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hi everybody  i hope you can hear me  uh give me a thumbs up if you can good i  see you guys let's see i see them coming  through  um  so we're gonna get started um  um  uh okay so getting back to what we've  been doing um you know we we've been  talking about space complexity  measures how much memory the algorithm  requires or various problems require  and  we uh defined the  uh space complexity classes uh space f  of n and non-deterministic space f of n  uh the polynomial space and  non-determining space non-deterministic  space classes  and gave some examples and so on and  today we're going to pick up where we  left off last time  one of the examples which is going to be  an important one for us is concerns this  latter dfa problem so i'm going to go  over that again give a little bit more  emphasis to the  uh  space analysis uh which i got some  questions about last time  and then we're going to move on from  there and prove savage's theorem and  then talk about complete problems for p  space um it showed that this problem  tqbf which we introduced last time is  actually a piece based complete problem  but  all in due course um  a little bit of a review so we defined  we defined what we mean by a turing  machine to run in a certain amount of  space  that means it uses at most f of n if  it's running in space f of n uses at  most f of n cells tape cells  on every input of length n and  similarly a non-deterministic turing  machine does the same  but what's  in addition to that the  non-deterministic machine has to halt on  every branch of its computation  and each branch of its computation has  to use at most that bound that bounded  amount of tape cells  um  so we're going to be talking about  non-deterministic space computation  today as well  it's going to be  relevant to us  um  so we defined the classes as i mentioned  um  and the polynomial and uh  the p space and non-deterministic piece  based classes and this is how they uh  see how we believe they relate to one  another  um the classes co np and np as well  and of course  uh as i mentioned last time  there are some very  major unsolved problems in this area so  everything could conceivably collapse  down to p  which would be of course very surprising  but we don't know how to prove otherwise  um  and the big theorem that we're going to  prove today is that polynomial space and  non-deterministic polynomial space  actually do collapse  down to each other  and  being the same class  so in contrast with the situation that  we believe to be the case for time  complexity where we believe the  non-deterministic converting  non-deterministic to deterministic gives  an exponential increase  uh for space complexity it only gives a  squaring income  increase as we'll see um  so uh  any  questions uh on any of this um  or we will just march into a little  review of this latter problem  um  so  re reviewing  some of the notation and and let me  emphasize that so the big theorem we're  going to be proving today is that p  space and n p space are equal um  and also we're going to be talking about  uh  piece based completeness  but uh  both of those involve proving theorems  um in the in the first case uh savage's  theorem that converting  non-deterministic to deterministic  spaces of squaring and in the second  case proving that tqbf is p-space  complete  both of those theorems can be thought of  as generalizations  of the of of um  this  um  theorem here that the latter dfa problem  can be done in in  the deterministic uh polynomial space or  n squared space  um  so  it really pays to try to understand  how the proof of this theorem works  because in a sense this theorem is a  more concrete version of what we're  going to be seeing in those other two  theorems in a somewhat more abstract  form okay so  um i like understanding things in a more  concrete way first so that's why this is  a good example to start out with but  really in the end of the day it's the  same proof just repeated for those three  theorems so this is a really three pre  three for the price of one thief three  theorems one proof here so you're going  to be seeing the same proof  repeated three times  but in different levels of abstraction  okay  so  let's let's review again and  i know some of you got it but um  maybe some of you didn't and uh let's  just try to uh  be clear  on the algorithm  to solve the latter dfa problem so if  you remember first of all let me just  jump on ahead the latter problem is your  give a ladder first of all is a sequence  of strings  that change one symbol at a time  that perhaps connect go from one string  to another so you're going to go from  work to play  changing one symbol at a time so we gave  an example of this or you can easily  come up with an example of doing this  but uh  the computational problem is can you do  it can you get from this string to that  string and stay within a certain  language  so it might be the language of english  words or it might be the language of all  strings that some specific dfa  um uh  um  recognize um  uh recognizes so  it might be all of the string these  might all be strings that some dfa  accepts or you know  might be english words or some other  rule um  and so  uh  uh that's what we mean by um  trying to test if there's a ladder  um and so  uh the the latter problem is um well i  didn't i don't think i wrote down on the  lateral problem itself but the bounded  ladder problem is basically the same  idea you're given a dfa you're given the  strings u and v  and now  this is the bounded version of the  problem where i'm going to give you a  limited limit on the number of steps you  can take so i'm illustrating that here  so you're going to be given a b and you  want to say can i get from this string  to that string within b steps  okay  and we had a notation for writing that  going from u to v  if there is uh a ladder that connects u  to v within at most b steps  and so the bounded ladder problem which  i'm introducing because i'm going to be  aiming toward a recursive algorithm to  solve this problem  is can i get from  u to v by a ladder changing one symbol  at a time where each string along the  way is accepted by b  and only allowed b steps little b steps  okay  um  so that is the computational problem  that i'm going to be solving with the  algorithm that i'm going to describe um  so the algorithm i'm going to call bl  for bounded ladder problem um  and  here is the input  and uh the algorithm is first of all  going to look to see if b equals one if  i'm just trying to get  from u to v in a single step  in that case  it's a very simple problem because you  want to test obviously that you and v  are in the you know are accepted by the  automaton  um and they just have to differ in one  place and then you have a very simple  one-step ladder that takes u to v okay  so for the case b equals one it's very  simple  uh  for larger values of b we're going to  use we're going to solve the problem  recursively in terms of small smaller  values of b um  so  for b greater than 1  we're going to recursively test  you know if you're trying to solve the  problem can i get from u to v  instead we're going to try each possible  uh halfway through  we don't know that it's halfway through  so we're just going to try each possible  string  and we're going to test  can we get from u  the initial string  to that new string that  w in half the number of steps and can i  get to the final string v in half the  number of steps if i can do that  then i can get get from u to v  the total number of steps b  so i'm just going to try to do this  one w at a time for every possible w  this is going to be very expensive in  terms of time  but we're not worried about time right  now we're trying to cut down on the  amount of space that we're using and  this is going to be a big savings in  space  um  let's not worry about the division b  over 2 here all of the divisions and  we're going to be seeing this several  times going forward in the in the  lecture  we'll think of them rounding up and i'm  not going to cover up  make the notation look cumbersome by by  writing that every time  um okay so here we go here is  some candidate w string which is half  halfway through  um recursively test can i get from the  starting string to that w and from w to  that ending string  um  if i can if i find such a w then i  accept  and if i try all possible w  and i  never  manage to find a way  um to make both a top and the bottom  work then i know i cannot get from the  you know from the starting string to the  ending string within b steps and so i  reject okay and now i'm going to solve  the original unbounded  ladder problem by simply putting the  biggest possible bound into the bounded  ladder problem  and that's his value t  which is  gives the very trivial bound of the the  total number of possible strings that i  can write down  within my length m that i'm working with  so this is if sigma is the alphabet of  these strings  it's just sigma to the m that's all  possible strings of course that's going  to be a maximum size on the ladder um  uh  okay uh so now how much space does this  take and i think this is where people  got a little bit um  uh lost uh in  in the lecture last time so i'm gonna  try to animate this um i don't know if  that's gonna help or not but you're in  the end of the day you just have to kind  of have to think through how the re  how do you account for the cost of this  recursion  um  okay but the main thing  to start off you have to make sure you  understand the algorithm you know to get  from here to there we're going to try  all possible  midpoints and then  of the upper part and the lower part  recursively re using the space that's  the critical that's the way we're going  to get a saving by  solving this problem reusing the space  that we use to solve this problem  okay so  i'm going to i'm going to try to show  this to you on actually how the space  gets used on the turing machine you can  kind of think of here's the input and  then after that is going to be the stack  for the recursion if you're not that  familiar with how to implement recursion  doesn't really matter um but you can  just  think about what the algorithm needs to  keep track of  um  and so  as it's trying every possible w so you  know  just in order like an odometer um  just trying every possible  every possible string eventually maybe  it finds a string that's in the language  um that hears an english word one of the  first english words of length four that  you might run into um  and so now it makes sense actually to do  the recursion um  so that's all  every time you you're going to have to  have a register or a location on the  tape where you're going to be writing  down those different w's um so let's say  it's over here and and we're just going  to go through i hope that's not too  small for you to see  you know  that's really where that action is  happening um and finally maybe you got  to the string w now you're going to try  to do the recursion so here is um you  know as you recur  doing the recursion on the top half  again you're going to be cutting you're  going to be finding a new w  for the the intermediate uh  point just solving this uppers upper  problem where we're testing if i can get  from work to able later i'm going to  have to deal with getting from able to  play  um  good uh so  here again we're going to be trying  fixing able fixing that but the first w  we're going to try every possible way of  getting from work you know from the from  the start string to that to that middle  string um so we're going to try every  possible thing here eventually maybe we  find some string um some other string in  the language we get down to the string  book  um  and uh that's all going to get stored  you have to get you can't forget this  the string able but now we're going to  use more space to store those candidates  so that's a second version of w um  deeper in the recursion so here we're  going to be triangle  possible strings here again eventually  we get to some string book  um  and you know if that succeeds in getting  us  from  uh work to able via book now we're gonna  jump down to do the bottom half um to  see if i can get from able to play  um as a separate problem which gets  solved  in the same space so now um here we're  going to try all these possible  possibilities getting from able to play  maybe call is the right intermediate  string there  and so now  we're going to erase the book  and now we're going to solve the lower  sub problem in the same location  i hope this is helpful this is a lot of  work making these animations  um  and so  i hope so the point of all this is every  time  time we go down the level of the  recursion there's another register whose  size is big enough to hold one of one of  the strings  is needed  and that register gets reused um  uh  times as we're  um at uh  uh you know throughout as we're going  through through this recursion um  so anyway i hope that's helpful um  uh  anyway so each level of the recursion  adds another order  n  to record the w and so you have to how  many levels do we get  well the depth of the recursion is going  to be  how many times we end up having to  divide this picture in half to get until  we get down to one  and so the height of this  you know when we start off  is going to be um basically an  exponential  in m m is roughly the size of n  so when you take the log of that  you're going to get  um  you're going to  pull down the exponential so it's going  to be order m or which is again roughly  the same size as the input m is like  half the input because the whole input  is u and v m is just the size of um  and so  each level requires order n and the  depth is going to be order and deep  you know the log of the initial height  of this  ladder  and so the total space used is going to  be order n squared  okay so why don't we just take a minute  i'm happy to spend a little time going  through this either the algorithm's  correctness  um understanding the recursion or  understanding the space analysis if  there's any questions that you feel you  can ask that would be clarifying for you  um  jump in we'll we'll we'll i'll i'll set  aside a few minutes just to answer  questions here  um  so i've got a question here in step five  why do we reject  if all fail  instead of just one fails  um  well here  so remember what we're trying to do  we're trying to say can i get from u to  v in b steps  the way i'm going to be doing that  is trying every possible intermediate  string w  if i find some w  which does not work  that doesn't mean that there's not some  other w which might work all i need is  one w  for which i can get from u to w and half  the number of steps and w to v and half  the number of steps so i'm going to try  every possible w if any one of them is  good  then i can accept if any one of them  succeeds where i can get from u to w and  half the steps and from w to v and half  the steps  then i know i can get from u to v in in  the full number of steps  um so  i only need to find one  if i if one particular one doesn't work  i'll just go on to the next one  um  i okay here's a this is a good question  um  uh do we have to save the word book  so once we succeed in getting from work  to able let's say via book  do we need to save that word book  anywhere no  all we need to remember is that we've  succeeded in getting from work to able  we don't need to remember book anymore  we just  you know  remember that we've succeeded and that  is by virtue of where we are in the  algorithm so we but you know if we have  succeeded then we move on to the second  uh recursion second call recursive call  um so we found some way to do it so we  found some um  some intermediate point which succeeds  for this one so we move on to that one  we don't have to keep any of that work  anymore all you have to do is remember  yes i can get from work to able and half  the number of steps now i need what all  that's left is to get from able to play  in half the number of steps  i don't doesn't matter how i got to abel  in the first place so we don't have to  remember that  that was a good question though um  so i i think i i understand this to  before we replace  um the value for book  with call  the with the work involved to find coal  um  [Music]  yeah we have to check that we can get  from work to book and book to able  so we hang keep on to book while we're  working on the upper half  and only when we've finally succeeded in  getting from work to able let's say via  book then you can throw a book away  um  but while you're working on the upper  half you try book you try different you  know different strings of length four um  until one of them works  um  so  okay i'm not sure somebody's asking me  about breadth first search and depth  first search i'm not sure i see it um  this  i don't i'm not sure that's going to be  a helpful way of thinking about this so  i'm not going to answer that right now  but um  you know you can ask that offline later  if you want  why is the recursion  why is the recursion depth  log t instead of log m  okay well how high is this thing  initially  um  it's t high  but every time  we're doing a level we're calling the  recursion we're cutting t in half  first it's you know if you  you know i'm solving this in general for  b but we're starting off with b equal to  t t is the maximum size  so  initially this is going to be t and then  it's going to be t over 2 then t over 4.  so it's going to be log t levels before  we get down to 1.  um  yeah so somebody's asking can we think  of this as a memory stack yes this is  like that's where your typical  implementation of recursion is kind of  with a stack where you push when you  make a call and you pop when you return  from the call  um  is it possible that v can appear during  bl procedure on t  is it possible that v can appear  um i'm not sure what that means it can v  reappear  you know so i'm starting with u to v  is it possible that v might be one of  these intermediate strings yeah you know  um  uh this you're going to try every  possible intermediate stream  you know blindly including v is one of  them um  uh you know if you can reach v  more quickly  well great um  uh i guess what i have not dealt with  the issue of what happens if you  get to a  um  uh  yeah technically it's it's going to work  out because i'm allowing  you know the difference to be in at most  one place so even if you get there early  you're allowed to  not change anything and that still is a  legal step in the latter um  uh  yeah i don't see how to do this from a  bottom-up perspective somebody's asking  is there a bottom-up version of this  i don't  think so uh  but no i don't think so  all right why don't we move on  um  so now we're going to see this proof  again  but but this time we're going to be  proving  [Music]  that you can convert any  um nfa to a dfa with only a squaring  increase  so really  uh  well  let me just put that up put put that up  there um  uh so this is going to be savage's  theory that  among other things proves that p space  equals np space  uh so it says that  you can convert a non-deterministic  machine to a deterministic machine only  squaring the amount of space so you're  comfortable with this notation here  anything that you can do in f of n space  non-deterministically  you can do it f squared of n space  deterministically um  and we're going to accomplish that by  converting a  you know an ntm to a deterministic tm  but only squaring these space u's so n  is going to get converted to m  and now this proof is going to look very  similar to the proof from the previous  slide  um  it's the same proof  and um you know and and the fact from  the previous slide you know about ladder  really is implied  by this because you know we gave we had  an easy algorithm to show that the  latter problem is solvable in in  non-deterministic you know an np space  so ladder problem was easily shown to be  in here  um  if you remember you just guess  the  you basically guess the steps of the  ladder so non-deterministically you can  easily check can i get from the start to  the end  but  savage's theorem tells us that anything  you can do non-deterministically in  polynomial space you can do  deterministically in polynomial space so  what we showed in the previous slide  follows from this slide but this slide  is really just a generalization of the  same proof  okay  maybe i've said it too many times now um  so  we're going to  introduce a notation very similar to the  notation we had last time but now we're  going to be talking about  simulating this non-deterministic  machine with a deterministic machine  and we're going to have take two  configurations of this non-deterministic  machine c i and cj  and say  can i get from ci to cj in at most b  steps  i'm going to have a notation  very similar to like the notation for  the latter where i where i can get from  this word to that word in most p b steps  by a ladder here can i get from this  word to that if can i get from this  configuration to that configuration with  at most b steps of the turing machine's  operation  so these are two configurations now  um  of n  so can n go from this configuration ci  to that other configuration cj  but only taking b steps along the way  that's now the computational problem  that i'm going to solve for you with an  algorithm and it's going to be a  recursion exactly like the previous one  um  m gets its input the two configurations  c i and c j and one and the bound b and  want to check can i get from i to j  within b  okay so now the picture's a little  different um  but it's very similar so instead of a  ladder appearing here  it's really something that's basically a  tableau  for  um  for the machine n where i have a initial  configuration and an ending  configuration this would be hap this  would happen to be  um  uh  the starting point for the whole  procedure if you're testing whether n  accepts w but we would be solving this  in general for any config you know so in  that case i have the start configuration  of n on w and the accepting  configuration or an accepting  configuration but  in general what i  will be  solving is starting with any  configuration ci and going to any  configuration cj  so i want to  test can i get from ci to cj within at  most b steps  um so first of all if b is one  you can just check that directly um  you know and now again this is we're  operating deterministically to simulate  the non-deterministic machine  so this is the deterministic machine m  so m can easily if it's given two  configurations of n  and says can i get from the first one to  the second one in one step well that's  an easy check you just lay out those two  configurations  look at the ends transition function and  say is this a legal move for n  yes or no and you accept or reject  accordingly  now if b is larger than one  you're going to try all possible  intermediate  configurations  i'm calling them c mid this was like the  w from the previous  uh theorem  this is all possible strings c all  possible configuration c mid  and you know i don't you know a  configuration is just going to be a um  uh  so far so okay  this is all possible  look like my powerpoint correct that  seems okay um  [Music]  this is all possible configurations  which is just a string with a string of  tape symbols with a state symbol  appearing somewhere that's all it is um  you're going to try all possible  configurations as  candidate middle configurations and say  can i get from the upper one to the  middle the to this candidate middle one  and from that middle one to the to the  lower one within half the number of  steps each time  so um  and solving that problem recursively  um  okay  so i got a question here about the  possibility of looping forever  first of all if n is going to be looping  i don't have it i don't have to worry  about it because  um i'm starting off  i'm only i need to simulate machines  that always hold that are deciders  because i'm trying to show that any  language in here has to be accepted has  to be  uh has to be decided by some some  non-deterministic machine  so i'm not going to worry about machines  that are looping if they're looping  you know um  uh m may be misbehave in some way but  that's not going to be a problem for me  um so let's only let's keep life simple  think about the machine deciders only  um  okay so we're going to recursively test  here so that means i have a going to try  every possible middle see if i can get  from the  the start to the middle and the middle  to the end  um  uh if both of them work  for my after my i test them recursively  then i'm going to accept if not i'm  going to continue  and i reject if i try them all and none  of them have worked then i know there's  no way for n to get from this  configuration to that configuration in b  steps  okay  [Music]  and the overall picture i test whether n  accepts w as i mentioned by starting the  c  i is the start configuration and cj is  in the accept configuration  um  and now how big is t  because i need to calculate  a bound  on how  many how deep the  the the recursions are going to be um so  t  here is going to be the total number of  possible configurations um you know if  this is holding it never repeats the  configuration so this is going to be a  bound on how many steps n can be taken  and that's simply you know we calculated  this before it's the number of um  states times the number of head  positions times the number of tape  contents  tape contents and this is really going  to be the dominant  consideration anyway  and so  now each  recursive level and maybe i should have  emphasized this at the beginning how  wide is this picture it's big enough to  store a configuration  configuration is essentially a tape  contents  so that's going to be f of ny  so each recursive level stores one  configuration  now the w  costs f n space to write down  and the number of levels is going to be  the log  of the initial height  which is this so this is going to be the  dominating uh  part of it so the log of this is going  to be again order f of n  so this each one takes f of n space to  write down the depth of the recursion is  going to be order f of n so the total is  going to be order f squared of n and  that's how much space this uses  and that's the proof of savage is there  so  yeah so this is a good point there uh i  was thinking somebody asked can there be  multiple accepting configurations i  should have made the  i forgot to say this but and i was just  realizing it as i was explaining it one  of the things i should have you you can  enforce that there is just a single  accepting configuration  this is kind of a detail but so don't  worry about it if you  if you don't if you don't want to but  you can make sure that there's a single  accepting configuration by telling the  machine when it accepts it should erase  its tape and move its head into the  leftmost cell in the accept  configuration so there's just going to  be a single accepting configuration to  worry about instead of having to try  multiple  uh  possibilities which you could do in this  algorithm but it would just be annoying  to have to write that down that way so  we often assume there's just going to be  a single accepting configuration for  these machines  um  um  okay so this is  how do you know f of n um  so that's a that's actually a little bit  of a delicate issue uh  i mean if you could compute f of n  um the bound  which is for example if it's going to be  a polynomial bound you can just compute  f of n it's very easy to compute n  squared or n cubed  and so you can just  compute that and then  use that as  the size of the registers you're going  to be writing down  for  if you want to prove this in general for  f of n it's a little bit technical to  have to deal with it  and i'm going to have to refer you to  the book on that one the book tells you  how to do solve this for a general f of  n you basically have to  try every possible value from one until  it works um  and  i'm afraid that's going to  derail us trying to un decipher that so  let's not worry about that aspect of it  but it's not it's a it's a you can  handle general f of n here um you don't  need to put any conditions on f of n um  can we go over this term here  so we've seen this term once before when  we talked about lbas and and seeing that  lbas always  we can solve the alba problem this is  simply the number of different  configurations the machine can have  because the configuration is a state  it's a head position and a contents of  the tape  and this is the number of each of those  that that you can possibly have number  of states  a head position  you know the size of the tape of f of n  is f of n so this is that many different  head positions and this is the number of  d if d is the size of the tape alphabet  this is the number of contents tape  contents that you can have  um  okay  how is seeing if n accepts w with this  algorithm  convert non-deterministic turing machine  to some deterministic turing machine  well