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um welcome everyone welcome back to uh  theory of  computation and  um  just to recap uh where we are we  um  [Music]  have been looking at time complexity and  space complexity  and um we  just finished  approving what are called the hierarchy  theorems uh which  you know in a nutshell basically say  that  if you allow the computational model to  have a little bit more resource a little  bit more time a little bit more space  then  you can do more things  so  with certain conditions  so we proved that  last time it was a proof basically by a  diagonalization i don't know if you  recognized the diagonalization there but  um you know when you're encoding a  machine by an input  and then basically running all possible  different machines that's essentially a  diagonalization  so  today we're going to  build on that work  to give an example of what we call  a natural  intractable problem  we'll say a bit more about what that  means  and then we're going to talk about  something which is a different topic but  nevertheless related  having to do with oracles  and methods which  may or may not work  to make  to solve the p versus np problem which  of course is a  big open problem in the field  okay  so  the time and space hierarchy theorems  because we're going to be using those  today  um they say that if you give a little  bit more uh space here so if you're for  space constructable functions functions  that you can actually compute within the  amount of space that they uh specify  um you can act you can show that the  things that you can do in that much  space  is probably  is larger than what you can do in less  space  and you have can prove a similar  slightly weaker fact about the time  complexity classes so what that means is  that the uh  these  classes form a hierarchy so as you add  more time or let's say in this case  space from n squared to n cubed to enter  the fourth you get um  larger and larger classes which i'm kind  of illustrating here by putting putting  a dot there which shows that there's  something that we know that's new in  those classes as you as you go up um  these uh these different bounds  and this is going to be true for space  complexity and it's also going to be  true for time complexity  um  so  and one of the corollaries that we  pointed out last time um is that  uh  p space  is a properly includes um  non-deterministic log space nl so n l is  a proper subset of p space so this stuff  in p space  that um  is not in nl and remember this notation  here this means proper subset  um  one of the things that uh  a follow-on corollary that we didn't  mention last time but that's something  that you uh you should know  is that  um the tqbf problem you know our  piece-based complete problem is an  example of a problem that's in p space  obviously but we know it's also not in  nl  and in order to get that conclusion you  have to  look again at the proof that tqbf is  p-space complete  and observe  that the reductions that we gave in that  proof  can be carried out not only in  polynomial time but they can be carried  out in log space  and therefore if tqbf turned out to go  down to nl  um then  because everything in piece base is log  space reducible to tqbf that would bring  all of p space down to nl  but that we just proved is not the case  so therefore tqbf could not be an nl  okay and we're going to be using that  kind of reasoning  um  again in this in this lecture  uh  so just a quick check in um  these are a few  more or less easy maybe more or less  tricky um follow-ons uh that you can  conclude from the  uh time and space hierarchy theorems  plus some of the other things we've  proven along the way  and so just uh as a check of your  understanding  maybe these a little bit on the tricky  side so you have to read them carefully  which of these are known to be true  um based on the material  that we've presented and this is also  you know just material that's  you know the facts that we know to be  true in uh complexity theory  um  some launch that poll  and just just check off the ones  that  we can prove  okay  i'm going to close it down  so please  answer quickly if you're gonna okay one  two three and  uh  okay  well the two leading candidates uh are  correct  so it's uh and the two that are the  laggards here are in fact the ones that  are not true  uh so a and a and d are not true  um based on what we know  and um b and c are true so let's  understand first of all a we know is  false  because 2 to the n plus 1 is just 2  times 2 to the n  and so that's just uh these two bounds  differ only by a constant factor and so  in fact they're the same complexity  class and so you don't get proper  containment for a so that that we  absolutely know is false  um  d  um  well if we could prove that  then we would have solved a famous  problem because we don't know whether  even p equals p space  so if p equal p space then certainly p  space would equal np which is in between  the two and so uh we don't know how to  prove p space is different from np as  based on the current state of knowledge  of the field  um so this would not be something that  we know to be true uh based on what  things we've said  now b follows  directly from the time hierarchy theorem  because 2 to the 2n  is the square of 2 to the n  and that is asymptotically a larger um a  significantly larger bound and so uh  um you can prove that um time two to the  end is properly contained it  properly contains time two to the n  c is a little trickier um  because  you need to remember savage's theorem  now savage's theorem applies to space  but you also need to remember that what  you can do in time in non-deterministic  time n squared you can also do a  non-deterministic space n squared  which then in turn you can do in  deterministic space n to the fourth  um  which is properly contained within space  n to the fifth  so you can prove that p space properly  contains  not deterministic time n squared  okay it's just a bunch of containments  there that's perhaps a little the a and  c are perhaps so in a sense it may be  the most tricky of the of this group  okay um  so  let's move on so we're going to  introduce today two new classes um  and you know i actually i want to go  back to here you know what are we trying  to  what are we going to be trying to uh  accomplish in today's lecture  so we're going to be looking at provable  intractability  so our  a problem being intractable for us means  it's outside of p  so  we can't solve it in polynomial time  for our  for our um perspective we're going to  call that an  intractable problem  now this problem over here  um that's sitting in time 2 to the n but  not in the smaller classes so this is an  intractable problem that's outside of p  um  but  this example of a language  if you remember how the um  time hierarchy theorem or the space  hierarchy theorem was proved basically  this language itself is not an  interesting language for other than the  purpose that it serves to be in that  class and not in a lower class  but it's not a language that anyone  would care about and it's not even a  language that is easy to describe  it's just a language that some touring  machine  um  uh decides whether a turing machine is  specially designed to have the property  that its language is at a particular  complexity level  but otherwise there's no nice  description of that language it's not  like a to the end b to the n or some you  know um  uh  equivalence of two dfas or something  like that  so i would say that  that language is in a sense it serves  its purpose but it's not a natural  language that you really care about so  one of the go the go one of the goals of  today's lecture is to give an example of  a natural language a naturally occurring  language in a sense  um that's easy to describe where you can  prove that that language is intractable  is actually outside of p  so that's that that's a bit of  motivation where we're going um so along  the way we're going to prove we're going  to introduce these exponential  complexity classes  exponential time and exponential space  which are um exponentially bigger  than  polynomial time in polynomial space  classes so it's 2 to the n to the k in  both cases two to the two to a  polynomial  and um  the first five of these classes l  through p space we've already seen and  exponential time and exponential space  extend  the containments that we've already seen  so  you have to double check that you  understand why p space  is a  subset of exponential time but that's  because that you know as we showed  going from space to time you can do that  with an exponential increase uh that's  the cost of the simulation  and going from time to space you don't  need any increase at all anything that  you can do in a certain amount of time  you can do in that much space  so anything you can do in a certain  amount of space you can also do an  exponentially more amount of time  okay so those were  simple theorems that we proved right at  the very beginning  now the hierarchy theorems allow us to  conclude some separations among these  classes  so we already looked at this one nl  versus p space  and we saw that because nl is  by savage's theorem in deterministic log  squared space which is properly  contained  in  polynomial space you get a separation  between those two classes um provably so  and for similar reasons polynomial space  to exponential space you're going to get  a  a separation from the space hierarchy  theorem and polynomial time to  exponential time you get a provable  separation  by virtue of the hierarchy theory  um  now we're going to define  complete problems for these two classes  exponential time and exponential space  so we have exponential time complete  it's going to be  analogous to what we  showed before  which is that  it's a member of exponential time  and every problem in exponential time is  reducible to it let's say in polynomial  time though it's not going to really  turn out to be matter it could be in log  space  um some simple method of doing the  reduction is going to be good enough  let's say polynomial time it's a typical  definition here  and the same thing for exponential space  complete  we'll say it's exponent it's exponential  space complete if it's an exponential  space and anything else in exponential  space is polynomial time reducible to it  okay now but the important thing  is that  if something is exponential time  complete  you know it's outside of p  for the same reasons we've now seen  several times  namely that if an exponential time  complete problem  ended up being in p  then  because everything else in exponential  time is reducible to the complete  problem they would also be in p and so  exponential time and p would be equal  but we just said they're not equal  because of the hierarchy theorem  so the logic is the hierarchy theorem  separates the class  and then the complete problem inherits  the difficulty of the larger class  so  the complete problem cannot be any lower  than the other problems in the class  because they're all reducible to it  okay um so the same thing is going to be  true for an exponential space complete  problem can't be even in p space  because exponential space and p space  are different  and and if it's not in p space it's not  going to be in p  and so in both cases  if you have a problem that's complete  for exponential space or exponential  time  we know that those problems are  intractable  and  our strategy then for giving a natural  intractable problem intractable problem  is to show it's complete for one of  these classes and it's actually going to  turn out to be an exponential space  complete problem that we're going to  give as our example  okay so that is the plan  um  i think it's a good time to let's just  take a few questions here  uh to make sure we're all on the same  page as what we're  uh as what we're  doing so let me just read i got a couple  questions already in here  [Music]  so this is a little bit of a side  comment that somebody's asking what's  interesting question basically  is it possible  that  we may not be able to prove  solve the p versus np problem that it's  just  it's not a  problem that one can answer from the  basic axioms in mathematics if i'm  interpreting the question correctly uh  there are certain problems in  mathematics um  you know i think i perhaps i mentioned  earlier in the term the problem of  whether there is a  set  whose size is in between  the integers and the real numbers you  know we know the real numbers are larger  in size than the integers  you know that was our first example of a  diagonalization and is there a problem  of size strictly in between the two  bigger than the integers  smaller than the real numbers so that's  a problem that was  posed a long time ago  it was one of hilbert's problems and was  eventually shown to be  unanswerable  using the basic axioms of mathematics  so the question is maybe p versus np is  in the same category  um  it could be  that could be true of any unsolved  problem in mathematics  but at least our experience has shown  that the kinds of problems that at least  have been shown to be unsolvable  you know from uh mathematical axioms  tend to involve infinities and very  large things things that are very far  from our intuitions  and  something as down to earth as p versus  np at least it would be very surprising  to me if that turned out to be un uh  unanswerable using our our mathematical  axioms but who knows  oh this is another good question do the  time and space  hierarchy theorems have  non-deterministic variants yes they do  they're much harder to prove however  we're not going to cover them but you  can also prove that non-deterministic  time  um  n-cubed properly includes  non-deterministic time n squared you're  not going to be responsible for that  don't worry  if you try to fig if you try to actually  prove that you'll see the  diagonalization  um  doesn't directly work um  and so uh you have to do something  fancier  you know it just uh people are asking  about which reduction method to use you  know  again you know  the kinds of reductions that we  encounter are always very simple so  we're just going to be working with very  weak notions of reductions  not interesting yet generally to  consider powerful kinds of reductions  like  um polynomial exponential time  reductions or things like that so it's  just not something that people really  think about much um  i mean i can talk about it at at length  offline but  let's just assume that our reduction  strength is something very low that log  space is going to be good enough to do  all of the reductions in this class  um  okay so let's move on then um so here's  and  here is the problem that we're going to  spend the next 20 minutes or so proving  to be uh exponential space complete um i  have to do a little introduction first  so  this is this is not the problem but this  is related to the problem so the problem  of testing if two regular expressions  are equivalent  do i just write down two regular  expressions do they generate the same  language  so that problem actually turns out to be  in p space  so it's not going to be exponential  space complete it's actually in p space  i don't think we're going to have to i  thought about presenting it in lecture  it's not that hard to show but it just  took too much time and doesn't really  introduce new methods it's a good  exercise actually using savages there um  but maybe we'll do it in recitation um  or if the lecture  miraculous miraculously ends earlier  i'll do it at the at the end but i don't  think we'll have time  um  so  that's a but that's a setup for  the  intractable problem that we're going to  talk about which is very related  now okay before we get to that so if i  have a regular expression  i'm going to enhance the kind the i'm  going to hand enhance our regular  expression  in one simple way by allowing exponents  or exponentiation  and that means if i regul if i have a  regular expression r  um  i can write r to the k to mean r  concatenated with itself k times sort of  we've been sort of informally using that  all the way along anyway like when we  talk about zero to the k one to the k  um so if we're going to formally allow  that when we write down regular  expressions  in some cases that might allow the  regular expression to be much smaller  especially if we're writing down k in  binary  because i can write r to the million  with just a few symbols  if i have exponentiation  but if i don't have exponentiation then  i have to write r concatenated with r  out a million times and i get a much  much longer and exponentially longer  expression if i don't have that exponent  as a way of describing regular regular  expressions and that's going to make a  big difference  so now the equivalence  a problem for regular expressions with  exponentiation that's what that little  up arrow means  well that signifies  now i'm giving you two regular  expressions but they're allowed to have  x the exponentiation operation  in addition to  the um  standard regular operations  um  so now  testing whether two of these regular  expressions that have exponentiation  that problem turns out to be exponential  space complete  okay  so here's the equivalence problem for  regular expressions with exponentiation  that's an exponential space complete  problem and as we pointed out that means  this problem is provably intractable  so there are  you know there's just no way in general  to solve that problem in polynomial time  that's proven that's known  um  so  we're going to go through the reduction  um i think it's going to be our last  reduction of the term of proving  problems complete for some class  but this one has each one of those has  their own kind of um  thing that makes it special  okay so first of all we have to show  that it's in exponential space and i'm  not that's really going to rely on this  other fact that we didn't prove so i'm  going to go over that very quickly  um but the interesting part is doing the  reduction so if i have something in  exponential space  then i can show that i can reduce it  to the equivalence problem  for regular expressions with  exponentiation  okay so quickly arguing  part one that we're in exponential space  basically what you do is you take your  two regular expressions that you want to  test to see if they're equivalent but  now they have exponentiation and as a  first step you get rid of the  exponentiation  you just expand the things out by  repeating  um the the parts that have the exponents  um  and of course that as i said that's  going to make the expression themselves  exponentially bigger  but now you run the p-space algorithm  on those two exponentially larger  expressions so the input to the p-space  algorithm is now exponential in the  original input size  but it's p-space in that enlarged input  so that's going to give you an  exponential space algorithm in the  original input size  because you know you expanded it to  become exponentially bigger and then you  run the p space algorithm on that  expanded  um  problem  okay  um so that gives you an exponential  space algorithm for  uh for this problem um  but now what we're gonna do  the interesting part is the reduction  so given some language in exponential  space  say decided  by some turing machine in that amount of  space 2 to the n to the k  we're going to give a reduction that  maps a  to  this equivalence problem  got it  that is the plan  so let's make sure we're all together on  the plan before we go ahead and carry  out that plan  so good good you know if you're  got you know we haven't we just sort of  set things up here in a sense um  for what we're going to be doing so  feel free to ask a question on just this  you know the plan  this part has it's going to get  technical because as doing these  reductions always is  there's a simulation involved and you  know you have to kind of  you know describe that simulation in its  own way so now we're going to be  simulating  in a certain sense  m on w  you know the the decider for this  exponential space  problem a  uh we're going to take m on w i'm going  to somehow have to express the fact that  m accepts w using this equivalence  problem for regular expressions with  exponentiation  okay um so no questions why don't we  keep when we move on then  so here's we're gonna we have i have  three slides on this  but they're kind of dense i'm sorry to  say  um  so here is  the plan as usual  um  we're gonna map a  with a polynomial time reduction  to the equivalence problem for regular  expressions with exponentiation  so that means we're going to have to  take an input  which may or may not be an a  and produce two regular expressions  with exponentiation  which are going to be equivalent  when  w is in a  or  when m accepts w  okay  so it's going to be  as these things always are  these are going to be in terms of the  computation history for mnw but in this  case it's going to turn out to be  convenient  convenient  to work with the rejecting  computation history for m on w  so remember  we have a turing machine m  um it's a decider  so that means it always holds  for strings in the language it ends up  at a q accept state for things not in  the language it ends up at a q reject  state  so a rejecting computation history is  the sequence of configurations the  machine goes through  from the start configuration  until it ends up at a configuration with  a reject state a rejecting configuration  okay  and we're going to make a  regular expression that describes all  strings except for that one  it's going to avoid describing a  rejecting computation history for m and  w otherwise it's going to  describe all possible strings  now if m  does not reject w so there is no  rejecting computation history namely m  accepts w by the way so if m accepts w  does not reject w it does not have a  rejecting computation history  what is r one describing  well it's describing in that case  everything  because there is no rejecting  computation history so it's describing  every other string besides  so that means it's describing all  strings  if there is no rejecting computation  history in the case that m accepts w  so what does that suggest we should use  for r2  r2 is going to be the regular expression  that just generates all strings  so we'll be testing whether r1 generates  all strings or not  which is the same as saying does  um m  except w  or not  okay so  r2 is going to be  you know i would like to say sigma star  but sigma is really the input to m so  and gamma is the tape alphabet for m so  we have a lot of greek letters to play  with so we're going to use delta  for the  alphabet  that we write the computation histories  in  and if you want to  get reminded what that delta is  you know a computation history can have  a tape alphabet symbol for m it can have  a state  symbol for m or it can have a delimiter  pound hashtag  so it's either  delta  this capital delta alphabet is a  tape alphabet symbol  or a state  something representing a state symbol or  a hashtag  that's just doubt so don't get you know  i always go i always feel bad if  somebody gets confused by something  that's supposed to be very simple don't  get confused by delta star this is just  all possible strings over the alphabet  delta  okay so what does r1 look so my job is  to do r1 r2 i already told you  r1 now has to describe all those strings  except for the rejecting computation  history  so everything that fails  to be a rejecting computation history so  it fails either because it started wrong  or it ended wrong or it's wrong  somewhere in the middle and by wrong i  mean it  fails to correctly describe the way the  machine operates  if it's ending up rejecting w  right so i'm going to describe all those  possible strings um  you know by breaking it down  in those into those three categories  starts wrong ends wrong or somewhere  computes wrong along the way  okay so  rejecting computation history looks  something like this  you know there's here's the start  configuration  as we you know  you know usually envision it you know  it's a start state looking at the first  symbol of the input and there's the rest  of the input  um so let me just write this out  um  this is a rejecting computation history  now so the start first configuration the  second one and so on and so on until we  end up at a rejecting computation is a  rejecting uh configuration  now  for convenience  i'm going to  uh insist  that all of these configurations are the  same  length  it's going to make my life easier  in doing the proof  because  then why can i do that um well i'm just  gonna take them you know because usually  you think of the configurations they  start small because they're just  basically length in but you know this is  using exponential space they're kind of  getting longer and longer let's just  pair them all out with blanks so that  they're all the same size  so as i've indicated over here we're  adding in a bunch of blanks that's going  to be a lot of blanks here right  to make sure they all have length 2 to  the end of the k which is the maximum  size of a configuration when you have  that much space  um  [Music]  all right  so  um  i'm going to construct so basically  that's my job i'm going to construct r1  so that it generates all all those  strings so i'm sort of wrote a little  box  around that thing i'm trying to that's  that's my  that's my to-do  and i'm also gonna it's gonna help me in  the coming slides because they're a  little bit dense  um  when i'm gonna draw this sort of reddish  pinkish box around something that means  that i'm going to try to describe all  strings except for that one  okay so that that one is um  uh  that's the one you know i  i want to avoid describing that one  because that's the rejecting computation  history but i don't want to describe  everything else that's my that's my  that's my wish  all right um so here's a check-in before  we move forward  but we can also maybe we should just  take some questions even before we  launch the check-in  how are we doing here  so  is our one describing  you know i'm not well r1 is a regular  expression so are we over here we're  talking about a this is just an ordinary  computation history but it ends with a  reject that's all a rejecting  computation history is just one that's a  little different at the end the machine  just ended up projecting instead of  accepting  otherwise everything has to be  you know  um  spelled out  in accordance with the rules of the  machine and the start configuration  um yeah we're assuming if you're in yeah  we're assuming one rejecting  uh  state  yeah that's the way we actually define  turing machines in the first place but  who's arguing yeah there's one there's  one reject state  we're all deterministic correct  um why do we need the padding because i  want to make these all the same size  all of these configurations that's going  to help me later in terms of describing  the invalid configurations the ones that  are not legal configurations legal  rejecting configurations  so just simply a matter of convenience  um but just accept it for now i just  want all of those configurations to be  the same length  in my  uh rejecting computation history  otherwise i'm not going to  you know i'm just encoding that  rejecting computation history in this  particular way um  so people asking about the detail is a  bad start yeah that's yet to come i have  two more slides on this  so i'll tell you about how we're going  to do those  so our bad start  uh that's a good question is our bad  start or all these are all the strings  that don't start this way we'll see it  in a second but our bad start or all the  things that don't start with the  you know they they start bad they  they're they're not a  uh  um they're not starting with the start  configuration uh they're starting with  some other junk  um  do we need only one rejecting  computation history what about the other  ones  you know this is a deterministic machine  so there's only going to be one if i  prescribe the lengths as i've done  there's going to be only one  rejecting computation history because  it's deterministic everything is going  to be forced from the beginning  should r1 be the not of those three  no i'm trying r one is describing all of  the strings except  except  this  uh  this one string so i'm capturing all the  different possible ways a string could  fail  to be the string and he could start  wrong it could be wrong along the middle  somewhere or so i have to union them  together  because i'm disgraced as in  as as i always believe negations are the  most confusing thing to everybody  including me um  so we're  describing all the things that are not  this string  we're trying to stay away from that one  we want to describe everything else  okay oh um  i think i better move on here i got a  lot of questions  talk to the tas  uh  all right check in  how big  is this rejecting computation history  anyway  interesting  there's a lesson here  i got a big burst of answers right at  the very beginning all wrong  but then the bright  the people who took a little bit more  time to think  uh started getting um the right answer  which is  so it's like we got a close election  here and folks though i have to report  i hope we don't have to do a recount  okay come on guys  answer up 10 seconds this is not super  hard  stop the count  yeah i think we better stop it that's  we're  on the edge  um  okay  three seconds  and polling  share results  well  um  the correct answer is in fact c  why is that because each configuration  is 2 to the n to the k  so that's how much space the machine has  exponential space  but the amount of time which is each one  of you know the  the number of configurations is going to  be the amount of time that's used  um is going to be exponentially more  than even than that  so it's going to be 2 to the 2 to the n  to the k  is how many steps the machine can run  and that's going to be how long the  computation history could be  so it's a very long thing  and when you think about it  um the regular expression we are  generating how big is that  the regular expression  and a lot of people playing off my  comments here uh  about  what the votes legal or not okay so  let's focus here so the  um  so this is  doubly exponentially large how big is  the regular expression that we're  generating well that has to be  produced in polynomial time so it's only  polynomially big so we have this  little teensy weensy relatively speaking  regular expression which is only poly  enter the k it's having to describe all  strings except for this this particular  string which is  2 to the 2 to the n to the k  so in a sense this string is that that  you know is related to that regular  expression is doubly exponentially  larger than that and that kind of  presents some of the challenge  in  uh doing the reduction in creating  constructing that regular expression  anyway okay so let's move on and start  doing this is the hard stuff  here is the bad start which is  it challenging enough uh even this  little piece is going to be a little bit  challenging to to describe  so just rewriting what from the previous  uh slide so we're trying to make r1  which is generating all those strings  except the rejecting computation history  frame on w  um it's in those three parts  right now i'm describing the bad start  piece  so that's going to describe all strings  that don't start  with this  c1 so let me write that out here it's  going to generate all strings that don't  start with  c1 c start or c1  which is as as specified looks like this  so any string that doesn't start with  these symbols  doesn't start exactly like this  should be  um  described  by bad start  that regular expression okay so that in  itself is going to be  further subdivided  and the  the reason for that is it's not that  it's not that hard to understand i'm  gonna bad start is gonna  get the  accomplish its goal by saying  well  anything that doesn't start this way  either it doesn't start with a q0 or  doesn't or doesn't have a w1 in the next  place or doesn't have a w2 in the next  place or somewhere along the way it has  a wrong  symbol each one of these guys is going  to be  about  one of those  symbols being wrong in some particular  place  okay so i'm going to show you what those  look like so right now i'm i'm going to  focus my  little um  uh  attention  on  describing all strings except for this  one and then everything you know all  strings that start with with something  except for this one  okay  so just remember delta is the alphabet  for the computation histories  um and there's some notation here delta  sub epsilon we've seen this before is  you're going to add in epsilon as a as  an allowed thing for uh delta so it's  all the symbols or or epsilon that now i  think thought of as a set here  and furthermore  it's going to be convenient to talk  about all the symbols in delta  except for some symbol so like at the  very beginning q0 i want to talk about  all the symbols except for the q0 symbol  because that's what i'm going to be used  to start off  um  you know our bad start it's going to be  anything except for q0  okay so let's just see how that looks  so here is s0 the very first part of our  bad start it's going to say  um i'm trying to color the sort of  the the active ingredient here in in the  pink color  uh so  delta  with q0 removed  followed by anything  so this little regular expression here  describes all strings that don't start  with a q0 so as i'm indicating over here  all strings that don't start with the q0  is what s0 describes so you have to  you have to understand that because it's  just going to build up from there  um  [Music]  so what do we want to say for s1 what's  going to be all strings that don't have  w1 in the second place  so i'm going to write that over here  so  s1 is  anything in the first place  i mean if the first place was wrong s0  took care of it so i'm just going to  keep my lights simple so if all i want  to do is describe all of the thing  places where  the second symbol is wrong namely it's  not w1  so anything in the first place  something besides w1 and this in the  next place  and then  anything at all afterwards those are all  strings that don't have  location so i'll write it over here like  that  now s2 simile's going to take  now i'm going to since i have  exponentiation let's use that for  convenience  um  delta delta or just delta squared  so anything in the first two places  then not w2  in the next place and then anything  so that's going to capture this part so  this is what these different s's do and  you can sort of get the idea so dot dot  this sn is going to describe everything  except for wn  in in that location which is going to be  the n plus first location actually  and now i have to continue on  doing that for the  blanks  so now if you think with me  um  let's just take a look how that's going  could go  so  so now  the next symbol which is  skipping over  the m plus one that i've already taken  care of  i want to say it's not a blank symbol  in this very first location after the  input  so again i'm describing these non these  strings which are not the start  configuration so it's not it could fail  because  it's not there's not a blank where  there's supposed to be a blank  um  and if i do suppose i do that for each  one of these guys  that would work  but  but what  think  this is actually not going to be a good  solution for us  because  there are exponentially many blanks over  here this is a hugely long configuration  and so there's exponentially many blanks  if i do it this way i'm going to end up  with an exponentially large regular  expression and that's not doable in  polynomial time  so i have a more complicated way of  getting the same effect which is i don't  really expect you to fully parse through  this right now in real time in lecture  but let me try to help you what i'm  going to do is  skip over these first initial n plus 1  places  and then a variable number of  places which is indicated by the next  piece here and and the way that works is  so these are all strings of length n  plus one through  the  the the the end of the configuration  um  and to understand that uh  you know it's almost  a little too technical to even try but  let's let's see  if i put delta to the seven that's all  strings of length seven but if i put  delta sub epsilon to the seven if you  think about what that means that's all  strings of length between between 0 and  7.  okay because i can either have an  epsilon in as my  or a symbol from from delta  and so that's what i'm doing over here  i'm getting a variable length space  spacer of deltas  um  that are going to then  end up at a certain location i'm going  to say at that place then i have a  non-blank because all i need to do is  describe the strings that fail to have a  blank somewhere in this range  so i've got to sort of have a kind of a  variable spacer out to that spot  where that missing blank might be  um  okay so that's what this describes  okay if you didn't get that don't worry  uh that is a technical point you can try  to think about it offline  and then at the very very end i'm going  to describe uh what happens make sure  that you know describe the strings that  fail to have a pound sign a hashtag in  that location  how i  describe all strings that don't start  right  okay that's a lot of work just to do  that little piece  uh fourthly the next two pieces are  easier  surprisingly  okay so  um you can jump in with a question um  but  uh maybe i should move push on um  so now i'm going to describe the bad  move and bad reject pieces um  and bad reject generates all strings  that don't contain the str the q reject  symbol  so that's going to certainly describe  all of the uh strings that don't end  correctly  and that's just simply  uh  the delta with the q reject symbol  removed  and then any string of those so that's  all strings that don't have q reject so  that's going to describe all strings uh  that um  don't end with a q reject plus some  other junk strings along the way but  that's all that's all  never  a problem to put in other strings that  you're going to might be capturing in  some some other part of the regular  expression that you know of bad strings  you just want to make sure you don't put  in that one uniquely good string which  is the rejecting computation history  and lastly  um we're going to use the notion of the  neighborhoods  this might be you might think this is  the hardest part but in fact not that  hard so these are all of the um  the strings that have somewhere along  the way  um  a violation  uh according to m's rules you want to  describe all of those as well i'm going  to do that in terms of the neighborhoods  but the neighborhoods now are going to  be  stretched out  they're not not you know we don't have a  tableau anymore so they're not  so easily visualizable but it's the same  idea the neighborhood so this is abc and  def but now it's an illegal neighborhood  def does not follow from abc  if all the neighborhoods are legal then  the whole  competition is  is a legitimate computation provided it  starts and ends correctly  so if it's not a legitimate computation  there's got to be an illegal  neighborhood somewhere and i'm going to  just describe all strings that have an  illegal neighborhood  and the interesting part is you have to  describe you have to place that  separator  between abc and def  so this is another place where we're  going to critically use the  exponentiation um  to  uh  and and the fact that all the  configurations are the same length  that's where we we're using that we know  exactly how far apart the bottom of the  2x3 neighborhood is from the top of the  2x3 neighborhood okay so we're going to  take a union over all  illegal 2x3 neighborhoods  neighborhood settings i should say and  there's only a fixed number of those for  the same reason that we had in the cook  eleventh theorem there's a fixed number  of those depending upon the machine  and now we're gonna have say we're gonna  start with anything  here's the top of the neighborhood here  is the separator that separates the top  from the bottom in the two consecutive  configurations here see i  going to see i plus one  inside my computation history  and then after that uh separator i put  in the the second part of the  neighborhood which is the def  okay  it's a little bit yeah you have to  really be comfortable with  the way we've been presenting these  other uh reductions up until now if to  really get this so  um anyway i think we're at the break  um  so we can just take questions uh during  the break if you have any  and um  i will otherwise uh see you in five  minutes  you know in my description back here so  let me just take this off um  um  for  bad reject uh it looks like i'm doing  kind of overkill and maybe doing  something wrong here i'm describing all  strings that don't never reject anywhere  but as long as i don't describe  the legitimate rejecting computation  history  i do describe all strings that don't end  correctly  um  i'm good  i could go through more effort  to make sure that i'm only describing  the very last configuration here  as being as not having the reject but  you know  [Music]  that would just be  more work and i don't need to do that  work so maybe it would be good just  understand why this is sufficient what  i've described here and it's not going  to cause me any problems  um  okay some  i'm getting a note from my uh from one  of my tas thomas  saying that the notion bad perhaps is  confusing because bad sounds like  rejecting  yes  i mean bad in the sense of not  describing a legal uh computation  history if you can think of another name  i'm happy to switch that for  future years  too late for now but yeah i don't mean  that rejecting i mean that it's not just  not it's ill it you know it's  i don't know what the right term is  illegal um  or  i i'm not sure what a good how are the  neighborhoods defined here what is the  tableau here  i think you do need to think about it  after lecture but um  the the tableau you can think of the  tableau now here just written out  linearly  they're all of the rows now instead of  nicely organized into a table they're  just appeared consecutively because i'm  just trying to describe i need to do it  to describe a string whether my regular  expression doesn't really make sense to  think about i mean you can fold it up  into a tableau if you like and then and  then abc and def will just line up but  here if you think about them written  consecutively this is exactly how far  apart they end up being  um  [Music]  are there only polynomially many illegal  neighborhoods  so we have to know that that's why i  kind of corrected myself it's not  illegal neighborhoods that we're told  because there the number of  neighborhoods in this picture is vast  but the neighbor number of neighborhood  settings  the way to set these values to abcdef  and there's only  i mean these are symbols that can appear  in the  in a configuration um  uh of the machine so these are there's  only a  fixed number of symbols that can appear  here um  that depend upon the the the definition  of the machine so it's not only  polynomial there's a constant number  of these things that only uh only the  depends on the machine so yeah you have  to think about what's going on i there's  a lot  this is a lot in the slide i  um  it history for reject it's a bad history  for rejecting somebody's suggesting  yeah  it's a bad history  what is  fake news maybe we should be fake fake  would be a good term nah that's not so  good i don't know um  yeah two by three the reason two by  three is two by three is the right so  somebody's asking why two by three two  by three is exactly the size as you need  to to say that if all the two by three  neighborhoods are correct everywhere in  the  computation history  then the whole history is going to be  consistent with the rules of m it's  going to be a legal representation of a  computation of m  um so if the string  which is  allegedly a computation history  has a bad neighborhood somewhere a bit  2x3 neighborhood somewhere then um well  if it's not a legal computation history  it's got to have a bad um  neighborhoods two by three neighborhoods  somewhere  anyway  um  okay let's move on um  to uh because i think we're out of time  here um  uh  our timer is up  five we're gonna  shift gears now anyway so if you got a  little lost  in the previous proof we're gonna talk  about something different and in some  ways a little bit i think a little  lighter less technical and that's about  oracles  okay so um  okay  what are oracles organs are a simple  thing um  but they're they're in a useful concept  uh for a number of reasons um  and  especially because they they're gonna  tell us something interesting about  methods which may or may not be useful  for proving the p versus mp question  when someday somebody hopefully does  that  so um what is an oracle  an oracle is free information you're  going to give to a turing machine which  might affect um the difficulty of  solving problems  and the way we're going to represent  that free information is  we're going to allow the turing machine  to test membership in some specified  language  without  charging  for  for that for the work involved  so um  i'm going to allow you to have any  language at all some language a and say  a turing machine with an oracle for a  is it isn't  written this way m with a superscript a  it's a machine that has a black box that  can answer questions is some string  which the machine can choose  in a or not  and it gets that answer  in one step effectively for free  um  [Music]  so you can imagine depending upon the  language that you're providing to the  machine  that may or may not be useful  so  for example suppose i give you an oracle  for the sat language  that can be very useful  it could be very useful for deciding sat  for example because now you don't have  to go through a brute force search to  solve sat you just ask the oracle and  the oracle is going to say  yes it's satisfiable or no it's not  satisfied  but you can use that to  to solve other languages too  quickly  because anything that you can do in np  you can reduce the sat so you can  convert it to a sad question which you  can then ship up to the oracle and the  oracle is going to tell you the answer  so then you know the word oracle already  sort of conveys something magical  we're not really going to be concerned  with the operation of the oracle so  don't ask me how does the oracle work or  what does it correspond to in reality it  doesn't it's just a mathematical device  which provides this free information to  the turing machine which enables it to  compute certain things  it turns out to be a useful concept  that's used in cryptography where you  might imagine the oracle could could  provide the factors to some number or  the password or to some system or  something free information and then what  can you do with that  so this is a notion that comes up in  other places  um  so  if we have a  an oracle  we can think of all of the things that  you can compute in polynomial time  relative to that oracle so that's what  we um  the terminology people usually use  sometimes called relativization or  computation relative to having this  extra information so p  with an a oracle is all of the language  that you can decide in polynomial time  if you have an oracle for a  um  [Music]  let's see  yeah if somebody's asking me is it  really free or does it cost one unit you  know it's even just setting up the  oracle and writing down the question to  the oracle is going to take you some  number of steps so you're not going to  be able to do an infinite number of  oracle calls in in zero time so charging  one step or zero steps not going to make  a difference because still have to  formulate the question  um  so  uh  the  as i pointed out p with a sad oracle so  all the things you do in polynomial time  with this satellite includes np  does it perhaps include other stuff  or are they or does it equal np  um  would have been a good check-in question  but i'm not going to ask that  the uh in fact it seems like it contains  other things too because cohen p  is also contained within  p  given a sad oracle  because  the sad article answers both yes or no  depending upon the the answer  so if the formula is unsatisfiable the  oracle is going to say no it's not in  the language  and now so you can do the complement of  the sat problem as well the  unsatisfiability problem so you can do  all of co and p in the same way  you can also define np  relative to some oracle so all the  things you can do in with a  non-deterministic turing machine where  all of the branches have separately  access  and they can ask multiple questions by  the way of the oracle  um  independently  um  so let's take let's do another a little  bit of a more challenging example uh the  main formula language which i hope you  remember from your homework  so those are all of the formulas  that do not have a shorter equivalent  formula  they are minimal  you cannot make a smaller formula that's  equivalent that gives you the same  boolean function  um  so you showed for example that that  language is in  uh p space as i recall  and there was some other yet another two  problems about that language um  the complement of the min formula  problem  is an np with a sad oracle  so mull that over for a second and then  we'll see why  [Music]  and that's because  here's an algorithm an np an an np with  a sad oracle  algorithm  so um  so in other words now i want to i  you know though  i want to kind of  implement that strategy which i argued  in the homework problem was not legal  but now that i have the sad oracle it's  going to turn make it possible where  before it was no was not possible  um  so let's just understand what i  mean by that  uh  so  if i'm trying to  uh do the  the non-minimal formulas namely the  formulas that do have a shorter  equivalent uh  uh uh formula  um so  i'm gonna guess that sort of formula  called psi  and now  the  the challenge before was testing whether  that shorter formula was actually  equivalent to fee  because that's not obviously doable in  polynomial time  but the equivalence problem for formulas  is a cohen p problem  or if you'd like to think about it the  other way any formula in equivalence is  an np problem because you just have to  the witness is the assignment on which  they disagree  so two formulas are equivalent if they  never disagree  and so that's a co-np problem  okay so a sad oracle can solve a coin  problem namely the equivalence of the  two formulas the input formula and the  one that you not deterministically  guessed  and if it turns out that they are  equivalent  a smaller formula is equivalent to the  input formula you know the input formula  is not minimal  and so you can accept  and if it gets the wrong formula it  turns out not to be equivalent um then  you reject on that branch of the  non-determinism just like we did before  and if the formula really was minimal  there's not going to be none of the  branches is going to find a shorter  equivalent formula  so  that's why this problem here is in np  with a sad oracle  okay so now i'm gonna we're gonna try to  investigate this on my uh we're getting  near the end of the of the lecture we're  gonna look at the um  you know problems like well  um  suppose i compare p with a sad oracle  and np with a sad oracle  could those  be the same  well there's reasons to believe those  are not the same  but  could there be any  a  where p with an a oracle  is the same as np with an aortal it  seems like no  but actually that's wrong  there is a language there are languages  for which np with that oracle and p  with that oracle are exactly the same  and then and that actually is an  interest it's not just a curiosity it  actually has relevance to  strategies for solving p versus np  um  which hopefully i'll have time to get to  um  can we think of an oracle like a hash  table  i think hashing is somehow different in  spirit  of sort of  um  i understand there's sort of some  similarity there but i don't see that  hashing is a way of um  uh you know finding a sort of a short  name  for for objects  um  which has a variety of different  purposes why you might want to do that  and so i don't really think it's the  same uh let's see in an oracle question  okay let's see how do we use sat oracle  to solve where the two formulas are  equivalent  so okay let's getting back to this point  how can we use the satellite oracle to  solve whether two formulas are  equivalent  well we can use a sad oracle to solve  any np problem  because it's reducible to sad  right you know in in other words if  p  you know with a sad oracle contains all  of np so you have to make sure you  understand that part  um  uh  but you know like if you have like the  clique problem you can reduce if i give  you a clique problem which is an np  problem  and i want to use the oracle  to  test if the formula if the graph has a  clique  i reduce that problem to a sad problem  using the cook leaven theory  and knowing that a clique of a certain  size is going to correspond to having a  formula which is satisfiable now i can  ask the oracle  and if i can do np problems i can do  co-np problems because p  is a deterministic class even though it  has an oracle it's still deterministic  it can invert the answer  something that non-deterministic  machines cannot necessarily do  okay so i don't know maybe that's  let's move on um  so there's an oracle where p  to the a equals np to the a which kind  of seems like  kind of amazing at some level because  here's an oracle where the  non-determinism if i give you that  oracle non-determinism non-determinism  doesn't help  um  and it's actually a language we've seen  it's tqbf  why is that well here's the whole proof  if i have an np with a tqbf oracle  let's just check each of these steps  i claim i can do that  with a  non-deterministic polynomial space  machine  which  that no longer has in oracle  the reason is that if i have polynomial  space  i can answer questions about tqbf  without needing an oracle i have enough  space just to answer the question  directly myself  and i use my non-determinism here to  simulate the non-determinism of  the np machine so every time the np  machine now  you know branches now deterministically  so do i every time one of those branches  asks the oracle a tqbf question i just  do my polynomial space algorithm to  solve that question myself  but now np space equals p space by  savage is there  and because tqbf  is piece based complete  for the very same reason  that  a sad oracle allows me to do every np  problem a tqbf problem allows me to do  every piece based problem  and so i get  np contained within p for tqbf oracle  and of course  you get the containment the other way  immediately  so they're equal  what does that have to do  with  um  [Music]  uh somebody said  you know  i'll i'll just i don't want to run out  of time so i'll take any questions at  the end  um  what has this got to do with the p  versus empty problem  okay so this is an actual this is a very  interesting connection  um  and  uh  so  you know remember  we just showed  through a combination of today's lecture  and yesterday's lecture uh and  and um  uh  uh i guess thursday's lecture  that  this problem this equivalence problem is  not in p space and therefore it's not in  p and therefore it's intractable  right we just that's what we just did  um  [Music]  we showed it's complete for a class  which is provably outside of p  probably bigger than p  um  well that's what that's the kind of  thing we would like to be able to do to  separate p and np we would like to show  that some other problem is not in p  some other problem is intractable namely  sat if we could do sat  then we're good  we've we've sal p and np  so we already have an example  of being able to do that could we use  the same method  um which is something people did try to  do many years ago to show that sat is  not in p  so what is that method really well that  method the guts of that method really  comes from the hierarchy theorem that's  where you're actually proving problems  are hard you're getting this problem  with the through the hierarchy  uh construction that's provably  um outside of p space  or and outside of p  um that's a diagonalization and if you  and you look carefully at what what's  going on there um so we're going to say  no  we're not going to be able to self-set  show stats outside of p in the same way  and the reason is suppose we could  um well the hierarchy theorems are  proved by diagonalization  and  you know  what i mean by that is  that  in the hierarchy theorem there's a  machine d which is simulating some other  machine m  so  um to remember what's going on there  remember that we made a machine  which is going to make its language  different from the language of every  machine that's running with less space  or with less time  right that's how d  was defined  it wants to make sure its language  cannot be done  in  in less space so it makes sure that its  language is different that it doesn't it  simulates the machines that use less  space and does something different from  what they do  well that  simulation um  if  we had an oracle  if we're trying to show that um  you know if we provide both the  simulator and the machine being  simulated with the same oracle the  simulation still works  every time the machine you're simulating  um  asks a question the simulator has the  same oracle so it can also ask the same  question  and can still do the simulation  so in other words if you could prove p  different from np  using basically a simulation which is  what what a diagonalization is  then you would be able to prove that p  is different from np for every oracle  so if you can prove p different mnp by a  diagonalization that would also  immediately prove that p is different  from np for every oracle  because the or the the the argument is  transparent to the oracle  if you just put the oracle down  everything still works  but  here is the big but  um  it can't be  that um we we know that p  a is  that we know this is false we just  exhibited an oracle for which they're  equal  it's not the case that p is different  from np for every oracle  sometimes they're equal for so many  oracles  so something that's just a basically a a  very straightforward diagonalization to  you know something that's at its core is  a diagonalization is not going to be  enough to solve p and np because  otherwise it would prove that they're  different for every oracle and and  sometimes they're not different for some  oracles so that that that's  you know  that's an important insight for what  kind of in the method will not be  adequate  to prove p different from np and this  comes up all the time  um you know people who are make proposed  hypothetical solutions that they're  trying to show p different from np one  of the very first things people ask is  well would that argument still work  um if you put an oracle there  and um  often it does which  points out there was a flaw  um anyway uh last check-in  so this is just a little test of your  knowledge about oracle's um  why don't we in our remaining minute  here  uh  let's say 30 seconds and then  we'll we'll we'll do a wrap on this and  i'll  point out which ones are right oh boy  we're all over the place of this one  uh  it is  you're liking them all looks well i  guess the one  the  the ones that are false are lagging  slightly  um  okay  um  okay let's let's conclude  um  did i  did i give you enough time there i share  results  um  uh so in fact um  uh  um  yeah so  having the an oracle for the compliment  is the same as having an oracle so the  this is certainly true  np said equal cohen p said  we have no reason to believe that would  be true and  we  we don't know it to be true so this is b  is  not a not a good choice then that's the  laggard here min formula  um  well is in p space and anything in p  space isn't  reducible to dq bf so this is certainly  true  and  same thing for np with tqbf and cohen p  with the tqbf once you have tqbf um  you're going to get  all of um  a p space and as we pointed out you're  going to this is gonna be cool as well  um  so uh why don't we end here  and i think we're this is the that  that's my last slide so oh no there's my  summary here this is what we've done  and i will uh  send you all off on your way um  how does the interaction between the  turing machine and the oracle look yeah  i didn't define exactly how the machine  interacts with an oracle you can imagine  having a separate tape where writes the  oracle question and then goes into a  special query state  you know you can formalize it however  you like they're all going to be any  reasonable way of formalizing it is  going to be  come up with the same notion in the end  um  it does show that p with a tqbf oracle  equals p space yes that is correct good  point  why do we need the oracle to be tqbf  wouldn't sat also work because it could  solve any np problem so you're asking  does p with a sad oracle equal np with a  sad oracle not known and and not  believed not to be true but we don't  don't have a compelling reason for that  but uh no one has any idea how to do  that because for example you know we  showed the um  uh the  complement of min formula  is an np with a sad oracle  um  but no one knows how to do because  there's sort of two levels of  non-determinism there there's guessing  the smaller formula and then guessing  again to check the uh equivalence  and they really can't be combined  because one of them sort of exit an  exist type guessing the other one is  kind of a for all type guessing  um  and so uh that's  no one knows how to do that in  polynomial time with the sad oracle  so  it generally believed that they're not  the same  um  in the check and why was b false b was  um  is  the same question does np  with the sat oracle equal coin p with a  sad oracle  i'm not saying it's false it's just not  known to be true  if you can  you know it doesn't follow from anything  that we've shown so far  and i think you know that would be  something that um  well i i guess it doesn't immediately  imply any famous open problem  so  you know uh i wouldn't necessarily you  wouldn't expect you to know that it's an  unsolved problem but it is  um  okay  could we have oracles for undecidable  languages absolutely  would it be helpful well if you're  trying to solve a solvent undecidable  problem it would be helpful  but people do study that  you know in fact the original concept of  oracle's  was um  presented in the um what was derived in  the compute computability theory  um  and a side note  you can talk about reducibility that i  don't want to go there too complicated  but um  what is not known to be true  again um  what is not known to be true is that np  with a sad oracle equals  cohen p with the sad oracle or equals p  with the sad oracle  nothing is known except the obvious  relations among those those are all on  unknown and just not known to be true  or false  um  is np with a sad oracle equal to np  probably not np with a sad oracle for  one thing contains cohen and p  because it's even more powerful we  pointed out that p with a certain oracle  contains with it it contains coin p  and so np with the satellite is going to  be at least as good  and so  it's going to contain cohen p and so  probably not going to be equal to np  unless  shockingly unexpected things happen in  our complexity world  you

