


1000
00:31:41,830 --> 00:31:43,269

 

1001
00:31:43,269 --> 00:31:45,509

 

1002
00:31:45,509 --> 00:31:47,750

 

1003
00:31:47,750 --> 00:31:49,430

 

1004
00:31:49,430 --> 00:31:51,909

 

1005
00:31:51,909 --> 00:31:53,750

 

1006
00:31:53,750 --> 00:31:55,590

 

1007
00:31:55,590 --> 00:31:57,430

 

1008
00:31:57,430 --> 00:31:58,950

 

1009
00:31:58,950 --> 00:32:00,950

 

1010
00:32:00,950 --> 00:32:02,870

 

1011
00:32:02,870 --> 00:32:07,269

 

1012
00:32:07,269 --> 00:32:09,190

 

1013
00:32:09,190 --> 00:32:10,630

 

1014
00:32:10,630 --> 00:32:14,470

 

1015
00:32:14,470 --> 00:32:17,590

 

1016
00:32:17,590 --> 00:32:20,470

 

1017
00:32:20,470 --> 00:32:20,480

 

1018
00:32:20,480 --> 00:32:22,389


1019
00:32:22,389 --> 00:32:24,070

 

1020
00:32:24,070 --> 00:32:26,310

 

1021
00:32:26,310 --> 00:32:28,470

 

1022
00:32:28,470 --> 00:32:30,230

 

1023
00:32:30,230 --> 00:32:32,549

 

1024
00:32:32,549 --> 00:32:34,070

 

1025
00:32:34,070 --> 00:32:35,909

 

1026
00:32:35,909 --> 00:32:35,919

 

1027
00:32:35,919 --> 00:32:37,190


1028
00:32:37,190 --> 00:32:39,029

 

1029
00:32:39,029 --> 00:32:39,039

 

1030
00:32:39,039 --> 00:32:39,830


1031
00:32:39,830 --> 00:32:41,269

 

1032
00:32:41,269 --> 00:32:43,590

 

1033
00:32:43,590 --> 00:32:43,600

 

1034
00:32:43,600 --> 00:32:44,470


1035
00:32:44,470 --> 00:32:46,149

 

1036
00:32:46,149 --> 00:32:48,470

 

1037
00:32:48,470 --> 00:32:49,750

 

1038
00:32:49,750 --> 00:32:57,350

 

1039
00:32:57,350 --> 00:32:59,190

 

1040
00:32:59,190 --> 00:33:00,470

 

1041
00:33:00,470 --> 00:33:02,870

 

1042
00:33:02,870 --> 00:33:05,750

 

1043
00:33:05,750 --> 00:33:08,149

 

1044
00:33:08,149 --> 00:33:10,710

 

1045
00:33:10,710 --> 00:33:12,230

 

1046
00:33:12,230 --> 00:33:13,669

 

1047
00:33:13,669 --> 00:33:13,679

 

1048
00:33:13,679 --> 00:33:14,630


1049
00:33:14,630 --> 00:33:16,870

 

1050
00:33:16,870 --> 00:33:19,269

 

1051
00:33:19,269 --> 00:33:19,279

 

1052
00:33:19,279 --> 00:33:20,470


1053
00:33:20,470 --> 00:33:22,070

 

1054
00:33:22,070 --> 00:33:23,590

 

1055
00:33:23,590 --> 00:33:28,149

 

1056
00:33:28,149 --> 00:33:30,149

 

1057
00:33:30,149 --> 00:33:32,070

 

1058
00:33:32,070 --> 00:33:33,750

 

1059
00:33:33,750 --> 00:33:35,669

 

1060
00:33:35,669 --> 00:33:35,679

 

1061
00:33:35,679 --> 00:33:36,870


1062
00:33:36,870 --> 00:33:41,509

 

1063
00:33:41,509 --> 00:33:43,509

 

1064
00:33:43,509 --> 00:33:46,070

 

1065
00:33:46,070 --> 00:33:48,149

 

1066
00:33:48,149 --> 00:33:49,750

 

1067
00:33:49,750 --> 00:33:51,350

 

1068
00:33:51,350 --> 00:33:52,549

 

1069
00:33:52,549 --> 00:33:54,549

 

1070
00:33:54,549 --> 00:33:56,710

 

1071
00:33:56,710 --> 00:33:57,990

 

1072
00:33:57,990 --> 00:33:59,590

 

1073
00:33:59,590 --> 00:34:01,110

 

1074
00:34:01,110 --> 00:34:02,549

 

1075
00:34:02,549 --> 00:34:05,509

 

1076
00:34:05,509 --> 00:34:08,629

 

1077
00:34:08,629 --> 00:34:10,550

 

1078
00:34:10,550 --> 00:34:11,589

 

1079
00:34:11,589 --> 00:34:11,599

 

1080
00:34:11,599 --> 00:34:12,869


1081
00:34:12,869 --> 00:34:15,510

 

1082
00:34:15,510 --> 00:34:15,520

 

1083
00:34:15,520 --> 00:34:17,510


1084
00:34:17,510 --> 00:34:19,829

 

1085
00:34:19,829 --> 00:34:23,669

 

1086
00:34:23,669 --> 00:34:24,710

 

1087
00:34:24,710 --> 00:34:24,720

 

1088
00:34:24,720 --> 00:34:25,669


1089
00:34:25,669 --> 00:34:27,990

 

1090
00:34:27,990 --> 00:34:29,510

 

1091
00:34:29,510 --> 00:34:31,270

 

1092
00:34:31,270 --> 00:34:31,280

 

1093
00:34:31,280 --> 00:34:32,470


1094
00:34:32,470 --> 00:34:35,829

 

1095
00:34:35,829 --> 00:34:38,710

 

1096
00:34:38,710 --> 00:34:40,790

 

1097
00:34:40,790 --> 00:34:43,669

 

1098
00:34:43,669 --> 00:34:43,679

 

1099
00:34:43,679 --> 00:34:46,470


1100
00:34:46,470 --> 00:34:51,109

 

1101
00:34:51,109 --> 00:34:52,470

 

1102
00:34:52,470 --> 00:34:52,480

 

1103
00:34:52,480 --> 00:34:53,510


1104
00:34:53,510 --> 00:34:56,869

 

1105
00:34:56,869 --> 00:35:01,349

 

1106
00:35:01,349 --> 00:35:03,270

 

1107
00:35:03,270 --> 00:35:04,390

 

1108
00:35:04,390 --> 00:35:05,670

 

1109
00:35:05,670 --> 00:35:05,680

 

1110
00:35:05,680 --> 00:35:06,790


1111
00:35:06,790 --> 00:35:11,190

 

1112
00:35:11,190 --> 00:35:13,190

 

1113
00:35:13,190 --> 00:35:13,200

 

1114
00:35:13,200 --> 00:35:14,550


1115
00:35:14,550 --> 00:35:24,069

 

1116
00:35:24,069 --> 00:35:25,670

 

1117
00:35:25,670 --> 00:35:28,069

 

1118
00:35:28,069 --> 00:35:29,430

 

1119
00:35:29,430 --> 00:35:31,190

 

1120
00:35:31,190 --> 00:35:34,390

 

1121
00:35:34,390 --> 00:35:37,589

 

1122
00:35:37,589 --> 00:35:39,910

 

1123
00:35:39,910 --> 00:35:41,109

 

1124
00:35:41,109 --> 00:35:43,990

 

1125
00:35:43,990 --> 00:35:44,000

 

1126
00:35:44,000 --> 00:35:46,950


1127
00:35:46,950 --> 00:35:48,550

 

1128
00:35:48,550 --> 00:35:50,790

 

1129
00:35:50,790 --> 00:35:50,800

 

1130
00:35:50,800 --> 00:35:53,430


1131
00:35:53,430 --> 00:35:55,990

 

1132
00:35:55,990 --> 00:35:58,870

 

1133
00:35:58,870 --> 00:36:00,550

 

1134
00:36:00,550 --> 00:36:01,990

 

1135
00:36:01,990 --> 00:36:02,000

 

1136
00:36:02,000 --> 00:36:04,790


1137
00:36:04,790 --> 00:36:07,750

 

1138
00:36:07,750 --> 00:36:07,760

 

1139
00:36:07,760 --> 00:36:09,829


1140
00:36:09,829 --> 00:36:12,710

 

1141
00:36:12,710 --> 00:36:14,790

 

1142
00:36:14,790 --> 00:36:14,800

 

1143
00:36:14,800 --> 00:36:15,750


1144
00:36:15,750 --> 00:36:16,870

 

1145
00:36:16,870 --> 00:36:18,950

 

1146
00:36:18,950 --> 00:36:18,960

 

1147
00:36:18,960 --> 00:36:20,069


1148
00:36:20,069 --> 00:36:21,589

 

1149
00:36:21,589 --> 00:36:25,030

 

1150
00:36:25,030 --> 00:36:26,630

 

1151
00:36:26,630 --> 00:36:28,550

 

1152
00:36:28,550 --> 00:36:29,589

 

1153
00:36:29,589 --> 00:36:31,190

 

1154
00:36:31,190 --> 00:36:33,109

 

1155
00:36:33,109 --> 00:36:34,470

 

1156
00:36:34,470 --> 00:36:35,910

 

1157
00:36:35,910 --> 00:36:38,150

 

1158
00:36:38,150 --> 00:36:39,589

 

1159
00:36:39,589 --> 00:36:41,109

 

1160
00:36:41,109 --> 00:36:42,790

 

1161
00:36:42,790 --> 00:36:42,800

 

1162
00:36:42,800 --> 00:36:43,990


1163
00:36:43,990 --> 00:36:44,000

 

1164
00:36:44,000 --> 00:36:46,550


1165
00:36:46,550 --> 00:36:48,310

 

1166
00:36:48,310 --> 00:36:50,390

 