n is the non-deterministic term so  n is  we're converting non-deterministic  turing machine n  to  deterministic turing machine m  so m is the deterministic simulator of n  that's what this whole m is  so if we can  do this for  you know for any end that we've proved  our theorem  um  okay why don't we defer i think you know  we're  i think i got to most of the questions  here if there's other things we can save  them for the um  coffee break which is coming soon i  think we have one more slide before that  um  okay so i'm going to define p space  completeness i think yeah and then i  think i think after that we have the uh  the break  so peace-based completeness is defined  and in very much inspired  uh similarly to np completeness  um  so  a problem is p space complete if it's in  p space  and every other member of p space is  reducible to it  in polynomial time  and we'll say a bit about why we choose  polynomial time  reducibility here um so here's a kind of  a picture of how  p space or have complete problems  of  you know uh  relate to  relate to com their complexity classes  so  you kind of think of a complete problem  for a class it's kind of the hardest  problem in that class because you can  convert you can reduce any other problem  in that class to the complete problem  so here are the np complete problems  sort of the hardest for np here the p  space complete problems kind of the  hardest for piece p space  if any if an  np complete problem goes into p that  pulls down all of n p to p if any piece  based complete problem goes into p it  pulls down all the p space into p by  following the chain of reductions um  because any piece based problem is  reducible to the complete problem which  in turn if it's in p  then everything goes into p so  if  you have a piece based complete problem  which is in p  then all the piece base goes into p  um so why do we use  polynomial time reducibility and say  instead of say polynomial space  reducibility  when we define this notion  it's a kind of a very reasonable  question  but if you think about it using  polynomial space reducibility would be a  terrible idea  um  because  and we've seen this phenomenon happen  before  every two problems in p space  are going to be p space reducible to one  another we haven't even defined that  notion yet but you can imagine what it  would be  because  a piece-based reduction can solve the  problem  um for a problem in p space and then it  can direct its answer anywhere that it  likes so in general when we think about  reductions  the reduction should be not capable of  solving the problems in the class  because if they could  then every two problems in the class  would be reducible to one another  and then all problems in the class would  be complete because everything would be  in the class would be reducible to any  one of the other problems  so it would not be an interesting notion  what you want to have happen is that the  reductions  should be  weaker  than the power of the class  and so um  uh  every  um you know  and if you if you look at the reductions  that we've defined so far they're  actually very simple  you know the only thing is yeah they  have to make sure that they can make it  that  they can make the output uh big enough  but actually constructing the output  they would do very simple  transformations in fact even polynomial  time is more than you typically need  there's even much more limited classes  that are capable of doing the reductions  as we'll see  um so having powerful reductions is  really not in the spirit of what  reductions are all about you want very  very simple  transform transformations um to be the  reductions anyway i hope that's helpful  so what what we're going to be aiming  for in the second part of the lecture is  showing that tqbf is based complete um  and let me give ahead as a check in  uh  uh  so this is our first check-in um coming  a little late in the lecture  suppose  we have proven that as we will that tqbf  is piece based complete  uh what can we cl conclude if tqbf is  actually  not necessarily in p only goes to np  that's and this is irrelevant to a  question that's coming in from the chat  but i'll answer that later so um suppose  tqbf ends up being in np and not in p  what what can we conclude remember if  if t  is in p  then p space equals p  suppose it goes to np  what happens then  check  you know there might be several uh  correct answers here  to check all that all that apply  all right so we're near the end  of the  pull so let me give you another  10 seconds and then we're gonna shut it  down  um  okay we all are we all in  closing it down  um  here are the results so  yes so uh first of all the most  reasonable solution the most reasonable  answer is b  uh which i think  uh  most of you have gotten that if  you know if a piece based complete  problem goes down to np well np is  capable of simulating the polynomial  time reduction  um and so  um  any other problem in p space would then  also be an np and p space would equal np  but note  if p space equals np they're also np  equals co np  um  because p space itself is closed on you  know is closed under complementation  so that was kind of a  little bit extra  fact that you could conclude from this  as well  um so let me let's move on then to our  uh coffee break  and we'll pick up the proof that tqbf is  piece based complete  after after that  okay  so  so was d true or not um  d was p equal np no d we cannot  conclude that p equals np  um if  um  from p space uh equal to np  um  so if tqbf is an np  it doesn't tell us anything it's [ __ ]  for all we know p equals np but uh from  the stuff that we've that we know so far  we cannot conclude that p equals np  um  oh and yeah so you can conclude oh i'm  sorry you you can con b and d  are both  correct here  um  let me just shut this thing off uh  so b and d are correct so if if a piece  based complete problem goes to np  then np equals p space and equals coin p  so the correct answer of b and d  sorry i i got myself confused so c but c  is not something you can conclude  or a  okay uh  so somebody said  somebody's asking me a fair question  you know  i say the reduction method should be  weaker than the class  uh  but you know for example even in the  case of p space p space might be equal  to p  and then um it wouldn't be weaker than  the class if we use polynomial time  reductions  but  um i think maybe i should say apparently  weaker  as far as we know it's weaker  but we believe it to be weaker  if it's true if p equals p space then  every problem in p is going to be p  space complete  it's just going to be a weird world if p  equals p space similar same thing for n  p and np  um  so so i'm getting a number of questions  also about other possible reducibilities  that are even weaker than polynomial  time reducibility um  so we're going to see uh um  very soon weaker reducing  you know complexity classes within p  so first of all p space seems to be  bigger than p we're also going to look  at log space but that's going to be um  actually in thursday's lecture those are  these are called classes  ins that seem to be inside well they're  inside p they may be proper we believe  they're properly inside p but we'll  we'll see that later  um  let me just see here  really should have so we're almost out  of time  uh let me put our  timer back in fact our timer is showing  us out of time  so um why don't we um  get going let me  move this um  back to  okay  continuing  t q d f is p space complete so first of  all let's remember t q b f  um  [Music]  uh these are all of the  quantified boolean formulas that are  true so t could be of stands for true  quantified boolean formulas  and remember we saw these examples  from the previous lecture  that you know here these are two  quantified build these are two qbfs uh  the first one is true the second one is  false and and  it's kind of interesting to think about  you know here they're exactly the same  except for the order of the quantifiers  and so what's really going on you know  here  uh just i think it's  good to understand these expressions  they come up everywhere in mathematics  uh these quantum  quantifiers um  you know in the upper one  uh when we say for every x there exists  a y  that y can depend  on the choice of x  you choose it make a different x you're  allowed to pick a different y  but this but the the lower expression  says there's a universal y there's some  one particular y that works for every x  so in a sense um  uh the lower statement is a stronger  statement  um you know whatever you have in the  interior in  the quantifier free part uh  so the lower one implies the upper one  happens that the lower one in this case  is false and the upper one is true but  in general the lower  in the you know when you have this uh  change of quantifiers like this the the  um  the lower one would imply the upper one  okay anyway that's sort of a side remark  so um let's get back to the proof that  tqbf is piece based complete that's what  our goal is  all right um  now as i mentioned  this is the same proof  you're going to be seeing it for the  third time today but  you know there's a certain amount of  it's sort of  the context changes  uh in each case  so now we want to show that tqbf is  p-space completed so it's one of these  hardest problems now but for p space  um  where satisfiability was like a hardest  problem for np  so we want to show  that every  language in p space is reducible to tqbf  um and so we're going to give  polynomial time reductions that map  some particular problem a which can be  done in space n to the k it's a  problem solvable in p space  we're going to show how um  f maps a to 2 q b f we have to construct  the f  so f is going to be a mapping that maps  strings which may or may not be an a so  strings w which may or may not be an a  to  these formulas these quantified formulas  um  so  uh  w is going to get mapped to  some formula fee sub mw it had exactly  the same  even symbols we used when the proof  proof of the cook 11th era about sat  being np-complete this is a very similar  proof  but you'll see that we have to do  something more in order to make it work  in this case  so if w  you know w is an a if and only if this  formula is going to be in tqbf in other  words if and only if this formula is  true  um  so  this formula  is going to kind of express the fact  that m accepts w  which means that w is an a because m is  the machine is the p space machine for a  so this uh formula says m accepts w  and it achieves that by  building in a simulation  of m on w  it kind of describes a simulation for m  w which ends up accepting  and if m does not accept w  that description is going to inevitably  be false  okay so let's just see how that's what  that's going to look like  so we're going to use the same idea that  we initial that we used for the cook 11  theorem that sat is np-complete this  notion of a tableau  so if you remember that we had it was  basically a table  which weighs was just simply a way of  writing down  an  a computation history for mw  okay so the rows are the steps of the  machine  the top row is the start configuration  the bottom row is let's say  some particular  accept configuration such as i just  described where the machine  clears its tapes and moves its head to  the left end so there's only one  accepting configuration we have to worry  about  okay and each of the rows here is a  configuration of m of m  okay  um  because m runs in space n to the k the  tableau kind of similarly to what i  described before has width into the k  so now you know  we're talking about polynomial time  machines  so the f which is the bound the space  bound is going to be some polynomial n  to the k  so the width of this  tableau the size of these configurations  are going to be n to the k  how  high is this tableau going to be  well that's going to be  limited by the possible running time of  the machine  which is  similar to what we saw before  it's going to be exponential in the  space bound  so it's going to be d to the n to the k  where d is the  essentially the tape alphabet of the  machine  okay so  are we all together on this  this is um very similar to the proof of  sat is mp complete  the  key difference there was m was  non-deterministic  um  which might be something to think about  later but  let's not focus on that right now here  this m is deterministic  and um  but the important difference was the  shape of the table the the the the size  of the tableau was very different in the  case of sat is np-complete we started  off with a polynomial time  non-deterministic machine so it only  could run for a polynomial number of  steps here is a polynomial space machine  which can run for an exponential number  of steps  that's a  that's going to be  an important difference here so let's  let's see why  uh  the reduction  has to construct  this  formula  fee sub mw  which  basically says that this tableau exists  now we already saw how to do that  when we  proved the cook 11 theorem that status  empty complete  remember we had all of those we had  variables for each cell  that told us what the contents of that  cell were was  and then we had a lot of logic here we  had a bunch of logic that said that  those  all the you know those neighborhoods  were correct which basically says that  the the tableau corresponds to a correct  uh computation of the machine  um so why don't we just do the same  thing  okay why don't we just  build our formula  using exactly the same process that we  used to build the formula when we had  sat being  np-complete  something goes wrong  we can't quite do that  the problem is that if you remember the  formula that we built before was really  about as big as the tableau is  because it had some logic for each one  of the cells it had a set of some of the  had variables for each one of the cells  and had some logic you know for each of  those neighborhoods basically so he says  that each of the cells  does the right thing  um  so it was a pretty big formula but it  was still polynomial  the problem is that tableau  is now into the k by  d to the n to the k  that's an exponentially big object  so if your formula is going to be as big  as the tableau  there's no way you can hope to produce  that formula in polynomial time  and that's the problem  the formula is going to be too big  remember we're trying to get a  polynomial time reduction  from this language a  so we have an input to a  you know a string that might be an a  which is  we're simulating the machine  and then you know  the size of the tab below relative to w  is going to be something enormous  and so if the formula is as big as the  tableau there's no way to produce that  formula in polynomial time so this  is not going to work  let's try again  so now we have here  uh  now so remember this notation from ci to  cj in b steps  okay so we're going to give a general  way  of constructing formulas  which  express this fact that i can get from  uh  configuration i of m to configuration j  of m in b steps  whatever that b is b is going to be some  bound and i want to know  can i get from this configuration to  that configuration and i want to write  that down as a formula which is going to  express that fact and it'll be either  true or false  and i'm going to  give you a recursive construction for  this formula so i'm going to build that  formula  for a value b  out of formulas  uh for smaller values of b  okay so this is going to be a way of  constructing that formula in terms of  other formulas that i'm going to build  and there's going to be a basis for the  recursion when b equals one  okay so that's the big picture so let's  see how does this formula look  um  so let's  let's not worry about the case for b  equal one right now this is the case for  larger values of b  so i'm gonna the fact of that i can get  from c i to c j within b steps  i'm gonna write this down in the in this  way  uh and let's try to  uh  unpack that and see what it's saying  um  you know without worrying about  how we gonna carry this out let's just  try to understand at a high level the  semantics of this thing what he's trying  to say to you  it's gonna say well i can get from ci to  cj in b steps  so m  can get from c i to c j and b steps  if there is some  other configuration c mid  i'm calling some other configure i'm  calling it c mid  very much inspired by  the previous proof  of savage's theorem uh where there was c  mid was that intermediate configuration  so now instead of trying them all i'm  saying does there exist one  where i can get from c i to c mid and  half the number of steps and from c mid  to c j in half the number of steps  so if i can build these two config these  two formulas  then i can combine them  with this sort of extra stuff out here  and then and them together and put an  exist quantifier that says does there  exist some configuration some way to  find a configuration such that it works  as these formulas require  if i can do that  then i'm going to be good because then i  can  um  well good at least i'll be good in terms  of making something that is going to  work um  so first of all  let's understand what i mean by writing  down uh does there exist c mid it's  really  if you you know thinking back to the way  we did the koch 11 theorem we  represented these configurations by  variables which  were indicator variables for each of the  cells and we're going to do exactly the  same thing so we're going to have a  bunch of boolean variables which are  going to represent  some configuration  so  more more  you know formally or in more detail  what does does there exist a c minute  really is an assignment to all of those  variables that represent um the cells of  the contents of the cells of those  uh of that  configuration okay so now  let's see how to  you know how does how will the recursion  work  so um  to get this value here i'm going to  do the recursion further so does there  exist a c mid and now  um  for  getting from ci to this c mid can is  there some other c mid this is like  another value of w from the previous  slide where i'm getting again i'm  cutting the number of steps in half  again so i'm going from p over two steps  to b over four steps  and i'm going to do the same thing over  here  so i'm just uh unraveling  the  uh  the construction of this formula  in terms of um  building built by building it up  recursively  so  um  and then i'm just going to keep doing  that until i get down to the case where  b equals one  and when if i'm now trying to make a  formula  that's going to be say i can get from ci  to cj in just a single step  so this is really  talking about a tableau of height 1 or  high 2.  then i can just  directly write that down the way i now  the tableau is not very big so now i can  write that down using the neighborhoods  and so on that i did  in the cook leaven theorem proof  and this is how you put it all together  and now if you want to talk about the uh  getting  from  does does m accept w  so i initially say can i get from the  start configuration to the accept  configuration in t steps which is the  maximum running time of the machine so  again  you know if you followed me what  happened  um in savage's theorem it's the same  it's the same uh values  uh  now the thing is is we have to  understand how big this formula is and  if you think it through  if you think it through  um  there's a problem  because what's going on here  i'm expressing this formula  in terms of formulas  where um  the size of b is cut in half  but now there are two of them  so it's two formulas on half the value  of b  that's not going to be a recursion uh  that's going to work in our favor  um so let's just see what happens  so each recursive level uh  doubles the number of formulas  right going here we have two formulas  here we're going to have four formulas  and and so on so the number of formulas  doubles each time so the length of the  thing um that we're writing down is  doubling in size  each time we go to down the recursion  that's going to be okay if we don't go  too many levels but unfortunately we are  going quite a few levels because the  number of levels is going to be log of  this initial  um exponential size thing so it's going  to be n to the k levels and so if you're  doubling something n to the k times  you're going to end up with an  exponentially sized formula and  again  we failed  okay so let's i have a check in on this  but i'd like maybe we should spend a  couple of minutes just making trying to  understand what's going on here  because the next slide is my really my  last slide  and it's going to fix this problem  but let's make sure we understand what  the problem is before we try to fix it  um  why can we no longer write over each  layer of the recursion as we did in  ladder oh that's kind of a cool question  what does it even mean to write over the  different well  you know  uh  so that that's kind of an interesting  question here so what in a sense that's  going to be the solution  we're going to  uh  um  you know reuse things in a certain way  but but for but  i want you to understand that this  method  itself  does not work  because this recursion here where i'm  writing the formula i'm building the  formula for b out of  formulas for smaller values of b um  if i if i  do it that way  i'm going to end up  if i do it as it's described in this  procedure here i'm going to end up with  an exponentially big formula  and that's not  good enough  so if you cut the formula in half each  time even though you have two formulas  won't the length be the same  um  i'm not cutting the formula in here from  cutting the value of b in half  so you have to say b is initially an  exponentially big value  so we're going to end up with an  exponentially big formula  so it's not really cutting the formula  in half cutting the b in half and b it  starts out big  you know  i mean that b is this b is initially  this value here d to the n to the k  i'm worried not too many questions here  i feeling that's  that's that's probably not a good sign  uh  so if you're if you're if you're  well i mean if you're hopelessly  confused maybe i can't fix it um quickly  so anyway why don't we move on  and see how to  repair  this how to fix this problem um  and that is going to be by a trick  um  in fact i know  the people who were involved with coming  up with this this was actually this  proof was done originally at mit  um  and uh many years ago in the 1970s  and uh the folks who were involved with  it called it the abbreviation trick so  that's what we're going to do on the  next slide  oh no this is a check in first  why shouldn't we be surprised that this  construction fails  um  a well we can't  just the notion of defining a quantified  boolean formula by by using recursion is  just  uh  not allowed  so you just can't you can't define  formulas that way it doesn't use the for  all quantifier anywhere  or because we know that tqbf is not in p  okay see what you um  what  what do you think why can't why should  we not be surprised uh  i guess i could put in could have put a  d in there not surprised because  you're not you don't know what's going  on  so that's another reason not to be  surprised  but um anyway  um  hopefully you have some a glimmer of  what what's happening here  and when we just uh  almost finished  so i'm gonna  shut this down in a second  last call  okay ending  all right  so in fact the right answer is  b  um  i mean one should be suspicious  that if you're not  you know if you're not you're not  there's no for all appearing in this  construction anywhere um so really what  we're doing is we're constructing a  formula that has only exist quantifiers  so it's a satisfiability problem  so really what we just did  was we constructed  a um  we we did in a more complicated way the  cook levin construction because you end  up with a sad  sad formula only with exist quantifiers  and so uh really try one and try two  with the same  so it's not surprising that you end up  with an exponentially um big  formula as a result i don't know what a  lot of you answered we know that tqbf is  in p is not in p  we don't know that i don't know where  you  what's happening with you guys  uh  uh but no uh maybe that was a protest  vote uh but  anyway no we did we don't know that tq  and what does that got to do with  anything anyway so anyway that's  let's let's see how we solve this um in  our remaining a few minutes here  and  so here is the solution  he remember  this part where we're saying we're  trying to find c mid  um does there exist a c mid such i can  get from c i to c mid and half number  steps and c mid to c j and half the  number of steps  i'm expressing one formula in terms of  two formulas  that's where the blow up is occurring  because these two are then in terms of  gonna each it  so these two are gonna become four  become eight  and that's not good can i express this  fact in terms of just one formula  and there's kind of a little bit in the  spirit of someone your suggestions can  we kind of  reuse things in a way and that's what  we're actually kind of kind of to do um  so here's another way of saying the same  thing but with just a single formula and  it uses a for all  and the idea behind it is that a an and  is kind of like a for all  or for all is kind of like an end  just like an exist is kind of like an or  so when you're saying does something  exist you know it's is it this thing or  that thing or that thing or that thing  and when you're one saying for all it  has this thing and that thing and that  thing and that thing  so ands and paroles are very much  related  and so we're going to convert this and  into a furrow we're going to say  for all configurations cg and ch  that are either set to cic mid or to c  mid cj  the formula c g c h i can get from c g  to c h and half the number of steps so  you have to think about what this is  meaning here um  uh  and i also want to make sure that you  don't feel i'm cheating you because well  first of all  so now we have just a single formula  we're going to go down to the case b  equals 1 as we did before  you have to make sure that  saying  this restricted for all is is not a  cheat when you have for all  x is an s like we have over here  is equivalent to saying for all x  if x happens to be an s then the other  thing follows and this implication can  be expressed using hands and ors  um  and  as before the initial  um  starting point is going to be with going  from c start to c accept and t steps  and so the analysis  that we get is that  uh  because there's no longer a blow up each  recursive level just adds  um  this stuff here in front  um the exists c mid and the four and  this for all part so that's going to be  adding order n  to the to the  to the formula instead of multiplying  because we have two formulas  um and so now the total number of levels  is order n to the k as before so the  size is going to be n to the k times n  to the k um  uh or order into the 2k  um i actually had a brief check in here  um which i'd like you to do just you  know in our remaining few seconds  um  does this construction depend on m being  deterministic  um  so let me just launch that  uh i want you guys to get your check-in  credit here  but  in fact if you  uh  you have to think this through um  where  that formula says  that the tableau  you you get a tableau  is that going to matter depending upon  whether it's deterministic or  non-deterministic it's actually  um  well it's the kind of running 50 50 here  um  i want to just pick something because  i'd like to just uh close this out and  just get to our last slide  um so if you don't don't follow don't  worry  um but it's actually an interesting  point that um  the fact that this  um so i'm going to end this  all in  uh  so in fact the right answer is um it  still would work if it's  non-deterministic and this would give  you an alternative way of proving  savage's theorem so really this all  comes down to  this proof which implies savage's  theorem and that in turn implies  the latter dfa problem so anyway that's  side note not critical for understanding  really you can take those as all  separate results and and that's good  that's good enough  all right so um  um  coming back oops uh coming back  this is uh  this is what we did  and  um so each recursive level the size of  the qbf is not the same somebody's  asking is it the same at each recursive  level now we had to add in  um  uh  let's just see what  each recursion so you know we have this  recur  this is recursively built here but now  we have to add this part in front  and this part here in front  so the quantifier  you know  which is quantified over a bunch of  variables representing the configuration  that gets added on at each level  so it's not just it stays the same  um  there's stuff that's getting added in  but what's important is that doesn't  blow up exponentially  this stuff stuff gets added in every  time but not multiplied  um  okay so we're done here so anybody you  can all take off i think many of you  already have  bye-bye  thank you  you