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um welcome everyone welcome back to uh  theory of  computation and  um  just to recap uh where we are we  um  [Music]  have been looking at time complexity and  space complexity  and um we  just finished  approving what are called the hierarchy  theorems uh which  you know in a nutshell basically say  that  if you allow the computational model to  have a little bit more resource a little  bit more time a little bit more space  then  you can do more things  so  with certain conditions  so we proved that  last time it was a proof basically by a  diagonalization i don't know if you  recognized the diagonalization there but  um you know when you're encoding a  machine by an input  and then basically running all possible  different machines that's essentially a  diagonalization  so  today we're going to  build on that work  to give an example of what we call  a natural  intractable problem  we'll say a bit more about what that  means  and then we're going to talk about  something which is a different topic but  nevertheless related  having to do with oracles  and methods which  may or may not work  to make  to solve the p versus np problem which  of course is a  big open problem in the field  okay  so  the time and space hierarchy theorems  because we're going to be using those  today  um they say that if you give a little  bit more uh space here so if you're for  space constructable functions functions  that you can actually compute within the  amount of space that they uh specify  um you can act you can show that the  things that you can do in that much  space  is probably  is larger than what you can do in less  space  and you have can prove a similar  slightly weaker fact about the time  complexity classes so what that means is  that the uh  these  classes form a hierarchy so as you add  more time or let's say in this case  space from n squared to n cubed to enter  the fourth you get um  larger and larger classes which i'm kind  of illustrating here by putting putting  a dot there which shows that there's  something that we know that's new in  those classes as you as you go up um  these uh these different bounds  and this is going to be true for space  complexity and it's also going to be  true for time complexity  um  so  and one of the corollaries that we  pointed out last time um is that  uh  p space  is a properly includes um  non-deterministic log space nl so n l is  a proper subset of p space so this stuff  in p space  that um  is not in nl and remember this notation  here this means proper subset  um  one of the things that uh  a follow-on corollary that we didn't  mention last time but that's something  that you uh you should know  is that  um the tqbf problem you know our  piece-based complete problem is an  example of a problem that's in p space  obviously but we know it's also not in  nl  and in order to get that conclusion you  have to  look again at the proof that tqbf is  p-space complete  and observe  that the reductions that we gave in that  proof  can be carried out not only in  polynomial time but they can be carried  out in log space  and therefore if tqbf turned out to go  down to nl  um then  because everything in piece base is log  space reducible to tqbf that would bring  all of p space down to nl  but that we just proved is not the case  so therefore tqbf could not be an nl  okay and we're going to be using that  kind of reasoning  um  again in this in this lecture  uh  so just a quick check in um  these are a few  more or less easy maybe more or less  tricky um follow-ons uh that you can  conclude from the  uh time and space hierarchy theorems  plus some of the other things we've  proven along the way  and so just uh as a check of your  understanding  maybe these a little bit on the tricky  side so you have to read them carefully  which of these are known to be true  um based on the material  that we've presented and this is also  you know just material that's  you know the facts that we know to be  true in uh complexity theory  um  some launch that poll  and just just check off the ones  that  we can prove  okay  i'm going to close it down  so please  answer quickly if you're gonna okay one  two three and  uh  okay  well the two leading candidates uh are  correct  so it's uh and the two that are the  laggards here are in fact the ones that  are not true  uh so a and a and d are not true  um based on what we know  and um b and c are true so let's  understand first of all a we know is  false  because 2 to the n plus 1 is just 2  times 2 to the n  and so that's just uh these two bounds  differ only by a constant factor and so  in fact they're the same complexity  class and so you don't get proper  containment for a so that that we  absolutely know is false  um  d  um  well if we could prove that  then we would have solved a famous  problem because we don't know whether  even p equals p space  so if p equal p space then certainly p  space would equal np which is in between  the two and so uh we don't know how to  prove p space is different from np as  based on the current state of knowledge  of the field  um so this would not be something that  we know to be true uh based on what  things we've said  now b follows  directly from the time hierarchy theorem  because 2 to the 2n  is the square of 2 to the n  and that is asymptotically a larger um a  significantly larger bound and so uh  um you can prove that um time two to the  end is properly contained it  properly contains time two to the n  c is a little trickier um  because  you need to remember savage's theorem  now savage's theorem applies to space  but you also need to remember that what  you can do in time in non-deterministic  time n squared you can also do a  non-deterministic space n squared  which then in turn you can do in  deterministic space n to the fourth  um  which is properly contained within space  n to the fifth  so you can prove that p space properly  contains  not deterministic time n squared  okay it's just a bunch of containments  there that's perhaps a little the a and  c are perhaps so in a sense it may be  the most tricky of the of this group  okay um  so  let's move on so we're going to  introduce today two new classes um  and you know i actually i want to go  back to here you know what are we trying  to  what are we going to be trying to uh  accomplish in today's lecture  so we're going to be looking at provable  intractability  so our  a problem being intractable for us means  it's outside of p  so  we can't solve it in polynomial time  for our  for our um perspective we're going to  call that an  intractable problem  now this problem over here  um that's sitting in time 2 to the n but  not in the smaller classes so this is an  intractable problem that's outside of p  um  but  this example of a language  if you remember how the um  time hierarchy theorem or the space  hierarchy theorem was proved basically  this language itself is not an  interesting language for other than the  purpose that it serves to be in that  class and not in a lower class  but it's not a language that anyone  would care about and it's not even a  language that is easy to describe  it's just a language that some touring  machine  um  uh decides whether a turing machine is  specially designed to have the property  that its language is at a particular  complexity level  but otherwise there's no nice  description of that language it's not  like a to the end b to the n or some you  know um  uh  equivalence of two dfas or something  like that  so i would say that  that language is in a sense it serves  its purpose but it's not a natural  language that you really care about so  one of the go the go one of the goals of  today's lecture is to give an example of  a natural language a naturally occurring  language in a sense  um that's easy to describe where you can  prove that that language is intractable  is actually outside of p  so that's that that's a bit of  motivation where we're going um so along  the way we're going to prove we're going  to introduce these exponential  complexity classes  exponential time and exponential space  which are um exponentially bigger  than  polynomial time in polynomial space  classes so it's 2 to the n to the k in  both cases two to the two to a  polynomial  and um  the first five of these classes l  through p space we've already seen and  exponential time and exponential space  extend  the containments that we've already seen  so  you have to double check that you  understand why p space  is a  subset of exponential time but that's  because that you know as we showed  going from space to time you can do that  with an exponential increase uh that's  the cost of the simulation  and going from time to space you don't  need any increase at all anything that  you can do in a certain amount of time  you can do in that much space  so anything you can do in a certain  amount of space you can also do an  exponentially more amount of time  okay so those were  simple theorems that we proved right at  the very beginning  now the hierarchy theorems allow us to  conclude some separations among these  classes  so we already looked at this one nl  versus p space  and we saw that because nl is  by savage's theorem in deterministic log  squared space which is properly  contained  in  polynomial space you get a separation  between those two classes um provably so  and for similar reasons polynomial space  to exponential space you're going to get  a  a separation from the space hierarchy  theorem and polynomial time to  exponential time you get a provable  separation  by virtue of the hierarchy theory  um  now we're going to define  complete problems for these two classes  exponential time and exponential space  so we have exponential time complete  it's going to be  analogous to what we  showed before  which is that  it's a member of exponential time  and every problem in exponential time is  reducible to it let's say in polynomial  time though it's not going to really  turn out to be matter it could be in log  space  um some simple method of doing the  reduction is going to be good enough  let's say polynomial time it's a typical  definition here  and the same thing for exponential space  complete  we'll say it's exponent it's exponential  space complete if it's an exponential  space and anything else in exponential  space is polynomial time reducible to it  okay now but the important thing  is that  if something is exponential time  complete  you know it's outside of p  for the same reasons we've now seen  several times  namely that if an exponential time  complete problem  ended up being in p  then  because everything else in exponential  time is reducible to the complete  problem they would also be in p and so  exponential time and p would be equal  but we just said they're not equal  because of the hierarchy theorem  so the logic is the hierarchy theorem  separates the class  and then the complete problem inherits  the difficulty of the larger class  so  the complete problem cannot be any lower  than the other problems in the class  because they're all reducible to it  okay um so the same thing is going to be  true for an exponential space complete  problem can't be even in p space  because exponential space and p space  are different  and and if it's not in p space it's not  going to be in p  and so in both cases  if you have a problem that's complete  for exponential space or exponential  time  we know that those problems are  intractable  and  our strategy then for giving a natural  intractable problem intractable problem  is to show it's complete for one of  these classes and it's actually going to  turn out to be an exponential space  complete problem that we're going to  give as our example  okay so that is the plan  um  i think it's a good time to let's just  take a few questions here  uh to make sure we're all on the same  page as what we're  uh as what we're  doing so let me just read i got a couple  questions already in here  [Music]  so this is a little bit of a side  comment that somebody's asking what's  interesting question basically  is it possible  that  we may not be able to prove  solve the p versus np problem that it's  just  it's not a  problem that one can answer from the  basic axioms in mathematics if i'm  interpreting the question correctly uh  there are certain problems in  mathematics um  you know i think i perhaps i mentioned  earlier in the term the problem of  whether there is a  set  whose size is in between  the integers and the real numbers you  know we know the real numbers are larger  in size than the integers  you know that was our first example of a  diagonalization and is there a problem  of size strictly in between the two  bigger than the integers  smaller than the real numbers so that's  a problem that was  posed a long time ago  it was one of hilbert's problems and was  eventually shown to be  unanswerable  using the basic axioms of mathematics  so the question is maybe p versus np is  in the same category  um  it could be  that could be true of any unsolved  problem in mathematics  but at least our experience has shown  that the kinds of problems that at least  have been shown to be unsolvable  you know from uh mathematical axioms  tend to involve infinities and very  large things things that are very far  from our intuitions  and  something as down to earth as p versus  np at least it would be very surprising  to me if that turned out to be un uh  unanswerable using our our mathematical  axioms but who knows  oh this is another good question do the  time and space  hierarchy theorems have  non-deterministic variants yes they do  they're much harder to prove however  we're not going to cover them but you  can also prove that non-deterministic  time  um  n-cubed properly includes  non-deterministic time n squared you're  not going to be responsible for that  don't worry  if you try to fig if you try to actually  prove that you'll see the  diagonalization  um  doesn't directly work um  and so uh you have to do something  fancier  you know it just uh people are asking  about which reduction method to use you  know  again you know  the kinds of reductions that we  encounter are always very simple so  we're just going to be working with very  weak notions of reductions  not interesting yet generally to  consider powerful kinds of reductions  like  um polynomial exponential time  reductions or things like that so it's  just not something that people really  think about much um  i mean i can talk about it at at length  offline but  let's just assume that our reduction  strength is something very low that log  space is going to be good enough to do  all of the reductions in this class  um  okay so let's move on then um so here's  and  here is the problem that we're going to  spend the next 20 minutes or so proving  to be uh exponential space complete um i  have to do a little introduction first  so  this is this is not the problem but this  is related to the problem so the problem  of testing if two regular expressions  are equivalent  do i just write down two regular  expressions do they generate the same  language  so that problem actually turns out to be  in p space  so it's not going to be exponential  space complete it's actually in p space  i don't think we're going to have to i  thought about presenting it in lecture  it's not that hard to show but it just  took too much time and doesn't really  introduce new methods it's a good  exercise actually using savages there um  but maybe we'll do it in recitation um  or if the lecture  miraculous miraculously ends earlier  i'll do it at the at the end but i don't  think we'll have time  um  so  that's a but that's a setup for  the  intractable problem that we're going to  talk about which is very related  now okay before we get to that so if i  have a regular expression  i'm going to enhance the kind the i'm  going to hand enhance our regular  expression  in one simple way by allowing exponents  or exponentiation  and that means if i regul if i have a  regular expression r  um  i can write r to the k to mean r  concatenated with itself k times sort of  we've been sort of informally using that  all the way along anyway like when we  talk about zero to the k one to the k  um so if we're going to formally allow  that when we write down regular  expressions  in some cases that might allow the  regular expression to be much smaller  especially if we're writing down k in  binary  because i can write r to the million  with just a few symbols  if i have exponentiation  but if i don't have exponentiation then  i have to write r concatenated with r  out a million times and i get a much  much longer and exponentially longer  expression if i don't have that exponent  as a way of describing regular regular  expressions and that's going to make a  big difference  so now the equivalence  a problem for regular expressions with  exponentiation that's what that little  up arrow means  well that signifies  now i'm giving you two regular  expressions but they're allowed to have  x the exponentiation operation  in addition to  the um  standard regular operations  um  so now  testing whether two of these regular  expressions that have exponentiation  that problem turns out to be exponential  space complete  okay  so here's the equivalence problem for  regular expressions with exponentiation  that's an exponential space complete  problem and as we pointed out that means  this problem is provably intractable  so there are  you know there's just no way in general  to solve that problem in polynomial time  that's proven that's known  um  so  we're going to go through the reduction  um i think it's going to be our last  reduction of the term of proving  problems complete for some class  but this one has each one of those has  their own kind of um  thing that makes it special  okay so first of all we have to show  that it's in exponential space and i'm  not that's really going to rely on this  other fact that we didn't prove so i'm  going to go over that very quickly  um but the interesting part is doing the  reduction so if i have something in  exponential space  then i can show that i can reduce it  to the equivalence problem  for regular expressions with  exponentiation  okay so quickly arguing  part one that we're in exponential space  basically what you do is you take your  two regular expressions that you want to  test to see if they're equivalent but  now they have exponentiation and as a  first step you get rid of the  exponentiation  you just expand the things out by  repeating  um the the parts that have the exponents  um  and of course that as i said that's  going to make the expression themselves  exponentially bigger  but now you run the p-space algorithm  on those two exponentially larger  expressions so the input to the p-space  algorithm is now exponential in the  original input size  but it's p-space in that enlarged input  so that's going to give you an  exponential space algorithm in the  original input size  because you know you expanded it to  become exponentially bigger and then you  run the p space algorithm on that  expanded  um  problem  okay  um so that gives you an exponential  space algorithm for  uh for this problem um  but now what we're gonna do  the interesting part is the reduction  so given some language in exponential  space  say decided  by some turing machine in that amount of  space 2 to the n to the k  we're going to give a reduction that  maps a  to  this equivalence problem  got it  that is the plan  so let's make sure we're all together on  the plan before we go ahead and carry  out that plan  so good good you know if you're  got you know we haven't we just sort of  set things up here in a sense um  for what we're going to be doing so  feel free to ask a question on just this  you know the plan  this part has it's going to get  technical because as doing these  reductions always is  there's a simulation involved and you  know you have to kind of  you know describe that simulation in its  own way so now we're going to be  simulating  in a certain sense  m on w  you know the the decider for this  exponential space  problem a  uh we're going to take m on w i'm going  to somehow have to express the fact that  m accepts w using this equivalence  problem for regular expressions with  exponentiation  okay um so no questions why don't we  keep when we move on then  so here's we're gonna we have i have  three slides on this  but they're kind of dense i'm sorry to  say  um  so here is  the plan as usual  um  we're gonna map a  with a polynomial time reduction  to the equivalence problem for regular  expressions with exponentiation  so that means we're going to have to  take an input  which may or may not be an a  and produce two regular expressions  with exponentiation  which are going to be equivalent  when  w is in a  or  when m accepts w  okay  so it's going to be  as these things always are  these are going to be in terms of the  computation history for mnw but in this  case it's going to turn out to be  convenient  convenient  to work with the rejecting  computation history for m on w  so remember  we have a turing machine m  um it's a decider  so that means it always holds  for strings in the language it ends up  at a q accept state for things not in  the language it ends up at a q reject  state  so a rejecting computation history is  the sequence of configurations the  machine goes through  from the start configuration  until it ends up at a configuration with  a reject state a rejecting configuration  okay  and we're going to make a  regular expression that describes all  strings except for that one  it's going to avoid describing a  rejecting computation history for m and  w otherwise it's going to  describe all possible strings  now if m  does not reject w so there is no  rejecting computation history namely m  accepts w by the way so if m accepts w  does not reject w it does not have a  rejecting computation history  what is r one describing  well it's describing in that case  everything  because there is no rejecting  computation history so it's describing  every other string besides  so that means it's describing all  strings  if there is no rejecting computation  history in the case that m accepts w  so what does that suggest we should use  for r2  r2 is going to be the regular expression  that just generates all strings  so we'll be testing whether r1 generates  all strings or not  which is the same as saying does  um m  except w  or not  okay so  r2 is going to be  you know i would like to say sigma star  but sigma is really the input to m so  and gamma is the tape alphabet for m so  we have a lot of greek letters to play  with so we're going to use delta  for the  alphabet  that we write the computation histories  in  and if you want to  get reminded what that delta is  you know a computation history can have  a tape alphabet symbol for m it can have  a state  symbol for m or it can have a delimiter  pound hashtag  so it's either  delta  this capital delta alphabet is a  tape alphabet symbol  or a state  something representing a state symbol or  a hashtag  that's just doubt so don't get you know  i always go i always feel bad if  somebody gets confused by something  that's supposed to be very simple don't  get confused by delta star this is just  all possible strings over the alphabet  delta  okay so what does r1 look so my job is  to do r1 r2 i already told you  r1 now has to describe all those strings  except for the rejecting computation  history  so everything that fails  to be a rejecting computation history so  it fails either because it started wrong  or it ended wrong or it's wrong  somewhere in the middle and by wrong i  mean it  fails to correctly describe the way the  machine operates  if it's ending up rejecting w  right so i'm going to describe all those  possible strings um  you know by breaking it down  in those into those three categories  starts wrong ends wrong or somewhere  computes wrong along the way  okay so  rejecting computation history looks  something like this  you know there's here's the start  configuration  as we you know  you know usually envision it you know  it's a start state looking at the first  symbol of the input and there's the rest  of the input  um so let me just write this out  um  this is a rejecting computation history  now so the start first configuration the  second one and so on and so on until we  end up at a rejecting computation is a  rejecting uh configuration  now  for convenience  i'm going to  uh insist  that all of these configurations are the  same  length  it's going to make my life easier  in doing the proof  because  then why can i do that um well i'm just  gonna take them you know because usually  you think of the configurations they  start small because they're just  basically length in but you know this is  using exponential space they're kind of  getting longer and longer let's just  pair them all out with blanks so that  they're all the same size  so as i've indicated over here we're  adding in a bunch of blanks that's going  to be a lot of blanks here right  to make sure they all have length 2 to  the end of the k which is the maximum  size of a configuration when you have  that much space  um  [Music]  all right  so  um  i'm going to construct so basically  that's my job i'm going to construct r1  so that it generates all all those  strings so i'm sort of wrote a little  box  around that thing i'm trying to that's  that's my  that's my to-do  and i'm also gonna it's gonna help me in  the coming slides because they're a  little bit dense  um  when i'm gonna draw this sort of reddish  pinkish box around something that means  that i'm going to try to describe all  strings except for that one  okay so that that one is um  uh  that's the one you know i  i want to avoid describing that one  because that's the rejecting computation  history but i don't want to describe  everything else that's my that's my  that's my wish  all right um so here's a check-in before  we move forward  but we can also maybe we should just  take some questions even before we  launch the check-in  how are we doing here  so  is our one describing  you know i'm not well r1 is a regular  expression so are we over here we're  talking about a this is just an ordinary  computation history but it ends with a  reject that's all a rejecting  computation history is just one that's a  little different at the end the machine  just ended up projecting instead of  accepting  otherwise everything has to be  you know  um  spelled out  in accordance with the rules of the  machine and the start configuration  um yeah we're assuming if you're in yeah  we're assuming one rejecting  uh  state  yeah that's the way we actually define  turing machines in the first place but  who's arguing yeah there's one there's  one reject state  we're all deterministic correct  um why do we need the padding because i  want to make these all the same size  all of these configurations that's going  to help me later in terms of describing  the invalid configurations the ones that  are not legal configurations legal  rejecting configurations  so just simply a matter of convenience  um but just accept it for now i just  want all of those configurations to be  the same length  in my  uh rejecting computation history  otherwise i'm not going to  you know i'm just encoding that  rejecting computation history in this  particular way um  so people asking about the detail is a  bad start yeah that's yet to come i have  two more slides on this  so i'll tell you about how we're going  to do those  so our bad start  uh that's a good question is our bad  start or all these are all the strings  that don't start this way we'll see it  in a second but our bad start or all the  things that don't start with the  you know they they start bad they  they're they're not a  uh  um they're not starting with the start  configuration uh they're starting with  some other junk  um  do we need only one rejecting  computation history what about the other  ones  you know this is a deterministic machine  so there's only going to be one if i  prescribe the lengths as i've done  there's going to be only one  rejecting computation history because  it's deterministic everything is going  to be forced from the beginning  should r1 be the not of those three  no i'm trying r one is describing all of  the strings except  except  this  uh  this one string so i'm capturing all the  different possible ways a string could  fail  to be the string and he could start  wrong it could be wrong along the middle  somewhere or so i have to union them  together  because i'm disgraced as in  as as i always believe negations are the  most confusing thing to everybody  including me um  so we're  describing all the things that are not  this string  we're trying to stay away from that one  we want to describe everything else  okay oh um  i think i better move on here i got a  lot of questions  talk to the tas  uh  all right check in  how big  is this rejecting computation history  anyway  interesting  there's a lesson here  i got a big burst of answers right at  the very beginning all wrong  but then the bright  the people who took a little bit more  time to think  uh started getting um the right answer  which is  so it's like we got a close election  here and folks though i have to report  i hope we don't have to do a recount  okay come on guys  answer up 10 seconds this is not super  hard  stop the count  yeah i think we better stop it that's  we're  on the edge  um  okay  three seconds  and polling  share results  well  um  the correct answer is in fact c  why is that because each configuration  is 2 to the n to the k  so that's how much space the machine has  exponential space  but the amount of time which is each one  of you know the  the number of configurations is going to  be the amount of time that's used  um is going to be exponentially more  than even than that  so it's going to be 2 to the 2 to the n  to the k  is how many steps the machine can run  and that's going to be how long the  computation history could be  so it's a very long thing  and when you think about it  um the regular expression we are  generating how big is that  the regular expression  and a lot of people playing off my  comments here uh  about  what the votes legal or not okay so  let's focus here so the  um  so this is  doubly exponentially large how big is  the regular expression that we're  generating well that has to be  produced in polynomial time so it's only  polynomially big so we have this  little teensy weensy relatively speaking  regular expression which is only poly  enter the k it's having to describe all  strings except for this this particular  string which is  2 to the 2 to the n to the k  so in a sense this string is that that  you know is related to that regular  expression is doubly exponentially  larger than that and that kind of  presents some of the challenge  in  uh doing the reduction in creating  constructing that regular expression  anyway okay so let's move on and start  doing this is the hard stuff  here is the bad start which is  it challenging enough uh even this  little piece is going to be a little bit  challenging to to describe  so just rewriting what from the previous  uh slide so we're trying to make r1  which is generating all those strings  except the rejecting computation history  frame on w  um it's in those three parts  right now i'm describing the bad start  piece  so that's going to describe all strings  that don't start  with this  c1 so let me write that out here it's  going to generate all strings that don't  start with  c1 c start or c1  which is as as specified looks like this  so any string that doesn't start with  these symbols  doesn't start exactly like this  should be  um  described  by bad start  that regular expression okay so that in  itself is going to be  further subdivided  and the  the reason for that is it's not that  it's not that hard to understand i'm  gonna bad start is gonna  get the  accomplish its goal by saying  well  anything that doesn't start this way  either it doesn't start with a q0 or  doesn't or doesn't have a w1 in the next  place or doesn't have a w2 in the next  place or somewhere along the way it has  a wrong  symbol each one of these guys is going  to be  about  one of those  symbols being wrong in some particular  place  okay so i'm going to show you what those  look like so right now i'm i'm going to  focus my  little um  uh  attention  on  describing all strings except for this  one and then everything you know all  strings that start with with something  except for this one  okay  so just remember delta is the alphabet  for the computation histories  um and there's some notation here delta  sub epsilon we've seen this before is  you're going to add in epsilon as a as  an allowed thing for uh delta so it's  all the symbols or or epsilon that now i  think thought of as a set here  and furthermore  it's going to be convenient to talk  about all the symbols in delta  except for some symbol so like at the  very beginning q0 i want to talk about  all the symbols except for the q0 symbol  because that's what i'm going to be used  to start off  um  you know our bad start it's going to be  anything except for q0  okay so let's just see how that looks  so here is s0 the very first part of our  bad start it's going to say  um i'm trying to color the sort of  the the active ingredient here in in the  pink color  uh so  delta  with q0 removed  followed by anything  so this little regular expression here  describes all strings that don't start  with a q0 so as i'm indicating over here  all strings that don't start with the q0  is what s0 describes so you have to  you have to understand that because it's  just going to build up from there  um  [Music]  so what do we want to say for s1 what's  going to be all strings that don't have  w1 in the second place  so i'm going to write that over here  so  s1 is  anything in the first place  i mean if the first place was wrong s0  took care of it so i'm just going to  keep my lights simple so if all i want  to do is describe all of the thing  places where  the second symbol is wrong namely it's  not w1  so anything in the first place  something besides w1 and this in the  next place  and then  anything at all afterwards those are all  strings that don't have  location so i'll write it over here like  that  now s2 simile's going to take  now i'm going to since i have  exponentiation let's use that for  convenience  um  delta delta or just delta squared  so anything in the first two places  then not w2  in the next place and then anything  so that's going to capture this part so  this is what these different s's do and  you can sort of get the idea so dot dot  this sn is going to describe everything  except for wn  in in that location which is going to be  the n plus first location actually  and now i have to continue on  doing that for the  blanks  so now if you think with me  um  let's just take a look how that's going  could go  so  so now  the next symbol which is  skipping over  the m plus one that i've already taken  care of  i want to say it's not a blank symbol  in this very first location after the  input  so again i'm describing these non these  strings which are not the start  configuration so it's not it could fail  because  it's not there's not a blank where  there's supposed to be a blank  um  and if i do suppose i do that for each  one of these guys  that would work  but  but what  think  this is actually not going to be a good  solution for us  because  there are exponentially many blanks over  here this is a hugely long configuration  and so there's exponentially many blanks  if i do it this way i'm going to end up  with an exponentially large regular  expression and that's not doable in  polynomial time  so i have a more complicated way of  getting the same effect which is i don't  really expect you to fully parse through  this right now in real time in lecture  but let me try to help you what i'm  going to do is  skip over these first initial n plus 1  places  and then a variable number of  places which is indicated by the next  piece here and and the way that works is  so these are all strings of length n  plus one through  the  the the the end of the configuration  um  and to understand that uh  you know it's almost  a little too technical to even try but  let's let's see  if i put delta to the seven that's all  strings of length seven but if i put  delta sub epsilon to the seven if you  think about what that means that's all  strings of length between between 0 and  7.  