1167
00:36:50,390 --> 00:36:50,400

 

1168
00:36:50,400 --> 00:36:51,109


1169
00:36:51,109 --> 00:36:52,790

 

1170
00:36:52,790 --> 00:36:53,990

 

1171
00:36:53,990 --> 00:36:54,000

 

1172
00:36:54,000 --> 00:36:55,430


1173
00:36:55,430 --> 00:36:55,440

 

1174
00:36:55,440 --> 00:36:56,310


1175
00:36:56,310 --> 00:36:59,349

 

1176
00:36:59,349 --> 00:37:02,470

 

1177
00:37:02,470 --> 00:37:03,829

 

1178
00:37:03,829 --> 00:37:05,510

 

1179
00:37:05,510 --> 00:37:06,790

 

1180
00:37:06,790 --> 00:37:09,910

 

1181
00:37:09,910 --> 00:37:13,349

 

1182
00:37:13,349 --> 00:37:15,589

 

1183
00:37:15,589 --> 00:37:15,599

 

1184
00:37:15,599 --> 00:37:17,109


1185
00:37:17,109 --> 00:37:17,119

 

1186
00:37:17,119 --> 00:37:18,630


1187
00:37:18,630 --> 00:37:21,109

 

1188
00:37:21,109 --> 00:37:23,109

 

1189
00:37:23,109 --> 00:37:25,510

 

1190
00:37:25,510 --> 00:37:28,069

 

1191
00:37:28,069 --> 00:37:29,589

 

1192
00:37:29,589 --> 00:37:32,390

 

1193
00:37:32,390 --> 00:37:32,400

 

1194
00:37:32,400 --> 00:37:33,750


1195
00:37:33,750 --> 00:37:36,470

 

1196
00:37:36,470 --> 00:37:40,470

 

1197
00:37:40,470 --> 00:37:42,470

 

1198
00:37:42,470 --> 00:37:43,990

 

1199
00:37:43,990 --> 00:37:46,710

 

1200
00:37:46,710 --> 00:37:47,750

 

1201
00:37:47,750 --> 00:37:49,430

 

1202
00:37:49,430 --> 00:37:52,470

 

1203
00:37:52,470 --> 00:37:54,310

 

1204
00:37:54,310 --> 00:37:56,230

 

1205
00:37:56,230 --> 00:37:58,470

 

1206
00:37:58,470 --> 00:38:00,630

 

1207
00:38:00,630 --> 00:38:02,829

 

1208
00:38:02,829 --> 00:38:06,710

 

1209
00:38:06,710 --> 00:38:09,589

 

1210
00:38:09,589 --> 00:38:09,599

 

1211
00:38:09,599 --> 00:38:10,390


1212
00:38:10,390 --> 00:38:13,349

 

1213
00:38:13,349 --> 00:38:15,750

 

1214
00:38:15,750 --> 00:38:18,069

 

1215
00:38:18,069 --> 00:38:19,670

 

1216
00:38:19,670 --> 00:38:22,069

 

1217
00:38:22,069 --> 00:38:23,829

 

1218
00:38:23,829 --> 00:38:25,430

 

1219
00:38:25,430 --> 00:38:27,589

 

1220
00:38:27,589 --> 00:38:29,589

 

1221
00:38:29,589 --> 00:38:31,589

 

1222
00:38:31,589 --> 00:38:34,390

 

1223
00:38:34,390 --> 00:38:34,400

 

1224
00:38:34,400 --> 00:38:35,270


1225
00:38:35,270 --> 00:38:37,270

 

1226
00:38:37,270 --> 00:38:39,430

 

1227
00:38:39,430 --> 00:38:41,430

 

1228
00:38:41,430 --> 00:38:42,950

 

1229
00:38:42,950 --> 00:38:42,960

 

1230
00:38:42,960 --> 00:38:45,670


1231
00:38:45,670 --> 00:38:48,069

 

1232
00:38:48,069 --> 00:38:50,550

 

1233
00:38:50,550 --> 00:38:50,560

 

1234
00:38:50,560 --> 00:38:52,230


1235
00:38:52,230 --> 00:38:52,240

 

1236
00:38:52,240 --> 00:38:52,560


1237
00:38:52,560 --> 00:38:52,570

 

1238
00:38:52,570 --> 00:38:55,589


1239
00:38:55,589 --> 00:38:57,349

 

1240
00:38:57,349 --> 00:38:58,950

 

1241
00:38:58,950 --> 00:39:01,349

 

1242
00:39:01,349 --> 00:39:04,150

 

1243
00:39:04,150 --> 00:39:06,150

 

1244
00:39:06,150 --> 00:39:07,670

 

1245
00:39:07,670 --> 00:39:10,630

 

1246
00:39:10,630 --> 00:39:12,230

 

1247
00:39:12,230 --> 00:39:14,390

 

1248
00:39:14,390 --> 00:39:14,400

 

1249
00:39:14,400 --> 00:39:15,349


1250
00:39:15,349 --> 00:39:15,359

 

1251
00:39:15,359 --> 00:39:16,550


1252
00:39:16,550 --> 00:39:18,710

 

1253
00:39:18,710 --> 00:39:20,710

 

1254
00:39:20,710 --> 00:39:22,870

 

1255
00:39:22,870 --> 00:39:24,150

 

1256
00:39:24,150 --> 00:39:25,990

 

1257
00:39:25,990 --> 00:39:27,750

 

1258
00:39:27,750 --> 00:39:31,510

 

1259
00:39:31,510 --> 00:39:32,870

 

1260
00:39:32,870 --> 00:39:34,230

 

1261
00:39:34,230 --> 00:39:37,430

 

1262
00:39:37,430 --> 00:39:38,710

 

1263
00:39:38,710 --> 00:39:40,950

 

1264
00:39:40,950 --> 00:39:43,829

 

1265
00:39:43,829 --> 00:39:43,839

 

1266
00:39:43,839 --> 00:39:45,990


1267
00:39:45,990 --> 00:39:48,710

 

1268
00:39:48,710 --> 00:39:51,750

 

1269
00:39:51,750 --> 00:39:52,790

 

1270
00:39:52,790 --> 00:39:54,550

 

1271
00:39:54,550 --> 00:39:55,910

 

1272
00:39:55,910 --> 00:39:57,510

 

1273
00:39:57,510 --> 00:39:58,950

 

1274
00:39:58,950 --> 00:39:58,960

 

1275
00:39:58,960 --> 00:39:59,990


1276
00:39:59,990 --> 00:40:00,000

 

1277
00:40:00,000 --> 00:40:01,030


1278
00:40:01,030 --> 00:40:04,230

 

1279
00:40:04,230 --> 00:40:07,349

 

1280
00:40:07,349 --> 00:40:09,670

 

1281
00:40:09,670 --> 00:40:11,190

 

1282
00:40:11,190 --> 00:40:13,910

 

1283
00:40:13,910 --> 00:40:16,390

 

1284
00:40:16,390 --> 00:40:18,870

 

1285
00:40:18,870 --> 00:40:20,630

 

1286
00:40:20,630 --> 00:40:22,710

 

1287
00:40:22,710 --> 00:40:22,720

 

1288
00:40:22,720 --> 00:40:27,270


1289
00:40:27,270 --> 00:40:27,280

 

1290
00:40:27,280 --> 00:40:28,150


1291
00:40:28,150 --> 00:40:30,230

 

1292
00:40:30,230 --> 00:40:31,990

 

1293
00:40:31,990 --> 00:40:34,230

 

1294
00:40:34,230 --> 00:40:36,870

 

1295
00:40:36,870 --> 00:40:39,030

 

1296
00:40:39,030 --> 00:40:40,790

 

1297
00:40:40,790 --> 00:40:42,550

 

1298
00:40:42,550 --> 00:40:44,710

 

1299
00:40:44,710 --> 00:40:44,720

 

1300
00:40:44,720 --> 00:40:46,630


1301
00:40:46,630 --> 00:40:46,640

 

1302
00:40:46,640 --> 00:40:47,829


1303
00:40:47,829 --> 00:40:47,839

 

1304
00:40:47,839 --> 00:40:48,710


1305
00:40:48,710 --> 00:40:48,720

 

1306
00:40:48,720 --> 00:40:51,990


1307
00:40:51,990 --> 00:40:53,670

 

1308
00:40:53,670 --> 00:40:56,150

 

1309
00:40:56,150 --> 00:40:58,470

 

1310
00:40:58,470 --> 00:41:00,950

 

1311
00:41:00,950 --> 00:41:02,309

 

1312
00:41:02,309 --> 00:41:02,319

 

1313
00:41:02,319 --> 00:41:04,230


1314
00:41:04,230 --> 00:41:06,710

 

1315
00:41:06,710 --> 00:41:08,309

 

1316
00:41:08,309 --> 00:41:11,270

 

1317
00:41:11,270 --> 00:41:13,190

 

1318
00:41:13,190 --> 00:41:13,200

 

1319
00:41:13,200 --> 00:41:14,630


1320
00:41:14,630 --> 00:41:16,870

 

1321
00:41:16,870 --> 00:41:18,790

 

1322
00:41:18,790 --> 00:41:21,190

 

1323
00:41:21,190 --> 00:41:25,430

 

1324
00:41:25,430 --> 00:41:27,990

 

1325
00:41:27,990 --> 00:41:29,670

 

1326
00:41:29,670 --> 00:41:32,230

 

1327
00:41:32,230 --> 00:41:33,270

 

1328
00:41:33,270 --> 00:41:35,349

 

1329
00:41:35,349 --> 00:41:35,359

 

1330
00:41:35,359 --> 00:41:36,630


1331
00:41:36,630 --> 00:41:38,390

 

1332
00:41:38,390 --> 00:41:41,030

 

1333
00:41:41,030 --> 00:41:43,109

 

1334
00:41:43,109 --> 00:41:44,150

 