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hi everybody  i hope you can hear me  uh give me a thumbs up if you can good i  see you guys let's see i see them coming  through  um  so we're gonna get started um  um  uh okay so getting back to what we've  been doing um you know we we've been  talking about space complexity  measures how much memory the algorithm  requires or various problems require  and  we uh defined the  uh space complexity classes uh space f  of n and non-deterministic space f of n  uh the polynomial space and  non-determining space non-deterministic  space classes  and gave some examples and so on and  today we're going to pick up where we  left off last time  one of the examples which is going to be  an important one for us is concerns this  latter dfa problem so i'm going to go  over that again give a little bit more  emphasis to the  uh  space analysis uh which i got some  questions about last time  and then we're going to move on from  there and prove savage's theorem and  then talk about complete problems for p  space um it showed that this problem  tqbf which we introduced last time is  actually a piece based complete problem  but  all in due course um  a little bit of a review so we defined  we defined what we mean by a turing  machine to run in a certain amount of  space  that means it uses at most f of n if  it's running in space f of n uses at  most f of n cells tape cells  on every input of length n and  similarly a non-deterministic turing  machine does the same  but what's  in addition to that the  non-deterministic machine has to halt on  every branch of its computation  and each branch of its computation has  to use at most that bound that bounded  amount of tape cells  um  so we're going to be talking about  non-deterministic space computation  today as well  it's going to be  relevant to us  um  so we defined the classes as i mentioned  um  and the polynomial and uh  the p space and non-deterministic piece  based classes and this is how they uh  see how we believe they relate to one  another  um the classes co np and np as well  and of course  uh as i mentioned last time  there are some very  major unsolved problems in this area so  everything could conceivably collapse  down to p  which would be of course very surprising  but we don't know how to prove otherwise  um  and the big theorem that we're going to  prove today is that polynomial space and  non-deterministic polynomial space  actually do collapse  down to each other  and  being the same class  so in contrast with the situation that  we believe to be the case for time  complexity where we believe the  non-deterministic converting  non-deterministic to deterministic gives  an exponential increase  uh for space complexity it only gives a  squaring income  increase as we'll see um  so uh  any  questions uh on any of this um  or we will just march into a little  review of this latter problem  um  so  re reviewing  some of the notation and and let me  emphasize that so the big theorem we're  going to be proving today is that p  space and n p space are equal um  and also we're going to be talking about  uh  piece based completeness  but uh  both of those involve proving theorems  um in the in the first case uh savage's  theorem that converting  non-deterministic to deterministic  spaces of squaring and in the second  case proving that tqbf is p-space  complete  both of those theorems can be thought of  as generalizations  of the of of um  this  um  theorem here that the latter dfa problem  can be done in in  the deterministic uh polynomial space or  n squared space  um  so  it really pays to try to understand  how the proof of this theorem works  because in a sense this theorem is a  more concrete version of what we're  going to be seeing in those other two  theorems in a somewhat more abstract  form okay so  um i like understanding things in a more  concrete way first so that's why this is  a good example to start out with but  really in the end of the day it's the  same proof just repeated for those three  theorems so this is a really three pre  three for the price of one thief three  theorems one proof here so you're going  to be seeing the same proof  repeated three times  but in different levels of abstraction  okay  so  let's let's review again and  i know some of you got it but um  maybe some of you didn't and uh let's  just try to uh  be clear  on the algorithm  to solve the latter dfa problem so if  you remember first of all let me just  jump on ahead the latter problem is your  give a ladder first of all is a sequence  of strings  that change one symbol at a time  that perhaps connect go from one string  to another so you're going to go from  work to play  changing one symbol at a time so we gave  an example of this or you can easily  come up with an example of doing this  but uh  the computational problem is can you do  it can you get from this string to that  string and stay within a certain  language  so it might be the language of english  words or it might be the language of all  strings that some specific dfa  um uh  um  recognize um  uh recognizes so  it might be all of the string these  might all be strings that some dfa  accepts or you know  might be english words or some other  rule um  and so  uh  uh that's what we mean by um  trying to test if there's a ladder  um and so  uh the the latter problem is um well i  didn't i don't think i wrote down on the  lateral problem itself but the bounded  ladder problem is basically the same  idea you're given a dfa you're given the  strings u and v  and now  this is the bounded version of the  problem where i'm going to give you a  limited limit on the number of steps you  can take so i'm illustrating that here  so you're going to be given a b and you  want to say can i get from this string  to that string within b steps  okay  and we had a notation for writing that  going from u to v  if there is uh a ladder that connects u  to v within at most b steps  and so the bounded ladder problem which  i'm introducing because i'm going to be  aiming toward a recursive algorithm to  solve this problem  is can i get from  u to v by a ladder changing one symbol  at a time where each string along the  way is accepted by b  and only allowed b steps little b steps  okay  um  so that is the computational problem  that i'm going to be solving with the  algorithm that i'm going to describe um  so the algorithm i'm going to call bl  for bounded ladder problem um  and  here is the input  and uh the algorithm is first of all  going to look to see if b equals one if  i'm just trying to get  from u to v in a single step  in that case  it's a very simple problem because you  want to test obviously that you and v  are in the you know are accepted by the  automaton  um and they just have to differ in one  place and then you have a very simple  one-step ladder that takes u to v okay  so for the case b equals one it's very  simple  uh  for larger values of b we're going to  use we're going to solve the problem  recursively in terms of small smaller  values of b um  so  for b greater than 1  we're going to recursively test  you know if you're trying to solve the  problem can i get from u to v  instead we're going to try each possible  uh halfway through  we don't know that it's halfway through  so we're just going to try each possible  string  and we're going to test  can we get from u  the initial string  to that new string that  w in half the number of steps and can i  get to the final string v in half the  number of steps if i can do that  then i can get get from u to v  the total number of steps b  so i'm just going to try to do this  one w at a time for every possible w  this is going to be very expensive in  terms of time  but we're not worried about time right  now we're trying to cut down on the  amount of space that we're using and  this is going to be a big savings in  space  um  let's not worry about the division b  over 2 here all of the divisions and  we're going to be seeing this several  times going forward in the in the  lecture  we'll think of them rounding up and i'm  not going to cover up  make the notation look cumbersome by by  writing that every time  um okay so here we go here is  some candidate w string which is half  halfway through  um recursively test can i get from the  starting string to that w and from w to  that ending string  um  if i can if i find such a w then i  accept  and if i try all possible w  and i  never  manage to find a way  um to make both a top and the bottom  work then i know i cannot get from the  you know from the starting string to the  ending string within b steps and so i  reject okay and now i'm going to solve  the original unbounded  ladder problem by simply putting the  biggest possible bound into the bounded  ladder problem  and that's his value t  which is  gives the very trivial bound of the the  total number of possible strings that i  can write down  within my length m that i'm working with  so this is if sigma is the alphabet of  these strings  it's just sigma to the m that's all  possible strings of course that's going  to be a maximum size on the ladder um  uh  okay uh so now how much space does this  take and i think this is where people  got a little bit um  uh lost uh in  in the lecture last time so i'm gonna  try to animate this um i don't know if  that's gonna help or not but you're in  the end of the day you just have to kind  of have to think through how the re  how do you account for the cost of this  recursion  um  okay but the main thing  to start off you have to make sure you  understand the algorithm you know to get  from here to there we're going to try  all possible  midpoints and then  of the upper part and the lower part  recursively re using the space that's  the critical that's the way we're going  to get a saving by  solving this problem reusing the space  that we use to solve this problem  okay so  i'm going to i'm going to try to show  this to you on actually how the space  gets used on the turing machine you can  kind of think of here's the input and  then after that is going to be the stack  for the recursion if you're not that  familiar with how to implement recursion  doesn't really matter um but you can  just  think about what the algorithm needs to  keep track of  um  and so  as it's trying every possible w so you  know  just in order like an odometer um  just trying every possible  every possible string eventually maybe  it finds a string that's in the language  um that hears an english word one of the  first english words of length four that  you might run into um  and so now it makes sense actually to do  the recursion um  so that's all  every time you you're going to have to  have a register or a location on the  tape where you're going to be writing  down those different w's um so let's say  it's over here and and we're just going  to go through i hope that's not too  small for you to see  you know  that's really where that action is  happening um and finally maybe you got  to the string w now you're going to try  to do the recursion so here is um you  know as you recur  doing the recursion on the top half  again you're going to be cutting you're  going to be finding a new w  for the the intermediate uh  point just solving this uppers upper  problem where we're testing if i can get  from work to able later i'm going to  have to deal with getting from able to  play  um  good uh so  here again we're going to be trying  fixing able fixing that but the first w  we're going to try every possible way of  getting from work you know from the from  the start string to that to that middle  string um so we're going to try every  possible thing here eventually maybe we  find some string um some other string in  the language we get down to the string  book  um  and uh that's all going to get stored  you have to get you can't forget this  the string able but now we're going to  use more space to store those candidates  so that's a second version of w um  deeper in the recursion so here we're  going to be triangle  possible strings here again eventually  we get to some string book  um  and you know if that succeeds in getting  us  from  uh work to able via book now we're gonna  jump down to do the bottom half um to  see if i can get from able to play  um as a separate problem which gets  solved  in the same space so now um here we're  going to try all these possible  possibilities getting from able to play  maybe call is the right intermediate  string there  and so now  we're going to erase the book  and now we're going to solve the lower  sub problem in the same location  i hope this is helpful this is a lot of  work making these animations  um  and so  i hope so the point of all this is every  time  time we go down the level of the  recursion there's another register whose  size is big enough to hold one of one of  the strings  is needed  and that register gets reused um  uh  times as we're  um at uh  uh you know throughout as we're going  through through this recursion um  so anyway i hope that's helpful um  uh  anyway so each level of the recursion  adds another order  n  to record the w and so you have to how  many levels do we get  well the depth of the recursion is going  to be  how many times we end up having to  divide this picture in half to get until  we get down to one  and so the height of this  you know when we start off  is going to be um basically an  exponential  in m m is roughly the size of n  so when you take the log of that  you're going to get  um  you're going to  pull down the exponential so it's going  to be order m or which is again roughly  the same size as the input m is like  half the input because the whole input  is u and v m is just the size of um  and so  each level requires order n and the  depth is going to be order and deep  you know the log of the initial height  of this  ladder  and so the total space used is going to  be order n squared  okay so why don't we just take a minute  i'm happy to spend a little time going  through this either the algorithm's  correctness  um understanding the recursion or  understanding the space analysis if  there's any questions that you feel you  can ask that would be clarifying for you  um  jump in we'll we'll we'll i'll i'll set  aside a few minutes just to answer  questions here  um  so i've got a question here in step five  why do we reject  if all fail  instead of just one fails  um  well here  so remember what we're trying to do  we're trying to say can i get from u to  v in b steps  the way i'm going to be doing that  is trying every possible intermediate  string w  if i find some w  which does not work  that doesn't mean that there's not some  other w which might work all i need is  one w  for which i can get from u to w and half  the number of steps and w to v and half  the number of steps so i'm going to try  every possible w if any one of them is  good  then i can accept if any one of them  succeeds where i can get from u to w and  half the steps and from w to v and half  the steps  then i know i can get from u to v in in  the full number of steps  um so  i only need to find one  if i if one particular one doesn't work  i'll just go on to the next one  um  i okay here's a this is a good question  um  uh do we have to save the word book  so once we succeed in getting from work  to able let's say via book  do we need to save that word book  anywhere no  all we need to remember is that we've  succeeded in getting from work to able  we don't need to remember book anymore  we just  you know  remember that we've succeeded and that  is by virtue of where we are in the  algorithm so we but you know if we have  succeeded then we move on to the second  uh recursion second call recursive call  um so we found some way to do it so we  found some um  some intermediate point which succeeds  for this one so we move on to that one  we don't have to keep any of that work  anymore all you have to do is remember  yes i can get from work to able and half  the number of steps now i need what all  that's left is to get from able to play  in half the number of steps  i don't doesn't matter how i got to abel  in the first place so we don't have to  remember that  that was a good question though um  so i i think i i understand this to  before we replace  um the value for book  with call  the with the work involved to find coal  um  [Music]  yeah we have to check that we can get  from work to book and book to able  so we hang keep on to book while we're  working on the upper half  and only when we've finally succeeded in  getting from work to able let's say via  book then you can throw a book away  um  but while you're working on the upper  half you try book you try different you  know different strings of length four um  until one of them works  um  so  okay i'm not sure somebody's asking me  about breadth first search and depth  first search i'm not sure i see it um  this  i don't i'm not sure that's going to be  a helpful way of thinking about this so  i'm not going to answer that right now  but um  you know you can ask that offline later  if you want  why is the recursion  why is the recursion depth  log t instead of log m  okay well how high is this thing  initially  um  it's t high  but every time  we're doing a level we're calling the  recursion we're cutting t in half  first it's you know if you  you know i'm solving this in general for  b but we're starting off with b equal to  t t is the maximum size  so  initially this is going to be t and then  it's going to be t over 2 then t over 4.  so it's going to be log t levels before  we get down to 1.  