okay because i can either have an  epsilon in as my  or a symbol from from delta  and so that's what i'm doing over here  i'm getting a variable length space  spacer of deltas  um  that are going to then  end up at a certain location i'm going  to say at that place then i have a  non-blank because all i need to do is  describe the strings that fail to have a  blank somewhere in this range  so i've got to sort of have a kind of a  variable spacer out to that spot  where that missing blank might be  um  okay so that's what this describes  okay if you didn't get that don't worry  uh that is a technical point you can try  to think about it offline  and then at the very very end i'm going  to describe uh what happens make sure  that you know describe the strings that  fail to have a pound sign a hashtag in  that location  how i  describe all strings that don't start  right  okay that's a lot of work just to do  that little piece  uh fourthly the next two pieces are  easier  surprisingly  okay so  um you can jump in with a question um  but  uh maybe i should move push on um  so now i'm going to describe the bad  move and bad reject pieces um  and bad reject generates all strings  that don't contain the str the q reject  symbol  so that's going to certainly describe  all of the uh strings that don't end  correctly  and that's just simply  uh  the delta with the q reject symbol  removed  and then any string of those so that's  all strings that don't have q reject so  that's going to describe all strings uh  that um  don't end with a q reject plus some  other junk strings along the way but  that's all that's all  never  a problem to put in other strings that  you're going to might be capturing in  some some other part of the regular  expression that you know of bad strings  you just want to make sure you don't put  in that one uniquely good string which  is the rejecting computation history  and lastly  um we're going to use the notion of the  neighborhoods  this might be you might think this is  the hardest part but in fact not that  hard so these are all of the um  the strings that have somewhere along  the way  um  a violation  uh according to m's rules you want to  describe all of those as well i'm going  to do that in terms of the neighborhoods  but the neighborhoods now are going to  be  stretched out  they're not not you know we don't have a  tableau anymore so they're not  so easily visualizable but it's the same  idea the neighborhood so this is abc and  def but now it's an illegal neighborhood  def does not follow from abc  if all the neighborhoods are legal then  the whole  competition is  is a legitimate computation provided it  starts and ends correctly  so if it's not a legitimate computation  there's got to be an illegal  neighborhood somewhere and i'm going to  just describe all strings that have an  illegal neighborhood  and the interesting part is you have to  describe you have to place that  separator  between abc and def  so this is another place where we're  going to critically use the  exponentiation um  to  uh  and and the fact that all the  configurations are the same length  that's where we we're using that we know  exactly how far apart the bottom of the  2x3 neighborhood is from the top of the  2x3 neighborhood okay so we're going to  take a union over all  illegal 2x3 neighborhoods  neighborhood settings i should say and  there's only a fixed number of those for  the same reason that we had in the cook  eleventh theorem there's a fixed number  of those depending upon the machine  and now we're gonna have say we're gonna  start with anything  here's the top of the neighborhood here  is the separator that separates the top  from the bottom in the two consecutive  configurations here see i  going to see i plus one  inside my computation history  and then after that uh separator i put  in the the second part of the  neighborhood which is the def  okay  it's a little bit yeah you have to  really be comfortable with  the way we've been presenting these  other uh reductions up until now if to  really get this so  um anyway i think we're at the break  um  so we can just take questions uh during  the break if you have any  and um  i will otherwise uh see you in five  minutes  you know in my description back here so  let me just take this off um  um  for  bad reject uh it looks like i'm doing  kind of overkill and maybe doing  something wrong here i'm describing all  strings that don't never reject anywhere  but as long as i don't describe  the legitimate rejecting computation  history  i do describe all strings that don't end  correctly  um  i'm good  i could go through more effort  to make sure that i'm only describing  the very last configuration here  as being as not having the reject but  you know  [Music]  that would just be  more work and i don't need to do that  work so maybe it would be good just  understand why this is sufficient what  i've described here and it's not going  to cause me any problems  um  okay some  i'm getting a note from my uh from one  of my tas thomas  saying that the notion bad perhaps is  confusing because bad sounds like  rejecting  yes  i mean bad in the sense of not  describing a legal uh computation  history if you can think of another name  i'm happy to switch that for  future years  too late for now but yeah i don't mean  that rejecting i mean that it's not just  not it's ill it you know it's  i don't know what the right term is  illegal um  or  i i'm not sure what a good how are the  neighborhoods defined here what is the  tableau here  i think you do need to think about it  after lecture but um  the the tableau you can think of the  tableau now here just written out  linearly  they're all of the rows now instead of  nicely organized into a table they're  just appeared consecutively because i'm  just trying to describe i need to do it  to describe a string whether my regular  expression doesn't really make sense to  think about i mean you can fold it up  into a tableau if you like and then and  then abc and def will just line up but  here if you think about them written  consecutively this is exactly how far  apart they end up being  um  [Music]  are there only polynomially many illegal  neighborhoods  so we have to know that that's why i  kind of corrected myself it's not  illegal neighborhoods that we're told  because there the number of  neighborhoods in this picture is vast  but the neighbor number of neighborhood  settings  the way to set these values to abcdef  and there's only  i mean these are symbols that can appear  in the  in a configuration um  uh of the machine so these are there's  only a  fixed number of symbols that can appear  here um  that depend upon the the the definition  of the machine so it's not only  polynomial there's a constant number  of these things that only uh only the  depends on the machine so yeah you have  to think about what's going on i there's  a lot  this is a lot in the slide i  um  it history for reject it's a bad history  for rejecting somebody's suggesting  yeah  it's a bad history  what is  fake news maybe we should be fake fake  would be a good term nah that's not so  good i don't know um  yeah two by three the reason two by  three is two by three is the right so  somebody's asking why two by three two  by three is exactly the size as you need  to to say that if all the two by three  neighborhoods are correct everywhere in  the  computation history  then the whole history is going to be  consistent with the rules of m it's  going to be a legal representation of a  computation of m  um so if the string  which is  allegedly a computation history  has a bad neighborhood somewhere a bit  2x3 neighborhood somewhere then um well  if it's not a legal computation history  it's got to have a bad um  neighborhoods two by three neighborhoods  somewhere  anyway  um  okay let's move on um  to uh because i think we're out of time  here um  uh  our timer is up  five we're gonna  shift gears now anyway so if you got a  little lost  in the previous proof we're gonna talk  about something different and in some  ways a little bit i think a little  lighter less technical and that's about  oracles  okay so um  okay  what are oracles organs are a simple  thing um  but they're they're in a useful concept  uh for a number of reasons um  and  especially because they they're gonna  tell us something interesting about  methods which may or may not be useful  for proving the p versus mp question  when someday somebody hopefully does  that  so um what is an oracle  an oracle is free information you're  going to give to a turing machine which  might affect um the difficulty of  solving problems  and the way we're going to represent  that free information is  we're going to allow the turing machine  to test membership in some specified  language  without  charging  for  for that for the work involved  so um  i'm going to allow you to have any  language at all some language a and say  a turing machine with an oracle for a  is it isn't  written this way m with a superscript a  it's a machine that has a black box that  can answer questions is some string  which the machine can choose  in a or not  and it gets that answer  in one step effectively for free  um  [Music]  so you can imagine depending upon the  language that you're providing to the  machine  that may or may not be useful  so  for example suppose i give you an oracle  for the sat language  that can be very useful  it could be very useful for deciding sat  for example because now you don't have  to go through a brute force search to  solve sat you just ask the oracle and  the oracle is going to say  yes it's satisfiable or no it's not  satisfied  but you can use that to  to solve other languages too  quickly  because anything that you can do in np  you can reduce the sat so you can  convert it to a sad question which you  can then ship up to the oracle and the  oracle is going to tell you the answer  so then you know the word oracle already  sort of conveys something magical  we're not really going to be concerned  with the operation of the oracle so  don't ask me how does the oracle work or  what does it correspond to in reality it  doesn't it's just a mathematical device  which provides this free information to  the turing machine which enables it to  compute certain things  it turns out to be a useful concept  that's used in cryptography where you  might imagine the oracle could could  provide the factors to some number or  the password or to some system or  something free information and then what  can you do with that  so this is a notion that comes up in  other places  um  so  if we have a  an oracle  we can think of all of the things that  you can compute in polynomial time  relative to that oracle so that's what  we um  the terminology people usually use  sometimes called relativization or  computation relative to having this  extra information so p  with an a oracle is all of the language  that you can decide in polynomial time  if you have an oracle for a  um  [Music]  let's see  yeah if somebody's asking me is it  really free or does it cost one unit you  know it's even just setting up the  oracle and writing down the question to  the oracle is going to take you some  number of steps so you're not going to  be able to do an infinite number of  oracle calls in in zero time so charging  one step or zero steps not going to make  a difference because still have to  formulate the question  um  so  uh  the  as i pointed out p with a sad oracle so  all the things you do in polynomial time  with this satellite includes np  does it perhaps include other stuff  or are they or does it equal np  um  would have been a good check-in question  but i'm not going to ask that  the uh in fact it seems like it contains  other things too because cohen p  is also contained within  p  given a sad oracle  because  the sad article answers both yes or no  depending upon the the answer  so if the formula is unsatisfiable the  oracle is going to say no it's not in  the language  and now so you can do the complement of  the sat problem as well the  unsatisfiability problem so you can do  all of co and p in the same way  you can also define np  relative to some oracle so all the  things you can do in with a  non-deterministic turing machine where  all of the branches have separately  access  and they can ask multiple questions by  the way of the oracle  um  independently  um  so let's take let's do another a little  bit of a more challenging example uh the  main formula language which i hope you  remember from your homework  so those are all of the formulas  that do not have a shorter equivalent  formula  they are minimal  you cannot make a smaller formula that's  equivalent that gives you the same  boolean function  um  so you showed for example that that  language is in  uh p space as i recall  and there was some other yet another two  problems about that language um  the complement of the min formula  problem  is an np with a sad oracle  so mull that over for a second and then  we'll see why  [Music]  and that's because  here's an algorithm an np an an np with  a sad oracle  algorithm  so um  so in other words now i want to i  you know though  i want to kind of  implement that strategy which i argued  in the homework problem was not legal  but now that i have the sad oracle it's  going to turn make it possible where  before it was no was not possible  um  so let's just understand what i  mean by that  uh  so  if i'm trying to  uh do the  the non-minimal formulas namely the  formulas that do have a shorter  equivalent uh  uh uh formula  um so  i'm gonna guess that sort of formula  called psi  and now  the  the challenge before was testing whether  that shorter formula was actually  equivalent to fee  because that's not obviously doable in  polynomial time  but the equivalence problem for formulas  is a cohen p problem  or if you'd like to think about it the  other way any formula in equivalence is  an np problem because you just have to  the witness is the assignment on which  they disagree  so two formulas are equivalent if they  never disagree  and so that's a co-np problem  okay so a sad oracle can solve a coin  problem namely the equivalence of the  two formulas the input formula and the  one that you not deterministically  guessed  and if it turns out that they are  equivalent  a smaller formula is equivalent to the  input formula you know the input formula  is not minimal  and so you can accept  and if it gets the wrong formula it  turns out not to be equivalent um then  you reject on that branch of the  non-determinism just like we did before  and if the formula really was minimal  there's not going to be none of the  branches is going to find a shorter  equivalent formula  so  that's why this problem here is in np  with a sad oracle  okay so now i'm gonna we're gonna try to  investigate this on my uh we're getting  near the end of the of the lecture we're  gonna look at the um  you know problems like well  um  suppose i compare p with a sad oracle  and np with a sad oracle  could those  be the same  well there's reasons to believe those  are not the same  but  could there be any  a  where p with an a oracle  is the same as np with an aortal it  seems like no  but actually that's wrong  there is a language there are languages  for which np with that oracle and p  with that oracle are exactly the same  and then and that actually is an  interest it's not just a curiosity it  actually has relevance to  strategies for solving p versus np  um  which hopefully i'll have time to get to  um  can we think of an oracle like a hash  table  i think hashing is somehow different in  spirit  of sort of  um  i understand there's sort of some  similarity there but i don't see that  hashing is a way of um  uh you know finding a sort of a short  name  for for objects  um  which has a variety of different  purposes why you might want to do that  and so i don't really think it's the  same uh let's see in an oracle question  okay let's see how do we use sat oracle  to solve where the two formulas are  equivalent  so okay let's getting back to this point  how can we use the satellite oracle to  solve whether two formulas are  equivalent  well we can use a sad oracle to solve  any np problem  because it's reducible to sad  right you know in in other words if  p  you know with a sad oracle contains all  of np so you have to make sure you  understand that part  um  uh  but you know like if you have like the  clique problem you can reduce if i give  you a clique problem which is an np  problem  and i want to use the oracle  to  test if the formula if the graph has a  clique  i reduce that problem to a sad problem  using the cook leaven theory  and knowing that a clique of a certain  size is going to correspond to having a  formula which is satisfiable now i can  ask the oracle  and if i can do np problems i can do  co-np problems because p  is a deterministic class even though it  has an oracle it's still deterministic  it can invert the answer  something that non-deterministic  machines cannot necessarily do  okay so i don't know maybe that's  let's move on um  so there's an oracle where p  to the a equals np to the a which kind  of seems like  kind of amazing at some level because  here's an oracle where the  non-determinism if i give you that  oracle non-determinism non-determinism  doesn't help  um  and it's actually a language we've seen  it's tqbf  why is that well here's the whole proof  if i have an np with a tqbf oracle  let's just check each of these steps  i claim i can do that  with a  non-deterministic polynomial space  machine  which  that no longer has in oracle  the reason is that if i have polynomial  space  i can answer questions about tqbf  without needing an oracle i have enough  space just to answer the question  directly myself  and i use my non-determinism here to  simulate the non-determinism of  the np machine so every time the np  machine now  you know branches now deterministically  so do i every time one of those branches  asks the oracle a tqbf question i just  do my polynomial space algorithm to  solve that question myself  but now np space equals p space by  savage is there  and because tqbf  is piece based complete  for the very same reason  that  a sad oracle allows me to do every np  problem a tqbf problem allows me to do  every piece based problem  and so i get  np contained within p for tqbf oracle  and of course  you get the containment the other way  immediately  so they're equal  what does that have to do  with  um  [Music]  uh somebody said  you know  i'll i'll just i don't want to run out  of time so i'll take any questions at  the end  um  what has this got to do with the p  versus empty problem  okay so this is an actual this is a very  interesting connection  um  and  uh  so  you know remember  we just showed  through a combination of today's lecture  and yesterday's lecture uh and  and um  uh  uh i guess thursday's lecture  that  this problem this equivalence problem is  not in p space and therefore it's not in  p and therefore it's intractable  right we just that's what we just did  um  [Music]  we showed it's complete for a class  which is provably outside of p  probably bigger than p  um  well that's what that's the kind of  thing we would like to be able to do to  separate p and np we would like to show  that some other problem is not in p  some other problem is intractable namely  sat if we could do sat  then we're good  we've we've sal p and np  so we already have an example  of being able to do that could we use  the same method  um which is something people did try to  do many years ago to show that sat is  not in p  so what is that method really well that  method the guts of that method really  comes from the hierarchy theorem that's  where you're actually proving problems  are hard you're getting this problem  with the through the hierarchy  uh construction that's provably  um outside of p space  or and outside of p  um that's a diagonalization and if you  and you look carefully at what what's  going on there um so we're going to say  no  we're not going to be able to self-set  show stats outside of p in the same way  and the reason is suppose we could  um well the hierarchy theorems are  proved by diagonalization  and  you know  what i mean by that is  that  in the hierarchy theorem there's a  machine d which is simulating some other  machine m  so  um to remember what's going on there  remember that we made a machine  which is going to make its language  different from the language of every  machine that's running with less space  or with less time  right that's how d  was defined  it wants to make sure its language  cannot be done  in  in less space so it makes sure that its  language is different that it doesn't it  simulates the machines that use less  space and does something different from  what they do  well that  simulation um  if  we had an oracle  if we're trying to show that um  you know if we provide both the  simulator and the machine being  simulated with the same oracle the  simulation still works  every time the machine you're simulating  um  asks a question the simulator has the  same oracle so it can also ask the same  question  and can still do the simulation  so in other words if you could prove p  different from np  using basically a simulation which is  what what a diagonalization is  then you would be able to prove that p  is different from np for every oracle  so if you can prove p different mnp by a  diagonalization that would also  immediately prove that p is different  from np for every oracle  because the or the the the argument is  transparent to the oracle  if you just put the oracle down  everything still works  but  here is the big but  um  it can't be  that um we we know that p  a is  that we know this is false we just  exhibited an oracle for which they're  equal  it's not the case that p is different  from np for every oracle  sometimes they're equal for so many  oracles  so something that's just a basically a a  very straightforward diagonalization to  you know something that's at its core is  a diagonalization is not going to be  enough to solve p and np because  otherwise it would prove that they're  different for every oracle and and  sometimes they're not different for some  oracles so that that that's  you know  that's an important insight for what  kind of in the method will not be  adequate  to prove p different from np and this  comes up all the time  um you know people who are make proposed  hypothetical solutions that they're  trying to show p different from np one  of the very first things people ask is  well would that argument still work  um if you put an oracle there  and um  often it does which  points out there was a flaw  um anyway uh last check-in  so this is just a little test of your  knowledge about oracle's um  why don't we in our remaining minute  here  uh  let's say 30 seconds and then  we'll we'll we'll do a wrap on this and  i'll  point out which ones are right oh boy  we're all over the place of this one  uh  it is  you're liking them all looks well i  guess the one  the  the ones that are false are lagging  slightly  um  okay  um  okay let's let's conclude  um  did i  did i give you enough time there i share  results  um  uh so in fact um  uh  um  yeah so  having the an oracle for the compliment  is the same as having an oracle so the  this is certainly true  np said equal cohen p said  we have no reason to believe that would  be true and  we  we don't know it to be true so this is b  is  not a not a good choice then that's the  laggard here min formula  um  well is in p space and anything in p  space isn't  reducible to dq bf so this is certainly  true  and  same thing for np with tqbf and cohen p  with the tqbf once you have tqbf um  you're going to get  all of um  a p space and as we pointed out you're  going to this is gonna be cool as well  um  so uh why don't we end here  and i think we're this is the that  that's my last slide so oh no there's my  summary here this is what we've done  and i will uh  send you all off on your way um  how does the interaction between the  turing machine and the oracle look yeah  i didn't define exactly how the machine  interacts with an oracle you can imagine  having a separate tape where writes the  oracle question and then goes into a  special query state  you know you can formalize it however  you like they're all going to be any  reasonable way of formalizing it is  going to be  come up with the same notion in the end  um  it does show that p with a tqbf oracle  equals p space yes that is correct good  point  why do we need the oracle to be tqbf  wouldn't sat also work because it could  solve any np problem so you're asking  does p with a sad oracle equal np with a  sad oracle not known and and not  believed not to be true but we don't  don't have a compelling reason for that  but uh no one has any idea how to do  that because for example you know we  showed the um  uh the  complement of min formula  is an np with a sad oracle  um  but no one knows how to do because  there's sort of two levels of  non-determinism there there's guessing  the smaller formula and then guessing  again to check the uh equivalence  and they really can't be combined  because one of them sort of exit an  exist type guessing the other one is  kind of a for all type guessing  um  and so uh that's  no one knows how to do that in  polynomial time with the sad oracle  so  it generally believed that they're not  the same  um  in the check and why was b false b was  um  is  the same question does np  with the sat oracle equal coin p with a  sad oracle  i'm not saying it's false it's just not  known to be true  if you can  you know it doesn't follow from anything  that we've shown so far  and i think you know that would be  something that um  well i i guess it doesn't immediately  imply any famous open problem  so  you know uh i wouldn't necessarily you  wouldn't expect you to know that it's an  unsolved problem but it is  um  okay  could we have oracles for undecidable  languages absolutely  would it be helpful well if you're  trying to solve a solvent undecidable  problem it would be helpful  but people do study that  you know in fact the original concept of  oracle's  was um  presented in the um what was derived in  the compute computability theory  um  and a side note  you can talk about reducibility that i  don't want to go there too complicated  but um  what is not known to be true  again um  what is not known to be true is that np  with a sad oracle equals  cohen p with the sad oracle or equals p  with the sad oracle  nothing is known except the obvious  relations among those those are all on  unknown and just not known to be true  or false  um  is np with a sad oracle equal to np  probably not np with a sad oracle for  one thing contains cohen and p  because it's even more powerful we  pointed out that p with a certain oracle  contains with it it contains coin p  and so np with the satellite is going to  be at least as good  and so  it's going to contain cohen p and so  probably not going to be equal to np  unless  shockingly unexpected things happen in  our complexity world  you
 