1335
00:41:44,150 --> 00:41:46,950

 

1336
00:41:46,950 --> 00:41:50,870

 

1337
00:41:50,870 --> 00:41:50,880

 

1338
00:41:50,880 --> 00:41:52,550


1339
00:41:52,550 --> 00:41:54,950

 

1340
00:41:54,950 --> 00:41:57,190

 

1341
00:41:57,190 --> 00:41:59,270

 

1342
00:41:59,270 --> 00:42:02,870

 

1343
00:42:02,870 --> 00:42:04,470

 

1344
00:42:04,470 --> 00:42:06,309

 

1345
00:42:06,309 --> 00:42:09,109

 

1346
00:42:09,109 --> 00:42:09,119

 

1347
00:42:09,119 --> 00:42:10,950


1348
00:42:10,950 --> 00:42:13,190

 

1349
00:42:13,190 --> 00:42:15,190

 

1350
00:42:15,190 --> 00:42:17,030

 

1351
00:42:17,030 --> 00:42:18,870

 

1352
00:42:18,870 --> 00:42:20,150

 

1353
00:42:20,150 --> 00:42:23,109

 

1354
00:42:23,109 --> 00:42:24,630

 

1355
00:42:24,630 --> 00:42:25,990

 

1356
00:42:25,990 --> 00:42:27,349

 

1357
00:42:27,349 --> 00:42:27,359

 

1358
00:42:27,359 --> 00:42:29,109


1359
00:42:29,109 --> 00:42:29,119

 

1360
00:42:29,119 --> 00:42:30,710


1361
00:42:30,710 --> 00:42:32,069

 

1362
00:42:32,069 --> 00:42:34,309

 

1363
00:42:34,309 --> 00:42:36,870

 

1364
00:42:36,870 --> 00:42:36,880

 

1365
00:42:36,880 --> 00:42:38,790


1366
00:42:38,790 --> 00:42:40,470

 

1367
00:42:40,470 --> 00:42:45,109

 

1368
00:42:45,109 --> 00:42:45,119

 

1369
00:42:45,119 --> 00:42:45,910


1370
00:42:45,910 --> 00:42:45,920

 

1371
00:42:45,920 --> 00:42:48,230


1372
00:42:48,230 --> 00:42:52,230

 

1373
00:42:52,230 --> 00:42:54,790

 

1374
00:42:54,790 --> 00:42:57,589

 

1375
00:42:57,589 --> 00:43:00,069

 

1376
00:43:00,069 --> 00:43:01,750

 

1377
00:43:01,750 --> 00:43:01,760

 

1378
00:43:01,760 --> 00:43:02,790


1379
00:43:02,790 --> 00:43:02,800

 

1380
00:43:02,800 --> 00:43:04,790


1381
00:43:04,790 --> 00:43:07,190

 

1382
00:43:07,190 --> 00:43:09,349

 

1383
00:43:09,349 --> 00:43:10,950

 

1384
00:43:10,950 --> 00:43:13,589

 

1385
00:43:13,589 --> 00:43:15,510

 

1386
00:43:15,510 --> 00:43:18,150

 

1387
00:43:18,150 --> 00:43:18,160

 

1388
00:43:18,160 --> 00:43:21,990


1389
00:43:21,990 --> 00:43:22,000

 

1390
00:43:22,000 --> 00:43:23,829


1391
00:43:23,829 --> 00:43:25,270

 

1392
00:43:25,270 --> 00:43:26,470

 

1393
00:43:26,470 --> 00:43:29,190

 

1394
00:43:29,190 --> 00:43:32,230

 

1395
00:43:32,230 --> 00:43:32,240

 

1396
00:43:32,240 --> 00:43:34,630


1397
00:43:34,630 --> 00:43:35,589

 

1398
00:43:35,589 --> 00:43:37,109

 

1399
00:43:37,109 --> 00:43:39,510

 

1400
00:43:39,510 --> 00:43:42,710

 

1401
00:43:42,710 --> 00:43:44,550

 

1402
00:43:44,550 --> 00:43:47,349

 

1403
00:43:47,349 --> 00:43:49,349

 

1404
00:43:49,349 --> 00:43:51,750

 

1405
00:43:51,750 --> 00:43:53,510

 

1406
00:43:53,510 --> 00:43:56,470

 

1407
00:43:56,470 --> 00:43:58,230

 

1408
00:43:58,230 --> 00:44:00,630

 

1409
00:44:00,630 --> 00:44:02,630

 

1410
00:44:02,630 --> 00:44:02,640

 

1411
00:44:02,640 --> 00:44:05,190


1412
00:44:05,190 --> 00:44:06,710

 

1413
00:44:06,710 --> 00:44:08,870

 

1414
00:44:08,870 --> 00:44:12,870

 

1415
00:44:12,870 --> 00:44:15,109

 

1416
00:44:15,109 --> 00:44:19,030

 

1417
00:44:19,030 --> 00:44:21,670

 

1418
00:44:21,670 --> 00:44:26,309

 

1419
00:44:26,309 --> 00:44:27,670

 

1420
00:44:27,670 --> 00:44:31,670

 

1421
00:44:31,670 --> 00:44:33,550

 

1422
00:44:33,550 --> 00:44:36,069

 

1423
00:44:36,069 --> 00:44:38,710

 

1424
00:44:38,710 --> 00:44:40,630

 

1425
00:44:40,630 --> 00:44:42,630

 

1426
00:44:42,630 --> 00:44:45,030

 

1427
00:44:45,030 --> 00:44:49,190

 

1428
00:44:49,190 --> 00:44:50,710

 

1429
00:44:50,710 --> 00:44:52,150

 

1430
00:44:52,150 --> 00:44:55,109

 

1431
00:44:55,109 --> 00:44:57,030

 

1432
00:44:57,030 --> 00:45:00,550

 

1433
00:45:00,550 --> 00:45:01,750

 

1434
00:45:01,750 --> 00:45:03,670

 

1435
00:45:03,670 --> 00:45:06,550

 

1436
00:45:06,550 --> 00:45:08,309

 

1437
00:45:08,309 --> 00:45:10,069

 

1438
00:45:10,069 --> 00:45:12,309

 

1439
00:45:12,309 --> 00:45:13,750

 

1440
00:45:13,750 --> 00:45:15,990

 

1441
00:45:15,990 --> 00:45:16,000

 

1442
00:45:16,000 --> 00:45:17,030


1443
00:45:17,030 --> 00:45:18,710

 

1444
00:45:18,710 --> 00:45:20,870

 

1445
00:45:20,870 --> 00:45:23,829

 

1446
00:45:23,829 --> 00:45:23,839

 

1447
00:45:23,839 --> 00:45:25,270


1448
00:45:25,270 --> 00:45:27,910

 

1449
00:45:27,910 --> 00:45:27,920

 

1450
00:45:27,920 --> 00:45:32,950


1451
00:45:32,950 --> 00:45:35,109

 

1452
00:45:35,109 --> 00:45:36,230

 

1453
00:45:36,230 --> 00:45:37,589

 

1454
00:45:37,589 --> 00:45:39,990

 

1455
00:45:39,990 --> 00:45:42,150

 

1456
00:45:42,150 --> 00:45:43,990

 

1457
00:45:43,990 --> 00:45:47,270

 

1458
00:45:47,270 --> 00:45:48,390

 

1459
00:45:48,390 --> 00:45:50,309

 

1460
00:45:50,309 --> 00:45:51,589

 

1461
00:45:51,589 --> 00:45:57,030

 

1462
00:45:57,030 --> 00:45:58,550

 

1463
00:45:58,550 --> 00:46:00,710

 

1464
00:46:00,710 --> 00:46:02,710

 

1465
00:46:02,710 --> 00:46:02,720

 

1466
00:46:02,720 --> 00:46:04,069


1467
00:46:04,069 --> 00:46:04,079

 

1468
00:46:04,079 --> 00:46:05,109


1469
00:46:05,109 --> 00:46:07,030

 

1470
00:46:07,030 --> 00:46:11,510

 

1471
00:46:11,510 --> 00:46:15,670

 

1472
00:46:15,670 --> 00:46:17,589

 

1473
00:46:17,589 --> 00:46:33,190

 

1474
00:46:33,190 --> 00:46:33,200

 

1475
00:46:33,200 --> 00:46:34,630


1476
00:46:34,630 --> 00:46:37,020

 

1477
00:46:37,020 --> 00:46:37,030

 

1478
00:46:37,030 --> 00:46:40,870


1479
00:46:40,870 --> 00:46:43,190

 

1480
00:46:43,190 --> 00:46:43,200

 

1481
00:46:43,200 --> 00:46:44,230


1482
00:46:44,230 --> 00:46:45,670

 

1483
00:46:45,670 --> 00:46:47,020

 

1484
00:46:47,020 --> 00:46:47,030

 

1485
00:46:47,030 --> 00:46:49,670


1486
00:46:49,670 --> 00:46:50,870

 

1487
00:46:50,870 --> 00:46:52,950

 

1488
00:46:52,950 --> 00:46:54,470

 

1489
00:46:54,470 --> 00:46:56,950

 

1490
00:46:56,950 --> 00:47:00,309

 

1491
00:47:00,309 --> 00:47:02,069

 

1492
00:47:02,069 --> 00:47:04,550

 

1493
00:47:04,550 --> 00:47:06,309

 

1494
00:47:06,309 --> 00:47:09,750

 

1495
00:47:09,750 --> 00:47:12,069

 

1496
00:47:12,069 --> 00:47:14,230

 

1497
00:47:14,230 --> 00:47:17,349

 

1498
00:47:17,349 --> 00:47:19,829

 

1499
00:47:19,829 --> 00:47:23,430

 

1500
00:47:23,430 --> 00:47:25,829

 

1501
00:47:25,829 --> 00:47:27,829

 