um  yeah so somebody's asking can we think  of this as a memory stack yes this is  like that's where your typical  implementation of recursion is kind of  with a stack where you push when you  make a call and you pop when you return  from the call  um  is it possible that v can appear during  bl procedure on t  is it possible that v can appear  um i'm not sure what that means it can v  reappear  you know so i'm starting with u to v  is it possible that v might be one of  these intermediate strings yeah you know  um  uh this you're going to try every  possible intermediate stream  you know blindly including v is one of  them um  uh you know if you can reach v  more quickly  well great um  uh i guess what i have not dealt with  the issue of what happens if you  get to a  um  uh  yeah technically it's it's going to work  out because i'm allowing  you know the difference to be in at most  one place so even if you get there early  you're allowed to  not change anything and that still is a  legal step in the latter um  uh  yeah i don't see how to do this from a  bottom-up perspective somebody's asking  is there a bottom-up version of this  i don't  think so uh  but no i don't think so  all right why don't we move on  um  so now we're going to see this proof  again  but but this time we're going to be  proving  [Music]  that you can convert any  um nfa to a dfa with only a squaring  increase  so really  uh  well  let me just put that up put put that up  there um  uh so this is going to be savage's  theory that  among other things proves that p space  equals np space  uh so it says that  you can convert a non-deterministic  machine to a deterministic machine only  squaring the amount of space so you're  comfortable with this notation here  anything that you can do in f of n space  non-deterministically  you can do it f squared of n space  deterministically um  and we're going to accomplish that by  converting a  you know an ntm to a deterministic tm  but only squaring these space u's so n  is going to get converted to m  and now this proof is going to look very  similar to the proof from the previous  slide  um  it's the same proof  and um you know and and the fact from  the previous slide you know about ladder  really is implied  by this because you know we gave we had  an easy algorithm to show that the  latter problem is solvable in in  non-deterministic you know an np space  so ladder problem was easily shown to be  in here  um  if you remember you just guess  the  you basically guess the steps of the  ladder so non-deterministically you can  easily check can i get from the start to  the end  but  savage's theorem tells us that anything  you can do non-deterministically in  polynomial space you can do  deterministically in polynomial space so  what we showed in the previous slide  follows from this slide but this slide  is really just a generalization of the  same proof  okay  maybe i've said it too many times now um  so  we're going to  introduce a notation very similar to the  notation we had last time but now we're  going to be talking about  simulating this non-deterministic  machine with a deterministic machine  and we're going to have take two  configurations of this non-deterministic  machine c i and cj  and say  can i get from ci to cj in at most b  steps  i'm going to have a notation  very similar to like the notation for  the latter where i where i can get from  this word to that word in most p b steps  by a ladder here can i get from this  word to that if can i get from this  configuration to that configuration with  at most b steps of the turing machine's  operation  so these are two configurations now  um  of n  so can n go from this configuration ci  to that other configuration cj  but only taking b steps along the way  that's now the computational problem  that i'm going to solve for you with an  algorithm and it's going to be a  recursion exactly like the previous one  um  m gets its input the two configurations  c i and c j and one and the bound b and  want to check can i get from i to j  within b  okay so now the picture's a little  different um  but it's very similar so instead of a  ladder appearing here  it's really something that's basically a  tableau  for  um  for the machine n where i have a initial  configuration and an ending  configuration this would be hap this  would happen to be  um  uh  the starting point for the whole  procedure if you're testing whether n  accepts w but we would be solving this  in general for any config you know so in  that case i have the start configuration  of n on w and the accepting  configuration or an accepting  configuration but  in general what i  will be  solving is starting with any  configuration ci and going to any  configuration cj  so i want to  test can i get from ci to cj within at  most b steps  um so first of all if b is one  you can just check that directly um  you know and now again this is we're  operating deterministically to simulate  the non-deterministic machine  so this is the deterministic machine m  so m can easily if it's given two  configurations of n  and says can i get from the first one to  the second one in one step well that's  an easy check you just lay out those two  configurations  look at the ends transition function and  say is this a legal move for n  yes or no and you accept or reject  accordingly  now if b is larger than one  you're going to try all possible  intermediate  configurations  i'm calling them c mid this was like the  w from the previous  uh theorem  this is all possible strings c all  possible configuration c mid  and you know i don't you know a  configuration is just going to be a um  uh  so far so okay  this is all possible  look like my powerpoint correct that  seems okay um  [Music]  this is all possible configurations  which is just a string with a string of  tape symbols with a state symbol  appearing somewhere that's all it is um  you're going to try all possible  configurations as  candidate middle configurations and say  can i get from the upper one to the  middle the to this candidate middle one  and from that middle one to the to the  lower one within half the number of  steps each time  so um  and solving that problem recursively  um  okay  so i got a question here about the  possibility of looping forever  first of all if n is going to be looping  i don't have it i don't have to worry  about it because  um i'm starting off  i'm only i need to simulate machines  that always hold that are deciders  because i'm trying to show that any  language in here has to be accepted has  to be  uh has to be decided by some some  non-deterministic machine  so i'm not going to worry about machines  that are looping if they're looping  you know um  uh m may be misbehave in some way but  that's not going to be a problem for me  um so let's only let's keep life simple  think about the machine deciders only  um  okay so we're going to recursively test  here so that means i have a going to try  every possible middle see if i can get  from the  the start to the middle and the middle  to the end  um  uh if both of them work  for my after my i test them recursively  then i'm going to accept if not i'm  going to continue  and i reject if i try them all and none  of them have worked then i know there's  no way for n to get from this  configuration to that configuration in b  steps  okay  [Music]  and the overall picture i test whether n  accepts w as i mentioned by starting the  c  i is the start configuration and cj is  in the accept configuration  um  and now how big is t  because i need to calculate  a bound  on how  many how deep the  the the recursions are going to be um so  t  here is going to be the total number of  possible configurations um you know if  this is holding it never repeats the  configuration so this is going to be a  bound on how many steps n can be taken  and that's simply you know we calculated  this before it's the number of um  states times the number of head  positions times the number of tape  contents  tape contents and this is really going  to be the dominant  consideration anyway  and so  now each  recursive level and maybe i should have  emphasized this at the beginning how  wide is this picture it's big enough to  store a configuration  configuration is essentially a tape  contents  so that's going to be f of ny  so each recursive level stores one  configuration  now the w  costs f n space to write down  and the number of levels is going to be  the log  of the initial height  which is this so this is going to be the  dominating uh  part of it so the log of this is going  to be again order f of n  so this each one takes f of n space to  write down the depth of the recursion is  going to be order f of n so the total is  going to be order f squared of n and  that's how much space this uses  and that's the proof of savage is there  so  yeah so this is a good point there uh i  was thinking somebody asked can there be  multiple accepting configurations i  should have made the  i forgot to say this but and i was just  realizing it as i was explaining it one  of the things i should have you you can  enforce that there is just a single  accepting configuration  this is kind of a detail but so don't  worry about it if you  if you don't if you don't want to but  you can make sure that there's a single  accepting configuration by telling the  machine when it accepts it should erase  its tape and move its head into the  leftmost cell in the accept  configuration so there's just going to  be a single accepting configuration to  worry about instead of having to try  multiple  uh  possibilities which you could do in this  algorithm but it would just be annoying  to have to write that down that way so  we often assume there's just going to be  a single accepting configuration for  these machines  um  um  okay so this is  how do you know f of n um  so that's a that's actually a little bit  of a delicate issue uh  i mean if you could compute f of n  um the bound  which is for example if it's going to be  a polynomial bound you can just compute  f of n it's very easy to compute n  squared or n cubed  and so you can just  compute that and then  use that as  the size of the registers you're going  to be writing down  for  if you want to prove this in general for  f of n it's a little bit technical to  have to deal with it  and i'm going to have to refer you to  the book on that one the book tells you  how to do solve this for a general f of  n you basically have to  try every possible value from one until  it works um  and  i'm afraid that's going to  derail us trying to un decipher that so  let's not worry about that aspect of it  but it's not it's a it's a you can  handle general f of n here um you don't  need to put any conditions on f of n um  can we go over this term here  so we've seen this term once before when  we talked about lbas and and seeing that  lbas always  we can solve the alba problem this is  simply the number of different  configurations the machine can have  because the configuration is a state  it's a head position and a contents of  the tape  and this is the number of each of those  that that you can possibly have number  of states  a head position  you know the size of the tape of f of n  is f of n so this is that many different  head positions and this is the number of  d if d is the size of the tape alphabet  this is the number of contents tape  contents that you can have  um  okay  how is seeing if n accepts w with this  algorithm  convert non-deterministic turing machine  to some deterministic turing machine  well n is the non-deterministic term so  n is  we're converting non-deterministic  turing machine n  to  deterministic turing machine m  so m is the deterministic simulator of n  that's what this whole m is  so if we can  do this for  you know for any end that we've proved  our theorem  um  okay why don't we defer i think you know  we're  i think i got to most of the questions  here if there's other things we can save  them for the um  coffee break which is coming soon i  think we have one more slide before that  um  okay so i'm going to define p space  completeness i think yeah and then i  think i think after that we have the uh  the break  so peace-based completeness is defined  and in very much inspired  uh similarly to np completeness  um  so  a problem is p space complete if it's in  p space  and every other member of p space is  reducible to it  in polynomial time  and we'll say a bit about why we choose  polynomial time  reducibility here um so here's a kind of  a picture of how  p space or have complete problems  of  you know uh  relate to  relate to com their complexity classes  so  you kind of think of a complete problem  for a class it's kind of the hardest  problem in that class because you can  convert you can reduce any other problem  in that class to the complete problem  so here are the np complete problems  sort of the hardest for np here the p  space complete problems kind of the  hardest for piece p space  if any if an  np complete problem goes into p that  pulls down all of n p to p if any piece  based complete problem goes into p it  pulls down all the p space into p by  following the chain of reductions um  because any piece based problem is  reducible to the complete problem which  in turn if it's in p  then everything goes into p so  if  you have a piece based complete problem  which is in p  then all the piece base goes into p  um so why do we use  polynomial time reducibility and say  instead of say polynomial space  reducibility  when we define this notion  it's a kind of a very reasonable  question  but if you think about it using  polynomial space reducibility would be a  terrible idea  um  because  and we've seen this phenomenon happen  before  every two problems in p space  are going to be p space reducible to one  another we haven't even defined that  notion yet but you can imagine what it  would be  because  a piece-based reduction can solve the  problem  um for a problem in p space and then it  can direct its answer anywhere that it  likes so in general when we think about  reductions  the reduction should be not capable of  solving the problems in the class  because if they could  then every two problems in the class  would be reducible to one another  and then all problems in the class would  be complete because everything would be  in the class would be reducible to any  one of the other problems  so it would not be an interesting notion  what you want to have happen is that the  reductions  should be  weaker  than the power of the class  and so um  uh  every  um you know  and if you if you look at the reductions  that we've defined so far they're  actually very simple  you know the only thing is yeah they  have to make sure that they can make it  that  they can make the output uh big enough  but actually constructing the output  they would do very simple  transformations in fact even polynomial  time is more than you typically need  there's even much more limited classes  that are capable of doing the reductions  as we'll see  um so having powerful reductions is  really not in the spirit of what  reductions are all about you want very  very simple  transform transformations um to be the  reductions anyway i hope that's helpful  so what what we're going to be aiming  for in the second part of the lecture is  showing that tqbf is based complete um  and let me give ahead as a check in  uh  uh  so this is our first check-in um coming  a little late in the lecture  suppose  we have proven that as we will that tqbf  is piece based complete  uh what can we cl conclude if tqbf is  actually  not necessarily in p only goes to np  that's and this is irrelevant to a  question that's coming in from the chat  but i'll answer that later so um suppose  tqbf ends up being in np and not in p  what what can we conclude remember if  if t  is in p  then p space equals p  suppose it goes to np  what happens then  check  you know there might be several uh  correct answers here  to check all that all that apply  all right so we're near the end  of the  pull so let me give you another  10 seconds and then we're gonna shut it  down  um  okay we all are we all in  closing it down  um  here are the results so  yes so uh first of all the most  reasonable solution the most reasonable  answer is b  uh which i think  uh  most of you have gotten that if  you know if a piece based complete  problem goes down to np well np is  capable of simulating the polynomial  time reduction  um and so  um  any other problem in p space would then  also be an np and p space would equal np  but note  if p space equals np they're also np  equals co np  um  because p space itself is closed on you  know is closed under complementation  so that was kind of a  little bit extra  fact that you could conclude from this  as well  um so let me let's move on then to our  uh coffee break  and we'll pick up the proof that tqbf is  piece based complete  after after that  okay  so  so was d true or not um  d was p equal np no d we cannot  conclude that p equals np  um if  um  from p space uh equal to np  um  so if tqbf is an np  it doesn't tell us anything it's [ __ ]  for all we know p equals np but uh from  the stuff that we've that we know so far  we cannot conclude that p equals np  um  oh and yeah so you can conclude oh i'm  sorry you you can con b and d  are both  correct here  um  let me just shut this thing off uh  so b and d are correct so if if a piece  based complete problem goes to np  then np equals p space and equals coin p  so the correct answer of b and d  sorry i i got myself confused so c but c  is not something you can conclude  or a  okay uh  so somebody said  somebody's asking me a fair question  you know  i say the reduction method should be  weaker than the class  uh  but you know for example even in the  case of p space p space might be equal  to p  and then um it wouldn't be weaker than  the class if we use polynomial time  reductions  but  um i think maybe i should say apparently  weaker  as far as we know it's weaker  but we believe it to be weaker  if it's true if p equals p space then  every problem in p is going to be p  space complete  it's just going to be a weird world if p  equals p space similar same thing for n  p and np  um  so so i'm getting a number of questions  also about other possible reducibilities  that are even weaker than polynomial  time reducibility um  so we're going to see uh um  very soon weaker reducing  you know complexity classes within p  so first of all p space seems to be  bigger than p we're also going to look  at log space but that's going to be um  actually in thursday's lecture those are  these are called classes  ins that seem to be inside well they're  inside p they may be proper we believe  they're properly inside p but we'll  we'll see that later  um  let me just see here  really should have so we're almost out  of time  uh let me put our  timer back in fact our timer is showing  us out of time  so um why don't we um  get going let me  move this um  back to  okay  continuing  t q d f is p space complete so first of  all let's remember t q b f  um  [Music]  uh these are all of the  quantified boolean formulas that are  true so t could be of stands for true  quantified boolean formulas  and remember we saw these examples  from the previous lecture  that you know here these are two  quantified build these are two qbfs uh  the first one is true the second one is  false and and  it's kind of interesting to think about  you know here they're exactly the same  except for the order of the quantifiers  and so what's really going on you know  here  uh just i think it's  good to understand these expressions  they come up everywhere in mathematics  uh these quantum  quantifiers um  you know in the upper one  uh when we say for every x there exists  a y  that y can depend  on the choice of x  you choose it make a different x you're  allowed to pick a different y  but this but the the lower expression  says there's a universal y there's some  one particular y that works for every x  so in a sense um  uh the lower statement is a stronger  statement  um you know whatever you have in the  interior in  the quantifier free part uh  so the lower one implies the upper one  happens that the lower one in this case  is false and the upper one is true but  in general the lower  in the you know when you have this uh  change of quantifiers like this the the  um  the lower one would imply the upper one  okay anyway that's sort of a side remark  so um let's get back to the proof that  tqbf is piece based complete that's what  our goal is  all right um  now as i mentioned  this is the same proof  you're going to be seeing it for the  third time today but  you know there's a certain amount of  it's sort of  the context changes  uh in each case  so now we want to show that tqbf is  p-space completed so it's one of these  hardest problems now but for p space  um  where satisfiability was like a hardest  problem for np  so we want to show  that every  language in p space is reducible to tqbf  um and so we're going to give  polynomial time reductions that map  some particular problem a which can be  done in space n to the k it's a  problem solvable in p space  we're going to show how um  f maps a to 2 q b f we have to