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337
00:10:47,670 --> 00:10:50,389

 

338
00:10:50,389 --> 00:10:53,350

 

339
00:10:53,350 --> 00:10:53,360

 

340
00:10:53,360 --> 00:10:54,230


341
00:10:54,230 --> 00:10:56,389

 

342
00:10:56,389 --> 00:10:59,190

 

343
00:10:59,190 --> 00:11:01,269

 

344
00:11:01,269 --> 00:11:01,279

 

345
00:11:01,279 --> 00:11:03,750


346
00:11:03,750 --> 00:11:06,870

 

347
00:11:06,870 --> 00:11:08,550

 

348
00:11:08,550 --> 00:11:10,550

 

349
00:11:10,550 --> 00:11:12,870

 

350
00:11:12,870 --> 00:11:12,880

 

351
00:11:12,880 --> 00:11:14,550


352
00:11:14,550 --> 00:11:17,829

 

353
00:11:17,829 --> 00:11:17,839

 

354
00:11:17,839 --> 00:11:18,550


355
00:11:18,550 --> 00:11:20,630

 

356
00:11:20,630 --> 00:11:23,750

 

357
00:11:23,750 --> 00:11:24,790

 

358
00:11:24,790 --> 00:11:27,269

 

359
00:11:27,269 --> 00:11:30,150

 

360
00:11:30,150 --> 00:11:32,870

 

361
00:11:32,870 --> 00:11:35,190

 

362
00:11:35,190 --> 00:11:37,030

 

363
00:11:37,030 --> 00:11:38,949

 

364
00:11:38,949 --> 00:11:40,710

 

365
00:11:40,710 --> 00:11:42,310

 

366
00:11:42,310 --> 00:11:44,069

 

367
00:11:44,069 --> 00:11:45,430

 

368
00:11:45,430 --> 00:11:47,509

 

369
00:11:47,509 --> 00:11:49,990

 

370
00:11:49,990 --> 00:11:51,670

 

371
00:11:51,670 --> 00:11:53,750

 

372
00:11:53,750 --> 00:11:57,269

 

373
00:11:57,269 --> 00:11:59,670

 

374
00:11:59,670 --> 00:12:01,670

 

375
00:12:01,670 --> 00:12:01,680

 

376
00:12:01,680 --> 00:12:02,870


377
00:12:02,870 --> 00:12:05,190

 

378
00:12:05,190 --> 00:12:06,710

 

379
00:12:06,710 --> 00:12:10,550

 

380
00:12:10,550 --> 00:12:13,350

 

381
00:12:13,350 --> 00:12:15,350

 

382
00:12:15,350 --> 00:12:15,360

 

383
00:12:15,360 --> 00:12:16,710


384
00:12:16,710 --> 00:12:16,720

 

385
00:12:16,720 --> 00:12:17,509


386
00:12:17,509 --> 00:12:20,230

 

387
00:12:20,230 --> 00:12:24,230

 

388
00:12:24,230 --> 00:12:26,550

 

389
00:12:26,550 --> 00:12:28,550

 

390
00:12:28,550 --> 00:12:28,560

 

391
00:12:28,560 --> 00:12:29,430


392
00:12:29,430 --> 00:12:31,269

 

393
00:12:31,269 --> 00:12:33,190

 

394
00:12:33,190 --> 00:12:35,030

 

395
00:12:35,030 --> 00:12:35,040

 

396
00:12:35,040 --> 00:12:36,629


397
00:12:36,629 --> 00:12:39,910

 

398
00:12:39,910 --> 00:12:39,920

 

399
00:12:39,920 --> 00:12:41,750


400
00:12:41,750 --> 00:12:44,230

 

401
00:12:44,230 --> 00:12:47,269

 

402
00:12:47,269 --> 00:12:49,030

 

403
00:12:49,030 --> 00:12:51,430

 

404
00:12:51,430 --> 00:12:53,269

 

405
00:12:53,269 --> 00:12:55,509

 

406
00:12:55,509 --> 00:12:56,790

 

407
00:12:56,790 --> 00:12:59,590

 

408
00:12:59,590 --> 00:13:01,829

 

409
00:13:01,829 --> 00:13:04,710

 

410
00:13:04,710 --> 00:13:07,190

 

411
00:13:07,190 --> 00:13:08,310

 

412
00:13:08,310 --> 00:13:10,790

 

413
00:13:10,790 --> 00:13:10,800

 

414
00:13:10,800 --> 00:13:11,670


415
00:13:11,670 --> 00:13:13,670

 

416
00:13:13,670 --> 00:13:15,030

 

417
00:13:15,030 --> 00:13:16,870

 

418
00:13:16,870 --> 00:13:18,629

 

419
00:13:18,629 --> 00:13:20,310

 

420
00:13:20,310 --> 00:13:20,320

 

421
00:13:20,320 --> 00:13:21,190


422
00:13:21,190 --> 00:13:22,790

 

423
00:13:22,790 --> 00:13:24,230

 

424
00:13:24,230 --> 00:13:26,550

 

425
00:13:26,550 --> 00:13:30,870

 

426
00:13:30,870 --> 00:13:34,389

 

427
00:13:34,389 --> 00:13:35,910

 

428
00:13:35,910 --> 00:13:38,389

 

429
00:13:38,389 --> 00:13:38,399

 

430
00:13:38,399 --> 00:13:39,829


431
00:13:39,829 --> 00:13:43,990

 

432
00:13:43,990 --> 00:13:45,990

 

433
00:13:45,990 --> 00:13:49,750

 

434
00:13:49,750 --> 00:13:52,550

 

435
00:13:52,550 --> 00:13:54,069

 

436
00:13:54,069 --> 00:13:58,069

 

437
00:13:58,069 --> 00:13:58,079

 

438
00:13:58,079 --> 00:13:59,110


439
00:13:59,110 --> 00:14:01,030

 

440
00:14:01,030 --> 00:14:03,110

 

441
00:14:03,110 --> 00:14:05,910

 

442
00:14:05,910 --> 00:14:08,470

 

443
00:14:08,470 --> 00:14:11,670

 

444
00:14:11,670 --> 00:14:15,030

 

445
00:14:15,030 --> 00:14:17,590

 

446
00:14:17,590 --> 00:14:19,509

 

447
00:14:19,509 --> 00:14:22,629

 

448
00:14:22,629 --> 00:14:25,750

 

449
00:14:25,750 --> 00:14:25,760

 

450
00:14:25,760 --> 00:14:26,629


451
00:14:26,629 --> 00:14:29,509

 

452
00:14:29,509 --> 00:14:31,030

 

453
00:14:31,030 --> 00:14:35,030

 

454
00:14:35,030 --> 00:14:37,269

 

455
00:14:37,269 --> 00:14:39,590

 

456
00:14:39,590 --> 00:14:41,829

 

457
00:14:41,829 --> 00:14:43,509

 

458
00:14:43,509 --> 00:14:44,470

 

459
00:14:44,470 --> 00:14:46,470

 

460
00:14:46,470 --> 00:14:47,829

 

461
00:14:47,829 --> 00:14:49,990

 

462
00:14:49,990 --> 00:14:51,990

 

463
00:14:51,990 --> 00:14:53,829

 

464
00:14:53,829 --> 00:14:53,839

 

465
00:14:53,839 --> 00:14:54,710


466
00:14:54,710 --> 00:14:56,389

 

467
00:14:56,389 --> 00:14:56,399

 

468
00:14:56,399 --> 00:14:59,030


469
00:14:59,030 --> 00:14:59,040

 

470
00:14:59,040 --> 00:14:59,750


471
00:14:59,750 --> 00:15:03,590

 

472
00:15:03,590 --> 00:15:06,389

 

473
00:15:06,389 --> 00:15:08,790

 

474
00:15:08,790 --> 00:15:10,389

 

475
00:15:10,389 --> 00:15:13,430

 

476
00:15:13,430 --> 00:15:14,870

 

477
00:15:14,870 --> 00:15:17,670

 

478
00:15:17,670 --> 00:15:21,350

 

479
00:15:21,350 --> 00:15:21,360

 

480
00:15:21,360 --> 00:15:22,389


481
00:15:22,389 --> 00:15:24,949

 

482
00:15:24,949 --> 00:15:26,710

 

483
00:15:26,710 --> 00:15:28,470

 

484
00:15:28,470 --> 00:15:31,030

 

485
00:15:31,030 --> 00:15:32,790

 

486
00:15:32,790 --> 00:15:34,710

 

487
00:15:34,710 --> 00:15:36,510

 

488
00:15:36,510 --> 00:15:36,520

 

489
00:15:36,520 --> 00:15:45,189


490
00:15:45,189 --> 00:15:46,629

 

491
00:15:46,629 --> 00:15:48,150

 

492
00:15:48,150 --> 00:15:50,150

 

493
00:15:50,150 --> 00:15:51,829

 

494
00:15:51,829 --> 00:15:51,839

 

495
00:15:51,839 --> 00:15:53,110


496
00:15:53,110 --> 00:15:55,269

 

497
00:15:55,269 --> 00:15:57,350

 

498
00:15:57,350 --> 00:15:57,360

 

499
00:15:57,360 --> 00:15:58,310


500
00:15:58,310 --> 00:15:59,749

 

501
00:15:59,749 --> 00:16:01,110

 

502
00:16:01,110 --> 00:16:02,949

 

503
00:16:02,949 --> 00:16:06,550

 

504
00:16:06,550 --> 00:16:07,990

 

505
00:16:07,990 --> 00:16:10,069

 

506
00:16:10,069 --> 00:16:11,670

 

507
00:16:11,670 --> 00:16:13,430

 

508
00:16:13,430 --> 00:16:15,990

 

509
00:16:15,990 --> 00:16:16,000

 

510
00:16:16,000 --> 00:16:16,870


511
00:16:16,870 --> 00:16:19,749

 

512
00:16:19,749 --> 00:16:22,310

 

513
00:16:22,310 --> 00:16:24,150

 

514
00:16:24,150 --> 00:16:26,069

 

515
00:16:26,069 --> 00:16:27,670

 

516
00:16:27,670 --> 00:16:29,749

 

517
00:16:29,749 --> 00:16:32,470

 

518
00:16:32,470 --> 00:16:34,150

 

519
00:16:34,150 --> 00:16:36,069

 

520
00:16:36,069 --> 00:16:37,590

 

521
00:16:37,590 --> 00:16:39,430

 

522
00:16:39,430 --> 00:16:41,509

 

523
00:16:41,509 --> 00:16:44,389

 

524
00:16:44,389 --> 00:16:44,399

 

525
00:16:44,399 --> 00:16:46,310


526
00:16:46,310 --> 00:16:49,110

 

527
00:16:49,110 --> 00:16:51,189

 

528
00:16:51,189 --> 00:16:52,710

 

529
00:16:52,710 --> 00:16:52,720

 

530
00:16:52,720 --> 00:16:55,110


531
00:16:55,110 --> 00:16:56,470

 

532
00:16:56,470 --> 00:16:57,910

 

533
00:16:57,910 --> 00:16:59,990

 

534
00:16:59,990 --> 00:17:01,990

 

535
00:17:01,990 --> 00:17:04,150

 

536
00:17:04,150 --> 00:17:07,110

 

537
00:17:07,110 --> 00:17:09,270

 

538
00:17:09,270 --> 00:17:11,669

 

539
00:17:11,669 --> 00:17:13,350

 

540
00:17:13,350 --> 00:17:14,949

 

541
00:17:14,949 --> 00:17:14,959

 

542
00:17:14,959 --> 00:17:15,669


543
00:17:15,669 --> 00:17:17,590

 

544
00:17:17,590 --> 00:17:19,829

 

545
00:17:19,829 --> 00:17:22,549

 

546
00:17:22,549 --> 00:17:24,789

 

547
00:17:24,789 --> 00:17:26,710

 

548
00:17:26,710 --> 00:17:28,789

 

549
00:17:28,789 --> 00:17:30,230

 

550
00:17:30,230 --> 00:17:31,350

 

551
00:17:31,350 --> 00:17:34,310

 

552
00:17:34,310 --> 00:17:35,830

 

553
00:17:35,830 --> 00:17:37,350

 

554
00:17:37,350 --> 00:17:39,669

 

555
00:17:39,669 --> 00:17:39,679

 

556
00:17:39,679 --> 00:17:40,549


557
00:17:40,549 --> 00:17:40,559

 

558
00:17:40,559 --> 00:17:41,909


559
00:17:41,909 --> 00:17:43,830

 

560
00:17:43,830 --> 00:17:45,669

 

561
00:17:45,669 --> 00:17:47,029

 

562
00:17:47,029 --> 00:17:48,870

 

563
00:17:48,870 --> 00:17:50,630

 

564
00:17:50,630 --> 00:17:52,310

 

565
00:17:52,310 --> 00:17:52,320

 

566
00:17:52,320 --> 00:17:54,789


567
00:17:54,789 --> 00:17:54,799

 

568
00:17:54,799 --> 00:17:55,990


569
00:17:55,990 --> 00:17:58,470

 

570
00:17:58,470 --> 00:18:01,270

 

571
00:18:01,270 --> 00:18:01,280

 

572
00:18:01,280 --> 00:18:02,870


573
00:18:02,870 --> 00:18:04,950

 

574
00:18:04,950 --> 00:18:07,350

 

575
00:18:07,350 --> 00:18:07,360

 

576
00:18:07,360 --> 00:18:08,390


577
00:18:08,390 --> 00:18:09,590

 

578
00:18:09,590 --> 00:18:10,950

 

579
00:18:10,950 --> 00:18:13,270

 

580
00:18:13,270 --> 00:18:15,190

 

581
00:18:15,190 --> 00:18:17,350

 

582
00:18:17,350 --> 00:18:18,870

 

583
00:18:18,870 --> 00:18:20,630

 

584
00:18:20,630 --> 00:18:20,640

 

585
00:18:20,640 --> 00:18:21,830


586
00:18:21,830 --> 00:18:24,390

 

587
00:18:24,390 --> 00:18:26,230

 

588
00:18:26,230 --> 00:18:27,750

 

589
00:18:27,750 --> 00:18:29,990

 

590
00:18:29,990 --> 00:18:32,630

 

591
00:18:32,630 --> 00:18:34,310

 

592
00:18:34,310 --> 00:18:36,150

 

593
00:18:36,150 --> 00:18:38,470

 

594
00:18:38,470 --> 00:18:40,230

 

595
00:18:40,230 --> 00:18:42,950

 

596
00:18:42,950 --> 00:18:42,960

 

597
00:18:42,960 --> 00:18:45,110


598
00:18:45,110 --> 00:18:48,549

 

599
00:18:48,549 --> 00:18:48,559

 

600
00:18:48,559 --> 00:18:49,350


601
00:18:49,350 --> 00:18:51,110

 

602
00:18:51,110 --> 00:18:54,870

 

603
00:18:54,870 --> 00:18:58,950

 

604
00:18:58,950 --> 00:19:01,029

 

605
00:19:01,029 --> 00:19:01,039

 

606
00:19:01,039 --> 00:19:03,750


607
00:19:03,750 --> 00:19:05,750

 

608
00:19:05,750 --> 00:19:07,590

 

609
00:19:07,590 --> 00:19:09,909

 

610
00:19:09,909 --> 00:19:11,430

 

611
00:19:11,430 --> 00:19:12,789

 

612
00:19:12,789 --> 00:19:14,310

 

613
00:19:14,310 --> 00:19:14,320

 

614
00:19:14,320 --> 00:19:15,510


615
00:19:15,510 --> 00:19:17,510

 

616
00:19:17,510 --> 00:19:18,789

 

617
00:19:18,789 --> 00:19:19,990

 

618
00:19:19,990 --> 00:19:22,789

 

619
00:19:22,789 --> 00:19:24,070

 

620
00:19:24,070 --> 00:19:26,070

 

621
00:19:26,070 --> 00:19:28,470

 

622
00:19:28,470 --> 00:19:29,990

 

623
00:19:29,990 --> 00:19:32,230

 

624
00:19:32,230 --> 00:19:35,750

 

625
00:19:35,750 --> 00:19:38,549

 

626
00:19:38,549 --> 00:19:40,150

 

627
00:19:40,150 --> 00:19:42,950

 

628
00:19:42,950 --> 00:19:44,630

 

629
00:19:44,630 --> 00:19:46,950

 

630
00:19:46,950 --> 00:19:46,960

 

631
00:19:46,960 --> 00:19:48,390


632
00:19:48,390 --> 00:19:48,400

 

633
00:19:48,400 --> 00:19:49,909


634
00:19:49,909 --> 00:19:52,710

 

635
00:19:52,710 --> 00:19:52,720

 

636
00:19:52,720 --> 00:19:53,669


637
00:19:53,669 --> 00:19:55,750

 

638
00:19:55,750 --> 00:19:58,870

 

639
00:19:58,870 --> 00:20:01,430

 

640
00:20:01,430 --> 00:20:03,990

 

641
00:20:03,990 --> 00:20:06,789

 

642
00:20:06,789 --> 00:20:08,470

 

643
00:20:08,470 --> 00:20:08,480

 

644
00:20:08,480 --> 00:20:10,149


645
00:20:10,149 --> 00:20:13,750

 

646
00:20:13,750 --> 00:20:15,430

 

647
00:20:15,430 --> 00:20:17,350

 

648
00:20:17,350 --> 00:20:20,070

 

649
00:20:20,070 --> 00:20:20,080

 

650
00:20:20,080 --> 00:20:20,870


651
00:20:20,870 --> 00:20:23,510

 

652
00:20:23,510 --> 00:20:25,990

 

653
00:20:25,990 --> 00:20:27,830

 

654
00:20:27,830 --> 00:20:29,430

 

655
00:20:29,430 --> 00:20:33,430

 

656
00:20:33,430 --> 00:20:35,590

 

657
00:20:35,590 --> 00:20:37,510

 

658
00:20:37,510 --> 00:20:37,520

 

659
00:20:37,520 --> 00:20:38,710


660
00:20:38,710 --> 00:20:41,110

 

661
00:20:41,110 --> 00:20:44,950

 

662
00:20:44,950 --> 00:20:47,029

 

663
00:20:47,029 --> 00:20:47,039

 

664
00:20:47,039 --> 00:20:48,630


665
00:20:48,630 --> 00:20:51,350

 

666
00:20:51,350 --> 00:20:53,350

 

667
00:20:53,350 --> 00:20:55,430

 

668
00:20:55,430 --> 00:20:57,590

 