1502
00:47:27,829 --> 00:47:29,430

 

1503
00:47:29,430 --> 00:47:31,670

 

1504
00:47:31,670 --> 00:47:33,430

 

1505
00:47:33,430 --> 00:47:33,440

 

1506
00:47:33,440 --> 00:47:36,309


1507
00:47:36,309 --> 00:47:38,549

 

1508
00:47:38,549 --> 00:47:42,390

 

1509
00:47:42,390 --> 00:47:43,589

 

1510
00:47:43,589 --> 00:47:45,589

 

1511
00:47:45,589 --> 00:47:48,230

 

1512
00:47:48,230 --> 00:47:49,990

 

1513
00:47:49,990 --> 00:47:50,000

 

1514
00:47:50,000 --> 00:47:50,829


1515
00:47:50,829 --> 00:47:50,839

 

1516
00:47:50,839 --> 00:47:52,470


1517
00:47:52,470 --> 00:47:53,910

 

1518
00:47:53,910 --> 00:47:55,829

 

1519
00:47:55,829 --> 00:47:57,670

 

1520
00:47:57,670 --> 00:47:59,430

 

1521
00:47:59,430 --> 00:48:01,190

 

1522
00:48:01,190 --> 00:48:04,710

 

1523
00:48:04,710 --> 00:48:06,950

 

1524
00:48:06,950 --> 00:48:06,960

 

1525
00:48:06,960 --> 00:48:07,829


1526
00:48:07,829 --> 00:48:09,829

 

1527
00:48:09,829 --> 00:48:11,910

 

1528
00:48:11,910 --> 00:48:14,390

 

1529
00:48:14,390 --> 00:48:17,190

 

1530
00:48:17,190 --> 00:48:17,200

 

1531
00:48:17,200 --> 00:48:18,069


1532
00:48:18,069 --> 00:48:20,870

 

1533
00:48:20,870 --> 00:48:22,390

 

1534
00:48:22,390 --> 00:48:26,150

 

1535
00:48:26,150 --> 00:48:28,630

 

1536
00:48:28,630 --> 00:48:30,150

 

1537
00:48:30,150 --> 00:48:32,790

 

1538
00:48:32,790 --> 00:48:34,150

 

1539
00:48:34,150 --> 00:48:35,589

 

1540
00:48:35,589 --> 00:48:38,230

 

1541
00:48:38,230 --> 00:48:42,309

 

1542
00:48:42,309 --> 00:48:45,750

 

1543
00:48:45,750 --> 00:48:45,760

 

1544
00:48:45,760 --> 00:48:46,870


1545
00:48:46,870 --> 00:48:49,589

 

1546
00:48:49,589 --> 00:48:50,710

 

1547
00:48:50,710 --> 00:48:53,109

 

1548
00:48:53,109 --> 00:48:59,270

 

1549
00:48:59,270 --> 00:49:01,349

 

1550
00:49:01,349 --> 00:49:03,190

 

1551
00:49:03,190 --> 00:49:05,109

 

1552
00:49:05,109 --> 00:49:06,790

 

1553
00:49:06,790 --> 00:49:09,589

 

1554
00:49:09,589 --> 00:49:12,150

 

1555
00:49:12,150 --> 00:49:14,150

 

1556
00:49:14,150 --> 00:49:16,950

 

1557
00:49:16,950 --> 00:49:18,630

 

1558
00:49:18,630 --> 00:49:21,670

 

1559
00:49:21,670 --> 00:49:24,230

 

1560
00:49:24,230 --> 00:49:26,390

 

1561
00:49:26,390 --> 00:49:29,190

 

1562
00:49:29,190 --> 00:49:30,470

 

1563
00:49:30,470 --> 00:49:30,480

 

1564
00:49:30,480 --> 00:49:31,910


1565
00:49:31,910 --> 00:49:37,270

 

1566
00:49:37,270 --> 00:49:39,349

 

1567
00:49:39,349 --> 00:49:39,359

 

1568
00:49:39,359 --> 00:49:40,470


1569
00:49:40,470 --> 00:49:41,510

 

1570
00:49:41,510 --> 00:49:41,520

 

1571
00:49:41,520 --> 00:49:42,950


1572
00:49:42,950 --> 00:49:46,069

 

1573
00:49:46,069 --> 00:49:48,309

 

1574
00:49:48,309 --> 00:49:50,710

 

1575
00:49:50,710 --> 00:49:50,720

 

1576
00:49:50,720 --> 00:49:51,589


1577
00:49:51,589 --> 00:49:53,910

 

1578
00:49:53,910 --> 00:49:55,670

 

1579
00:49:55,670 --> 00:49:58,150

 

1580
00:49:58,150 --> 00:49:59,510

 

1581
00:49:59,510 --> 00:50:00,950

 

1582
00:50:00,950 --> 00:50:04,790

 

1583
00:50:04,790 --> 00:50:06,230

 

1584
00:50:06,230 --> 00:50:06,240

 

1585
00:50:06,240 --> 00:50:08,710


1586
00:50:08,710 --> 00:50:11,510

 

1587
00:50:11,510 --> 00:50:14,870

 

1588
00:50:14,870 --> 00:50:14,880

 

1589
00:50:14,880 --> 00:50:16,950


1590
00:50:16,950 --> 00:50:18,549

 

1591
00:50:18,549 --> 00:50:22,150

 

1592
00:50:22,150 --> 00:50:22,160

 

1593
00:50:22,160 --> 00:50:22,870


1594
00:50:22,870 --> 00:50:27,109

 

1595
00:50:27,109 --> 00:50:29,430

 

1596
00:50:29,430 --> 00:50:31,750

 

1597
00:50:31,750 --> 00:50:34,069

 

1598
00:50:34,069 --> 00:50:39,510

 

1599
00:50:39,510 --> 00:50:39,520

 

1600
00:50:39,520 --> 00:50:40,710


1601
00:50:40,710 --> 00:50:43,270

 

1602
00:50:43,270 --> 00:50:43,280

 

1603
00:50:43,280 --> 00:50:44,870


1604
00:50:44,870 --> 00:50:47,589

 

1605
00:50:47,589 --> 00:50:49,109

 

1606
00:50:49,109 --> 00:50:51,349

 

1607
00:50:51,349 --> 00:50:51,359

 

1608
00:50:51,359 --> 00:50:53,510


1609
00:50:53,510 --> 00:50:54,870

 

1610
00:50:54,870 --> 00:50:56,309

 

1611
00:50:56,309 --> 00:50:59,270

 

1612
00:50:59,270 --> 00:51:01,109

 

1613
00:51:01,109 --> 00:51:02,630

 

1614
00:51:02,630 --> 00:51:02,640

 

1615
00:51:02,640 --> 00:51:04,549


1616
00:51:04,549 --> 00:51:06,630

 

1617
00:51:06,630 --> 00:51:08,549

 

1618
00:51:08,549 --> 00:51:09,990

 

1619
00:51:09,990 --> 00:51:11,910

 

1620
00:51:11,910 --> 00:51:13,430

 

1621
00:51:13,430 --> 00:51:16,950

 

1622
00:51:16,950 --> 00:51:20,950

 

1623
00:51:20,950 --> 00:51:22,390

 

1624
00:51:22,390 --> 00:51:24,150

 

1625
00:51:24,150 --> 00:51:26,470

 

1626
00:51:26,470 --> 00:51:28,470

 

1627
00:51:28,470 --> 00:51:31,349

 

1628
00:51:31,349 --> 00:51:31,359

 

1629
00:51:31,359 --> 00:51:33,109


1630
00:51:33,109 --> 00:51:34,470

 

1631
00:51:34,470 --> 00:51:36,950

 

1632
00:51:36,950 --> 00:51:39,109

 

1633
00:51:39,109 --> 00:51:41,349

 

1634
00:51:41,349 --> 00:51:43,589

 

1635
00:51:43,589 --> 00:51:45,510

 

1636
00:51:45,510 --> 00:51:47,270

 

1637
00:51:47,270 --> 00:51:47,280

 

1638
00:51:47,280 --> 00:51:49,109


1639
00:51:49,109 --> 00:51:49,119

 

1640
00:51:49,119 --> 00:51:50,390


1641
00:51:50,390 --> 00:51:52,390

 

1642
00:51:52,390 --> 00:51:54,470

 

1643
00:51:54,470 --> 00:51:54,480

 

1644
00:51:54,480 --> 00:51:58,309


1645
00:51:58,309 --> 00:52:01,109

 

1646
00:52:01,109 --> 00:52:02,470

 

1647
00:52:02,470 --> 00:52:05,190

 

1648
00:52:05,190 --> 00:52:09,270

 

1649
00:52:09,270 --> 00:52:12,309

 

1650
00:52:12,309 --> 00:52:15,270

 

1651
00:52:15,270 --> 00:52:15,280

 

1652
00:52:15,280 --> 00:52:17,190


1653
00:52:17,190 --> 00:52:19,829

 

1654
00:52:19,829 --> 00:52:21,270

 

1655
00:52:21,270 --> 00:52:21,280

 

1656
00:52:21,280 --> 00:52:22,630


1657
00:52:22,630 --> 00:52:28,230

 

1658
00:52:28,230 --> 00:52:31,670

 

1659
00:52:31,670 --> 00:52:31,680

 

1660
00:52:31,680 --> 00:52:35,270


1661
00:52:35,270 --> 00:52:35,280

 

1662
00:52:35,280 --> 00:52:38,230


1663
00:52:38,230 --> 00:52:41,910

 

1664
00:52:41,910 --> 00:52:44,150

 

1665
00:52:44,150 --> 00:52:46,390

 

1666
00:52:46,390 --> 00:52:48,470

 

1667
00:52:48,470 --> 00:52:51,589

 

1668
00:52:51,589 --> 00:52:53,030

 

1669
00:52:53,030 --> 00:52:54,549

 