construct  the f  so f is going to be a mapping that maps  strings which may or may not be an a so  strings w which may or may not be an a  to  these formulas these quantified formulas  um  so  uh  w is going to get mapped to  some formula fee sub mw it had exactly  the same  even symbols we used when the proof  proof of the cook 11th era about sat  being np-complete this is a very similar  proof  but you'll see that we have to do  something more in order to make it work  in this case  so if w  you know w is an a if and only if this  formula is going to be in tqbf in other  words if and only if this formula is  true  um  so  this formula  is going to kind of express the fact  that m accepts w  which means that w is an a because m is  the machine is the p space machine for a  so this uh formula says m accepts w  and it achieves that by  building in a simulation  of m on w  it kind of describes a simulation for m  w which ends up accepting  and if m does not accept w  that description is going to inevitably  be false  okay so let's just see how that's what  that's going to look like  so we're going to use the same idea that  we initial that we used for the cook 11  theorem that sat is np-complete this  notion of a tableau  so if you remember that we had it was  basically a table  which weighs was just simply a way of  writing down  an  a computation history for mw  okay so the rows are the steps of the  machine  the top row is the start configuration  the bottom row is let's say  some particular  accept configuration such as i just  described where the machine  clears its tapes and moves its head to  the left end so there's only one  accepting configuration we have to worry  about  okay and each of the rows here is a  configuration of m of m  okay  um  because m runs in space n to the k the  tableau kind of similarly to what i  described before has width into the k  so now you know  we're talking about polynomial time  machines  so the f which is the bound the space  bound is going to be some polynomial n  to the k  so the width of this  tableau the size of these configurations  are going to be n to the k  how  high is this tableau going to be  well that's going to be  limited by the possible running time of  the machine  which is  similar to what we saw before  it's going to be exponential in the  space bound  so it's going to be d to the n to the k  where d is the  essentially the tape alphabet of the  machine  okay so  are we all together on this  this is um very similar to the proof of  sat is mp complete  the  key difference there was m was  non-deterministic  um  which might be something to think about  later but  let's not focus on that right now here  this m is deterministic  and um  but the important difference was the  shape of the table the the the the size  of the tableau was very different in the  case of sat is np-complete we started  off with a polynomial time  non-deterministic machine so it only  could run for a polynomial number of  steps here is a polynomial space machine  which can run for an exponential number  of steps  that's a  that's going to be  an important difference here so let's  let's see why  uh  the reduction  has to construct  this  formula  fee sub mw  which  basically says that this tableau exists  now we already saw how to do that  when we  proved the cook 11 theorem that status  empty complete  remember we had all of those we had  variables for each cell  that told us what the contents of that  cell were was  and then we had a lot of logic here we  had a bunch of logic that said that  those  all the you know those neighborhoods  were correct which basically says that  the the tableau corresponds to a correct  uh computation of the machine  um so why don't we just do the same  thing  okay why don't we just  build our formula  using exactly the same process that we  used to build the formula when we had  sat being  np-complete  something goes wrong  we can't quite do that  the problem is that if you remember the  formula that we built before was really  about as big as the tableau is  because it had some logic for each one  of the cells it had a set of some of the  had variables for each one of the cells  and had some logic you know for each of  those neighborhoods basically so he says  that each of the cells  does the right thing  um  so it was a pretty big formula but it  was still polynomial  the problem is that tableau  is now into the k by  d to the n to the k  that's an exponentially big object  so if your formula is going to be as big  as the tableau  there's no way you can hope to produce  that formula in polynomial time  and that's the problem  the formula is going to be too big  remember we're trying to get a  polynomial time reduction  from this language a  so we have an input to a  you know a string that might be an a  which is  we're simulating the machine  and then you know  the size of the tab below relative to w  is going to be something enormous  and so if the formula is as big as the  tableau there's no way to produce that  formula in polynomial time so this  is not going to work  let's try again  so now we have here  uh  now so remember this notation from ci to  cj in b steps  okay so we're going to give a general  way  of constructing formulas  which  express this fact that i can get from  uh  configuration i of m to configuration j  of m in b steps  whatever that b is b is going to be some  bound and i want to know  can i get from this configuration to  that configuration and i want to write  that down as a formula which is going to  express that fact and it'll be either  true or false  and i'm going to  give you a recursive construction for  this formula so i'm going to build that  formula  for a value b  out of formulas  uh for smaller values of b  okay so this is going to be a way of  constructing that formula in terms of  other formulas that i'm going to build  and there's going to be a basis for the  recursion when b equals one  okay so that's the big picture so let's  see how does this formula look  um  so let's  let's not worry about the case for b  equal one right now this is the case for  larger values of b  so i'm gonna the fact of that i can get  from c i to c j within b steps  i'm gonna write this down in the in this  way  uh and let's try to  uh  unpack that and see what it's saying  um  you know without worrying about  how we gonna carry this out let's just  try to understand at a high level the  semantics of this thing what he's trying  to say to you  it's gonna say well i can get from ci to  cj in b steps  so m  can get from c i to c j and b steps  if there is some  other configuration c mid  i'm calling some other configure i'm  calling it c mid  very much inspired by  the previous proof  of savage's theorem uh where there was c  mid was that intermediate configuration  so now instead of trying them all i'm  saying does there exist one  where i can get from c i to c mid and  half the number of steps and from c mid  to c j in half the number of steps  so if i can build these two config these  two formulas  then i can combine them  with this sort of extra stuff out here  and then and them together and put an  exist quantifier that says does there  exist some configuration some way to  find a configuration such that it works  as these formulas require  if i can do that  then i'm going to be good because then i  can  um  well good at least i'll be good in terms  of making something that is going to  work um  so first of all  let's understand what i mean by writing  down uh does there exist c mid it's  really  if you you know thinking back to the way  we did the koch 11 theorem we  represented these configurations by  variables which  were indicator variables for each of the  cells and we're going to do exactly the  same thing so we're going to have a  bunch of boolean variables which are  going to represent  some configuration  so  more more  you know formally or in more detail  what does does there exist a c minute  really is an assignment to all of those  variables that represent um the cells of  the contents of the cells of those  uh of that  configuration okay so now  let's see how to  you know how does how will the recursion  work  so um  to get this value here i'm going to  do the recursion further so does there  exist a c mid and now  um  for  getting from ci to this c mid can is  there some other c mid this is like  another value of w from the previous  slide where i'm getting again i'm  cutting the number of steps in half  again so i'm going from p over two steps  to b over four steps  and i'm going to do the same thing over  here  so i'm just uh unraveling  the  uh  the construction of this formula  in terms of um  building built by building it up  recursively  so  um  and then i'm just going to keep doing  that until i get down to the case where  b equals one  and when if i'm now trying to make a  formula  that's going to be say i can get from ci  to cj in just a single step  so this is really  talking about a tableau of height 1 or  high 2.  then i can just  directly write that down the way i now  the tableau is not very big so now i can  write that down using the neighborhoods  and so on that i did  in the cook leaven theorem proof  and this is how you put it all together  and now if you want to talk about the uh  getting  from  does does m accept w  so i initially say can i get from the  start configuration to the accept  configuration in t steps which is the  maximum running time of the machine so  again  you know if you followed me what  happened  um in savage's theorem it's the same  it's the same uh values  uh  now the thing is is we have to  understand how big this formula is and  if you think it through  if you think it through  um  there's a problem  because what's going on here  i'm expressing this formula  in terms of formulas  where um  the size of b is cut in half  but now there are two of them  so it's two formulas on half the value  of b  that's not going to be a recursion uh  that's going to work in our favor  um so let's just see what happens  so each recursive level uh  doubles the number of formulas  right going here we have two formulas  here we're going to have four formulas  and and so on so the number of formulas  doubles each time so the length of the  thing um that we're writing down is  doubling in size  each time we go to down the recursion  that's going to be okay if we don't go  too many levels but unfortunately we are  going quite a few levels because the  number of levels is going to be log of  this initial  um exponential size thing so it's going  to be n to the k levels and so if you're  doubling something n to the k times  you're going to end up with an  exponentially sized formula and  again  we failed  okay so let's i have a check in on this  but i'd like maybe we should spend a  couple of minutes just making trying to  understand what's going on here  because the next slide is my really my  last slide  and it's going to fix this problem  but let's make sure we understand what  the problem is before we try to fix it  um  why can we no longer write over each  layer of the recursion as we did in  ladder oh that's kind of a cool question  what does it even mean to write over the  different well  you know  uh  so that that's kind of an interesting  question here so what in a sense that's  going to be the solution  we're going to  uh  um  you know reuse things in a certain way  but but for but  i want you to understand that this  method  itself  does not work  because this recursion here where i'm  writing the formula i'm building the  formula for b out of  formulas for smaller values of b um  if i if i  do it that way  i'm going to end up  if i do it as it's described in this  procedure here i'm going to end up with  an exponentially big formula  and that's not  good enough  so if you cut the formula in half each  time even though you have two formulas  won't the length be the same  um  i'm not cutting the formula in here from  cutting the value of b in half  so you have to say b is initially an  exponentially big value  so we're going to end up with an  exponentially big formula  so it's not really cutting the formula  in half cutting the b in half and b it  starts out big  you know  i mean that b is this b is initially  this value here d to the n to the k  i'm worried not too many questions here  i feeling that's  that's that's probably not a good sign  uh  so if you're if you're if you're  well i mean if you're hopelessly  confused maybe i can't fix it um quickly  so anyway why don't we move on  and see how to  repair  this how to fix this problem um  and that is going to be by a trick  um  in fact i know  the people who were involved with coming  up with this this was actually this  proof was done originally at mit  um  and uh many years ago in the 1970s  and uh the folks who were involved with  it called it the abbreviation trick so  that's what we're going to do on the  next slide  oh no this is a check in first  why shouldn't we be surprised that this  construction fails  um  a well we can't  just the notion of defining a quantified  boolean formula by by using recursion is  just  uh  not allowed  so you just can't you can't define  formulas that way it doesn't use the for  all quantifier anywhere  or because we know that tqbf is not in p  okay see what you um  what  what do you think why can't why should  we not be surprised uh  i guess i could put in could have put a  d in there not surprised because  you're not you don't know what's going  on  so that's another reason not to be  surprised  but um anyway  um  hopefully you have some a glimmer of  what what's happening here  and when we just uh  almost finished  so i'm gonna  shut this down in a second  last call  okay ending  all right  so in fact the right answer is  b  um  i mean one should be suspicious  that if you're not  you know if you're not you're not  there's no for all appearing in this  construction anywhere um so really what  we're doing is we're constructing a  formula that has only exist quantifiers  so it's a satisfiability problem  so really what we just did  was we constructed  a um  we we did in a more complicated way the  cook levin construction because you end  up with a sad  sad formula only with exist quantifiers  and so uh really try one and try two  with the same  so it's not surprising that you end up  with an exponentially um big  formula as a result i don't know what a  lot of you answered we know that tqbf is  in p is not in p  we don't know that i don't know where  you  what's happening with you guys  uh  uh but no uh maybe that was a protest  vote uh but  anyway no we did we don't know that tq  and what does that got to do with  anything anyway so anyway that's  let's let's see how we solve this um in  our remaining a few minutes here  and  so here is the solution  he remember  this part where we're saying we're  trying to find c mid  um does there exist a c mid such i can  get from c i to c mid and half number  steps and c mid to c j and half the  number of steps  i'm expressing one formula in terms of  two formulas  that's where the blow up is occurring  because these two are then in terms of  gonna each it  so these two are gonna become four  become eight  and that's not good can i express this  fact in terms of just one formula  and there's kind of a little bit in the  spirit of someone your suggestions can  we kind of  reuse things in a way and that's what  we're actually kind of kind of to do um  so here's another way of saying the same  thing but with just a single formula and  it uses a for all  and the idea behind it is that a an and  is kind of like a for all  or for all is kind of like an end  just like an exist is kind of like an or  so when you're saying does something  exist you know it's is it this thing or  that thing or that thing or that thing  and when you're one saying for all it  has this thing and that thing and that  thing and that thing  so ands and paroles are very much  related  and so we're going to convert this and  into a furrow we're going to say  for all configurations cg and ch  that are either set to cic mid or to c  mid cj  the formula c g c h i can get from c g  to c h and half the number of steps so  you have to think about what this is  meaning here um  uh  and i also want to make sure that you  don't feel i'm cheating you because well  first of all  so now we have just a single formula  we're going to go down to the case b  equals 1 as we did before  you have to make sure that  saying  this restricted for all is is not a  cheat when you have for all  x is an s like we have over here  is equivalent to saying for all x  if x happens to be an s then the other  thing follows and this implication can  be expressed using hands and ors  um  and  as before the initial  um  starting point is going to be with going  from c start to c accept and t steps  and so the analysis  that we get is that  uh  because there's no longer a blow up each  recursive level just adds  um  this stuff here in front  um the exists c mid and the four and  this for all part so that's going to be  adding order n  to the to the  to the formula instead of multiplying  because we have two formulas  um and so now the total number of levels  is order n to the k as before so the  size is going to be n to the k times n  to the k um  uh or order into the 2k  um i actually had a brief check in here  um which i'd like you to do just you  know in our remaining few seconds  um  does this construction depend on m being  deterministic  um  so let me just launch that  uh i want you guys to get your check-in  credit here  but  in fact if you  uh  you have to think this through um  where  that formula says  that the tableau  you you get a tableau  is that going to matter depending upon  whether it's deterministic or  non-deterministic it's actually  um  well it's the kind of running 50 50 here  um  i want to just pick something because  i'd like to just uh close this out and  just get to our last slide  um so if you don't don't follow don't  worry  um but it's actually an interesting  point that um  the fact that this  um so i'm going to end this  all in  uh  so in fact the right answer is um it  still would work if it's  non-deterministic and this would give  you an alternative way of proving  savage's theorem so really this all  comes down to  this proof which implies savage's  theorem and that in turn implies  the latter dfa problem so anyway that's  side note not critical for understanding  really you can take those as all  separate results and and that's good  that's good enough  all right so um  um  coming back oops uh coming back  this is uh  this is what we did  and  um so each recursive level the size of  the qbf is not the same somebody's  asking is it the same at each recursive  level now we had to add in  um  uh  let's just see what  each recursion so you know we have this  recur  this is recursively built here but now  we have to add this part in front  and this part here in front  so the quantifier  you know  which is quantified over a bunch of  variables representing the configuration  that gets added on at each level  so it's not just it stays the same  um  there's stuff that's getting added in  but what's important is that doesn't  blow up exponentially  this stuff stuff gets added in every  time but not multiplied  um  okay so we're done here so anybody you  can all take off i think many of you  already have  bye-bye  thank you  you
 