669
00:20:57,590 --> 00:21:01,029

 

670
00:21:01,029 --> 00:21:03,510

 

671
00:21:03,510 --> 00:21:06,230

 

672
00:21:06,230 --> 00:21:09,510

 

673
00:21:09,510 --> 00:21:12,070

 

674
00:21:12,070 --> 00:21:13,750

 

675
00:21:13,750 --> 00:21:15,669

 

676
00:21:15,669 --> 00:21:18,549

 

677
00:21:18,549 --> 00:21:21,190

 

678
00:21:21,190 --> 00:21:23,510

 

679
00:21:23,510 --> 00:21:25,669

 

680
00:21:25,669 --> 00:21:27,590

 

681
00:21:27,590 --> 00:21:29,430

 

682
00:21:29,430 --> 00:21:31,590

 

683
00:21:31,590 --> 00:21:34,710

 

684
00:21:34,710 --> 00:21:36,789

 

685
00:21:36,789 --> 00:21:38,710

 

686
00:21:38,710 --> 00:21:40,789

 

687
00:21:40,789 --> 00:21:40,799

 

688
00:21:40,799 --> 00:21:41,750


689
00:21:41,750 --> 00:21:44,149

 

690
00:21:44,149 --> 00:21:45,750

 

691
00:21:45,750 --> 00:21:48,070

 

692
00:21:48,070 --> 00:21:50,950

 

693
00:21:50,950 --> 00:21:54,310

 

694
00:21:54,310 --> 00:21:54,320

 

695
00:21:54,320 --> 00:21:56,549


696
00:21:56,549 --> 00:21:57,909

 

697
00:21:57,909 --> 00:22:00,149

 

698
00:22:00,149 --> 00:22:02,070

 

699
00:22:02,070 --> 00:22:04,630

 

700
00:22:04,630 --> 00:22:08,149

 

701
00:22:08,149 --> 00:22:09,590

 

702
00:22:09,590 --> 00:22:11,750

 

703
00:22:11,750 --> 00:22:13,990

 

704
00:22:13,990 --> 00:22:17,110

 

705
00:22:17,110 --> 00:22:17,120

 

706
00:22:17,120 --> 00:22:19,110


707
00:22:19,110 --> 00:22:19,120

 

708
00:22:19,120 --> 00:22:20,310


709
00:22:20,310 --> 00:22:22,070

 

710
00:22:22,070 --> 00:22:24,230

 

711
00:22:24,230 --> 00:22:26,230

 

712
00:22:26,230 --> 00:22:28,470

 

713
00:22:28,470 --> 00:22:30,630

 

714
00:22:30,630 --> 00:22:34,230

 

715
00:22:34,230 --> 00:22:36,230

 

716
00:22:36,230 --> 00:22:38,149

 

717
00:22:38,149 --> 00:22:40,390

 

718
00:22:40,390 --> 00:22:42,070

 

719
00:22:42,070 --> 00:22:44,070

 

720
00:22:44,070 --> 00:22:46,470

 

721
00:22:46,470 --> 00:22:48,630

 

722
00:22:48,630 --> 00:22:50,390

 

723
00:22:50,390 --> 00:22:51,830

 

724
00:22:51,830 --> 00:22:54,789

 

725
00:22:54,789 --> 00:22:57,110

 

726
00:22:57,110 --> 00:22:58,549

 

727
00:22:58,549 --> 00:22:58,559

 

728
00:22:58,559 --> 00:23:00,630


729
00:23:00,630 --> 00:23:03,110

 

730
00:23:03,110 --> 00:23:06,870

 

731
00:23:06,870 --> 00:23:08,390

 

732
00:23:08,390 --> 00:23:09,990

 

733
00:23:09,990 --> 00:23:12,149

 

734
00:23:12,149 --> 00:23:14,390

 

735
00:23:14,390 --> 00:23:15,590

 

736
00:23:15,590 --> 00:23:15,600

 

737
00:23:15,600 --> 00:23:17,190


738
00:23:17,190 --> 00:23:19,669

 

739
00:23:19,669 --> 00:23:19,679

 

740
00:23:19,679 --> 00:23:20,710


741
00:23:20,710 --> 00:23:23,750

 

742
00:23:23,750 --> 00:23:23,760

 

743
00:23:23,760 --> 00:23:25,029


744
00:23:25,029 --> 00:23:27,590

 

745
00:23:27,590 --> 00:23:30,310

 

746
00:23:30,310 --> 00:23:33,029

 

747
00:23:33,029 --> 00:23:36,470

 

748
00:23:36,470 --> 00:23:39,110

 

749
00:23:39,110 --> 00:23:41,830

 

750
00:23:41,830 --> 00:23:44,230

 

751
00:23:44,230 --> 00:23:47,430

 

752
00:23:47,430 --> 00:23:50,230

 

753
00:23:50,230 --> 00:23:51,510

 

754
00:23:51,510 --> 00:23:53,669

 

755
00:23:53,669 --> 00:23:55,750

 

756
00:23:55,750 --> 00:23:57,110

 

757
00:23:57,110 --> 00:23:58,710

 

758
00:23:58,710 --> 00:24:00,710

 

759
00:24:00,710 --> 00:24:00,720

 

760
00:24:00,720 --> 00:24:01,990


761
00:24:01,990 --> 00:24:02,000

 

762
00:24:02,000 --> 00:24:04,789


763
00:24:04,789 --> 00:24:04,799

 

764
00:24:04,799 --> 00:24:06,870


765
00:24:06,870 --> 00:24:06,880

 

766
00:24:06,880 --> 00:24:07,830


767
00:24:07,830 --> 00:24:09,590

 

768
00:24:09,590 --> 00:24:12,070

 

769
00:24:12,070 --> 00:24:15,190

 

770
00:24:15,190 --> 00:24:17,029

 

771
00:24:17,029 --> 00:24:20,230

 

772
00:24:20,230 --> 00:24:22,630

 

773
00:24:22,630 --> 00:24:22,640

 

774
00:24:22,640 --> 00:24:23,830


775
00:24:23,830 --> 00:24:25,269

 

776
00:24:25,269 --> 00:24:27,269

 

777
00:24:27,269 --> 00:24:30,390

 

778
00:24:30,390 --> 00:24:32,390

 

779
00:24:32,390 --> 00:24:33,669

 

780
00:24:33,669 --> 00:24:33,679

 

781
00:24:33,679 --> 00:24:34,710


782
00:24:34,710 --> 00:24:38,549

 

783
00:24:38,549 --> 00:24:39,990

 

784
00:24:39,990 --> 00:24:44,830

 

785
00:24:44,830 --> 00:24:47,029

 

786
00:24:47,029 --> 00:24:50,310

 

787
00:24:50,310 --> 00:24:52,149

 

788
00:24:52,149 --> 00:24:54,630

 

789
00:24:54,630 --> 00:24:56,470

 

790
00:24:56,470 --> 00:24:59,750

 

791
00:24:59,750 --> 00:25:01,990

 

792
00:25:01,990 --> 00:25:05,110

 

793
00:25:05,110 --> 00:25:07,110

 

794
00:25:07,110 --> 00:25:08,630

 

795
00:25:08,630 --> 00:25:10,710

 

796
00:25:10,710 --> 00:25:12,549

 

797
00:25:12,549 --> 00:25:14,630

 

798
00:25:14,630 --> 00:25:16,149

 

799
00:25:16,149 --> 00:25:18,310

 

800
00:25:18,310 --> 00:25:19,750

 

801
00:25:19,750 --> 00:25:19,760

 

802
00:25:19,760 --> 00:25:21,909


803
00:25:21,909 --> 00:25:24,070

 

804
00:25:24,070 --> 00:25:26,630

 

805
00:25:26,630 --> 00:25:28,549

 

806
00:25:28,549 --> 00:25:30,789

 

807
00:25:30,789 --> 00:25:32,470

 

808
00:25:32,470 --> 00:25:34,390

 

809
00:25:34,390 --> 00:25:37,110

 

810
00:25:37,110 --> 00:25:39,750

 

811
00:25:39,750 --> 00:25:41,110

 

812
00:25:41,110 --> 00:25:41,120

 

813
00:25:41,120 --> 00:25:46,070


814
00:25:46,070 --> 00:25:48,149

 

815
00:25:48,149 --> 00:25:49,830

 

816
00:25:49,830 --> 00:25:52,149

 

817
00:25:52,149 --> 00:25:54,070

 

818
00:25:54,070 --> 00:25:56,710

 

819
00:25:56,710 --> 00:25:56,720

 

820
00:25:56,720 --> 00:25:57,750


821
00:25:57,750 --> 00:25:57,760

 

822
00:25:57,760 --> 00:26:00,710


823
00:26:00,710 --> 00:26:02,070

 

824
00:26:02,070 --> 00:26:03,669

 

825
00:26:03,669 --> 00:26:03,679

 

826
00:26:03,679 --> 00:26:04,870


827
00:26:04,870 --> 00:26:07,190

 

828
00:26:07,190 --> 00:26:09,350

 

829
00:26:09,350 --> 00:26:10,950

 

830
00:26:10,950 --> 00:26:13,430

 

831
00:26:13,430 --> 00:26:14,630

 

832
00:26:14,630 --> 00:26:16,230

 

833
00:26:16,230 --> 00:26:18,230

 

834
00:26:18,230 --> 00:26:21,269

 

835
00:26:21,269 --> 00:26:24,470

 

836
00:26:24,470 --> 00:26:26,789

 

837
00:26:26,789 --> 00:26:26,799

 

838
00:26:26,799 --> 00:26:27,990


839
00:26:27,990 --> 00:26:29,830

 

840
00:26:29,830 --> 00:26:29,840

 

841
00:26:29,840 --> 00:26:31,510


842
00:26:31,510 --> 00:26:40,070

 

843
00:26:40,070 --> 00:26:40,080

 

844
00:26:40,080 --> 00:26:41,430


845
00:26:41,430 --> 00:26:43,269

 

846
00:26:43,269 --> 00:26:45,510

 

847
00:26:45,510 --> 00:26:46,789

 

848
00:26:46,789 --> 00:26:50,310

 

849
00:26:50,310 --> 00:26:51,669

 

850
00:26:51,669 --> 00:26:51,679

 

851
00:26:51,679 --> 00:26:52,870


852
00:26:52,870 --> 00:26:52,880

 

853
00:26:52,880 --> 00:26:54,070


854
00:26:54,070 --> 00:26:56,390

 

855
00:26:56,390 --> 00:26:59,029

 

856
00:26:59,029 --> 00:27:00,950

 

857
00:27:00,950 --> 00:27:03,269

 

858
00:27:03,269 --> 00:27:05,669

 

859
00:27:05,669 --> 00:27:07,909

 

860
00:27:07,909 --> 00:27:09,590

 

861
00:27:09,590 --> 00:27:11,750

 

862
00:27:11,750 --> 00:27:13,350

 

863
00:27:13,350 --> 00:27:13,360

 

864
00:27:13,360 --> 00:27:15,269


865
00:27:15,269 --> 00:27:17,669

 

866
00:27:17,669 --> 00:27:19,269

 

867
00:27:19,269 --> 00:27:20,870

 

868
00:27:20,870 --> 00:27:22,789

 

869
00:27:22,789 --> 00:27:25,190

 

870
00:27:25,190 --> 00:27:28,789

 

871
00:27:28,789 --> 00:27:28,799

 

872
00:27:28,799 --> 00:27:29,750


873
00:27:29,750 --> 00:27:31,190

 

874
00:27:31,190 --> 00:27:34,149

 

875
00:27:34,149 --> 00:27:38,389

 

876
00:27:38,389 --> 00:27:41,510

 

877
00:27:41,510 --> 00:27:43,750

 

878
00:27:43,750 --> 00:27:45,909

 

879
00:27:45,909 --> 00:27:50,389

 

880
00:27:50,389 --> 00:27:51,990

 

881
00:27:51,990 --> 00:27:55,269

 

882
00:27:55,269 --> 00:27:57,669

 

883
00:27:57,669 --> 00:28:00,389

 

884
00:28:00,389 --> 00:28:02,630

 

885
00:28:02,630 --> 00:28:05,190

 

886
00:28:05,190 --> 00:28:09,190

 

887
00:28:09,190 --> 00:28:11,110

 

888
00:28:11,110 --> 00:28:11,120

 

889
00:28:11,120 --> 00:28:12,470


890
00:28:12,470 --> 00:28:14,549

 

891
00:28:14,549 --> 00:28:17,190

 

892
00:28:17,190 --> 00:28:19,110

 

893
00:28:19,110 --> 00:28:21,510

 

894
00:28:21,510 --> 00:28:21,520

 

895
00:28:21,520 --> 00:28:22,710


896
00:28:22,710 --> 00:28:24,149

 

897
00:28:24,149 --> 00:28:27,269

 

898
00:28:27,269 --> 00:28:28,789

 

899
00:28:28,789 --> 00:28:30,710

 

900
00:28:30,710 --> 00:28:33,350

 

901
00:28:33,350 --> 00:28:36,310

 

902
00:28:36,310 --> 00:28:38,950

 

903
00:28:38,950 --> 00:28:40,710

 

904
00:28:40,710 --> 00:28:43,269

 

905
00:28:43,269 --> 00:28:45,750

 

906
00:28:45,750 --> 00:28:47,669

 

907
00:28:47,669 --> 00:28:48,950

 

908
00:28:48,950 --> 00:28:49,990

 

909
00:28:49,990 --> 00:28:52,230

 

910
00:28:52,230 --> 00:28:54,389

 

911
00:28:54,389 --> 00:28:57,750

 

912
00:28:57,750 --> 00:29:00,789

 

913
00:29:00,789 --> 00:29:02,149

 

914
00:29:02,149 --> 00:29:04,710

 

915
00:29:04,710 --> 00:29:05,909

 

916
00:29:05,909 --> 00:29:05,919

 

917
00:29:05,919 --> 00:29:07,110


918
00:29:07,110 --> 00:29:09,669

 

919
00:29:09,669 --> 00:29:09,679

 

920
00:29:09,679 --> 00:29:10,549


921
00:29:10,549 --> 00:29:13,190

 

922
00:29:13,190 --> 00:29:16,630

 

923
00:29:16,630 --> 00:29:18,549

 

924
00:29:18,549 --> 00:29:21,110

 

925
00:29:21,110 --> 00:29:22,230

 

926
00:29:22,230 --> 00:29:24,950

 

927
00:29:24,950 --> 00:29:26,789

 

928
00:29:26,789 --> 00:29:28,630

 

929
00:29:28,630 --> 00:29:28,640

 

930
00:29:28,640 --> 00:29:29,750


931
00:29:29,750 --> 00:29:33,350

 

932
00:29:33,350 --> 00:29:34,950

 

933
00:29:34,950 --> 00:29:36,310

 

934
00:29:36,310 --> 00:29:38,789

 

935
00:29:38,789 --> 00:29:40,549

 

936
00:29:40,549 --> 00:29:42,230

 

937
00:29:42,230 --> 00:29:43,750

 

938
00:29:43,750 --> 00:29:45,430

 

939
00:29:45,430 --> 00:29:47,029

 

940
00:29:47,029 --> 00:29:49,669

 

941
00:29:49,669 --> 00:29:51,830

 

942
00:29:51,830 --> 00:29:51,840

 

943
00:29:51,840 --> 00:29:56,310


944
00:29:56,310 --> 00:29:59,990

 

945
00:29:59,990 --> 00:30:04,230

 

946
00:30:04,230 --> 00:30:07,350

 

947
00:30:07,350 --> 00:30:09,750

 

948
00:30:09,750 --> 00:30:09,760

 

949
00:30:09,760 --> 00:30:10,789


950
00:30:10,789 --> 00:30:13,830

 

951
00:30:13,830 --> 00:30:16,310

 

952
00:30:16,310 --> 00:30:19,190

 

953
00:30:19,190 --> 00:30:21,190

 

954
00:30:21,190 --> 00:30:23,029

 

955
00:30:23,029 --> 00:30:24,870

 

956
00:30:24,870 --> 00:30:27,990

 

957
00:30:27,990 --> 00:30:29,909

 

958
00:30:29,909 --> 00:30:35,669

 

959
00:30:35,669 --> 00:30:37,269

 

960
00:30:37,269 --> 00:30:40,070

 

961
00:30:40,070 --> 00:30:42,470

 

962
00:30:42,470 --> 00:30:44,630

 

963
00:30:44,630 --> 00:30:47,190

 

964
00:30:47,190 --> 00:30:51,190

 

965
00:30:51,190 --> 00:30:52,549

 

966
00:30:52,549 --> 00:30:54,230

 

967
00:30:54,230 --> 00:30:55,990

 

968
00:30:55,990 --> 00:30:57,350

 

969
00:30:57,350 --> 00:30:57,360

 

970
00:30:57,360 --> 00:30:58,789


971
00:30:58,789 --> 00:31:01,509

 

972
00:31:01,509 --> 00:31:04,230

 

973
00:31:04,230 --> 00:31:05,830

 

974
00:31:05,830 --> 00:31:07,509

 

975
00:31:07,509 --> 00:31:08,789

 

976
00:31:08,789 --> 00:31:11,350

 

977
00:31:11,350 --> 00:31:11,360

 

978
00:31:11,360 --> 00:31:13,269


979
00:31:13,269 --> 00:31:15,110

 

980
00:31:15,110 --> 00:31:18,230

 

981
00:31:18,230 --> 00:31:20,630

 

982
00:31:20,630 --> 00:31:22,630

 

983
00:31:22,630 --> 00:31:26,230

 

984
00:31:26,230 --> 00:31:26,240

 

985
00:31:26,240 --> 00:31:27,029


986
00:31:27,029 --> 00:31:28,950

 

987
00:31:28,950 --> 00:31:30,630

 

988
00:31:30,630 --> 00:31:32,470

 

989
00:31:32,470 --> 00:31:35,430

 

990
00:31:35,430 --> 00:31:35,440

 

991
00:31:35,440 --> 00:31:36,389


992
00:31:36,389 --> 00:31:36,399

 

993
00:31:36,399 --> 00:31:38,630


994
00:31:38,630 --> 00:31:41,110

 

995
00:31:41,110 --> 00:31:42,950

 

996
00:31:42,950 --> 00:31:42,960

 

997
00:31:42,960 --> 00:31:44,950


998
00:31:44,950 --> 00:31:47,909

 

999
00:31:47,909 --> 00:31:50,149

 

1000
00:31:50,149 --> 00:31:51,669

 

1001
00:31:51,669 --> 00:31:53,110

 

1002
00:31:53,110 --> 00:31:54,870

 

1003
00:31:54,870 --> 00:31:56,549

 

1004
00:31:56,549 --> 00:31:58,389

 

1005
00:31:58,389 --> 00:32:00,470

 

1006
00:32:00,470 --> 00:32:02,630

 

1007
00:32:02,630 --> 00:32:04,710

 

1008
00:32:04,710 --> 00:32:06,470

 

1009
00:32:06,470 --> 00:32:09,509

 

1010
00:32:09,509 --> 00:32:11,909

 

1011
00:32:11,909 --> 00:32:13,509

 

1012
00:32:13,509 --> 00:32:15,590

 

1013
00:32:15,590 --> 00:32:19,029

 

1014
00:32:19,029 --> 00:32:19,039

 

1015
00:32:19,039 --> 00:32:19,470


1016
00:32:19,470 --> 00:32:19,480

 

1017
00:32:19,480 --> 00:32:21,269


1018
00:32:21,269 --> 00:32:22,230

 

1019
00:32:22,230 --> 00:32:22,240

 

1020
00:32:22,240 --> 00:32:23,110


1021
00:32:23,110 --> 00:32:23,120

 

1022
00:32:23,120 --> 00:32:24,230


1023
00:32:24,230 --> 00:32:25,830

 

1024
00:32:25,830 --> 00:32:28,149

 

1025
00:32:28,149 --> 00:32:30,230

 

1026
00:32:30,230 --> 00:32:33,029

 

1027
00:32:33,029 --> 00:32:33,039

 

1028
00:32:33,039 --> 00:32:35,430


1029
00:32:35,430 --> 00:32:37,990

 

1030
00:32:37,990 --> 00:32:41,750

 

1031
00:32:41,750 --> 00:32:43,350

 

1032
00:32:43,350 --> 00:32:45,269

 

1033
00:32:45,269 --> 00:32:46,630

 

1034
00:32:46,630 --> 00:32:47,990

 

1035
00:32:47,990 --> 00:32:48,000

 

1036
00:32:48,000 --> 00:32:49,269


1037
00:32:49,269 --> 00:32:51,190

 

1038
00:32:51,190 --> 00:32:54,549

 

1039
00:32:54,549 --> 00:32:57,029

 

1040
00:32:57,029 --> 00:33:02,149

 

1041
00:33:02,149 --> 00:33:07,669

 

1042
00:33:07,669 --> 00:33:07,679

 

1043
00:33:07,679 --> 00:33:08,710


1044
00:33:08,710 --> 00:33:10,549

 

1045
00:33:10,549 --> 00:33:12,470

 

1046
00:33:12,470 --> 00:33:14,070

 

1047
00:33:14,070 --> 00:33:15,430

 

1048
00:33:15,430 --> 00:33:17,750

 

1049
00:33:17,750 --> 00:33:20,470

 

1050
00:33:20,470 --> 00:33:23,029

 

1051
00:33:23,029 --> 00:33:24,950

 

1052
00:33:24,950 --> 00:33:26,549

 

1053
00:33:26,549 --> 00:33:28,070

 

1054
00:33:28,070 --> 00:33:30,230

 

1055
00:33:30,230 --> 00:33:35,830

 

1056
00:33:35,830 --> 00:33:35,840

 

1057
00:33:35,840 --> 00:33:36,549


1058
00:33:36,549 --> 00:33:39,350

 

1059
00:33:39,350 --> 00:33:41,990

 

1060
00:33:41,990 --> 00:33:44,470

 

1061
00:33:44,470 --> 00:33:46,470

 

1062
00:33:46,470 --> 00:33:48,149

 

1063
00:33:48,149 --> 00:33:49,750

 

1064
00:33:49,750 --> 00:33:51,909

 

1065
00:33:51,909 --> 00:33:53,669

 

1066
00:33:53,669 --> 00:33:55,029

 

1067
00:33:55,029 --> 00:33:55,039

 

1068
00:33:55,039 --> 00:33:55,990


1069
00:33:55,990 --> 00:33:58,310

 

1070
00:33:58,310 --> 00:33:59,269

 

1071
00:33:59,269 --> 00:33:59,279

 

1072
00:33:59,279 --> 00:34:00,870


1073
00:34:00,870 --> 00:34:02,789

 

1074
00:34:02,789 --> 00:34:03,990

 

1075
00:34:03,990 --> 00:34:07,190

 

1076
00:34:07,190 --> 00:34:09,669

 

1077
00:34:09,669 --> 00:34:11,829

 

1078
00:34:11,829 --> 00:34:11,839

 

1079
00:34:11,839 --> 00:34:12,869


1080
00:34:12,869 --> 00:34:12,879

 

1081
00:34:12,879 --> 00:34:13,750


1082
00:34:13,750 --> 00:34:15,109

 

1083
00:34:15,109 --> 00:34:17,829

 

1084
00:34:17,829 --> 00:34:19,829

 

1085
00:34:19,829 --> 00:34:21,349

 

1086
00:34:21,349 --> 00:34:23,750

 

1087
00:34:23,750 --> 00:34:26,389

 

1088
00:34:26,389 --> 00:34:28,790

 

1089
00:34:28,790 --> 00:34:30,790

 

1090
00:34:30,790 --> 00:34:33,909

 

1091
00:34:33,909 --> 00:34:37,669

 

1092
00:34:37,669 --> 00:34:40,230

 

1093
00:34:40,230 --> 00:34:42,710

 

1094
00:34:42,710 --> 00:34:44,950

 

1095
00:34:44,950 --> 00:34:47,669

 

1096
00:34:47,669 --> 00:34:49,750

 

1097
00:34:49,750 --> 00:34:51,190

 

1098
00:34:51,190 --> 00:34:52,230

 

1099
00:34:52,230 --> 00:34:55,349

 

1100
00:34:55,349 --> 00:34:57,190

 

1101
00:34:57,190 --> 00:34:58,550

 

1102
00:34:58,550 --> 00:35:00,150

 

1103
00:35:00,150 --> 00:35:06,390

 

1104
00:35:06,390 --> 00:35:07,910

 

1105
00:35:07,910 --> 00:35:10,470

 

1106
00:35:10,470 --> 00:35:12,630

 

1107
00:35:12,630 --> 00:35:13,750

 

1108
00:35:13,750 --> 00:35:16,790

 

1109
00:35:16,790 --> 00:35:18,950

 

1110
00:35:18,950 --> 00:35:20,630

 

1111
00:35:20,630 --> 00:35:23,270

 

1112
00:35:23,270 --> 00:35:26,150

 

1113
00:35:26,150 --> 00:35:28,310

 

1114
00:35:28,310 --> 00:35:31,190

 

1115
00:35:31,190 --> 00:35:33,109

 

1116
00:35:33,109 --> 00:35:34,950

 

1117
00:35:34,950 --> 00:35:34,960

 

1118
00:35:34,960 --> 00:35:35,750


1119
00:35:35,750 --> 00:35:38,230

 

1120
00:35:38,230 --> 00:35:40,390

 

1121
00:35:40,390 --> 00:35:42,550

 

1122
00:35:42,550 --> 00:35:42,560

 

1123
00:35:42,560 --> 00:35:45,829


1124
00:35:45,829 --> 00:35:47,270

 

1125
00:35:47,270 --> 00:35:49,349

 

1126
00:35:49,349 --> 00:35:49,359

 

1127
00:35:49,359 --> 00:35:50,950


1128
00:35:50,950 --> 00:35:52,790

 

1129
00:35:52,790 --> 00:35:54,630

 

1130
00:35:54,630 --> 00:35:56,710

 

1131
00:35:56,710 --> 00:35:58,390

 

1132
00:35:58,390 --> 00:35:59,990

 

1133
00:35:59,990 --> 00:36:01,270

 

1134
00:36:01,270 --> 00:36:06,230

 

1135
00:36:06,230 --> 00:36:08,870

 

1136
00:36:08,870 --> 00:36:11,190

 

1137
00:36:11,190 --> 00:36:14,550

 

1138
00:36:14,550 --> 00:36:14,560

 

1139
00:36:14,560 --> 00:36:15,990


1140
00:36:15,990 --> 00:36:16,000

 

1141
00:36:16,000 --> 00:36:16,790


1142
00:36:16,790 --> 00:36:16,800

 

1143
00:36:16,800 --> 00:36:17,589


1144
00:36:17,589 --> 00:36:19,990

 

1145
00:36:19,990 --> 00:36:22,230

 

1146
00:36:22,230 --> 00:36:22,240

 

1147
00:36:22,240 --> 00:36:23,109


1148
00:36:23,109 --> 00:36:24,870

 

1149
00:36:24,870 --> 00:36:27,670

 

1150
00:36:27,670 --> 00:36:29,829

 

1151
00:36:29,829 --> 00:36:29,839

 

1152
00:36:29,839 --> 00:36:31,589


1153
00:36:31,589 --> 00:36:33,589

 

1154
00:36:33,589 --> 00:36:35,829

 

1155
00:36:35,829 --> 00:36:38,069

 

1156
00:36:38,069 --> 00:36:41,349

 

1157
00:36:41,349 --> 00:36:42,230

 

1158
00:36:42,230 --> 00:36:44,630

 

1159
00:36:44,630 --> 00:36:46,950

 

1160
00:36:46,950 --> 00:36:48,710

 

1161
00:36:48,710 --> 00:36:50,790

 

1162
00:36:50,790 --> 00:36:55,349

 

1163
00:36:55,349 --> 00:36:57,190

 

1164
00:36:57,190 --> 00:36:58,550

 

1165
00:36:58,550 --> 00:37:00,069

 

1166
00:37:00,069 --> 00:37:00,079

 

1167
00:37:00,079 --> 00:37:01,270


1168
00:37:01,270 --> 00:37:09,030

 

1169
00:37:09,030 --> 00:37:10,390

 

1170
00:37:10,390 --> 00:37:12,710

 

1171
00:37:12,710 --> 00:37:12,720

 

1172
00:37:12,720 --> 00:37:14,950


1173
00:37:14,950 --> 00:37:14,960

 