1670
00:52:54,549 --> 00:52:55,910

 

1671
00:52:55,910 --> 00:52:57,430

 

1672
00:52:57,430 --> 00:52:57,440

 

1673
00:52:57,440 --> 00:52:58,390


1674
00:52:58,390 --> 00:53:02,150

 

1675
00:53:02,150 --> 00:53:04,630

 

1676
00:53:04,630 --> 00:53:06,230

 

1677
00:53:06,230 --> 00:53:06,240

 

1678
00:53:06,240 --> 00:53:07,109


1679
00:53:07,109 --> 00:53:10,710

 

1680
00:53:10,710 --> 00:53:10,720

 

1681
00:53:10,720 --> 00:53:11,510


1682
00:53:11,510 --> 00:53:11,520

 

1683
00:53:11,520 --> 00:53:13,030


1684
00:53:13,030 --> 00:53:14,950

 

1685
00:53:14,950 --> 00:53:17,109

 

1686
00:53:17,109 --> 00:53:18,710

 

1687
00:53:18,710 --> 00:53:20,470

 

1688
00:53:20,470 --> 00:53:22,870

 

1689
00:53:22,870 --> 00:53:25,270

 

1690
00:53:25,270 --> 00:53:28,230

 

1691
00:53:28,230 --> 00:53:28,240

 

1692
00:53:28,240 --> 00:53:29,109


1693
00:53:29,109 --> 00:53:29,119

 

1694
00:53:29,119 --> 00:53:31,190


1695
00:53:31,190 --> 00:53:32,870

 

1696
00:53:32,870 --> 00:53:36,150

 

1697
00:53:36,150 --> 00:53:38,390

 

1698
00:53:38,390 --> 00:53:39,829

 

1699
00:53:39,829 --> 00:53:42,390

 

1700
00:53:42,390 --> 00:53:42,400

 

1701
00:53:42,400 --> 00:53:43,670


1702
00:53:43,670 --> 00:53:43,680

 

1703
00:53:43,680 --> 00:53:44,790


1704
00:53:44,790 --> 00:53:46,549

 

1705
00:53:46,549 --> 00:53:48,870

 

1706
00:53:48,870 --> 00:53:48,880

 

1707
00:53:48,880 --> 00:53:50,150


1708
00:53:50,150 --> 00:53:52,950

 

1709
00:53:52,950 --> 00:53:52,960

 

1710
00:53:52,960 --> 00:53:53,829


1711
00:53:53,829 --> 00:53:55,510

 

1712
00:53:55,510 --> 00:53:57,349

 

1713
00:53:57,349 --> 00:54:00,710

 

1714
00:54:00,710 --> 00:54:02,710

 

1715
00:54:02,710 --> 00:54:05,670

 

1716
00:54:05,670 --> 00:54:07,190

 

1717
00:54:07,190 --> 00:54:07,200

 

1718
00:54:07,200 --> 00:54:08,829


1719
00:54:08,829 --> 00:54:11,430

 

1720
00:54:11,430 --> 00:54:12,870

 

1721
00:54:12,870 --> 00:54:15,990

 

1722
00:54:15,990 --> 00:54:17,589

 

1723
00:54:17,589 --> 00:54:19,430

 

1724
00:54:19,430 --> 00:54:19,440

 

1725
00:54:19,440 --> 00:54:20,230


1726
00:54:20,230 --> 00:54:22,829

 

1727
00:54:22,829 --> 00:54:25,109

 

1728
00:54:25,109 --> 00:54:27,910

 

1729
00:54:27,910 --> 00:54:27,920

 

1730
00:54:27,920 --> 00:54:28,950


1731
00:54:28,950 --> 00:54:28,960

 

1732
00:54:28,960 --> 00:54:30,069


1733
00:54:30,069 --> 00:54:33,349

 

1734
00:54:33,349 --> 00:54:35,109

 

1735
00:54:35,109 --> 00:54:35,119

 

1736
00:54:35,119 --> 00:54:37,109


1737
00:54:37,109 --> 00:54:40,549

 

1738
00:54:40,549 --> 00:54:43,109

 

1739
00:54:43,109 --> 00:54:44,549

 

1740
00:54:44,549 --> 00:54:46,630

 

1741
00:54:46,630 --> 00:54:50,390

 

1742
00:54:50,390 --> 00:54:51,829

 

1743
00:54:51,829 --> 00:54:53,430

 

1744
00:54:53,430 --> 00:54:54,789

 

1745
00:54:54,789 --> 00:54:58,230

 

1746
00:54:58,230 --> 00:54:59,670

 

1747
00:54:59,670 --> 00:55:01,109

 

1748
00:55:01,109 --> 00:55:02,870

 

1749
00:55:02,870 --> 00:55:04,870

 

1750
00:55:04,870 --> 00:55:07,430

 

1751
00:55:07,430 --> 00:55:08,789

 

1752
00:55:08,789 --> 00:55:10,309

 

1753
00:55:10,309 --> 00:55:12,549

 

1754
00:55:12,549 --> 00:55:15,030

 

1755
00:55:15,030 --> 00:55:16,870

 

1756
00:55:16,870 --> 00:55:19,030

 

1757
00:55:19,030 --> 00:55:19,040

 

1758
00:55:19,040 --> 00:55:21,910


1759
00:55:21,910 --> 00:55:21,920

 

1760
00:55:21,920 --> 00:55:23,670


1761
00:55:23,670 --> 00:55:25,589

 

1762
00:55:25,589 --> 00:55:27,430

 

1763
00:55:27,430 --> 00:55:29,430

 

1764
00:55:29,430 --> 00:55:31,430

 

1765
00:55:31,430 --> 00:55:33,589

 

1766
00:55:33,589 --> 00:55:35,190

 

1767
00:55:35,190 --> 00:55:38,789

 

1768
00:55:38,789 --> 00:55:38,799

 

1769
00:55:38,799 --> 00:55:39,589


1770
00:55:39,589 --> 00:55:39,599

 

1771
00:55:39,599 --> 00:55:40,870


1772
00:55:40,870 --> 00:55:43,670

 

1773
00:55:43,670 --> 00:55:45,430

 

1774
00:55:45,430 --> 00:55:48,870

 

1775
00:55:48,870 --> 00:55:51,829

 

1776
00:55:51,829 --> 00:55:55,910

 

1777
00:55:55,910 --> 00:55:57,750

 

1778
00:55:57,750 --> 00:55:57,760

 

1779
00:55:57,760 --> 00:55:58,630


1780
00:55:58,630 --> 00:55:58,640

 

1781
00:55:58,640 --> 00:55:59,829


1782
00:55:59,829 --> 00:56:03,829

 

1783
00:56:03,829 --> 00:56:05,510

 

1784
00:56:05,510 --> 00:56:07,910

 

1785
00:56:07,910 --> 00:56:10,150

 

1786
00:56:10,150 --> 00:56:11,990

 

1787
00:56:11,990 --> 00:56:14,789

 

1788
00:56:14,789 --> 00:56:17,430

 

1789
00:56:17,430 --> 00:56:19,510

 

1790
00:56:19,510 --> 00:56:19,520

 

1791
00:56:19,520 --> 00:56:22,549


1792
00:56:22,549 --> 00:56:23,990

 

1793
00:56:23,990 --> 00:56:26,069

 

1794
00:56:26,069 --> 00:56:27,430

 

1795
00:56:27,430 --> 00:56:30,230

 

1796
00:56:30,230 --> 00:56:32,549

 

1797
00:56:32,549 --> 00:56:32,559

 

1798
00:56:32,559 --> 00:56:33,750


1799
00:56:33,750 --> 00:56:37,589

 

1800
00:56:37,589 --> 00:56:39,030

 

1801
00:56:39,030 --> 00:56:41,109

 

1802
00:56:41,109 --> 00:56:43,349

 

1803
00:56:43,349 --> 00:56:47,190

 

1804
00:56:47,190 --> 00:56:49,109

 

1805
00:56:49,109 --> 00:56:50,549

 

1806
00:56:50,549 --> 00:56:51,990

 

1807
00:56:51,990 --> 00:56:53,349

 

1808
00:56:53,349 --> 00:56:53,359

 

1809
00:56:53,359 --> 00:56:55,510


1810
00:56:55,510 --> 00:56:55,520

 

1811
00:56:55,520 --> 00:56:57,030


1812
00:56:57,030 --> 00:56:59,030

 

1813
00:56:59,030 --> 00:56:59,040

 

1814
00:56:59,040 --> 00:57:00,470


1815
00:57:00,470 --> 00:57:01,990

 

1816
00:57:01,990 --> 00:57:03,910

 

1817
00:57:03,910 --> 00:57:05,109

 

1818
00:57:05,109 --> 00:57:06,789

 

1819
00:57:06,789 --> 00:57:08,630

 

1820
00:57:08,630 --> 00:57:10,549

 

1821
00:57:10,549 --> 00:57:12,630

 

1822
00:57:12,630 --> 00:57:17,190

 

1823
00:57:17,190 --> 00:57:17,200

 

1824
00:57:17,200 --> 00:57:18,309


1825
00:57:18,309 --> 00:57:19,990

 

1826
00:57:19,990 --> 00:57:21,990

 

1827
00:57:21,990 --> 00:57:23,829

 

1828
00:57:23,829 --> 00:57:26,470

 

1829
00:57:26,470 --> 00:57:28,069

 

1830
00:57:28,069 --> 00:57:30,069

 

1831
00:57:30,069 --> 00:57:34,789

 

1832
00:57:34,789 --> 00:57:36,630

 

1833
00:57:36,630 --> 00:57:37,910

 

1834
00:57:37,910 --> 00:57:40,950

 

1835
00:57:40,950 --> 00:57:42,710

 

1836
00:57:42,710 --> 00:57:44,630

 