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399
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401
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402
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403
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440
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441
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442
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443
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444
00:14:33,189 --> 00:14:37,829

 

445
00:14:37,829 --> 00:14:40,629

 

446
00:14:40,629 --> 00:14:42,550

 

447
00:14:42,550 --> 00:14:44,790

 

448
00:14:44,790 --> 00:14:47,189

 

449
00:14:47,189 --> 00:14:49,590

 

450
00:14:49,590 --> 00:14:52,310

 

451
00:14:52,310 --> 00:14:53,750

 

452
00:14:53,750 --> 00:14:53,760

 

453
00:14:53,760 --> 00:14:54,550


454
00:14:54,550 --> 00:14:54,560

 

455
00:14:54,560 --> 00:14:55,990


456
00:14:55,990 --> 00:15:00,310

 

457
00:15:00,310 --> 00:15:02,069

 

458
00:15:02,069 --> 00:15:03,670

 

459
00:15:03,670 --> 00:15:07,829

 

460
00:15:07,829 --> 00:15:11,509

 

461
00:15:11,509 --> 00:15:13,670

 

462
00:15:13,670 --> 00:15:15,670

 

463
00:15:15,670 --> 00:15:18,629

 

464
00:15:18,629 --> 00:15:21,030

 

465
00:15:21,030 --> 00:15:21,040

 

466
00:15:21,040 --> 00:15:21,910


467
00:15:21,910 --> 00:15:24,629

 

468
00:15:24,629 --> 00:15:24,639

 

469
00:15:24,639 --> 00:15:25,509


470
00:15:25,509 --> 00:15:25,519

 

471
00:15:25,519 --> 00:15:26,470


472
00:15:26,470 --> 00:15:29,350

 

473
00:15:29,350 --> 00:15:31,749

 

474
00:15:31,749 --> 00:15:33,749

 

475
00:15:33,749 --> 00:15:36,470

 

476
00:15:36,470 --> 00:15:36,480

 

477
00:15:36,480 --> 00:15:37,430


478
00:15:37,430 --> 00:15:40,949

 

479
00:15:40,949 --> 00:15:42,470

 

480
00:15:42,470 --> 00:15:44,550

 

481
00:15:44,550 --> 00:15:46,150

 

482
00:15:46,150 --> 00:15:47,670

 

483
00:15:47,670 --> 00:15:49,110

 

484
00:15:49,110 --> 00:15:51,509

 

485
00:15:51,509 --> 00:15:53,110

 

486
00:15:53,110 --> 00:15:55,269

 

487
00:15:55,269 --> 00:15:57,350

 

488
00:15:57,350 --> 00:15:59,110

 

489
00:15:59,110 --> 00:15:59,120

 

490
00:15:59,120 --> 00:16:00,310


491
00:16:00,310 --> 00:16:05,269

 

492
00:16:05,269 --> 00:16:07,509

 

493
00:16:07,509 --> 00:16:07,519

 

494
00:16:07,519 --> 00:16:08,550


495
00:16:08,550 --> 00:16:10,550

 

496
00:16:10,550 --> 00:16:14,150

 

497
00:16:14,150 --> 00:16:17,110

 

498
00:16:17,110 --> 00:16:18,790

 

499
00:16:18,790 --> 00:16:20,710

 

500
00:16:20,710 --> 00:16:24,150

 

501
00:16:24,150 --> 00:16:24,160

 

502
00:16:24,160 --> 00:16:26,150


503
00:16:26,150 --> 00:16:27,670

 

504
00:16:27,670 --> 00:16:29,910

 

505
00:16:29,910 --> 00:16:32,310

 

506
00:16:32,310 --> 00:16:35,430

 

507
00:16:35,430 --> 00:16:38,949

 

508
00:16:38,949 --> 00:16:38,959

 

509
00:16:38,959 --> 00:16:39,670


510
00:16:39,670 --> 00:16:43,509

 

511
00:16:43,509 --> 00:16:45,670

 

512
00:16:45,670 --> 00:16:45,680

 

513
00:16:45,680 --> 00:16:46,389


514
00:16:46,389 --> 00:16:49,030

 

515
00:16:49,030 --> 00:16:51,269

 

516
00:16:51,269 --> 00:16:52,949

 

517
00:16:52,949 --> 00:16:53,749

 

518
00:16:53,749 --> 00:16:55,910

 

519
00:16:55,910 --> 00:16:58,790

 

520
00:16:58,790 --> 00:17:00,949

 

521
00:17:00,949 --> 00:17:02,790

 

522
00:17:02,790 --> 00:17:04,630

 

523
00:17:04,630 --> 00:17:07,270

 

524
00:17:07,270 --> 00:17:07,280

 

525
00:17:07,280 --> 00:17:08,549


526
00:17:08,549 --> 00:17:11,350

 

527
00:17:11,350 --> 00:17:14,309

 

528
00:17:14,309 --> 00:17:15,829

 

529
00:17:15,829 --> 00:17:15,839

 

530
00:17:15,839 --> 00:17:18,710


531
00:17:18,710 --> 00:17:20,549

 

532
00:17:20,549 --> 00:17:22,150

 

533
00:17:22,150 --> 00:17:24,549

 

534
00:17:24,549 --> 00:17:26,789

 

535
00:17:26,789 --> 00:17:28,630

 

536
00:17:28,630 --> 00:17:32,549

 

537
00:17:32,549 --> 00:17:33,830

 

538
00:17:33,830 --> 00:17:36,470

 

539
00:17:36,470 --> 00:17:39,029

 

540
00:17:39,029 --> 00:17:40,950

 

541
00:17:40,950 --> 00:17:43,510

 

542
00:17:43,510 --> 00:17:43,520

 

543
00:17:43,520 --> 00:17:44,549


544
00:17:44,549 --> 00:17:46,150

 

545
00:17:46,150 --> 00:17:48,870

 

546
00:17:48,870 --> 00:17:51,430

 

547
00:17:51,430 --> 00:17:53,669

 

548
00:17:53,669 --> 00:17:55,510

 

549
00:17:55,510 --> 00:17:55,520

 

550
00:17:55,520 --> 00:17:57,110


551
00:17:57,110 --> 00:17:59,590

 

552
00:17:59,590 --> 00:18:01,190

 

553
00:18:01,190 --> 00:18:02,630

 

554
00:18:02,630 --> 00:18:04,950

 

555
00:18:04,950 --> 00:18:04,960

 

556
00:18:04,960 --> 00:18:05,909


557
00:18:05,909 --> 00:18:08,390

 

558
00:18:08,390 --> 00:18:09,750

 

559
00:18:09,750 --> 00:18:11,590

 

560
00:18:11,590 --> 00:18:11,600

 

561
00:18:11,600 --> 00:18:13,430


562
00:18:13,430 --> 00:18:16,230

 

563
00:18:16,230 --> 00:18:18,390

 