1174
00:37:14,960 --> 00:37:16,150


1175
00:37:16,150 --> 00:37:18,310

 

1176
00:37:18,310 --> 00:37:20,550

 

1177
00:37:20,550 --> 00:37:24,390

 

1178
00:37:24,390 --> 00:37:25,829

 

1179
00:37:25,829 --> 00:37:27,109

 

1180
00:37:27,109 --> 00:37:28,870

 

1181
00:37:28,870 --> 00:37:33,910

 

1182
00:37:33,910 --> 00:37:35,190

 

1183
00:37:35,190 --> 00:37:36,950

 

1184
00:37:36,950 --> 00:37:39,750

 

1185
00:37:39,750 --> 00:37:45,510

 

1186
00:37:45,510 --> 00:37:47,270

 

1187
00:37:47,270 --> 00:37:50,550

 

1188
00:37:50,550 --> 00:37:50,560

 

1189
00:37:50,560 --> 00:37:53,510


1190
00:37:53,510 --> 00:37:57,270

 

1191
00:37:57,270 --> 00:37:59,190

 

1192
00:37:59,190 --> 00:37:59,200

 

1193
00:37:59,200 --> 00:38:00,150


1194
00:38:00,150 --> 00:38:01,430

 

1195
00:38:01,430 --> 00:38:01,440

 

1196
00:38:01,440 --> 00:38:03,589


1197
00:38:03,589 --> 00:38:03,599

 

1198
00:38:03,599 --> 00:38:05,109


1199
00:38:05,109 --> 00:38:09,829

 

1200
00:38:09,829 --> 00:38:12,630

 

1201
00:38:12,630 --> 00:38:17,270

 

1202
00:38:17,270 --> 00:38:17,280

 

1203
00:38:17,280 --> 00:38:18,390


1204
00:38:18,390 --> 00:38:18,400

 

1205
00:38:18,400 --> 00:38:20,470


1206
00:38:20,470 --> 00:38:22,950

 

1207
00:38:22,950 --> 00:38:25,190

 

1208
00:38:25,190 --> 00:38:27,670

 

1209
00:38:27,670 --> 00:38:29,510

 

1210
00:38:29,510 --> 00:38:31,670

 

1211
00:38:31,670 --> 00:38:33,990

 

1212
00:38:33,990 --> 00:38:35,270

 

1213
00:38:35,270 --> 00:38:36,710

 

1214
00:38:36,710 --> 00:38:38,790

 

1215
00:38:38,790 --> 00:38:40,790

 

1216
00:38:40,790 --> 00:38:42,150

 

1217
00:38:42,150 --> 00:38:44,310

 

1218
00:38:44,310 --> 00:38:45,430

 

1219
00:38:45,430 --> 00:38:47,349

 

1220
00:38:47,349 --> 00:38:49,750

 

1221
00:38:49,750 --> 00:38:52,150

 

1222
00:38:52,150 --> 00:38:55,510

 

1223
00:38:55,510 --> 00:38:58,829

 

1224
00:38:58,829 --> 00:39:02,150

 

1225
00:39:02,150 --> 00:39:06,710

 

1226
00:39:06,710 --> 00:39:08,870

 

1227
00:39:08,870 --> 00:39:10,790

 

1228
00:39:10,790 --> 00:39:13,109

 

1229
00:39:13,109 --> 00:39:13,119

 

1230
00:39:13,119 --> 00:39:15,430


1231
00:39:15,430 --> 00:39:18,790

 

1232
00:39:18,790 --> 00:39:21,270

 

1233
00:39:21,270 --> 00:39:21,280

 

1234
00:39:21,280 --> 00:39:22,790


1235
00:39:22,790 --> 00:39:24,310

 

1236
00:39:24,310 --> 00:39:26,630

 

1237
00:39:26,630 --> 00:39:28,550

 

1238
00:39:28,550 --> 00:39:31,030

 

1239
00:39:31,030 --> 00:39:32,950

 

1240
00:39:32,950 --> 00:39:35,589

 

1241
00:39:35,589 --> 00:39:38,069

 

1242
00:39:38,069 --> 00:39:40,230

 

1243
00:39:40,230 --> 00:39:43,829

 

1244
00:39:43,829 --> 00:39:46,630

 

1245
00:39:46,630 --> 00:39:48,310

 

1246
00:39:48,310 --> 00:39:50,870

 

1247
00:39:50,870 --> 00:39:53,510

 

1248
00:39:53,510 --> 00:39:55,430

 

1249
00:39:55,430 --> 00:39:57,430

 

1250
00:39:57,430 --> 00:39:59,190

 

1251
00:39:59,190 --> 00:40:01,109

 

1252
00:40:01,109 --> 00:40:01,119

 

1253
00:40:01,119 --> 00:40:02,310


1254
00:40:02,310 --> 00:40:04,230

 

1255
00:40:04,230 --> 00:40:06,470

 

1256
00:40:06,470 --> 00:40:09,030

 

1257
00:40:09,030 --> 00:40:12,550

 

1258
00:40:12,550 --> 00:40:14,550

 

1259
00:40:14,550 --> 00:40:17,829

 

1260
00:40:17,829 --> 00:40:19,510

 

1261
00:40:19,510 --> 00:40:21,990

 

1262
00:40:21,990 --> 00:40:25,270

 

1263
00:40:25,270 --> 00:40:27,670

 

1264
00:40:27,670 --> 00:40:29,510

 

1265
00:40:29,510 --> 00:40:31,109

 

1266
00:40:31,109 --> 00:40:33,670

 

1267
00:40:33,670 --> 00:40:35,829

 

1268
00:40:35,829 --> 00:40:37,910

 

1269
00:40:37,910 --> 00:40:37,920

 

1270
00:40:37,920 --> 00:40:38,870


1271
00:40:38,870 --> 00:40:41,349

 

1272
00:40:41,349 --> 00:40:43,430

 

1273
00:40:43,430 --> 00:40:45,589

 

1274
00:40:45,589 --> 00:40:47,829

 

1275
00:40:47,829 --> 00:40:49,829

 

1276
00:40:49,829 --> 00:40:51,430

 

1277
00:40:51,430 --> 00:40:54,309

 

1278
00:40:54,309 --> 00:40:57,109

 

1279
00:40:57,109 --> 00:40:59,190

 

1280
00:40:59,190 --> 00:41:00,790

 

1281
00:41:00,790 --> 00:41:03,829

 

1282
00:41:03,829 --> 00:41:05,030

 

1283
00:41:05,030 --> 00:41:05,040

 

1284
00:41:05,040 --> 00:41:06,390


1285
00:41:06,390 --> 00:41:06,400

 

1286
00:41:06,400 --> 00:41:07,750


1287
00:41:07,750 --> 00:41:09,510

 

1288
00:41:09,510 --> 00:41:13,109

 

1289
00:41:13,109 --> 00:41:14,870

 

1290
00:41:14,870 --> 00:41:17,430

 

1291
00:41:17,430 --> 00:41:18,390

 

1292
00:41:18,390 --> 00:41:20,550

 

1293
00:41:20,550 --> 00:41:22,710

 

1294
00:41:22,710 --> 00:41:24,950

 

1295
00:41:24,950 --> 00:41:26,550

 

1296
00:41:26,550 --> 00:41:29,990

 

1297
00:41:29,990 --> 00:41:30,000

 

1298
00:41:30,000 --> 00:41:31,430


1299
00:41:31,430 --> 00:41:33,990

 

1300
00:41:33,990 --> 00:41:36,710

 

1301
00:41:36,710 --> 00:41:39,589

 

1302
00:41:39,589 --> 00:41:41,670

 

1303
00:41:41,670 --> 00:41:44,950

 

1304
00:41:44,950 --> 00:41:46,829

 

1305
00:41:46,829 --> 00:41:49,910

 

1306
00:41:49,910 --> 00:41:51,430

 

1307
00:41:51,430 --> 00:41:51,440

 

1308
00:41:51,440 --> 00:41:52,550


1309
00:41:52,550 --> 00:41:54,470

 

1310
00:41:54,470 --> 00:41:56,550

 

1311
00:41:56,550 --> 00:41:56,560

 

1312
00:41:56,560 --> 00:41:58,710


1313
00:41:58,710 --> 00:41:59,910

 

1314
00:41:59,910 --> 00:42:02,870

 

1315
00:42:02,870 --> 00:42:04,230

 

1316
00:42:04,230 --> 00:42:05,589

 

1317
00:42:05,589 --> 00:42:05,599

 

1318
00:42:05,599 --> 00:42:06,630


1319
00:42:06,630 --> 00:42:06,640

 

1320
00:42:06,640 --> 00:42:07,750


1321
00:42:07,750 --> 00:42:07,760

 

1322
00:42:07,760 --> 00:42:09,270


1323
00:42:09,270 --> 00:42:11,430

 

1324
00:42:11,430 --> 00:42:13,670

 

1325
00:42:13,670 --> 00:42:17,030

 

1326
00:42:17,030 --> 00:42:19,270

 

1327
00:42:19,270 --> 00:42:19,280

 

1328
00:42:19,280 --> 00:42:20,790


1329
00:42:20,790 --> 00:42:23,109

 

1330
00:42:23,109 --> 00:42:25,270

 

1331
00:42:25,270 --> 00:42:27,829

 

1332
00:42:27,829 --> 00:42:30,069

 

1333
00:42:30,069 --> 00:42:32,390

 

1334
00:42:32,390 --> 00:42:35,670

 

1335
00:42:35,670 --> 00:42:39,109

 

1336
00:42:39,109 --> 00:42:41,670

 

1337
00:42:41,670 --> 00:42:43,109

 

1338
00:42:43,109 --> 00:42:44,550

 

1339
00:42:44,550 --> 00:42:47,589

 

1340
00:42:47,589 --> 00:42:49,670

 

1341
00:42:49,670 --> 00:42:52,069

 

1342
00:42:52,069 --> 00:42:55,270

 

1343
00:42:55,270 --> 00:42:56,630

 

1344
00:42:56,630 --> 00:42:58,550

 

1345
00:42:58,550 --> 00:42:58,560

 

1346
00:42:58,560 --> 00:42:59,910


1347
00:42:59,910 --> 00:43:02,470

 

1348
00:43:02,470 --> 00:43:05,109

 

1349
00:43:05,109 --> 00:43:07,270

 

1350
00:43:07,270 --> 00:43:10,550

 

1351
00:43:10,550 --> 00:43:13,510

 

1352
00:43:13,510 --> 00:43:16,550

 

1353
00:43:16,550 --> 00:43:19,030

 

1354
00:43:19,030 --> 00:43:21,589

 

1355
00:43:21,589 --> 00:43:22,710

 

1356
00:43:22,710 --> 00:43:22,720

 

1357
00:43:22,720 --> 00:43:24,710


1358
00:43:24,710 --> 00:43:26,630

 

1359
00:43:26,630 --> 00:43:29,829

 

1360
00:43:29,829 --> 00:43:31,589

 

1361
00:43:31,589 --> 00:43:34,710

 

1362
00:43:34,710 --> 00:43:37,910

 

1363
00:43:37,910 --> 00:43:40,870

 

1364
00:43:40,870 --> 00:43:45,829

 

1365
00:43:45,829 --> 00:43:47,430

 

1366
00:43:47,430 --> 00:43:49,589

 

1367
00:43:49,589 --> 00:43:49,599

 

1368
00:43:49,599 --> 00:43:49,950


1369
00:43:49,950 --> 00:43:49,960

 

1370
00:43:49,960 --> 00:43:51,589


1371
00:43:51,589 --> 00:43:53,990

 

1372
00:43:53,990 --> 00:43:56,150

 

1373
00:43:56,150 --> 00:43:58,630

 

1374
00:43:58,630 --> 00:44:00,550

 

1375
00:44:00,550 --> 00:44:00,560

 

1376
00:44:00,560 --> 00:44:01,910


1377
00:44:01,910 --> 00:44:03,430

 

1378
00:44:03,430 --> 00:44:06,069

 

1379
00:44:06,069 --> 00:44:08,550

 

1380
00:44:08,550 --> 00:44:09,670

 

1381
00:44:09,670 --> 00:44:12,150

 

1382
00:44:12,150 --> 00:44:14,390

 

1383
00:44:14,390 --> 00:44:15,589

 

1384
00:44:15,589 --> 00:44:17,829

 

1385
00:44:17,829 --> 00:44:19,430

 

1386
00:44:19,430 --> 00:44:22,069

 

1387
00:44:22,069 --> 00:44:24,069

 

1388
00:44:24,069 --> 00:44:25,270

 

1389
00:44:25,270 --> 00:44:26,630

 

1390
00:44:26,630 --> 00:44:28,710

 

1391
00:44:28,710 --> 00:44:34,069

 

1392
00:44:34,069 --> 00:44:35,670

 

1393
00:44:35,670 --> 00:44:35,680

 

1394
00:44:35,680 --> 00:44:36,950


1395
00:44:36,950 --> 00:44:39,589

 

1396
00:44:39,589 --> 00:44:40,870

 

1397
00:44:40,870 --> 00:44:42,309

 

1398
00:44:42,309 --> 00:44:42,319

 

1399
00:44:42,319 --> 00:44:43,829


1400
00:44:43,829 --> 00:44:43,839

 

1401
00:44:43,839 --> 00:44:44,870


1402
00:44:44,870 --> 00:44:47,589

 

1403
00:44:47,589 --> 00:44:50,630

 

1404
00:44:50,630 --> 00:44:53,190

 

1405
00:44:53,190 --> 00:44:55,430

 

1406
00:44:55,430 --> 00:44:57,510

 

1407
00:44:57,510 --> 00:44:59,349

 

1408
00:44:59,349 --> 00:45:02,150

 

1409
00:45:02,150 --> 00:45:04,630

 

1410
00:45:04,630 --> 00:45:06,790

 

1411
00:45:06,790 --> 00:45:08,790

 

1412
00:45:08,790 --> 00:45:12,470

 

1413
00:45:12,470 --> 00:45:14,790

 

1414
00:45:14,790 --> 00:45:16,829

 

1415
00:45:16,829 --> 00:45:16,839

 

1416
00:45:16,839 --> 00:45:18,470


1417
00:45:18,470 --> 00:45:21,990

 

1418
00:45:21,990 --> 00:45:22,000

 

1419
00:45:22,000 --> 00:45:23,829


1420
00:45:23,829 --> 00:45:25,270

 

1421
00:45:25,270 --> 00:45:26,470

 

1422
00:45:26,470 --> 00:45:26,480

 

1423
00:45:26,480 --> 00:45:27,829


1424
00:45:27,829 --> 00:45:29,510

 

1425
00:45:29,510 --> 00:45:32,470

 

1426
00:45:32,470 --> 00:45:33,670

 

1427
00:45:33,670 --> 00:45:35,430

 

1428
00:45:35,430 --> 00:45:38,150

 

1429
00:45:38,150 --> 00:45:41,190

 

1430
00:45:41,190 --> 00:45:43,109

 

1431
00:45:43,109 --> 00:45:43,119

 

1432
00:45:43,119 --> 00:45:44,150


1433
00:45:44,150 --> 00:45:46,710

 

1434
00:45:46,710 --> 00:45:47,910

 

1435
00:45:47,910 --> 00:45:50,550

 

1436
00:45:50,550 --> 00:45:50,560

 

1437
00:45:50,560 --> 00:45:51,430


1438
00:45:51,430 --> 00:45:53,109

 

1439
00:45:53,109 --> 00:45:54,870

 

1440
00:45:54,870 --> 00:45:54,880

 

1441
00:45:54,880 --> 00:45:56,150


1442
00:45:56,150 --> 00:45:58,309

 

1443
00:45:58,309 --> 00:46:02,550

 

1444
00:46:02,550 --> 00:46:05,430

 

1445
00:46:05,430 --> 00:46:05,440

 

1446
00:46:05,440 --> 00:46:06,470


1447
00:46:06,470 --> 00:46:10,069

 

1448
00:46:10,069 --> 00:46:10,079

 

1449
00:46:10,079 --> 00:46:12,870


1450
00:46:12,870 --> 00:46:14,710

 

1451
00:46:14,710 --> 00:46:17,190

 

1452
00:46:17,190 --> 00:46:17,200

 

1453
00:46:17,200 --> 00:46:19,190


1454
00:46:19,190 --> 00:46:21,109

 

1455
00:46:21,109 --> 00:46:24,470

 

1456
00:46:24,470 --> 00:46:26,150

 

1457
00:46:26,150 --> 00:46:29,109

 

1458
00:46:29,109 --> 00:46:31,990

 

1459
00:46:31,990 --> 00:46:34,309

 

1460
00:46:34,309 --> 00:46:35,910

 

1461
00:46:35,910 --> 00:46:38,069

 

1462
00:46:38,069 --> 00:46:40,069

 

1463
00:46:40,069 --> 00:46:41,990

 

1464
00:46:41,990 --> 00:46:44,790

 

1465
00:46:44,790 --> 00:46:46,550

 

1466
00:46:46,550 --> 00:46:47,829

 

1467
00:46:47,829 --> 00:46:50,309

 

1468
00:46:50,309 --> 00:46:50,319

 

1469
00:46:50,319 --> 00:46:51,430


1470
00:46:51,430 --> 00:46:54,829

 

1471
00:46:54,829 --> 00:46:57,430

 

1472
00:46:57,430 --> 00:46:59,670

 

1473
00:46:59,670 --> 00:47:01,990

 

1474
00:47:01,990 --> 00:47:03,829

 

1475
00:47:03,829 --> 00:47:03,839

 

1476
00:47:03,839 --> 00:47:04,630


1477
00:47:04,630 --> 00:47:08,230

 

1478
00:47:08,230 --> 00:47:08,240

 

1479
00:47:08,240 --> 00:47:09,349


1480
00:47:09,349 --> 00:47:11,910

 

1481
00:47:11,910 --> 00:47:13,430

 

1482
00:47:13,430 --> 00:47:15,109

 

1483
00:47:15,109 --> 00:47:16,550

 

1484
00:47:16,550 --> 00:47:18,390

 

1485
00:47:18,390 --> 00:47:20,470

 

1486
00:47:20,470 --> 00:47:22,870

 

1487
00:47:22,870 --> 00:47:24,710

 

1488
00:47:24,710 --> 00:47:27,510

 

1489
00:47:27,510 --> 00:47:27,520

 

1490
00:47:27,520 --> 00:47:31,030


1491
00:47:31,030 --> 00:47:32,309

 

1492
00:47:32,309 --> 00:47:34,549

 

1493
00:47:34,549 --> 00:47:37,030

 

1494
00:47:37,030 --> 00:47:39,190

 

1495
00:47:39,190 --> 00:47:41,670

 

1496
00:47:41,670 --> 00:47:44,069

 

1497
00:47:44,069 --> 00:47:44,079

 

1498
00:47:44,079 --> 00:47:45,190


1499
00:47:45,190 --> 00:47:46,710

 

1500
00:47:46,710 --> 00:47:48,470

 

1501
00:47:48,470 --> 00:47:52,030

 

1502
00:47:52,030 --> 00:47:55,190

 

1503
00:47:55,190 --> 00:47:57,910

 

1504
00:47:57,910 --> 00:48:02,069

 

1505
00:48:02,069 --> 00:48:03,510

 

1506
00:48:03,510 --> 00:48:06,950

 

1507
00:48:06,950 --> 00:48:09,030

 

1508
00:48:09,030 --> 00:48:09,040

 

1509
00:48:09,040 --> 00:48:10,230


1510
00:48:10,230 --> 00:48:12,549

 

1511
00:48:12,549 --> 00:48:14,470

 

1512
00:48:14,470 --> 00:48:16,950

 

1513
00:48:16,950 --> 00:48:19,510

 

1514
00:48:19,510 --> 00:48:21,589

 

1515
00:48:21,589 --> 00:48:24,390

 

1516
00:48:24,390 --> 00:48:26,150

 

1517
00:48:26,150 --> 00:48:28,309

 

1518
00:48:28,309 --> 00:48:30,790

 

1519
00:48:30,790 --> 00:48:32,790

 

1520
00:48:32,790 --> 00:48:34,710

 

1521
00:48:34,710 --> 00:48:34,720

 

1522
00:48:34,720 --> 00:48:35,589


1523
00:48:35,589 --> 00:48:37,190

 

1524
00:48:37,190 --> 00:48:38,870

 

1525
00:48:38,870 --> 00:48:40,630

 

1526
00:48:40,630 --> 00:48:40,640

 

1527
00:48:40,640 --> 00:48:41,589


1528
00:48:41,589 --> 00:48:41,599

 

1529
00:48:41,599 --> 00:48:43,109


1530
00:48:43,109 --> 00:48:45,349

 

1531
00:48:45,349 --> 00:48:48,470

 

1532
00:48:48,470 --> 00:48:48,480

 

1533
00:48:48,480 --> 00:48:49,190


1534
00:48:49,190 --> 00:48:53,349

 

1535
00:48:53,349 --> 00:48:55,190

 

1536
00:48:55,190 --> 00:48:58,630

 

1537
00:48:58,630 --> 00:49:01,990

 

1538
00:49:01,990 --> 00:49:05,109

 

1539
00:49:05,109 --> 00:49:05,119

 

1540
00:49:05,119 --> 00:49:05,990


1541
00:49:05,990 --> 00:49:07,750

 

1542
00:49:07,750 --> 00:49:10,230

 

1543
00:49:10,230 --> 00:49:10,240

 

1544
00:49:10,240 --> 00:49:11,349


1545
00:49:11,349 --> 00:49:12,950

 

1546
00:49:12,950 --> 00:49:12,960

 

1547
00:49:12,960 --> 00:49:14,230


1548
00:49:14,230 --> 00:49:16,470

 

1549
00:49:16,470 --> 00:49:16,480

 

1550
00:49:16,480 --> 00:49:17,990


1551
00:49:17,990 --> 00:49:19,990

 

1552
00:49:19,990 --> 00:49:21,990

 

1553
00:49:21,990 --> 00:49:24,870

 

1554
00:49:24,870 --> 00:49:26,309

 

1555
00:49:26,309 --> 00:49:28,309

 

1556
00:49:28,309 --> 00:49:29,589

 

1557
00:49:29,589 --> 00:49:31,829

 

1558
00:49:31,829 --> 00:49:31,839

 

1559
00:49:31,839 --> 00:49:32,710


1560
00:49:32,710 --> 00:49:35,190

 

1561
00:49:35,190 --> 00:49:36,630

 

1562
00:49:36,630 --> 00:49:37,990

 

1563
00:49:37,990 --> 00:49:39,990

 

1564
00:49:39,990 --> 00:49:41,190

 

1565
00:49:41,190 --> 00:49:44,069

 

1566
00:49:44,069 --> 00:49:47,589

 

1567
00:49:47,589 --> 00:49:48,870

 

1568
00:49:48,870 --> 00:49:51,510

 

1569
00:49:51,510 --> 00:49:51,520

 

1570
00:49:51,520 --> 00:49:53,990


1571
00:49:53,990 --> 00:49:55,349

 

1572
00:49:55,349 --> 00:49:56,950

 

1573
00:49:56,950 --> 00:50:00,710

 

1574
00:50:00,710 --> 00:50:02,790

 

1575
00:50:02,790 --> 00:50:04,309

 

1576
00:50:04,309 --> 00:50:04,319

 

1577
00:50:04,319 --> 00:50:05,109


1578
00:50:05,109 --> 00:50:07,270

 

1579
00:50:07,270 --> 00:50:09,670

 

1580
00:50:09,670 --> 00:50:11,670

 

1581
00:50:11,670 --> 00:50:13,589

 

1582
00:50:13,589 --> 00:50:15,190

 

1583
00:50:15,190 --> 00:50:15,200

 

1584
00:50:15,200 --> 00:50:16,950


1585
00:50:16,950 --> 00:50:18,549

 

1586
00:50:18,549 --> 00:50:19,990

 

1587
00:50:19,990 --> 00:50:22,390

 

1588
00:50:22,390 --> 00:50:24,870

 

1589
00:50:24,870 --> 00:50:27,829

 

1590
00:50:27,829 --> 00:50:32,470

 

1591
00:50:32,470 --> 00:50:39,030

 

1592
00:50:39,030 --> 00:50:41,349

 

1593
00:50:41,349 --> 00:50:42,549

 

1594
00:50:42,549 --> 00:50:44,069

 

1595
00:50:44,069 --> 00:50:45,990

 

1596
00:50:45,990 --> 00:50:47,910

 

1597
00:50:47,910 --> 00:50:49,829

 

1598
00:50:49,829 --> 00:50:51,030

 

1599
00:50:51,030 --> 00:50:52,950

 

1600
00:50:52,950 --> 00:50:55,589

 

1601
00:50:55,589 --> 00:50:57,510

 

1602
00:50:57,510 --> 00:50:59,109

 

1603
00:50:59,109 --> 00:51:02,630

 

1604
00:51:02,630 --> 00:51:02,640

 

1605
00:51:02,640 --> 00:51:04,230


1606
00:51:04,230 --> 00:51:07,270

 

1607
00:51:07,270 --> 00:51:08,790

 

1608
00:51:08,790 --> 00:51:10,470

 

1609
00:51:10,470 --> 00:51:12,950

 

1610
00:51:12,950 --> 00:51:12,960

 

1611
00:51:12,960 --> 00:51:13,670


1612
00:51:13,670 --> 00:51:13,680

 

1613
00:51:13,680 --> 00:51:15,109


1614
00:51:15,109 --> 00:51:16,790

 

1615
00:51:16,790 --> 00:51:18,710

 

1616
00:51:18,710 --> 00:51:20,630

 

1617
00:51:20,630 --> 00:51:24,390

 

1618
00:51:24,390 --> 00:51:26,630

 

1619
00:51:26,630 --> 00:51:29,430

 

1620
00:51:29,430 --> 00:51:31,910

 

1621
00:51:31,910 --> 00:51:35,750

 

1622
00:51:35,750 --> 00:51:37,430

 

1623
00:51:37,430 --> 00:51:38,870

 

1624
00:51:38,870 --> 00:51:40,150

 

1625
00:51:40,150 --> 00:51:41,910

 

1626
00:51:41,910 --> 00:51:44,150

 

1627
00:51:44,150 --> 00:51:46,069

 

1628
00:51:46,069 --> 00:51:47,910

 

1629
00:51:47,910 --> 00:51:49,910

 

1630
00:51:49,910 --> 00:51:53,270

 

1631
00:51:53,270 --> 00:51:55,910

 

1632
00:51:55,910 --> 00:51:58,069

 

1633
00:51:58,069 --> 00:52:00,710

 

1634
00:52:00,710 --> 00:52:03,430

 

1635
00:52:03,430 --> 00:52:06,150

 

1636
00:52:06,150 --> 00:52:07,990

 

1637
00:52:07,990 --> 00:52:10,390

 

1638
00:52:10,390 --> 00:52:10,400

 

1639
00:52:10,400 --> 00:52:11,430


1640
00:52:11,430 --> 00:52:13,190

 

1641
00:52:13,190 --> 00:52:16,150

 

1642
00:52:16,150 --> 00:52:17,589

 

1643
00:52:17,589 --> 00:52:20,069

 

1644
00:52:20,069 --> 00:52:21,589

 

1645
00:52:21,589 --> 00:52:23,829

 

1646
00:52:23,829 --> 00:52:23,839

 

1647
00:52:23,839 --> 00:52:25,109


1648
00:52:25,109 --> 00:52:27,349

 

1649
00:52:27,349 --> 00:52:29,510

 

1650
00:52:29,510 --> 00:52:31,030

 

1651
00:52:31,030 --> 00:52:33,589

 

1652
00:52:33,589 --> 00:52:33,599

 

1653
00:52:33,599 --> 00:52:35,750


1654
00:52:35,750 --> 00:52:38,549

 

1655
00:52:38,549 --> 00:52:41,270

 

1656
00:52:41,270 --> 00:52:41,280

 

1657
00:52:41,280 --> 00:52:42,790


1658
00:52:42,790 --> 00:52:42,800

 

1659
00:52:42,800 --> 00:52:43,670


1660
00:52:43,670 --> 00:52:46,390

 

1661
00:52:46,390 --> 00:52:48,309

 

1662
00:52:48,309 --> 00:52:50,950

 

1663
00:52:50,950 --> 00:52:53,190

 