1837
00:57:44,630 --> 00:57:46,230

 

1838
00:57:46,230 --> 00:57:47,910

 

1839
00:57:47,910 --> 00:57:51,990

 

1840
00:57:51,990 --> 00:57:54,789

 

1841
00:57:54,789 --> 00:57:57,109

 

1842
00:57:57,109 --> 00:57:59,990

 

1843
00:57:59,990 --> 00:58:02,230

 

1844
00:58:02,230 --> 00:58:06,069

 

1845
00:58:06,069 --> 00:58:07,990

 

1846
00:58:07,990 --> 00:58:09,750

 

1847
00:58:09,750 --> 00:58:12,230

 

1848
00:58:12,230 --> 00:58:16,069

 

1849
00:58:16,069 --> 00:58:18,309

 

1850
00:58:18,309 --> 00:58:18,319

 

1851
00:58:18,319 --> 00:58:19,270


1852
00:58:19,270 --> 00:58:21,670

 

1853
00:58:21,670 --> 00:58:24,390

 

1854
00:58:24,390 --> 00:58:26,390

 

1855
00:58:26,390 --> 00:58:29,030

 

1856
00:58:29,030 --> 00:58:29,040

 

1857
00:58:29,040 --> 00:58:30,150


1858
00:58:30,150 --> 00:58:33,030

 

1859
00:58:33,030 --> 00:58:33,040

 

1860
00:58:33,040 --> 00:58:34,230


1861
00:58:34,230 --> 00:58:37,670

 

1862
00:58:37,670 --> 00:58:39,990

 

1863
00:58:39,990 --> 00:58:40,000

 

1864
00:58:40,000 --> 00:58:41,270


1865
00:58:41,270 --> 00:58:43,750

 

1866
00:58:43,750 --> 00:58:46,150

 

1867
00:58:46,150 --> 00:58:48,789

 

1868
00:58:48,789 --> 00:58:51,670

 

1869
00:58:51,670 --> 00:58:53,910

 

1870
00:58:53,910 --> 00:58:56,549

 

1871
00:58:56,549 --> 00:58:58,870

 

1872
00:58:58,870 --> 00:59:00,950

 

1873
00:59:00,950 --> 00:59:04,069

 

1874
00:59:04,069 --> 00:59:07,750

 

1875
00:59:07,750 --> 00:59:09,109

 

1876
00:59:09,109 --> 00:59:10,630

 

1877
00:59:10,630 --> 00:59:12,309

 

1878
00:59:12,309 --> 00:59:13,829

 

1879
00:59:13,829 --> 00:59:15,750

 

1880
00:59:15,750 --> 00:59:18,630

 

1881
00:59:18,630 --> 00:59:20,950

 

1882
00:59:20,950 --> 00:59:23,430

 

1883
00:59:23,430 --> 00:59:24,789

 

1884
00:59:24,789 --> 00:59:26,870

 

1885
00:59:26,870 --> 00:59:28,870

 

1886
00:59:28,870 --> 00:59:30,309

 

1887
00:59:30,309 --> 00:59:33,030

 

1888
00:59:33,030 --> 00:59:33,990

 

1889
00:59:33,990 --> 00:59:36,309

 

1890
00:59:36,309 --> 00:59:38,870

 

1891
00:59:38,870 --> 00:59:40,630

 

1892
00:59:40,630 --> 00:59:42,789

 

1893
00:59:42,789 --> 00:59:44,630

 

1894
00:59:44,630 --> 00:59:44,640

 

1895
00:59:44,640 --> 00:59:45,750


1896
00:59:45,750 --> 00:59:47,910

 

1897
00:59:47,910 --> 00:59:49,270

 

1898
00:59:49,270 --> 00:59:52,150

 

1899
00:59:52,150 --> 00:59:55,750

 

1900
00:59:55,750 --> 00:59:59,670

 

1901
00:59:59,670 --> 01:00:02,870

 

1902
01:00:02,870 --> 01:00:05,109

 

1903
01:00:05,109 --> 01:00:06,309

 

1904
01:00:06,309 --> 01:00:07,829

 

1905
01:00:07,829 --> 01:00:10,230

 

1906
01:00:10,230 --> 01:00:12,309

 

1907
01:00:12,309 --> 01:00:14,789

 

1908
01:00:14,789 --> 01:00:19,589

 

1909
01:00:19,589 --> 01:00:21,670

 

1910
01:00:21,670 --> 01:00:23,190

 

1911
01:00:23,190 --> 01:00:24,789

 

1912
01:00:24,789 --> 01:00:26,309

 

1913
01:00:26,309 --> 01:00:26,319

 

1914
01:00:26,319 --> 01:00:31,270


1915
01:00:31,270 --> 01:00:33,430

 

1916
01:00:33,430 --> 01:00:35,030

 

1917
01:00:35,030 --> 01:00:36,710

 

1918
01:00:36,710 --> 01:00:39,829

 

1919
01:00:39,829 --> 01:00:41,990

 

1920
01:00:41,990 --> 01:00:44,630

 

1921
01:00:44,630 --> 01:00:51,670

 

1922
01:00:51,670 --> 01:00:53,270

 

1923
01:00:53,270 --> 01:00:57,109

 

1924
01:00:57,109 --> 01:00:57,119

 

1925
01:00:57,119 --> 01:00:58,150


1926
01:00:58,150 --> 01:00:59,990

 

1927
01:00:59,990 --> 01:01:03,510

 

1928
01:01:03,510 --> 01:01:05,430

 

1929
01:01:05,430 --> 01:01:06,470

 

1930
01:01:06,470 --> 01:01:06,480

 

1931
01:01:06,480 --> 01:01:07,430


1932
01:01:07,430 --> 01:01:09,430

 

1933
01:01:09,430 --> 01:01:12,150

 

1934
01:01:12,150 --> 01:01:15,190

 

1935
01:01:15,190 --> 01:01:18,390

 

1936
01:01:18,390 --> 01:01:19,589

 

1937
01:01:19,589 --> 01:01:21,430

 

1938
01:01:21,430 --> 01:01:23,670

 

1939
01:01:23,670 --> 01:01:25,750

 

1940
01:01:25,750 --> 01:01:28,390

 

1941
01:01:28,390 --> 01:01:28,400

 

1942
01:01:28,400 --> 01:01:29,510


1943
01:01:29,510 --> 01:01:29,520

 

1944
01:01:29,520 --> 01:01:30,549


1945
01:01:30,549 --> 01:01:32,150

 

1946
01:01:32,150 --> 01:01:34,470

 

1947
01:01:34,470 --> 01:01:36,710

 

1948
01:01:36,710 --> 01:01:39,750

 

1949
01:01:39,750 --> 01:01:42,950

 

1950
01:01:42,950 --> 01:01:44,230

 

1951
01:01:44,230 --> 01:01:45,990

 

1952
01:01:45,990 --> 01:01:48,710

 

1953
01:01:48,710 --> 01:01:53,030

 

1954
01:01:53,030 --> 01:01:55,270

 

1955
01:01:55,270 --> 01:01:57,430

 

1956
01:01:57,430 --> 01:01:58,950

 

1957
01:01:58,950 --> 01:02:03,510

 

1958
01:02:03,510 --> 01:02:05,349

 

1959
01:02:05,349 --> 01:02:07,029

 

1960
01:02:07,029 --> 01:02:08,710

 

1961
01:02:08,710 --> 01:02:10,789

 

1962
01:02:10,789 --> 01:02:12,710

 

1963
01:02:12,710 --> 01:02:15,270

 

1964
01:02:15,270 --> 01:02:15,280

 

1965
01:02:15,280 --> 01:02:15,990


1966
01:02:15,990 --> 01:02:17,750

 

1967
01:02:17,750 --> 01:02:20,150

 

1968
01:02:20,150 --> 01:02:21,270

 

1969
01:02:21,270 --> 01:02:23,029

 

1970
01:02:23,029 --> 01:02:24,710

 

1971
01:02:24,710 --> 01:02:28,069

 

1972
01:02:28,069 --> 01:02:29,829

 

1973
01:02:29,829 --> 01:02:31,990

 

1974
01:02:31,990 --> 01:02:33,750

 

1975
01:02:33,750 --> 01:02:36,390

 

1976
01:02:36,390 --> 01:02:39,990

 

1977
01:02:39,990 --> 01:02:40,000

 

1978
01:02:40,000 --> 01:02:41,109


1979
01:02:41,109 --> 01:02:43,190

 

1980
01:02:43,190 --> 01:02:44,870

 

1981
01:02:44,870 --> 01:02:47,990

 

1982
01:02:47,990 --> 01:02:50,950

 

1983
01:02:50,950 --> 01:02:53,430

 

1984
01:02:53,430 --> 01:02:55,670

 

1985
01:02:55,670 --> 01:02:57,589

 

1986
01:02:57,589 --> 01:03:00,870

 

1987
01:03:00,870 --> 01:03:00,880

 

1988
01:03:00,880 --> 01:03:02,390


1989
01:03:02,390 --> 01:03:02,400

 

1990
01:03:02,400 --> 01:03:03,589


1991
01:03:03,589 --> 01:03:05,430

 

1992
01:03:05,430 --> 01:03:08,789

 

1993
01:03:08,789 --> 01:03:08,799

 

1994
01:03:08,799 --> 01:03:09,670


1995
01:03:09,670 --> 01:03:13,990

 

1996
01:03:13,990 --> 01:03:15,829

 

1997
01:03:15,829 --> 01:03:19,190

 

1998
01:03:19,190 --> 01:03:19,200

 

1999
01:03:19,200 --> 01:03:21,910


2000
01:03:21,910 --> 01:03:21,920

 

2001
01:03:21,920 --> 01:03:23,190


2002
01:03:23,190 --> 01:03:30,789

 

2003
01:03:30,789 --> 01:03:32,470

 