564
00:18:18,390 --> 00:18:20,150

 

565
00:18:20,150 --> 00:18:22,549

 

566
00:18:22,549 --> 00:18:22,559

 

567
00:18:22,559 --> 00:18:25,350


568
00:18:25,350 --> 00:18:27,909

 

569
00:18:27,909 --> 00:18:29,590

 

570
00:18:29,590 --> 00:18:32,710

 

571
00:18:32,710 --> 00:18:36,630

 

572
00:18:36,630 --> 00:18:38,710

 

573
00:18:38,710 --> 00:18:41,510

 

574
00:18:41,510 --> 00:18:45,669

 

575
00:18:45,669 --> 00:18:48,070

 

576
00:18:48,070 --> 00:18:51,830

 

577
00:18:51,830 --> 00:18:53,990

 

578
00:18:53,990 --> 00:18:56,390

 

579
00:18:56,390 --> 00:18:57,669

 

580
00:18:57,669 --> 00:19:00,630

 

581
00:19:00,630 --> 00:19:02,630

 

582
00:19:02,630 --> 00:19:04,950

 

583
00:19:04,950 --> 00:19:06,950

 

584
00:19:06,950 --> 00:19:06,960

 

585
00:19:06,960 --> 00:19:08,230


586
00:19:08,230 --> 00:19:10,870

 

587
00:19:10,870 --> 00:19:13,510

 

588
00:19:13,510 --> 00:19:15,830

 

589
00:19:15,830 --> 00:19:16,870

 

590
00:19:16,870 --> 00:19:19,590

 

591
00:19:19,590 --> 00:19:21,190

 

592
00:19:21,190 --> 00:19:22,870

 

593
00:19:22,870 --> 00:19:24,710

 

594
00:19:24,710 --> 00:19:27,110

 

595
00:19:27,110 --> 00:19:30,470

 

596
00:19:30,470 --> 00:19:30,480

 

597
00:19:30,480 --> 00:19:37,029


598
00:19:37,029 --> 00:19:39,190

 

599
00:19:39,190 --> 00:19:39,200

 

600
00:19:39,200 --> 00:19:40,950


601
00:19:40,950 --> 00:19:44,470

 

602
00:19:44,470 --> 00:19:46,789

 

603
00:19:46,789 --> 00:19:49,669

 

604
00:19:49,669 --> 00:19:51,110

 

605
00:19:51,110 --> 00:19:52,789

 

606
00:19:52,789 --> 00:19:55,029

 

607
00:19:55,029 --> 00:19:58,230

 

608
00:19:58,230 --> 00:20:00,470

 

609
00:20:00,470 --> 00:20:01,510

 

610
00:20:01,510 --> 00:20:02,310

 

611
00:20:02,310 --> 00:20:05,510

 

612
00:20:05,510 --> 00:20:07,909

 

613
00:20:07,909 --> 00:20:10,950

 

614
00:20:10,950 --> 00:20:13,590

 

615
00:20:13,590 --> 00:20:17,590

 

616
00:20:17,590 --> 00:20:20,789

 

617
00:20:20,789 --> 00:20:23,029

 

618
00:20:23,029 --> 00:20:25,110

 

619
00:20:25,110 --> 00:20:27,270

 

620
00:20:27,270 --> 00:20:28,789

 

621
00:20:28,789 --> 00:20:30,390

 

622
00:20:30,390 --> 00:20:32,630

 

623
00:20:32,630 --> 00:20:34,789

 

624
00:20:34,789 --> 00:20:37,590

 

625
00:20:37,590 --> 00:20:39,350

 

626
00:20:39,350 --> 00:20:41,830

 

627
00:20:41,830 --> 00:20:43,510

 

628
00:20:43,510 --> 00:20:45,270

 

629
00:20:45,270 --> 00:20:49,270

 

630
00:20:49,270 --> 00:20:52,710

 

631
00:20:52,710 --> 00:20:54,870

 

632
00:20:54,870 --> 00:20:58,149

 

633
00:20:58,149 --> 00:20:59,669

 

634
00:20:59,669 --> 00:21:02,470

 

635
00:21:02,470 --> 00:21:02,480

 

636
00:21:02,480 --> 00:21:02,810


637
00:21:02,810 --> 00:21:02,820

 

638
00:21:02,820 --> 00:21:04,149


639
00:21:04,149 --> 00:21:05,750

 

640
00:21:05,750 --> 00:21:08,549

 

641
00:21:08,549 --> 00:21:10,470

 

642
00:21:10,470 --> 00:21:14,070

 

643
00:21:14,070 --> 00:21:17,110

 

644
00:21:17,110 --> 00:21:19,750

 

645
00:21:19,750 --> 00:21:23,510

 

646
00:21:23,510 --> 00:21:23,520

 

647
00:21:23,520 --> 00:21:24,310


648
00:21:24,310 --> 00:21:25,830

 

649
00:21:25,830 --> 00:21:28,070

 

650
00:21:28,070 --> 00:21:31,029

 

651
00:21:31,029 --> 00:21:32,950

 

652
00:21:32,950 --> 00:21:32,960

 

653
00:21:32,960 --> 00:21:35,590


654
00:21:35,590 --> 00:21:35,600

 

655
00:21:35,600 --> 00:21:38,789


656
00:21:38,789 --> 00:21:40,310

 

657
00:21:40,310 --> 00:21:42,390

 

658
00:21:42,390 --> 00:21:46,950

 

659
00:21:46,950 --> 00:21:46,960

 

660
00:21:46,960 --> 00:21:49,350


661
00:21:49,350 --> 00:21:50,789

 

662
00:21:50,789 --> 00:21:52,390

 

663
00:21:52,390 --> 00:21:53,830

 

664
00:21:53,830 --> 00:21:55,750

 

665
00:21:55,750 --> 00:21:57,430

 

666
00:21:57,430 --> 00:21:58,549

 

667
00:21:58,549 --> 00:22:00,630

 

668
00:22:00,630 --> 00:22:02,789

 

669
00:22:02,789 --> 00:22:05,909

 

670
00:22:05,909 --> 00:22:08,390

 

671
00:22:08,390 --> 00:22:08,400

 

672
00:22:08,400 --> 00:22:10,070


673
00:22:10,070 --> 00:22:10,080

 

674
00:22:10,080 --> 00:22:11,110


675
00:22:11,110 --> 00:22:13,110

 

676
00:22:13,110 --> 00:22:14,789

 

677
00:22:14,789 --> 00:22:16,789

 

678
00:22:16,789 --> 00:22:19,909

 

679
00:22:19,909 --> 00:22:21,750

 

680
00:22:21,750 --> 00:22:23,590

 

681
00:22:23,590 --> 00:22:25,830

 

682
00:22:25,830 --> 00:22:28,789

 

683
00:22:28,789 --> 00:22:28,799

 

684
00:22:28,799 --> 00:22:30,549


685
00:22:30,549 --> 00:22:32,230

 

686
00:22:32,230 --> 00:22:35,190

 

687
00:22:35,190 --> 00:22:37,990

 

688
00:22:37,990 --> 00:22:40,789

 

689
00:22:40,789 --> 00:22:40,799

 

690
00:22:40,799 --> 00:22:46,789


691
00:22:46,789 --> 00:22:48,470

 

692
00:22:48,470 --> 00:22:50,230

 

693
00:22:50,230 --> 00:22:51,430

 

694
00:22:51,430 --> 00:22:53,350

 

695
00:22:53,350 --> 00:22:55,110

 

696
00:22:55,110 --> 00:22:56,950

 

697
00:22:56,950 --> 00:22:58,470

 

698
00:22:58,470 --> 00:22:58,480

 

699
00:22:58,480 --> 00:23:01,990


700
00:23:01,990 --> 00:23:04,789

 

701
00:23:04,789 --> 00:23:08,789

 

702
00:23:08,789 --> 00:23:11,750

 

703
00:23:11,750 --> 00:23:14,390

 

704
00:23:14,390 --> 00:23:14,400

 

705
00:23:14,400 --> 00:23:15,590


706
00:23:15,590 --> 00:23:18,470

 

707
00:23:18,470 --> 00:23:20,149

 

708
00:23:20,149 --> 00:23:22,870

 

709
00:23:22,870 --> 00:23:22,880

 

710
00:23:22,880 --> 00:23:24,149


711
00:23:24,149 --> 00:23:25,909

 

712
00:23:25,909 --> 00:23:27,750

 

713
00:23:27,750 --> 00:23:30,070

 

714
00:23:30,070 --> 00:23:32,070

 

715
00:23:32,070 --> 00:23:35,029

 

716
00:23:35,029 --> 00:23:36,470

 

717
00:23:36,470 --> 00:23:38,549

 

718
00:23:38,549 --> 00:23:40,710

 

719
00:23:40,710 --> 00:23:42,710

 

720
00:23:42,710 --> 00:23:45,110

 

721
00:23:45,110 --> 00:23:45,120

 

722
00:23:45,120 --> 00:23:47,190


723
00:23:47,190 --> 00:23:47,200

 

724
00:23:47,200 --> 00:23:50,230


725
00:23:50,230 --> 00:23:52,390

 

726
00:23:52,390 --> 00:23:54,549

 

727
00:23:54,549 --> 00:23:56,630

 

728
00:23:56,630 --> 00:23:58,870

 

729
00:23:58,870 --> 00:24:00,149

 

730
00:24:00,149 --> 00:24:02,070

 

731
00:24:02,070 --> 00:24:05,590

 

732
00:24:05,590 --> 00:24:05,600

 

733
00:24:05,600 --> 00:24:08,310


734
00:24:08,310 --> 00:24:10,230

 

735
00:24:10,230 --> 00:24:12,310

 

736
00:24:12,310 --> 00:24:14,310

 

737
00:24:14,310 --> 00:24:15,430

 

738
00:24:15,430 --> 00:24:17,190

 

739
00:24:17,190 --> 00:24:20,470

 

740
00:24:20,470 --> 00:24:22,230

 

741
00:24:22,230 --> 00:24:22,240

 

742
00:24:22,240 --> 00:24:23,590


743
00:24:23,590 --> 00:24:26,070

 

744
00:24:26,070 --> 00:24:26,080

 

745
00:24:26,080 --> 00:24:27,909


746
00:24:27,909 --> 00:24:30,070

 

747
00:24:30,070 --> 00:24:30,080

 

748
00:24:30,080 --> 00:24:31,360


749
00:24:31,360 --> 00:24:31,370

 

750
00:24:31,370 --> 00:24:33,990


751
00:24:33,990 --> 00:24:36,470

 

752
00:24:36,470 --> 00:24:40,470

 

753
00:24:40,470 --> 00:24:40,480

 

754
00:24:40,480 --> 00:24:41,510


755
00:24:41,510 --> 00:24:42,789

 

756
00:24:42,789 --> 00:24:42,799

 

757
00:24:42,799 --> 00:24:44,830


758
00:24:44,830 --> 00:24:44,840

 

759
00:24:44,840 --> 00:24:46,470


760
00:24:46,470 --> 00:24:48,549

 

761
00:24:48,549 --> 00:24:50,149

 

762
00:24:50,149 --> 00:24:51,430

 

763
00:24:51,430 --> 00:24:53,110

 

764
00:24:53,110 --> 00:24:54,870

 

765
00:24:54,870 --> 00:24:56,789

 

766
00:24:56,789 --> 00:24:58,950

 

767
00:24:58,950 --> 00:25:01,269

 

768
00:25:01,269 --> 00:25:04,470

 

769
00:25:04,470 --> 00:25:05,750

 

770
00:25:05,750 --> 00:25:07,190

 

771
00:25:07,190 --> 00:25:09,269

 

772
00:25:09,269 --> 00:25:09,279

 

773
00:25:09,279 --> 00:25:10,789


774
00:25:10,789 --> 00:25:13,350

 

775
00:25:13,350 --> 00:25:15,909

 

776
00:25:15,909 --> 00:25:17,669

 

777
00:25:17,669 --> 00:25:19,590

 

778
00:25:19,590 --> 00:25:23,909

 

779
00:25:23,909 --> 00:25:26,470

 

780
00:25:26,470 --> 00:25:28,870

 

781
00:25:28,870 --> 00:25:31,590

 

782
00:25:31,590 --> 00:25:33,430

 

783
00:25:33,430 --> 00:25:33,440

 

784
00:25:33,440 --> 00:25:34,950


785
00:25:34,950 --> 00:25:34,960

 

786
00:25:34,960 --> 00:25:35,750


787
00:25:35,750 --> 00:25:40,710

 

788
00:25:40,710 --> 00:25:43,590

 

789
00:25:43,590 --> 00:25:45,669

 

790
00:25:45,669 --> 00:25:47,269

 

791
00:25:47,269 --> 00:25:49,110

 

792
00:25:49,110 --> 00:25:50,870

 

793
00:25:50,870 --> 00:25:54,470

 

794
00:25:54,470 --> 00:25:56,789

 

795
00:25:56,789 --> 00:25:59,110

 

796
00:25:59,110 --> 00:26:00,390

 

797
00:26:00,390 --> 00:26:00,400

 

798
00:26:00,400 --> 00:26:01,350


799
00:26:01,350 --> 00:26:03,990

 

800
00:26:03,990 --> 00:26:04,000

 

801
00:26:04,000 --> 00:26:05,350


802
00:26:05,350 --> 00:26:06,870

 

803
00:26:06,870 --> 00:26:09,430

 

804
00:26:09,430 --> 00:26:11,750

 

805
00:26:11,750 --> 00:26:13,510

 

806
00:26:13,510 --> 00:26:13,520

 

807
00:26:13,520 --> 00:26:16,230


808
00:26:16,230 --> 00:26:17,990

 

809
00:26:17,990 --> 00:26:19,590

 

810
00:26:19,590 --> 00:26:20,710

 

811
00:26:20,710 --> 00:26:23,669

 

812
00:26:23,669 --> 00:26:25,269

 

813
00:26:25,269 --> 00:26:27,909

 

814
00:26:27,909 --> 00:26:29,510

 

815
00:26:29,510 --> 00:26:31,350

 

816
00:26:31,350 --> 00:26:31,360

 

817
00:26:31,360 --> 00:26:32,630


818
00:26:32,630 --> 00:26:35,590

 

819
00:26:35,590 --> 00:26:35,600

 

820
00:26:35,600 --> 00:26:37,029


821
00:26:37,029 --> 00:26:38,230

 

822
00:26:38,230 --> 00:26:40,630

 

823
00:26:40,630 --> 00:26:42,789

 

824
00:26:42,789 --> 00:26:44,630

 

825
00:26:44,630 --> 00:26:46,549

 

826
00:26:46,549 --> 00:26:49,269

 

827
00:26:49,269 --> 00:26:50,470

 

828
00:26:50,470 --> 00:26:52,549

 

829
00:26:52,549 --> 00:26:56,070

 

830
00:26:56,070 --> 00:26:57,110

 

831
00:26:57,110 --> 00:27:01,269

 

832
00:27:01,269 --> 00:27:01,279

 

833
00:27:01,279 --> 00:27:02,870


834
00:27:02,870 --> 00:27:05,029

 

835
00:27:05,029 --> 00:27:06,789

 

836
00:27:06,789 --> 00:27:09,029

 

837
00:27:09,029 --> 00:27:11,669

 

838
00:27:11,669 --> 00:27:13,990

 

839
00:27:13,990 --> 00:27:15,830

 

840
00:27:15,830 --> 00:27:17,990

 

841
00:27:17,990 --> 00:27:20,470

 

842
00:27:20,470 --> 00:27:20,480

 

843
00:27:20,480 --> 00:27:23,590


844
00:27:23,590 --> 00:27:26,230

 