1664
00:52:53,190 --> 00:52:55,990

 

1665
00:52:55,990 --> 00:52:58,390

 

1666
00:52:58,390 --> 00:52:58,400

 

1667
00:52:58,400 --> 00:53:00,150


1668
00:53:00,150 --> 00:53:02,630

 

1669
00:53:02,630 --> 00:53:02,640

 

1670
00:53:02,640 --> 00:53:04,230


1671
00:53:04,230 --> 00:53:04,240

 

1672
00:53:04,240 --> 00:53:05,829


1673
00:53:05,829 --> 00:53:07,270

 

1674
00:53:07,270 --> 00:53:09,750

 

1675
00:53:09,750 --> 00:53:11,750

 

1676
00:53:11,750 --> 00:53:14,549

 

1677
00:53:14,549 --> 00:53:18,150

 

1678
00:53:18,150 --> 00:53:18,500

 

1679
00:53:18,500 --> 00:53:18,510

 

1680
00:53:18,510 --> 00:53:19,750


1681
00:53:19,750 --> 00:53:21,510

 

1682
00:53:21,510 --> 00:53:23,349

 

1683
00:53:23,349 --> 00:53:24,950

 

1684
00:53:24,950 --> 00:53:27,670

 

1685
00:53:27,670 --> 00:53:28,870

 

1686
00:53:28,870 --> 00:53:30,870

 

1687
00:53:30,870 --> 00:53:30,880

 

1688
00:53:30,880 --> 00:53:34,230


1689
00:53:34,230 --> 00:53:36,470

 

1690
00:53:36,470 --> 00:53:39,109

 

1691
00:53:39,109 --> 00:53:41,349

 

1692
00:53:41,349 --> 00:53:43,829

 

1693
00:53:43,829 --> 00:53:45,349

 

1694
00:53:45,349 --> 00:53:45,359

 

1695
00:53:45,359 --> 00:53:46,549


1696
00:53:46,549 --> 00:53:46,559

 

1697
00:53:46,559 --> 00:53:47,510


1698
00:53:47,510 --> 00:53:49,270

 

1699
00:53:49,270 --> 00:53:52,309

 

1700
00:53:52,309 --> 00:53:54,150

 

1701
00:53:54,150 --> 00:53:57,030

 

1702
00:53:57,030 --> 00:53:58,870

 

1703
00:53:58,870 --> 00:54:01,349

 

1704
00:54:01,349 --> 00:54:03,430

 

1705
00:54:03,430 --> 00:54:07,030

 

1706
00:54:07,030 --> 00:54:08,829

 

1707
00:54:08,829 --> 00:54:10,950

 

1708
00:54:10,950 --> 00:54:10,960

 

1709
00:54:10,960 --> 00:54:12,549


1710
00:54:12,549 --> 00:54:14,870

 

1711
00:54:14,870 --> 00:54:16,390

 

1712
00:54:16,390 --> 00:54:18,790

 

1713
00:54:18,790 --> 00:54:20,390

 

1714
00:54:20,390 --> 00:54:22,630

 

1715
00:54:22,630 --> 00:54:24,870

 

1716
00:54:24,870 --> 00:54:26,470

 

1717
00:54:26,470 --> 00:54:26,480

 

1718
00:54:26,480 --> 00:54:27,910


1719
00:54:27,910 --> 00:54:30,150

 

1720
00:54:30,150 --> 00:54:32,790

 

1721
00:54:32,790 --> 00:54:35,270

 

1722
00:54:35,270 --> 00:54:36,789

 

1723
00:54:36,789 --> 00:54:39,109

 

1724
00:54:39,109 --> 00:54:40,470

 

1725
00:54:40,470 --> 00:54:42,309

 

1726
00:54:42,309 --> 00:54:44,470

 

1727
00:54:44,470 --> 00:54:47,270

 

1728
00:54:47,270 --> 00:54:49,109

 

1729
00:54:49,109 --> 00:54:51,510

 

1730
00:54:51,510 --> 00:54:54,470

 

1731
00:54:54,470 --> 00:54:54,480

 

1732
00:54:54,480 --> 00:54:54,980


1733
00:54:54,980 --> 00:54:54,990

 

1734
00:54:54,990 --> 00:54:56,630


1735
00:54:56,630 --> 00:54:59,750

 

1736
00:54:59,750 --> 00:54:59,760

 

1737
00:54:59,760 --> 00:55:01,030


1738
00:55:01,030 --> 00:55:03,109

 

1739
00:55:03,109 --> 00:55:04,789

 

1740
00:55:04,789 --> 00:55:07,030

 

1741
00:55:07,030 --> 00:55:08,150

 

1742
00:55:08,150 --> 00:55:11,349

 

1743
00:55:11,349 --> 00:55:13,270

 

1744
00:55:13,270 --> 00:55:13,280

 

1745
00:55:13,280 --> 00:55:14,470


1746
00:55:14,470 --> 00:55:19,030

 

1747
00:55:19,030 --> 00:55:20,710

 

1748
00:55:20,710 --> 00:55:23,349

 

1749
00:55:23,349 --> 00:55:25,430

 

1750
00:55:25,430 --> 00:55:27,990

 

1751
00:55:27,990 --> 00:55:29,910

 

1752
00:55:29,910 --> 00:55:31,190

 

1753
00:55:31,190 --> 00:55:33,109

 

1754
00:55:33,109 --> 00:55:34,549

 

1755
00:55:34,549 --> 00:55:37,349

 

1756
00:55:37,349 --> 00:55:38,710

 

1757
00:55:38,710 --> 00:55:41,190

 

1758
00:55:41,190 --> 00:55:43,829

 

1759
00:55:43,829 --> 00:55:45,589

 

1760
00:55:45,589 --> 00:55:47,510

 

1761
00:55:47,510 --> 00:55:48,470

 

1762
00:55:48,470 --> 00:55:51,670

 

1763
00:55:51,670 --> 00:55:51,680

 

1764
00:55:51,680 --> 00:55:54,230


1765
00:55:54,230 --> 00:55:56,870

 

1766
00:55:56,870 --> 00:55:59,109

 

1767
00:55:59,109 --> 00:55:59,119

 

1768
00:55:59,119 --> 00:56:00,789


1769
00:56:00,789 --> 00:56:03,750

 

1770
00:56:03,750 --> 00:56:04,789

 

1771
00:56:04,789 --> 00:56:07,589

 

1772
00:56:07,589 --> 00:56:09,670

 

1773
00:56:09,670 --> 00:56:17,430

 

1774
00:56:17,430 --> 00:56:19,270

 

1775
00:56:19,270 --> 00:56:21,190

 

1776
00:56:21,190 --> 00:56:22,870

 

1777
00:56:22,870 --> 00:56:25,190

 

1778
00:56:25,190 --> 00:56:27,589

 

1779
00:56:27,589 --> 00:56:30,230

 

1780
00:56:30,230 --> 00:56:30,240

 

1781
00:56:30,240 --> 00:56:31,109


1782
00:56:31,109 --> 00:56:33,510

 

1783
00:56:33,510 --> 00:56:36,549

 

1784
00:56:36,549 --> 00:56:38,309

 

1785
00:56:38,309 --> 00:56:40,230

 

1786
00:56:40,230 --> 00:56:41,829

 

1787
00:56:41,829 --> 00:56:45,190

 

1788
00:56:45,190 --> 00:56:46,230

 

1789
00:56:46,230 --> 00:56:49,430

 

1790
00:56:49,430 --> 00:56:51,670

 

1791
00:56:51,670 --> 00:56:55,030

 

1792
00:56:55,030 --> 00:56:57,349

 

1793
00:56:57,349 --> 00:57:00,069

 

1794
00:57:00,069 --> 00:57:01,349

 

1795
00:57:01,349 --> 00:57:01,359

 

1796
00:57:01,359 --> 00:57:02,870


1797
00:57:02,870 --> 00:57:02,880

 

1798
00:57:02,880 --> 00:57:04,069


1799
00:57:04,069 --> 00:57:04,079

 

1800
00:57:04,079 --> 00:57:06,150


1801
00:57:06,150 --> 00:57:09,270

 

1802
00:57:09,270 --> 00:57:10,950

 

1803
00:57:10,950 --> 00:57:12,549

 

1804
00:57:12,549 --> 00:57:12,559

 

1805
00:57:12,559 --> 00:57:13,750


1806
00:57:13,750 --> 00:57:15,270

 

1807
00:57:15,270 --> 00:57:16,950

 

1808
00:57:16,950 --> 00:57:18,870

 

1809
00:57:18,870 --> 00:57:20,309

 

1810
00:57:20,309 --> 00:57:23,670

 

1811
00:57:23,670 --> 00:57:24,789

 

1812
00:57:24,789 --> 00:57:25,990

 

1813
00:57:25,990 --> 00:57:27,829

 

1814
00:57:27,829 --> 00:57:27,839

 

1815
00:57:27,839 --> 00:57:29,030


1816
00:57:29,030 --> 00:57:33,670

 

1817
00:57:33,670 --> 00:57:33,680

 

1818
00:57:33,680 --> 00:57:36,390


1819
00:57:36,390 --> 00:57:38,829

 

1820
00:57:38,829 --> 00:57:41,510

 

1821
00:57:41,510 --> 00:57:44,230

 

1822
00:57:44,230 --> 00:57:47,109

 

1823
00:57:47,109 --> 00:57:47,119

 

1824
00:57:47,119 --> 00:57:47,910


1825
00:57:47,910 --> 00:57:49,349

 

1826
00:57:49,349 --> 00:57:50,630

 

1827
00:57:50,630 --> 00:57:53,030

 

1828
00:57:53,030 --> 00:57:55,510

 

1829
00:57:55,510 --> 00:57:57,510

 

1830
00:57:57,510 --> 00:57:57,520

 

1831
00:57:57,520 --> 00:58:00,069


1832
00:58:00,069 --> 00:58:02,870

 

1833
00:58:02,870 --> 00:58:05,109

 

1834
00:58:05,109 --> 00:58:06,870

 

1835
00:58:06,870 --> 00:58:10,230

 

1836
00:58:10,230 --> 00:58:11,510

 

1837
00:58:11,510 --> 00:58:12,710

 

1838
00:58:12,710 --> 00:58:14,549

 

1839
00:58:14,549 --> 00:58:16,630

 

1840
00:58:16,630 --> 00:58:19,670

 

1841
00:58:19,670 --> 00:58:19,680

 

1842
00:58:19,680 --> 00:58:21,589


1843
00:58:21,589 --> 00:58:21,599

 

1844
00:58:21,599 --> 00:58:22,630


1845
00:58:22,630 --> 00:58:22,640

 

1846
00:58:22,640 --> 00:58:24,230


1847
00:58:24,230 --> 00:58:24,240

 

1848
00:58:24,240 --> 00:58:24,950


1849
00:58:24,950 --> 00:58:28,870

 

1850
00:58:28,870 --> 00:58:30,789

 

1851
00:58:30,789 --> 00:58:32,069

 

1852
00:58:32,069 --> 00:58:34,950

 

1853
00:58:34,950 --> 00:58:38,230

 

1854
00:58:38,230 --> 00:58:39,910

 

1855
00:58:39,910 --> 00:58:42,630

 

1856
00:58:42,630 --> 00:58:44,870

 

1857
00:58:44,870 --> 00:58:48,230

 

1858
00:58:48,230 --> 00:58:50,470

 

1859
00:58:50,470 --> 00:58:52,470

 

1860
00:58:52,470 --> 00:58:53,990

 

1861
00:58:53,990 --> 00:58:58,710

 

1862
00:58:58,710 --> 00:58:58,720

 

1863
00:58:58,720 --> 00:58:59,180


1864
00:58:59,180 --> 00:58:59,190

 

1865
00:58:59,190 --> 00:59:00,950


1866
00:59:00,950 --> 00:59:03,510

 

1867
00:59:03,510 --> 00:59:04,789

 

1868
00:59:04,789 --> 00:59:04,799

 

1869
00:59:04,799 --> 00:59:05,750


1870
00:59:05,750 --> 00:59:10,150

 

1871
00:59:10,150 --> 00:59:10,160

 

1872
00:59:10,160 --> 00:59:11,109


1873
00:59:11,109 --> 00:59:13,109

 

1874
00:59:13,109 --> 00:59:15,589

 

1875
00:59:15,589 --> 00:59:17,589

 

1876
00:59:17,589 --> 00:59:20,230

 

1877
00:59:20,230 --> 00:59:22,230

 

1878
00:59:22,230 --> 00:59:24,230

 

1879
00:59:24,230 --> 00:59:26,710

 

1880
00:59:26,710 --> 00:59:28,309

 

1881
00:59:28,309 --> 00:59:30,150

 

1882
00:59:30,150 --> 00:59:30,160

 

1883
00:59:30,160 --> 00:59:31,270


1884
00:59:31,270 --> 00:59:33,030

 

1885
00:59:33,030 --> 00:59:35,109

 

1886
00:59:35,109 --> 00:59:35,119

 

1887
00:59:35,119 --> 00:59:36,150


1888
00:59:36,150 --> 00:59:39,109

 

1889
00:59:39,109 --> 00:59:41,030

 

1890
00:59:41,030 --> 00:59:42,870

 

1891
00:59:42,870 --> 00:59:44,470

 

1892
00:59:44,470 --> 00:59:46,710

 

1893
00:59:46,710 --> 00:59:48,870

 

1894
00:59:48,870 --> 00:59:51,270

 

1895
00:59:51,270 --> 00:59:53,109

 

1896
00:59:53,109 --> 00:59:54,630

 

1897
00:59:54,630 --> 00:59:56,150

 

1898
00:59:56,150 --> 00:59:58,470

 

1899
00:59:58,470 --> 01:00:00,870

 

1900
01:00:00,870 --> 01:00:02,549

 

1901
01:00:02,549 --> 01:00:04,309

 

1902
01:00:04,309 --> 01:00:05,829

 

1903
01:00:05,829 --> 01:00:07,510

 

1904
01:00:07,510 --> 01:00:09,030

 

1905
01:00:09,030 --> 01:00:11,030

 

1906
01:00:11,030 --> 01:00:13,430

 

1907
01:00:13,430 --> 01:00:15,670

 

1908
01:00:15,670 --> 01:00:17,750

 

1909
01:00:17,750 --> 01:00:19,430

 

1910
01:00:19,430 --> 01:00:21,829

 

1911
01:00:21,829 --> 01:00:24,710

 

1912
01:00:24,710 --> 01:00:24,720

 

1913
01:00:24,720 --> 01:00:25,829


1914
01:00:25,829 --> 01:00:25,839

 

1915
01:00:25,839 --> 01:00:27,510


1916
01:00:27,510 --> 01:00:31,510

 

1917
01:00:31,510 --> 01:00:32,789

 

1918
01:00:32,789 --> 01:00:34,309

 

1919
01:00:34,309 --> 01:00:37,270

 

1920
01:00:37,270 --> 01:00:39,589

 

1921
01:00:39,589 --> 01:00:41,430

 

1922
01:00:41,430 --> 01:00:43,430

 

1923
01:00:43,430 --> 01:00:45,670

 

1924
01:00:45,670 --> 01:00:47,829

 

1925
01:00:47,829 --> 01:00:50,230

 

1926
01:00:50,230 --> 01:00:52,630

 

1927
01:00:52,630 --> 01:00:54,630

 

1928
01:00:54,630 --> 01:00:57,030

 

1929
01:00:57,030 --> 01:00:57,040

 

1930
01:00:57,040 --> 01:00:57,520


1931
01:00:57,520 --> 01:00:57,530

 

1932
01:00:57,530 --> 01:00:58,789


1933
01:00:58,789 --> 01:01:05,030

 

1934
01:01:05,030 --> 01:01:06,630

 

1935
01:01:06,630 --> 01:01:09,829

 

1936
01:01:09,829 --> 01:01:11,430

 

1937
01:01:11,430 --> 01:01:12,870

 

1938
01:01:12,870 --> 01:01:13,990

 

1939
01:01:13,990 --> 01:01:15,670

 

1940
01:01:15,670 --> 01:01:16,870

 

1941
01:01:16,870 --> 01:01:20,710

 

1942
01:01:20,710 --> 01:01:22,950

 

1943
01:01:22,950 --> 01:01:24,230

 

1944
01:01:24,230 --> 01:01:26,069

 

1945
01:01:26,069 --> 01:01:26,079

 

1946
01:01:26,079 --> 01:01:27,510


1947
01:01:27,510 --> 01:01:27,520

 

1948
01:01:27,520 --> 01:01:28,870


1949
01:01:28,870 --> 01:01:28,880

 

1950
01:01:28,880 --> 01:01:30,630


1951
01:01:30,630 --> 01:01:30,640

 

1952
01:01:30,640 --> 01:01:31,990


1953
01:01:31,990 --> 01:01:34,950

 

1954
01:01:34,950 --> 01:01:36,230

 

1955
01:01:36,230 --> 01:01:39,349

 

1956
01:01:39,349 --> 01:01:41,750

 

1957
01:01:41,750 --> 01:01:45,270

 

1958
01:01:45,270 --> 01:01:45,280

 

1959
01:01:45,280 --> 01:01:46,470


1960
01:01:46,470 --> 01:01:47,670

 

1961
01:01:47,670 --> 01:01:50,309

 

1962
01:01:50,309 --> 01:01:53,510

 

1963
01:01:53,510 --> 01:01:55,670

 

1964
01:01:55,670 --> 01:01:58,069

 

1965
01:01:58,069 --> 01:01:58,079

 

1966
01:01:58,079 --> 01:01:58,870


1967
01:01:58,870 --> 01:02:00,470

 

1968
01:02:00,470 --> 01:02:00,480

 

1969
01:02:00,480 --> 01:02:01,510


1970
01:02:01,510 --> 01:02:04,069

 

1971
01:02:04,069 --> 01:02:06,950

 

1972
01:02:06,950 --> 01:02:09,829

 

1973
01:02:09,829 --> 01:02:11,829

 

1974
01:02:11,829 --> 01:02:12,950

 

1975
01:02:12,950 --> 01:02:15,670

 

1976
01:02:15,670 --> 01:02:17,150

 

1977
01:02:17,150 --> 01:02:18,789

 

1978
01:02:18,789 --> 01:02:21,430

 

1979
01:02:21,430 --> 01:02:24,309

 

1980
01:02:24,309 --> 01:02:26,230

 

1981
01:02:26,230 --> 01:02:27,430

 

1982
01:02:27,430 --> 01:02:29,430

 

1983
01:02:29,430 --> 01:02:32,069

 

1984
01:02:32,069 --> 01:02:32,079

 

1985
01:02:32,079 --> 01:02:33,190


1986
01:02:33,190 --> 01:02:34,950

 

1987
01:02:34,950 --> 01:02:37,109

 

1988
01:02:37,109 --> 01:02:37,119

 

1989
01:02:37,119 --> 01:02:38,710


1990
01:02:38,710 --> 01:02:38,720

 

1991
01:02:38,720 --> 01:02:41,270


1992
01:02:41,270 --> 01:02:41,280

 

1993
01:02:41,280 --> 01:02:42,390


1994
01:02:42,390 --> 01:02:44,309

 

1995
01:02:44,309 --> 01:02:46,549

 

1996
01:02:46,549 --> 01:02:49,430

 

1997
01:02:49,430 --> 01:02:53,430

 

1998
01:02:53,430 --> 01:02:56,390

 

1999
01:02:56,390 --> 01:02:58,950

 

2000
01:02:58,950 --> 01:02:58,960

 

2001
01:02:58,960 --> 01:03:01,589


2002
01:03:01,589 --> 01:03:03,109

 

2003
01:03:03,109 --> 01:03:05,029

 

2004
01:03:05,029 --> 01:03:06,950

 

2005
01:03:06,950 --> 01:03:08,870

 

2006
01:03:08,870 --> 01:03:08,880

 

2007
01:03:08,880 --> 01:03:09,750


2008
01:03:09,750 --> 01:03:11,589

 

2009
01:03:11,589 --> 01:03:13,029

 

2010
01:03:13,029 --> 01:03:15,349

 

2011
01:03:15,349 --> 01:03:17,270

 

2012
01:03:17,270 --> 01:03:23,029

 

2013
01:03:23,029 --> 01:03:25,029

 

2014
01:03:25,029 --> 01:03:25,039

 

2015
01:03:25,039 --> 01:03:26,390


2016
01:03:26,390 --> 01:03:31,430

 

2017
01:03:31,430 --> 01:03:33,109

 

2018
01:03:33,109 --> 01:03:36,250

 

2019
01:03:36,250 --> 01:03:36,260

 

2020
01:03:36,260 --> 01:03:38,549


2021
01:03:38,549 --> 01:03:41,829

 

2022
01:03:41,829 --> 01:03:45,270

 

2023
01:03:45,270 --> 01:03:46,789

 

2024
01:03:46,789 --> 01:03:46,799

 

2025
01:03:46,799 --> 01:03:48,230


2026
01:03:48,230 --> 01:03:49,829

 

2027
01:03:49,829 --> 01:03:53,349

 

2028
01:03:53,349 --> 01:03:54,789

 

2029
01:03:54,789 --> 01:03:56,870

 

2030
01:03:56,870 --> 01:03:58,789

 

2031
01:03:58,789 --> 01:04:00,470

 

2032
01:04:00,470 --> 01:04:02,309

 

2033
01:04:02,309 --> 01:04:04,950

 

2034
01:04:04,950 --> 01:04:07,670

 

2035
01:04:07,670 --> 01:04:07,680

 

2036
01:04:07,680 --> 01:04:10,069


2037
01:04:10,069 --> 01:04:11,990

 

2038
01:04:11,990 --> 01:04:13,270

 

2039
01:04:13,270 --> 01:04:13,280

 

2040
01:04:13,280 --> 01:04:14,150


2041
01:04:14,150 --> 01:04:14,160

 

2042
01:04:14,160 --> 01:04:15,510


2043
01:04:15,510 --> 01:04:17,589

 

2044
01:04:17,589 --> 01:04:19,349

 

2045
01:04:19,349 --> 01:04:21,190

 

2046
01:04:21,190 --> 01:04:23,270

 

2047
01:04:23,270 --> 01:04:25,190

 

2048
01:04:25,190 --> 01:04:27,190

 

2049
01:04:27,190 --> 01:04:28,870

 

2050
01:04:28,870 --> 01:04:30,710

 

2051
01:04:30,710 --> 01:04:32,150

 

2052
01:04:32,150 --> 01:04:33,990

 

2053
01:04:33,990 --> 01:04:34,000

 

2054
01:04:34,000 --> 01:04:35,109


2055
01:04:35,109 --> 01:04:37,109

 

2056
01:04:37,109 --> 01:04:38,710

 

2057
01:04:38,710 --> 01:04:41,270

 

2058
01:04:41,270 --> 01:04:43,349

 

2059
01:04:43,349 --> 01:04:45,829

 

2060
01:04:45,829 --> 01:04:48,549

 

2061
01:04:48,549 --> 01:04:51,349

 

2062
01:04:51,349 --> 01:04:52,710

 

2063
01:04:52,710 --> 01:04:55,829

 

2064
01:04:55,829 --> 01:04:58,470

 

2065
01:04:58,470 --> 01:05:00,789

 

2066
01:05:00,789 --> 01:05:05,190

 

2067
01:05:05,190 --> 01:05:07,430

 

2068
01:05:07,430 --> 01:05:09,670

 

2069
01:05:09,670 --> 01:05:14,390

 

2070
01:05:14,390 --> 01:05:17,349

 

2071
01:05:17,349 --> 01:05:19,750

 

2072
01:05:19,750 --> 01:05:22,390

 

2073
01:05:22,390 --> 01:05:24,309

 

2074
01:05:24,309 --> 01:05:24,319

 

2075
01:05:24,319 --> 01:05:25,589


2076
01:05:25,589 --> 01:05:26,789

 

2077
01:05:26,789 --> 01:05:26,799

 

2078
01:05:26,799 --> 01:05:27,990


2079
01:05:27,990 --> 01:05:29,829

 

2080
01:05:29,829 --> 01:05:32,710

 

2081
01:05:32,710 --> 01:05:34,470

 

2082
01:05:34,470 --> 01:05:39,029

 

2083
01:05:39,029 --> 01:05:41,029

 

2084
01:05:41,029 --> 01:05:43,349

 

2085
01:05:43,349 --> 01:05:44,630

 

2086
01:05:44,630 --> 01:05:47,109

 

2087
01:05:47,109 --> 01:05:49,750

 

2088
01:05:49,750 --> 01:05:51,190

 

2089
01:05:51,190 --> 01:05:52,710

 

2090
01:05:52,710 --> 01:05:54,150

 

2091
01:05:54,150 --> 01:05:54,160

 

2092
01:05:54,160 --> 01:05:55,190


2093
01:05:55,190 --> 01:05:57,829

 

2094
01:05:57,829 --> 01:06:02,829

 

2095
01:06:02,829 --> 01:06:06,549

 

2096
01:06:06,549 --> 01:06:09,750

 

2097
01:06:09,750 --> 01:06:12,870

 

2098
01:06:12,870 --> 01:06:16,230

 

2099
01:06:16,230 --> 01:06:19,029

 

2100
01:06:19,029 --> 01:06:19,039

 

2101
01:06:19,039 --> 01:06:20,150


2102
01:06:20,150 --> 01:06:22,549

 

2103
01:06:22,549 --> 01:06:25,829

 

2104
01:06:25,829 --> 01:06:27,430

 

2105
01:06:27,430 --> 01:06:28,549

 

2106
01:06:28,549 --> 01:06:30,470

 

2107
01:06:30,470 --> 01:06:33,990

 

2108
01:06:33,990 --> 01:06:34,000

 

2109
01:06:34,000 --> 01:06:35,349


2110
01:06:35,349 --> 01:06:37,430

 

2111
01:06:37,430 --> 01:06:37,440

 

2112
01:06:37,440 --> 01:06:39,109


2113
01:06:39,109 --> 01:06:41,270

 

2114
01:06:41,270 --> 01:06:43,589

 

2115
01:06:43,589 --> 01:06:45,589

 

2116
01:06:45,589 --> 01:06:47,829

 

2117
01:06:47,829 --> 01:06:51,109

 

2118
01:06:51,109 --> 01:06:54,150

 

2119
01:06:54,150 --> 01:06:56,630

 

2120
01:06:56,630 --> 01:06:58,390

 

2121
01:06:58,390 --> 01:07:00,630

 

2122
01:07:00,630 --> 01:07:03,029

 

2123
01:07:03,029 --> 01:07:06,069

 

2124
01:07:06,069 --> 01:07:06,079

 

2125
01:07:06,079 --> 01:07:07,829


2126
01:07:07,829 --> 01:07:10,470

 

2127
01:07:10,470 --> 01:07:10,480

 

2128
01:07:10,480 --> 01:07:11,589


2129
01:07:11,589 --> 01:07:13,430

 

2130
01:07:13,430 --> 01:07:13,440

 

2131
01:07:13,440 --> 01:07:14,870


2132
01:07:14,870 --> 01:07:16,870

 

2133
01:07:16,870 --> 01:07:16,880

 

2134
01:07:16,880 --> 01:07:17,829


2135
01:07:17,829 --> 01:07:19,430

 

2136
01:07:19,430 --> 01:07:19,440

 

2137
01:07:19,440 --> 01:07:20,710


2138
01:07:20,710 --> 01:07:22,069

 

2139
01:07:22,069 --> 01:07:24,150

 

2140
01:07:24,150 --> 01:07:27,029

 

2141
01:07:27,029 --> 01:07:29,589

 

2142
01:07:29,589 --> 01:07:29,599

 

2143
01:07:29,599 --> 01:07:30,470


2144
01:07:30,470 --> 01:07:32,230

 

2145
01:07:32,230 --> 01:07:32,240

 

2146
01:07:32,240 --> 01:07:33,349


2147
01:07:33,349 --> 01:07:35,750

 

2148
01:07:35,750 --> 01:07:37,910

 

2149
01:07:37,910 --> 01:07:39,270

 

2150
01:07:39,270 --> 01:07:42,470

 

2151
01:07:42,470 --> 01:07:44,390

 

2152
01:07:44,390 --> 01:07:45,829

 

2153
01:07:45,829 --> 01:07:45,839

 