2004
01:03:32,470 --> 01:03:34,150

 

2005
01:03:34,150 --> 01:03:37,430

 

2006
01:03:37,430 --> 01:03:37,440

 

2007
01:03:37,440 --> 01:03:38,630


2008
01:03:38,630 --> 01:03:38,640

 

2009
01:03:38,640 --> 01:03:40,630


2010
01:03:40,630 --> 01:03:42,390

 

2011
01:03:42,390 --> 01:03:46,309

 

2012
01:03:46,309 --> 01:03:48,230

 

2013
01:03:48,230 --> 01:03:50,390

 

2014
01:03:50,390 --> 01:03:51,510

 

2015
01:03:51,510 --> 01:03:53,349

 

2016
01:03:53,349 --> 01:03:53,359

 

2017
01:03:53,359 --> 01:03:54,390


2018
01:03:54,390 --> 01:03:54,400

 

2019
01:03:54,400 --> 01:03:55,270


2020
01:03:55,270 --> 01:03:57,910

 

2021
01:03:57,910 --> 01:04:01,190

 

2022
01:04:01,190 --> 01:04:03,510

 

2023
01:04:03,510 --> 01:04:06,549

 

2024
01:04:06,549 --> 01:04:08,870

 

2025
01:04:08,870 --> 01:04:11,029

 

2026
01:04:11,029 --> 01:04:12,870

 

2027
01:04:12,870 --> 01:04:15,589

 

2028
01:04:15,589 --> 01:04:18,470

 

2029
01:04:18,470 --> 01:04:21,109

 

2030
01:04:21,109 --> 01:04:24,390

 

2031
01:04:24,390 --> 01:04:24,400

 

2032
01:04:24,400 --> 01:04:25,589


2033
01:04:25,589 --> 01:04:25,599

 

2034
01:04:25,599 --> 01:04:27,349


2035
01:04:27,349 --> 01:04:29,589

 

2036
01:04:29,589 --> 01:04:31,430

 

2037
01:04:31,430 --> 01:04:33,430

 

2038
01:04:33,430 --> 01:04:34,710

 

2039
01:04:34,710 --> 01:04:35,829

 

2040
01:04:35,829 --> 01:04:37,670

 

2041
01:04:37,670 --> 01:04:37,680

 

2042
01:04:37,680 --> 01:04:38,470


2043
01:04:38,470 --> 01:04:40,710

 

2044
01:04:40,710 --> 01:04:43,910

 

2045
01:04:43,910 --> 01:04:46,069

 

2046
01:04:46,069 --> 01:04:48,789

 

2047
01:04:48,789 --> 01:04:51,109

 

2048
01:04:51,109 --> 01:04:53,670

 

2049
01:04:53,670 --> 01:04:56,150

 

2050
01:04:56,150 --> 01:04:58,390

 

2051
01:04:58,390 --> 01:05:00,710

 

2052
01:05:00,710 --> 01:05:04,789

 

2053
01:05:04,789 --> 01:05:05,829

 

2054
01:05:05,829 --> 01:05:07,029

 

2055
01:05:07,029 --> 01:05:09,109

 

2056
01:05:09,109 --> 01:05:10,309

 

2057
01:05:10,309 --> 01:05:12,549

 

2058
01:05:12,549 --> 01:05:14,470

 

2059
01:05:14,470 --> 01:05:15,349

 

2060
01:05:15,349 --> 01:05:18,230

 

2061
01:05:18,230 --> 01:05:20,069

 

2062
01:05:20,069 --> 01:05:21,430

 

2063
01:05:21,430 --> 01:05:23,750

 

2064
01:05:23,750 --> 01:05:26,150

 

2065
01:05:26,150 --> 01:05:30,309

 

2066
01:05:30,309 --> 01:05:30,319

 

2067
01:05:30,319 --> 01:05:31,190


2068
01:05:31,190 --> 01:05:33,750

 

2069
01:05:33,750 --> 01:05:35,190

 

2070
01:05:35,190 --> 01:05:37,109

 

2071
01:05:37,109 --> 01:05:41,589

 

2072
01:05:41,589 --> 01:05:45,109

 

2073
01:05:45,109 --> 01:05:47,990

 

2074
01:05:47,990 --> 01:05:52,069

 

2075
01:05:52,069 --> 01:05:54,870

 

2076
01:05:54,870 --> 01:05:56,470

 

2077
01:05:56,470 --> 01:05:58,630

 

2078
01:05:58,630 --> 01:06:00,630

 

2079
01:06:00,630 --> 01:06:02,069

 

2080
01:06:02,069 --> 01:06:05,430

 

2081
01:06:05,430 --> 01:06:07,589

 

2082
01:06:07,589 --> 01:06:13,589

 

2083
01:06:13,589 --> 01:06:13,599

 

2084
01:06:13,599 --> 01:06:15,190


2085
01:06:15,190 --> 01:06:19,190

 

2086
01:06:19,190 --> 01:06:19,200

 

2087
01:06:19,200 --> 01:06:24,950


2088
01:06:24,950 --> 01:06:27,190

 

2089
01:06:27,190 --> 01:06:29,829

 

2090
01:06:29,829 --> 01:06:31,510

 

2091
01:06:31,510 --> 01:06:33,270

 

2092
01:06:33,270 --> 01:06:34,710

 

2093
01:06:34,710 --> 01:06:36,309

 

2094
01:06:36,309 --> 01:06:37,990

 

2095
01:06:37,990 --> 01:06:46,630

 

2096
01:06:46,630 --> 01:06:48,309

 

2097
01:06:48,309 --> 01:06:50,309

 

2098
01:06:50,309 --> 01:06:52,230

 

2099
01:06:52,230 --> 01:06:52,240

 

2100
01:06:52,240 --> 01:06:53,990


2101
01:06:53,990 --> 01:06:54,000

 

2102
01:06:54,000 --> 01:06:55,910


2103
01:06:55,910 --> 01:06:55,920

 

2104
01:06:55,920 --> 01:06:56,950


2105
01:06:56,950 --> 01:06:58,870

 

2106
01:06:58,870 --> 01:07:01,349

 

2107
01:07:01,349 --> 01:07:01,359

 

2108
01:07:01,359 --> 01:07:05,190


2109
01:07:05,190 --> 01:07:05,200

 

2110
01:07:05,200 --> 01:07:06,390


2111
01:07:06,390 --> 01:07:09,190

 

2112
01:07:09,190 --> 01:07:09,200

 

2113
01:07:09,200 --> 01:07:10,069


2114
01:07:10,069 --> 01:07:11,670

 

2115
01:07:11,670 --> 01:07:13,990

 

2116
01:07:13,990 --> 01:07:15,910

 

2117
01:07:15,910 --> 01:07:17,510

 

2118
01:07:17,510 --> 01:07:19,109

 

2119
01:07:19,109 --> 01:07:20,309

 

2120
01:07:20,309 --> 01:07:20,319

 

2121
01:07:20,319 --> 01:07:22,630


2122
01:07:22,630 --> 01:07:24,230

 

2123
01:07:24,230 --> 01:07:25,750

 

2124
01:07:25,750 --> 01:07:28,069

 

2125
01:07:28,069 --> 01:07:29,190

 

2126
01:07:29,190 --> 01:07:30,549

 

2127
01:07:30,549 --> 01:07:32,789

 

2128
01:07:32,789 --> 01:07:32,799

 

2129
01:07:32,799 --> 01:07:34,630


2130
01:07:34,630 --> 01:07:34,640

 

2131
01:07:34,640 --> 01:07:36,549


2132
01:07:36,549 --> 01:07:38,309

 

2133
01:07:38,309 --> 01:07:39,670

 

2134
01:07:39,670 --> 01:07:41,270

 

2135
01:07:41,270 --> 01:07:42,549

 

2136
01:07:42,549 --> 01:07:43,829

 

2137
01:07:43,829 --> 01:07:45,670

 

2138
01:07:45,670 --> 01:07:45,680

 

2139
01:07:45,680 --> 01:07:47,109


2140
01:07:47,109 --> 01:07:48,789

 

2141
01:07:48,789 --> 01:07:50,870

 

2142
01:07:50,870 --> 01:07:52,630

 

2143
01:07:52,630 --> 01:07:52,640

 

2144
01:07:52,640 --> 01:07:57,510


2145
01:07:57,510 --> 01:07:59,109

 

2146
01:07:59,109 --> 01:08:00,789

 

2147
01:08:00,789 --> 01:08:02,789

 

2148
01:08:02,789 --> 01:08:07,109

 

2149
01:08:07,109 --> 01:08:08,950

 

2150
01:08:08,950 --> 01:08:11,349

 

2151
01:08:11,349 --> 01:08:13,190

 

2152
01:08:13,190 --> 01:08:13,200

 

2153
01:08:13,200 --> 01:08:14,390


2154
01:08:14,390 --> 01:08:17,349

 

2155
01:08:17,349 --> 01:08:20,309

 

2156
01:08:20,309 --> 01:08:22,870

 

2157
01:08:22,870 --> 01:08:25,590

 

2158
01:08:25,590 --> 01:08:28,229

 

2159
01:08:28,229 --> 01:08:30,709

 

2160
01:08:30,709 --> 01:08:32,550

 

2161
01:08:32,550 --> 01:08:35,030

 

2162
01:08:35,030 --> 01:08:36,550

 

2163
01:08:36,550 --> 01:08:38,950

 

2164
01:08:38,950 --> 01:08:40,550

 

2165
01:08:40,550 --> 01:08:40,560

 

2166
01:08:40,560 --> 01:08:42,390


2167
01:08:42,390 --> 01:08:42,400

 

2168
01:08:42,400 --> 01:08:43,430


2169
01:08:43,430 --> 01:08:44,870

 

2170
01:08:44,870 --> 01:08:46,709

 

2171
01:08:46,709 --> 01:08:46,719

 

2172
01:08:46,719 --> 01:08:47,669


2173
01:08:47,669 --> 01:08:47,679

 

2174
01:08:47,679 --> 01:08:48,789


2175
01:08:48,789 --> 01:08:50,309

 

2176
01:08:50,309 --> 01:08:52,229

 

2177
01:08:52,229 --> 01:08:54,309

 

2178
01:08:54,309 --> 01:08:56,070

 

2179
01:08:56,070 --> 01:08:56,080

 

2180
01:08:56,080 --> 01:08:56,870


2181
01:08:56,870 --> 01:08:58,789

 

2182
01:08:58,789 --> 01:09:00,070

 

2183
01:09:00,070 --> 01:09:02,630

 

2184
01:09:02,630 --> 01:09:05,430

 

2185
01:09:05,430 --> 01:09:07,749

 

2186
01:09:07,749 --> 01:09:11,430

 

2187
01:09:11,430 --> 01:09:12,789

 

2188
01:09:12,789 --> 01:09:12,799

 

2189
01:09:12,799 --> 01:09:13,590


2190
01:09:13,590 --> 01:09:13,600

 

2191
01:09:13,600 --> 01:09:15,510


2192
01:09:15,510 --> 01:09:17,829

 

2193
01:09:17,829 --> 01:09:20,390

 

2194
01:09:20,390 --> 01:09:20,400

 

2195
01:09:20,400 --> 01:09:21,349


2196
01:09:21,349 --> 01:09:25,669

 

2197
01:09:25,669 --> 01:09:27,269

 

2198
01:09:27,269 --> 01:09:30,390

 

2199
01:09:30,390 --> 01:09:33,189

 

2200
01:09:33,189 --> 01:09:36,950

 

2201
01:09:36,950 --> 01:09:39,189

 

2202
01:09:39,189 --> 01:09:41,990

 

2203
01:09:41,990 --> 01:09:44,630

 

2204
01:09:44,630 --> 01:09:46,390

 

2205
01:09:46,390 --> 01:09:48,470

 

2206
01:09:48,470 --> 01:09:48,480

 

2207
01:09:48,480 --> 01:09:50,309


2208
01:09:50,309 --> 01:09:52,309

 

2209
01:09:52,309 --> 01:09:54,229

 

2210
01:09:54,229 --> 01:09:55,669

 

2211
01:09:55,669 --> 01:09:58,070

 

2212
01:09:58,070 --> 01:10:01,350

 

2213
01:10:01,350 --> 01:10:02,550

 

2214
01:10:02,550 --> 01:10:03,830

 

2215
01:10:03,830 --> 01:10:05,910

 

2216
01:10:05,910 --> 01:10:07,430

 

2217
01:10:07,430 --> 01:10:07,440

 

2218
01:10:07,440 --> 01:10:11,030


2219
01:10:11,030 --> 01:10:13,590

 

2220
01:10:13,590 --> 01:10:16,390

 

2221
01:10:16,390 --> 01:10:18,790

 

2222
01:10:18,790 --> 01:10:21,350

 

2223
01:10:21,350 --> 01:10:22,310

 

2224
01:10:22,310 --> 01:10:24,149

 

2225
01:10:24,149 --> 01:10:26,790

 

2226
01:10:26,790 --> 01:10:26,800

 

2227
01:10:26,800 --> 01:10:27,669


2228
01:10:27,669 --> 01:10:29,110

 

2229
01:10:29,110 --> 01:10:30,550

 

2230
01:10:30,550 --> 01:10:30,560

 

2231
01:10:30,560 --> 01:10:31,510


2232
01:10:31,510 --> 01:10:31,520

 

2233
01:10:31,520 --> 01:10:32,550


2234
01:10:32,550 --> 01:10:35,189

 

2235
01:10:35,189 --> 01:10:36,470

 

2236
01:10:36,470 --> 01:10:40,550

 

2237
01:10:40,550 --> 01:10:42,470

 

2238
01:10:42,470 --> 01:10:45,910

 

2239
01:10:45,910 --> 01:10:47,750

 

2240
01:10:47,750 --> 01:10:50,550

 

2241
01:10:50,550 --> 01:10:50,560

 

2242
01:10:50,560 --> 01:10:51,350


2243
01:10:51,350 --> 01:10:53,110

 

2244
01:10:53,110 --> 01:10:57,189

 

2245
01:10:57,189 --> 01:11:01,830

 

2246
01:11:01,830 --> 01:11:01,840

 

2247
01:11:01,840 --> 01:11:03,830


2248
01:11:03,830 --> 01:11:03,840

 

2249
01:11:03,840 --> 01:11:08,870


2250
01:11:08,870 --> 01:11:10,550

 

2251
01:11:10,550 --> 01:11:13,350

 

2252
01:11:13,350 --> 01:11:14,709

 

2253
01:11:14,709 --> 01:11:16,709

 

2254
01:11:16,709 --> 01:11:19,830

 

2255
01:11:19,830 --> 01:11:21,830

 

2256
01:11:21,830 --> 01:11:24,070

 

2257
01:11:24,070 --> 01:11:26,390

 

2258
01:11:26,390 --> 01:11:29,669

 

2259
01:11:29,669 --> 01:11:32,070

 

2260
01:11:32,070 --> 01:11:32,080

 

2261
01:11:32,080 --> 01:11:33,430


2262
01:11:33,430 --> 01:11:35,669

 

2263
01:11:35,669 --> 01:11:38,709

 

2264
01:11:38,709 --> 01:11:41,189

 

2265
01:11:41,189 --> 01:11:44,229

 

2266
01:11:44,229 --> 01:11:46,390

 

2267
01:11:46,390 --> 01:11:48,630

 

2268
01:11:48,630 --> 01:11:50,310

 

2269
01:11:50,310 --> 01:11:53,430

 

2270
01:11:53,430 --> 01:11:54,790

 

2271
01:11:54,790 --> 01:11:56,950

 

2272
01:11:56,950 --> 01:11:57,910

 

2273
01:11:57,910 --> 01:12:00,790

 

2274
01:12:00,790 --> 01:12:03,189

 

2275
01:12:03,189 --> 01:12:06,550

 

2276
01:12:06,550 --> 01:12:06,560

 

2277
01:12:06,560 --> 01:12:07,270


2278
01:12:07,270 --> 01:12:10,070

 

2279
01:12:10,070 --> 01:12:12,229

 

2280
01:12:12,229 --> 01:12:14,149

 

2281
01:12:14,149 --> 01:12:16,790

 

2282
01:12:16,790 --> 01:12:18,709

 

2283
01:12:18,709 --> 01:12:20,870

 

2284
01:12:20,870 --> 01:12:23,030

 

2285
01:12:23,030 --> 01:12:25,430

 

2286
01:12:25,430 --> 01:12:27,110

 

2287
01:12:27,110 --> 01:12:29,750

 

2288
01:12:29,750 --> 01:12:31,590

 

2289
01:12:31,590 --> 01:12:33,510

 

2290
01:12:33,510 --> 01:12:35,990

 

2291
01:12:35,990 --> 01:12:38,870

 

2292
01:12:38,870 --> 01:12:38,880

 

2293
01:12:38,880 --> 01:12:42,870


2294
01:12:42,870 --> 01:12:42,880

 

2295
01:12:42,880 --> 01:12:48,830


2296
01:12:48,830 --> 01:12:51,590

 

2297
01:12:51,590 --> 01:12:54,470

 

2298
01:12:54,470 --> 01:12:56,870

 

2299
01:12:56,870 --> 01:12:59,430

 

2300
01:12:59,430 --> 01:13:01,110

 

2301
01:13:01,110 --> 01:13:02,390

 

2302
01:13:02,390 --> 01:13:03,750

 

2303
01:13:03,750 --> 01:13:03,760

 

2304
01:13:03,760 --> 01:13:04,390


2305
01:13:04,390 --> 01:13:06,709

 

2306
01:13:06,709 --> 01:13:08,630

 

2307
01:13:08,630 --> 01:13:10,870

 

2308
01:13:10,870 --> 01:13:16,630

 

2309
01:13:16,630 --> 01:13:18,709

 

2310
01:13:18,709 --> 01:13:19,990

 

2311
01:13:19,990 --> 01:13:21,350

 

2312
01:13:21,350 --> 01:13:23,830

 

2313
01:13:23,830 --> 01:13:25,110

 

2314
01:13:25,110 --> 01:13:27,270

 

2315
01:13:27,270 --> 01:13:29,350

 

2316
01:13:29,350 --> 01:13:30,830

 

2317
01:13:30,830 --> 01:13:33,110

 

2318
01:13:33,110 --> 01:13:35,669

 

2319
01:13:35,669 --> 01:13:38,470

 

2320
01:13:38,470 --> 01:13:40,390

 

2321
01:13:40,390 --> 01:13:42,870

 

2322
01:13:42,870 --> 01:13:45,430

 

2323
01:13:45,430 --> 01:13:48,950

 

2324
01:13:48,950 --> 01:13:50,310

 

2325
01:13:50,310 --> 01:13:52,390

 

2326
01:13:52,390 --> 01:13:54,870

 

2327
01:13:54,870 --> 01:13:56,709

 

2328
01:13:56,709 --> 01:13:59,830

 

2329
01:13:59,830 --> 01:14:01,590

 

2330
01:14:01,590 --> 01:14:03,510

 

2331
01:14:03,510 --> 01:14:05,270

 

2332
01:14:05,270 --> 01:14:05,280

 

2333
01:14:05,280 --> 01:14:06,310


2334
01:14:06,310 --> 01:14:07,510

 

2335
01:14:07,510 --> 01:14:09,590

 

2336
01:14:09,590 --> 01:14:29,590

 

2337
01:14:29,590 --> 01:14:29,600

 

2338
01:14:29,600 --> 01:14:31,679