2154
01:07:45,839 --> 01:07:50,870


2155
01:07:50,870 --> 01:07:52,630

 

2156
01:07:52,630 --> 01:07:54,230

 

2157
01:07:54,230 --> 01:07:55,589

 

2158
01:07:55,589 --> 01:07:55,599

 

2159
01:07:55,599 --> 01:07:58,390


2160
01:07:58,390 --> 01:08:00,309

 

2161
01:08:00,309 --> 01:08:02,789

 

2162
01:08:02,789 --> 01:08:06,069

 

2163
01:08:06,069 --> 01:08:08,470

 

2164
01:08:08,470 --> 01:08:08,480

 

2165
01:08:08,480 --> 01:08:09,589


2166
01:08:09,589 --> 01:08:11,990

 

2167
01:08:11,990 --> 01:08:13,349

 

2168
01:08:13,349 --> 01:08:15,430

 

2169
01:08:15,430 --> 01:08:15,440

 

2170
01:08:15,440 --> 01:08:17,349


2171
01:08:17,349 --> 01:08:17,359

 

2172
01:08:17,359 --> 01:08:19,749


2173
01:08:19,749 --> 01:08:21,749

 

2174
01:08:21,749 --> 01:08:23,749

 

2175
01:08:23,749 --> 01:08:25,990

 

2176
01:08:25,990 --> 01:08:26,000

 

2177
01:08:26,000 --> 01:08:27,030


2178
01:08:27,030 --> 01:08:29,110

 

2179
01:08:29,110 --> 01:08:29,120

 

2180
01:08:29,120 --> 01:08:29,829


2181
01:08:29,829 --> 01:08:32,229

 

2182
01:08:32,229 --> 01:08:32,239

 

2183
01:08:32,239 --> 01:08:33,430


2184
01:08:33,430 --> 01:08:35,669

 

2185
01:08:35,669 --> 01:08:41,269

 

2186
01:08:41,269 --> 01:08:43,030

 

2187
01:08:43,030 --> 01:08:44,550

 

2188
01:08:44,550 --> 01:08:46,789

 

2189
01:08:46,789 --> 01:08:49,590

 

2190
01:08:49,590 --> 01:08:51,189

 

2191
01:08:51,189 --> 01:08:53,590

 

2192
01:08:53,590 --> 01:08:55,269

 

2193
01:08:55,269 --> 01:08:56,630

 

2194
01:08:56,630 --> 01:08:58,709

 

2195
01:08:58,709 --> 01:08:59,910

 

2196
01:08:59,910 --> 01:09:02,950

 

2197
01:09:02,950 --> 01:09:05,030

 

2198
01:09:05,030 --> 01:09:07,829

 

2199
01:09:07,829 --> 01:09:11,030

 

2200
01:09:11,030 --> 01:09:13,349

 

2201
01:09:13,349 --> 01:09:14,870

 

2202
01:09:14,870 --> 01:09:16,630

 

2203
01:09:16,630 --> 01:09:18,070

 

2204
01:09:18,070 --> 01:09:20,070

 

2205
01:09:20,070 --> 01:09:22,630

 

2206
01:09:22,630 --> 01:09:24,950

 

2207
01:09:24,950 --> 01:09:24,960

 

2208
01:09:24,960 --> 01:09:26,229


2209
01:09:26,229 --> 01:09:28,709

 

2210
01:09:28,709 --> 01:09:32,309

 

2211
01:09:32,309 --> 01:09:38,070

 

2212
01:09:38,070 --> 01:09:44,630

 

2213
01:09:44,630 --> 01:09:49,189

 

2214
01:09:49,189 --> 01:09:52,229

 

2215
01:09:52,229 --> 01:09:53,110

 

2216
01:09:53,110 --> 01:09:55,110

 

2217
01:09:55,110 --> 01:09:55,120

 

2218
01:09:55,120 --> 01:09:56,550


2219
01:09:56,550 --> 01:09:56,560

 

2220
01:09:56,560 --> 01:09:58,550


2221
01:09:58,550 --> 01:10:02,550

 

2222
01:10:02,550 --> 01:10:04,790

 

2223
01:10:04,790 --> 01:10:04,800

 

2224
01:10:04,800 --> 01:10:06,149


2225
01:10:06,149 --> 01:10:08,709

 

2226
01:10:08,709 --> 01:10:10,550

 

2227
01:10:10,550 --> 01:10:11,910

 

2228
01:10:11,910 --> 01:10:14,229

 

2229
01:10:14,229 --> 01:10:15,910

 

2230
01:10:15,910 --> 01:10:18,630

 

2231
01:10:18,630 --> 01:10:21,430

 

2232
01:10:21,430 --> 01:10:22,790

 

2233
01:10:22,790 --> 01:10:24,310

 

2234
01:10:24,310 --> 01:10:26,709

 

2235
01:10:26,709 --> 01:10:29,830

 

2236
01:10:29,830 --> 01:10:31,750

 

2237
01:10:31,750 --> 01:10:34,550

 

2238
01:10:34,550 --> 01:10:36,870

 

2239
01:10:36,870 --> 01:10:38,550

 

2240
01:10:38,550 --> 01:10:40,310

 

2241
01:10:40,310 --> 01:10:42,950

 

2242
01:10:42,950 --> 01:10:44,709

 

2243
01:10:44,709 --> 01:10:44,719

 

2244
01:10:44,719 --> 01:10:45,750


2245
01:10:45,750 --> 01:10:48,070

 

2246
01:10:48,070 --> 01:10:50,790

 

2247
01:10:50,790 --> 01:10:54,070

 

2248
01:10:54,070 --> 01:10:55,590

 

2249
01:10:55,590 --> 01:10:59,110

 

2250
01:10:59,110 --> 01:11:00,390

 

2251
01:11:00,390 --> 01:11:01,990

 

2252
01:11:01,990 --> 01:11:02,000

 

2253
01:11:02,000 --> 01:11:03,110


2254
01:11:03,110 --> 01:11:06,070

 

2255
01:11:06,070 --> 01:11:07,669

 

2256
01:11:07,669 --> 01:11:07,679

 

2257
01:11:07,679 --> 01:11:08,470


2258
01:11:08,470 --> 01:11:08,480

 

2259
01:11:08,480 --> 01:11:08,880


2260
01:11:08,880 --> 01:11:08,890

 

2261
01:11:08,890 --> 01:11:10,310


2262
01:11:10,310 --> 01:11:12,390

 

2263
01:11:12,390 --> 01:11:13,510

 

2264
01:11:13,510 --> 01:11:15,430

 

2265
01:11:15,430 --> 01:11:17,669

 

2266
01:11:17,669 --> 01:11:18,709

 

2267
01:11:18,709 --> 01:11:18,719

 

2268
01:11:18,719 --> 01:11:19,669


2269
01:11:19,669 --> 01:11:20,790

 

2270
01:11:20,790 --> 01:11:23,750

 

2271
01:11:23,750 --> 01:11:25,750

 

2272
01:11:25,750 --> 01:11:27,510

 

2273
01:11:27,510 --> 01:11:27,520

 

2274
01:11:27,520 --> 01:11:28,709


2275
01:11:28,709 --> 01:11:28,719

 

2276
01:11:28,719 --> 01:11:31,590


2277
01:11:31,590 --> 01:11:31,600

 

2278
01:11:31,600 --> 01:11:33,910


2279
01:11:33,910 --> 01:11:33,920

 

2280
01:11:33,920 --> 01:11:34,950


2281
01:11:34,950 --> 01:11:36,390

 

2282
01:11:36,390 --> 01:11:38,550

 

2283
01:11:38,550 --> 01:11:40,470

 

2284
01:11:40,470 --> 01:11:42,470

 

2285
01:11:42,470 --> 01:11:43,910

 

2286
01:11:43,910 --> 01:11:43,920

 

2287
01:11:43,920 --> 01:11:44,790


2288
01:11:44,790 --> 01:11:47,030

 

2289
01:11:47,030 --> 01:11:47,040

 

2290
01:11:47,040 --> 01:11:48,630


2291
01:11:48,630 --> 01:11:51,189

 

2292
01:11:51,189 --> 01:11:53,189

 

2293
01:11:53,189 --> 01:11:55,510

 

2294
01:11:55,510 --> 01:11:58,870

 

2295
01:11:58,870 --> 01:11:58,880

 

2296
01:11:58,880 --> 01:11:59,380


2297
01:11:59,380 --> 01:11:59,390

 

2298
01:11:59,390 --> 01:12:01,510


2299
01:12:01,510 --> 01:12:03,830

 

2300
01:12:03,830 --> 01:12:06,470

 

2301
01:12:06,470 --> 01:12:08,550

 

2302
01:12:08,550 --> 01:12:08,560

 

2303
01:12:08,560 --> 01:12:11,590


2304
01:12:11,590 --> 01:12:13,990

 

2305
01:12:13,990 --> 01:12:16,070

 

2306
01:12:16,070 --> 01:12:18,470

 

2307
01:12:18,470 --> 01:12:21,590

 

2308
01:12:21,590 --> 01:12:23,830

 

2309
01:12:23,830 --> 01:12:25,990

 

2310
01:12:25,990 --> 01:12:27,430

 

2311
01:12:27,430 --> 01:12:29,910

 

2312
01:12:29,910 --> 01:12:31,910

 

2313
01:12:31,910 --> 01:12:34,070

 

2314
01:12:34,070 --> 01:12:36,390

 

2315
01:12:36,390 --> 01:12:38,229

 

2316
01:12:38,229 --> 01:12:40,709

 

2317
01:12:40,709 --> 01:12:42,550

 

2318
01:12:42,550 --> 01:12:45,030

 

2319
01:12:45,030 --> 01:12:47,350

 

2320
01:12:47,350 --> 01:12:50,149

 

2321
01:12:50,149 --> 01:12:51,669

 

2322
01:12:51,669 --> 01:12:53,510

 

2323
01:12:53,510 --> 01:12:55,990

 

2324
01:12:55,990 --> 01:12:59,110

 

2325
01:12:59,110 --> 01:13:02,310

 

2326
01:13:02,310 --> 01:13:04,630

 

2327
01:13:04,630 --> 01:13:07,430

 

2328
01:13:07,430 --> 01:13:09,110

 

2329
01:13:09,110 --> 01:13:11,270

 

2330
01:13:11,270 --> 01:13:11,280

 

2331
01:13:11,280 --> 01:13:11,990


2332
01:13:11,990 --> 01:13:13,910

 

2333
01:13:13,910 --> 01:13:16,070

 

2334
01:13:16,070 --> 01:13:18,390

 

2335
01:13:18,390 --> 01:13:20,390

 

2336
01:13:20,390 --> 01:13:22,630

 

2337
01:13:22,630 --> 01:13:22,640

 

2338
01:13:22,640 --> 01:13:24,390


2339
01:13:24,390 --> 01:13:25,430

 

2340
01:13:25,430 --> 01:13:27,990

 

2341
01:13:27,990 --> 01:13:28,000

 

2342
01:13:28,000 --> 01:13:28,790


2343
01:13:28,790 --> 01:13:30,390

 

2344
01:13:30,390 --> 01:13:33,830

 

2345
01:13:33,830 --> 01:13:35,030

 

2346
01:13:35,030 --> 01:13:35,040

 

2347
01:13:35,040 --> 01:13:36,709


2348
01:13:36,709 --> 01:13:39,110

 

2349
01:13:39,110 --> 01:13:41,270

 

2350
01:13:41,270 --> 01:13:43,110

 

2351
01:13:43,110 --> 01:13:44,950

 

2352
01:13:44,950 --> 01:13:47,510

 

2353
01:13:47,510 --> 01:13:51,669

 

2354
01:13:51,669 --> 01:13:53,189

 

2355
01:13:53,189 --> 01:13:54,550

 

2356
01:13:54,550 --> 01:13:56,149

 

2357
01:13:56,149 --> 01:13:57,510

 

2358
01:13:57,510 --> 01:13:57,520

 

2359
01:13:57,520 --> 01:13:59,510


2360
01:13:59,510 --> 01:14:01,990

 

2361
01:14:01,990 --> 01:14:04,390

 

2362
01:14:04,390 --> 01:14:06,950

 

2363
01:14:06,950 --> 01:14:08,950

 

2364
01:14:08,950 --> 01:14:11,830

 

2365
01:14:11,830 --> 01:14:12,830

 

2366
01:14:12,830 --> 01:14:14,870

 

2367
01:14:14,870 --> 01:14:14,880

 

2368
01:14:14,880 --> 01:14:15,669


2369
01:14:15,669 --> 01:14:17,990

 

2370
01:14:17,990 --> 01:14:20,950

 

2371
01:14:20,950 --> 01:14:22,870

 

2372
01:14:22,870 --> 01:14:24,950

 

2373
01:14:24,950 --> 01:14:27,110

 

2374
01:14:27,110 --> 01:14:29,189

 

2375
01:14:29,189 --> 01:14:31,350

 

2376
01:14:31,350 --> 01:14:31,360

 

2377
01:14:31,360 --> 01:14:32,470


2378
01:14:32,470 --> 01:14:34,310

 

2379
01:14:34,310 --> 01:14:36,149

 

2380
01:14:36,149 --> 01:14:36,159

 

2381
01:14:36,159 --> 01:14:37,110


2382
01:14:37,110 --> 01:14:39,270

 

2383
01:14:39,270 --> 01:14:42,310

 

2384
01:14:42,310 --> 01:14:43,830

 

2385
01:14:43,830 --> 01:14:46,229

 

2386
01:14:46,229 --> 01:14:48,470

 

2387
01:14:48,470 --> 01:14:50,390

 

2388
01:14:50,390 --> 01:14:55,590

 

2389
01:14:55,590 --> 01:14:57,270

 

2390
01:14:57,270 --> 01:14:59,110

 

2391
01:14:59,110 --> 01:15:01,270

 

2392
01:15:01,270 --> 01:15:03,430

 

2393
01:15:03,430 --> 01:15:06,550

 

2394
01:15:06,550 --> 01:15:10,630

 

2395
01:15:10,630 --> 01:15:11,910

 

2396
01:15:11,910 --> 01:15:15,189

 

2397
01:15:15,189 --> 01:15:15,199

 

2398
01:15:15,199 --> 01:15:16,630


2399
01:15:16,630 --> 01:15:18,709

 

2400
01:15:18,709 --> 01:15:18,719

 

2401
01:15:18,719 --> 01:15:20,950


2402
01:15:20,950 --> 01:15:22,470

 

2403
01:15:22,470 --> 01:15:26,310

 

2404
01:15:26,310 --> 01:15:27,590

 

2405
01:15:27,590 --> 01:15:29,669

 

2406
01:15:29,669 --> 01:15:31,430

 

2407
01:15:31,430 --> 01:15:31,440

 

2408
01:15:31,440 --> 01:15:32,870


2409
01:15:32,870 --> 01:15:34,630

 

2410
01:15:34,630 --> 01:15:38,630

 

2411
01:15:38,630 --> 01:15:40,229

 

2412
01:15:40,229 --> 01:15:40,239

 

2413
01:15:40,239 --> 01:15:41,430


2414
01:15:41,430 --> 01:15:43,590

 

2415
01:15:43,590 --> 01:15:45,750

 

2416
01:15:45,750 --> 01:15:47,590

 

2417
01:15:47,590 --> 01:15:49,350

 

2418
01:15:49,350 --> 01:15:51,510

 

2419
01:15:51,510 --> 01:15:53,030

 

2420
01:15:53,030 --> 01:15:54,550

 

2421
01:15:54,550 --> 01:15:56,630

 

2422
01:15:56,630 --> 01:15:59,510

 

2423
01:15:59,510 --> 01:16:00,470

 

2424
01:16:00,470 --> 01:16:02,790

 

2425
01:16:02,790 --> 01:16:05,590

 

2426
01:16:05,590 --> 01:16:05,600

 

2427
01:16:05,600 --> 01:16:06,790


2428
01:16:06,790 --> 01:16:09,590

 

2429
01:16:09,590 --> 01:16:11,510

 

2430
01:16:11,510 --> 01:16:14,790

 

2431
01:16:14,790 --> 01:16:16,470

 

2432
01:16:16,470 --> 01:16:18,229

 

2433
01:16:18,229 --> 01:16:20,790

 

2434
01:16:20,790 --> 01:16:23,430

 

2435
01:16:23,430 --> 01:16:25,910

 

2436
01:16:25,910 --> 01:16:27,189

 

2437
01:16:27,189 --> 01:16:29,270

 

2438
01:16:29,270 --> 01:16:31,189

 

2439
01:16:31,189 --> 01:16:35,590

 

2440
01:16:35,590 --> 01:16:37,590

 

2441
01:16:37,590 --> 01:16:41,510

 

2442
01:16:41,510 --> 01:16:43,270

 

2443
01:16:43,270 --> 01:16:43,280

 

2444
01:16:43,280 --> 01:16:44,070


2445
01:16:44,070 --> 01:16:44,080

 

2446
01:16:44,080 --> 01:16:46,149


2447
01:16:46,149 --> 01:16:49,110

 

2448
01:16:49,110 --> 01:16:52,070

 

2449
01:16:52,070 --> 01:16:52,080

 

2450
01:16:52,080 --> 01:16:52,870


2451
01:16:52,870 --> 01:16:54,950

 

2452
01:16:54,950 --> 01:16:57,430

 

2453
01:16:57,430 --> 01:16:57,440

 

2454
01:16:57,440 --> 01:16:58,950


2455
01:16:58,950 --> 01:17:00,630

 

2456
01:17:00,630 --> 01:17:02,550

 

2457
01:17:02,550 --> 01:17:03,830

 

2458
01:17:03,830 --> 01:17:03,840

 

2459
01:17:03,840 --> 01:17:05,189


2460
01:17:05,189 --> 01:17:07,590

 

2461
01:17:07,590 --> 01:17:07,600

 

2462
01:17:07,600 --> 01:17:09,510


2463
01:17:09,510 --> 01:17:09,520

 

2464
01:17:09,520 --> 01:17:10,709


2465
01:17:10,709 --> 01:17:10,719

 

2466
01:17:10,719 --> 01:17:12,070


2467
01:17:12,070 --> 01:17:12,080

 

2468
01:17:12,080 --> 01:17:15,430


2469
01:17:15,430 --> 01:17:18,709

 

2470
01:17:18,709 --> 01:17:18,719

 

2471
01:17:18,719 --> 01:17:20,229


2472
01:17:20,229 --> 01:17:21,189

 

2473
01:17:21,189 --> 01:17:23,910

 

2474
01:17:23,910 --> 01:17:23,920

 

2475
01:17:23,920 --> 01:17:26,229


2476
01:17:26,229 --> 01:17:26,239

 

2477
01:17:26,239 --> 01:17:29,350


2478
01:17:29,350 --> 01:17:33,110

 

2479
01:17:33,110 --> 01:17:33,120

 

2480
01:17:33,120 --> 01:17:36,550


2481
01:17:36,550 --> 01:17:36,560

 

2482
01:17:36,560 --> 01:17:37,430


2483
01:17:37,430 --> 01:17:39,430

 

2484
01:17:39,430 --> 01:17:41,350

 

2485
01:17:41,350 --> 01:17:43,270

 

2486
01:17:43,270 --> 01:17:45,990

 

2487
01:17:45,990 --> 01:17:49,350

 

2488
01:17:49,350 --> 01:17:50,790

 

2489
01:17:50,790 --> 01:17:52,709

 

2490
01:17:52,709 --> 01:17:52,719

 

2491
01:17:52,719 --> 01:17:53,430


2492
01:17:53,430 --> 01:17:55,750

 

2493
01:17:55,750 --> 01:17:55,760

 

2494
01:17:55,760 --> 01:17:57,030


2495
01:17:57,030 --> 01:17:59,590

 

2496
01:17:59,590 --> 01:18:02,149

 

2497
01:18:02,149 --> 01:18:02,159

 

2498
01:18:02,159 --> 01:18:03,030


2499
01:18:03,030 --> 01:18:05,030

 

2500
01:18:05,030 --> 01:18:06,630

 

2501
01:18:06,630 --> 01:18:08,630

 

2502
01:18:08,630 --> 01:18:08,640

 

2503
01:18:08,640 --> 01:18:09,669


2504
01:18:09,669 --> 01:18:09,679

 

2505
01:18:09,679 --> 01:18:10,709


2506
01:18:10,709 --> 01:18:13,270

 

2507
01:18:13,270 --> 01:18:18,229

 

2508
01:18:18,229 --> 01:18:19,510

 

2509
01:18:19,510 --> 01:18:21,430

 

2510
01:18:21,430 --> 01:18:24,790

 

2511
01:18:24,790 --> 01:18:26,550

 

2512
01:18:26,550 --> 01:18:26,560

 

2513
01:18:26,560 --> 01:18:27,430


2514
01:18:27,430 --> 01:18:30,550

 

2515
01:18:30,550 --> 01:18:33,510

 

2516
01:18:33,510 --> 01:18:35,590

 

2517
01:18:35,590 --> 01:18:38,229

 

2518
01:18:38,229 --> 01:18:41,030

 

2519
01:18:41,030 --> 01:18:43,750

 

2520
01:18:43,750 --> 01:18:45,270

 

2521
01:18:45,270 --> 01:18:46,709

 

2522
01:18:46,709 --> 01:18:48,870

 

2523
01:18:48,870 --> 01:18:50,950

 

2524
01:18:50,950 --> 01:18:53,110

 

2525
01:18:53,110 --> 01:18:54,709

 

2526
01:18:54,709 --> 01:18:56,709

 

2527
01:18:56,709 --> 01:18:58,709

 

2528
01:18:58,709 --> 01:19:00,229

 

2529
01:19:00,229 --> 01:19:01,669

 

2530
01:19:01,669 --> 01:19:02,709

 

2531
01:19:02,709 --> 01:19:05,110

 

2532
01:19:05,110 --> 01:19:05,120

 

2533
01:19:05,120 --> 01:19:06,390


2534
01:19:06,390 --> 01:19:08,870

 

2535
01:19:08,870 --> 01:19:11,430

 

2536
01:19:11,430 --> 01:19:11,440

 

2537
01:19:11,440 --> 01:19:13,830


2538
01:19:13,830 --> 01:19:16,070

 

2539
01:19:16,070 --> 01:19:17,669

 

2540
01:19:17,669 --> 01:19:21,270

 

2541
01:19:21,270 --> 01:19:24,070

 

2542
01:19:24,070 --> 01:19:27,110

 

2543
01:19:27,110 --> 01:19:29,110

 

2544
01:19:29,110 --> 01:19:31,270

 

2545
01:19:31,270 --> 01:19:33,750

 

2546
01:19:33,750 --> 01:19:35,990

 

2547
01:19:35,990 --> 01:19:38,310

 

2548
01:19:38,310 --> 01:19:39,830

 

2549
01:19:39,830 --> 01:19:42,470

 

2550
01:19:42,470 --> 01:19:45,910

 

2551
01:19:45,910 --> 01:19:45,920

 

2552
01:19:45,920 --> 01:19:46,950


2553
01:19:46,950 --> 01:19:48,870

 

2554
01:19:48,870 --> 01:19:49,990

 

2555
01:19:49,990 --> 01:19:52,470

 

2556
01:19:52,470 --> 01:19:54,790

 

2557
01:19:54,790 --> 01:19:58,870

 

2558
01:19:58,870 --> 01:20:00,470

 

2559
01:20:00,470 --> 01:20:02,229

 

2560
01:20:02,229 --> 01:20:04,310

 

2561
01:20:04,310 --> 01:20:06,550

 

2562
01:20:06,550 --> 01:20:06,560

 

2563
01:20:06,560 --> 01:20:07,750


2564
01:20:07,750 --> 01:20:10,870

 

2565
01:20:10,870 --> 01:20:12,310

 

2566
01:20:12,310 --> 01:20:15,669

 

2567
01:20:15,669 --> 01:20:15,679

 

2568
01:20:15,679 --> 01:20:16,830


2569
01:20:16,830 --> 01:20:19,270

 

2570
01:20:19,270 --> 01:20:20,950

 

2571
01:20:20,950 --> 01:20:20,960

 

2572
01:20:20,960 --> 01:20:23,110


2573
01:20:23,110 --> 01:20:25,830

 

2574
01:20:25,830 --> 01:20:25,840

 

2575
01:20:25,840 --> 01:20:26,950


2576
01:20:26,950 --> 01:20:26,960

 

2577
01:20:26,960 --> 01:20:28,629


2578
01:20:28,629 --> 01:20:31,189

 

2579
01:20:31,189 --> 01:20:33,270

 

2580
01:20:33,270 --> 01:20:35,590

 

2581
01:20:35,590 --> 01:20:37,030

 

2582
01:20:37,030 --> 01:20:38,709

 

2583
01:20:38,709 --> 01:20:39,669

 

2584
01:20:39,669 --> 01:20:41,270

 

2585
01:20:41,270 --> 01:20:43,590

 

2586
01:20:43,590 --> 01:20:44,709

 

2587
01:20:44,709 --> 01:20:47,430

 

2588
01:20:47,430 --> 01:20:49,350

 

2589
01:20:49,350 --> 01:20:52,070

 

2590
01:20:52,070 --> 01:20:52,080

 

2591
01:20:52,080 --> 01:20:53,350


2592
01:20:53,350 --> 01:20:56,070

 

2593
01:20:56,070 --> 01:20:58,550

 

2594
01:20:58,550 --> 01:21:01,910

 

2595
01:21:01,910 --> 01:21:01,920

 

2596
01:21:01,920 --> 01:21:03,270


2597
01:21:03,270 --> 01:21:03,280

 

2598
01:21:03,280 --> 01:21:04,709


2599
01:21:04,709 --> 01:21:06,310

 

2600
01:21:06,310 --> 01:21:08,310

 

2601
01:21:08,310 --> 01:21:09,669

 

2602
01:21:09,669 --> 01:21:11,590

 

2603
01:21:11,590 --> 01:21:13,430

 

2604
01:21:13,430 --> 01:21:15,669

 

2605
01:21:15,669 --> 01:21:18,229

 

2606
01:21:18,229 --> 01:21:18,239

 

2607
01:21:18,239 --> 01:21:19,189


2608
01:21:19,189 --> 01:21:20,709

 

2609
01:21:20,709 --> 01:21:23,669

 

2610
01:21:23,669 --> 01:21:26,629

 

2611
01:21:26,629 --> 01:21:26,639

 

2612
01:21:26,639 --> 01:21:27,590


2613
01:21:27,590 --> 01:21:29,830

 

2614
01:21:29,830 --> 01:21:31,830

 

2615
01:21:31,830 --> 01:21:33,350

 

2616
01:21:33,350 --> 01:21:37,110

 

2617
01:21:37,110 --> 01:21:39,270

 

2618
01:21:39,270 --> 01:21:40,950

 

2619
01:21:40,950 --> 01:21:43,030

 

2620
01:21:43,030 --> 01:21:45,110

 

2621
01:21:45,110 --> 01:21:47,350

 

2622
01:21:47,350 --> 01:21:48,870

 

2623
01:21:48,870 --> 01:21:51,350

 

2624
01:21:51,350 --> 01:21:54,390

 

2625
01:21:54,390 --> 01:21:56,870

 

2626
01:21:56,870 --> 01:21:58,550

 

2627
01:21:58,550 --> 01:21:58,560

 

2628
01:21:58,560 --> 01:22:04,790


2629
01:22:04,790 --> 01:22:08,629

 

2630
01:22:08,629 --> 01:22:11,430

 

2631
01:22:11,430 --> 01:22:14,870

 

2632
01:22:14,870 --> 01:22:16,629

 

2633
01:22:16,629 --> 01:22:18,470

 

2634
01:22:18,470 --> 01:22:20,950

 

2635
01:22:20,950 --> 01:22:22,550

 

2636
01:22:22,550 --> 01:22:24,629

 

2637
01:22:24,629 --> 01:22:25,669

 

2638
01:22:25,669 --> 01:22:27,830

 

2639
01:22:27,830 --> 01:22:30,070

 

2640
01:22:30,070 --> 01:22:30,080

 

2641
01:22:30,080 --> 01:22:32,550


2642
01:22:32,550 --> 01:22:34,629

 

2643
01:22:34,629 --> 01:22:49,510

 

2644
01:22:49,510 --> 01:22:49,520

 

2645
01:22:49,520 --> 01:22:51,600