845
00:27:26,230 --> 00:27:26,240

 

846
00:27:26,240 --> 00:27:27,350


847
00:27:27,350 --> 00:27:28,470

 

848
00:27:28,470 --> 00:27:32,149

 

849
00:27:32,149 --> 00:27:34,870

 

850
00:27:34,870 --> 00:27:37,669

 

851
00:27:37,669 --> 00:27:39,669

 

852
00:27:39,669 --> 00:27:41,190

 

853
00:27:41,190 --> 00:27:42,470

 

854
00:27:42,470 --> 00:27:45,510

 

855
00:27:45,510 --> 00:27:45,520

 

856
00:27:45,520 --> 00:27:48,950


857
00:27:48,950 --> 00:27:51,669

 

858
00:27:51,669 --> 00:27:55,110

 

859
00:27:55,110 --> 00:27:57,430

 

860
00:27:57,430 --> 00:27:59,110

 

861
00:27:59,110 --> 00:28:00,950

 

862
00:28:00,950 --> 00:28:03,029

 

863
00:28:03,029 --> 00:28:06,549

 

864
00:28:06,549 --> 00:28:08,630

 

865
00:28:08,630 --> 00:28:10,310

 

866
00:28:10,310 --> 00:28:10,320

 

867
00:28:10,320 --> 00:28:11,590


868
00:28:11,590 --> 00:28:11,600

 

869
00:28:11,600 --> 00:28:12,470


870
00:28:12,470 --> 00:28:12,480

 

871
00:28:12,480 --> 00:28:14,149


872
00:28:14,149 --> 00:28:17,750

 

873
00:28:17,750 --> 00:28:19,669

 

874
00:28:19,669 --> 00:28:22,149

 

875
00:28:22,149 --> 00:28:23,669

 

876
00:28:23,669 --> 00:28:23,679

 

877
00:28:23,679 --> 00:28:25,269


878
00:28:25,269 --> 00:28:25,279

 

879
00:28:25,279 --> 00:28:26,470


880
00:28:26,470 --> 00:28:27,750

 

881
00:28:27,750 --> 00:28:30,549

 

882
00:28:30,549 --> 00:28:33,350

 

883
00:28:33,350 --> 00:28:35,750

 

884
00:28:35,750 --> 00:28:37,510

 

885
00:28:37,510 --> 00:28:39,430

 

886
00:28:39,430 --> 00:28:41,510

 

887
00:28:41,510 --> 00:28:43,430

 

888
00:28:43,430 --> 00:28:46,470

 

889
00:28:46,470 --> 00:28:47,750

 

890
00:28:47,750 --> 00:28:49,350

 

891
00:28:49,350 --> 00:28:51,350

 

892
00:28:51,350 --> 00:28:54,149

 

893
00:28:54,149 --> 00:28:56,310

 

894
00:28:56,310 --> 00:28:59,269

 

895
00:28:59,269 --> 00:29:00,870

 

896
00:29:00,870 --> 00:29:05,190

 

897
00:29:05,190 --> 00:29:08,149

 

898
00:29:08,149 --> 00:29:10,630

 

899
00:29:10,630 --> 00:29:12,950

 

900
00:29:12,950 --> 00:29:15,190

 

901
00:29:15,190 --> 00:29:17,669

 

902
00:29:17,669 --> 00:29:19,510

 

903
00:29:19,510 --> 00:29:21,590

 

904
00:29:21,590 --> 00:29:23,190

 

905
00:29:23,190 --> 00:29:24,870

 

906
00:29:24,870 --> 00:29:27,110

 

907
00:29:27,110 --> 00:29:27,120

 

908
00:29:27,120 --> 00:29:28,470


909
00:29:28,470 --> 00:29:30,549

 

910
00:29:30,549 --> 00:29:36,870

 

911
00:29:36,870 --> 00:29:38,710

 

912
00:29:38,710 --> 00:29:38,720

 

913
00:29:38,720 --> 00:29:40,470


914
00:29:40,470 --> 00:29:44,470

 

915
00:29:44,470 --> 00:29:47,909

 

916
00:29:47,909 --> 00:29:47,919

 

917
00:29:47,919 --> 00:29:49,750


918
00:29:49,750 --> 00:29:49,760

 

919
00:29:49,760 --> 00:29:51,669


920
00:29:51,669 --> 00:29:53,909

 

921
00:29:53,909 --> 00:29:55,990

 

922
00:29:55,990 --> 00:29:57,269

 

923
00:29:57,269 --> 00:30:00,149

 

924
00:30:00,149 --> 00:30:02,630

 

925
00:30:02,630 --> 00:30:03,909

 

926
00:30:03,909 --> 00:30:07,350

 

927
00:30:07,350 --> 00:30:07,360

 

928
00:30:07,360 --> 00:30:11,190


929
00:30:11,190 --> 00:30:12,630

 

930
00:30:12,630 --> 00:30:14,950

 

931
00:30:14,950 --> 00:30:16,789

 

932
00:30:16,789 --> 00:30:18,330

 

933
00:30:18,330 --> 00:30:18,340

 

934
00:30:18,340 --> 00:30:19,510


935
00:30:19,510 --> 00:30:21,590

 

936
00:30:21,590 --> 00:30:23,990

 

937
00:30:23,990 --> 00:30:25,590

 

938
00:30:25,590 --> 00:30:27,990

 

939
00:30:27,990 --> 00:30:32,149

 

940
00:30:32,149 --> 00:30:33,990

 

941
00:30:33,990 --> 00:30:36,149

 

942
00:30:36,149 --> 00:30:38,070

 

943
00:30:38,070 --> 00:30:39,909

 

944
00:30:39,909 --> 00:30:41,590

 

945
00:30:41,590 --> 00:30:43,269

 

946
00:30:43,269 --> 00:30:45,110

 

947
00:30:45,110 --> 00:30:47,190

 

948
00:30:47,190 --> 00:30:51,909

 

949
00:30:51,909 --> 00:30:51,919

 

950
00:30:51,919 --> 00:30:54,870


951
00:30:54,870 --> 00:30:54,880

 

952
00:30:54,880 --> 00:30:56,149


953
00:30:56,149 --> 00:30:57,750

 

954
00:30:57,750 --> 00:31:03,110

 

955
00:31:03,110 --> 00:31:05,909

 

956
00:31:05,909 --> 00:31:07,590

 

957
00:31:07,590 --> 00:31:09,190

 

958
00:31:09,190 --> 00:31:11,750

 

959
00:31:11,750 --> 00:31:13,590

 

960
00:31:13,590 --> 00:31:16,149

 

961
00:31:16,149 --> 00:31:17,509

 

962
00:31:17,509 --> 00:31:19,830

 

963
00:31:19,830 --> 00:31:20,950

 

964
00:31:20,950 --> 00:31:23,909

 

965
00:31:23,909 --> 00:31:25,830

 

966
00:31:25,830 --> 00:31:27,590

 

967
00:31:27,590 --> 00:31:29,269

 

968
00:31:29,269 --> 00:31:31,350

 

969
00:31:31,350 --> 00:31:33,750

 

970
00:31:33,750 --> 00:31:35,590

 

971
00:31:35,590 --> 00:31:38,470

 

972
00:31:38,470 --> 00:31:41,830

 

973
00:31:41,830 --> 00:31:41,840

 

974
00:31:41,840 --> 00:31:44,230


975
00:31:44,230 --> 00:31:46,389

 

976
00:31:46,389 --> 00:31:49,029

 

977
00:31:49,029 --> 00:31:51,269

 

978
00:31:51,269 --> 00:31:52,230

 

979
00:31:52,230 --> 00:31:53,750

 

980
00:31:53,750 --> 00:31:54,870

 

981
00:31:54,870 --> 00:31:54,880

 

982
00:31:54,880 --> 00:31:55,909


983
00:31:55,909 --> 00:31:58,950

 

984
00:31:58,950 --> 00:32:01,909

 

985
00:32:01,909 --> 00:32:03,669

 

986
00:32:03,669 --> 00:32:05,110

 

987
00:32:05,110 --> 00:32:07,269

 

988
00:32:07,269 --> 00:32:09,190

 

989
00:32:09,190 --> 00:32:11,430

 

990
00:32:11,430 --> 00:32:13,830

 

991
00:32:13,830 --> 00:32:13,840

 

992
00:32:13,840 --> 00:32:15,190


993
00:32:15,190 --> 00:32:15,200

 

994
00:32:15,200 --> 00:32:16,550


995
00:32:16,550 --> 00:32:16,560

 

996
00:32:16,560 --> 00:32:18,470


997
00:32:18,470 --> 00:32:20,870

 

998
00:32:20,870 --> 00:32:23,830

 

999
00:32:23,830 --> 00:32:23,840

 

1000
00:32:23,840 --> 00:32:24,789


1001
00:32:24,789 --> 00:32:27,909

 

1002
00:32:27,909 --> 00:32:30,070

 

1003
00:32:30,070 --> 00:32:30,080

 

1004
00:32:30,080 --> 00:32:31,029


1005
00:32:31,029 --> 00:32:32,789

 

1006
00:32:32,789 --> 00:32:34,710

 

1007
00:32:34,710 --> 00:32:35,909

 

1008
00:32:35,909 --> 00:32:37,590

 

1009
00:32:37,590 --> 00:32:39,029

 

1010
00:32:39,029 --> 00:32:42,950

 

1011
00:32:42,950 --> 00:32:42,960

 

1012
00:32:42,960 --> 00:32:43,830


1013
00:32:43,830 --> 00:32:46,389

 

1014
00:32:46,389 --> 00:32:49,029

 

1015
00:32:49,029 --> 00:32:50,630

 

1016
00:32:50,630 --> 00:32:53,110

 

1017
00:32:53,110 --> 00:32:56,630

 

1018
00:32:56,630 --> 00:32:59,029

 

1019
00:32:59,029 --> 00:33:03,669

 

1020
00:33:03,669 --> 00:33:04,870

 

1021
00:33:04,870 --> 00:33:06,630

 

1022
00:33:06,630 --> 00:33:06,640

 

1023
00:33:06,640 --> 00:33:07,909


1024
00:33:07,909 --> 00:33:09,509

 

1025
00:33:09,509 --> 00:33:11,269

 

1026
00:33:11,269 --> 00:33:12,950

 

1027
00:33:12,950 --> 00:33:15,269

 

1028
00:33:15,269 --> 00:33:16,389

 

1029
00:33:16,389 --> 00:33:18,149

 

1030
00:33:18,149 --> 00:33:19,590

 

1031
00:33:19,590 --> 00:33:21,669

 

1032
00:33:21,669 --> 00:33:23,750

 

1033
00:33:23,750 --> 00:33:26,070

 

1034
00:33:26,070 --> 00:33:26,080

 

1035
00:33:26,080 --> 00:33:27,350


1036
00:33:27,350 --> 00:33:30,710

 

1037
00:33:30,710 --> 00:33:32,789

 

1038
00:33:32,789 --> 00:33:32,799

 

1039
00:33:32,799 --> 00:33:34,230


1040
00:33:34,230 --> 00:33:35,750

 

1041
00:33:35,750 --> 00:33:39,350

 

1042
00:33:39,350 --> 00:33:41,509

 

1043
00:33:41,509 --> 00:33:42,630

 

1044
00:33:42,630 --> 00:33:44,789

 

1045
00:33:44,789 --> 00:33:46,710

 

1046
00:33:46,710 --> 00:33:49,190

 

1047
00:33:49,190 --> 00:33:51,269

 

1048
00:33:51,269 --> 00:33:53,669

 

1049
00:33:53,669 --> 00:33:55,909

 

1050
00:33:55,909 --> 00:33:58,470

 

1051
00:33:58,470 --> 00:34:00,710

 

1052
00:34:00,710 --> 00:34:02,470

 

1053
00:34:02,470 --> 00:34:05,110

 

1054
00:34:05,110 --> 00:34:08,869

 

1055
00:34:08,869 --> 00:34:08,879

 

1056
00:34:08,879 --> 00:34:09,669


1057
00:34:09,669 --> 00:34:11,990

 

1058
00:34:11,990 --> 00:34:14,230

 

1059
00:34:14,230 --> 00:34:17,270

 

1060
00:34:17,270 --> 00:34:19,589

 

1061
00:34:19,589 --> 00:34:21,990

 

1062
00:34:21,990 --> 00:34:24,710

 

1063
00:34:24,710 --> 00:34:26,950

 

1064
00:34:26,950 --> 00:34:29,349

 

1065
00:34:29,349 --> 00:34:31,270

 

1066
00:34:31,270 --> 00:34:33,430

 

1067
00:34:33,430 --> 00:34:36,069

 

1068
00:34:36,069 --> 00:34:37,750

 

1069
00:34:37,750 --> 00:34:39,589

 

1070
00:34:39,589 --> 00:34:41,829

 

1071
00:34:41,829 --> 00:34:44,710

 

1072
00:34:44,710 --> 00:34:47,270

 

1073
00:34:47,270 --> 00:34:49,109

 

1074
00:34:49,109 --> 00:34:50,710

 

1075
00:34:50,710 --> 00:34:52,790

 

1076
00:34:52,790 --> 00:34:54,710

 

1077
00:34:54,710 --> 00:34:54,720

 

1078
00:34:54,720 --> 00:34:56,149


1079
00:34:56,149 --> 00:34:56,159

 

1080
00:34:56,159 --> 00:34:56,950


1081
00:34:56,950 --> 00:34:58,550

 

1082
00:34:58,550 --> 00:35:00,470

 

1083
00:35:00,470 --> 00:35:02,390

 

1084
00:35:02,390 --> 00:35:04,150

 

1085
00:35:04,150 --> 00:35:06,069

 

1086
00:35:06,069 --> 00:35:07,910

 

1087
00:35:07,910 --> 00:35:07,920

 

1088
00:35:07,920 --> 00:35:11,109


1089
00:35:11,109 --> 00:35:11,119

 

1090
00:35:11,119 --> 00:35:12,310


1091
00:35:12,310 --> 00:35:14,230

 

1092
00:35:14,230 --> 00:35:17,510

 

1093
00:35:17,510 --> 00:35:19,190

 

1094
00:35:19,190 --> 00:35:22,470

 

1095
00:35:22,470 --> 00:35:25,430

 

1096
00:35:25,430 --> 00:35:26,950

 

1097
00:35:26,950 --> 00:35:28,630

 

1098
00:35:28,630 --> 00:35:30,630

 

1099
00:35:30,630 --> 00:35:32,870

 

1100
00:35:32,870 --> 00:35:35,030

 

1101
00:35:35,030 --> 00:35:39,670

 

1102
00:35:39,670 --> 00:35:41,750

 

1103
00:35:41,750 --> 00:35:44,310

 

1104
00:35:44,310 --> 00:35:45,750

 

1105
00:35:45,750 --> 00:35:47,670

 

1106
00:35:47,670 --> 00:35:47,680

 

1107
00:35:47,680 --> 00:35:48,630


1108
00:35:48,630 --> 00:35:50,390

 

1109
00:35:50,390 --> 00:35:52,470

 

1110
00:35:52,470 --> 00:35:54,230

 

1111
00:35:54,230 --> 00:35:55,910

 

1112
00:35:55,910 --> 00:35:57,430

 

1113
00:35:57,430 --> 00:35:59,349

 

1114
00:35:59,349 --> 00:36:01,670

 

1115
00:36:01,670 --> 00:36:05,109

 

1116
00:36:05,109 --> 00:36:06,950

 

1117
00:36:06,950 --> 00:36:06,960

 

1118
00:36:06,960 --> 00:36:11,030


1119
00:36:11,030 --> 00:36:13,670

 

1120
00:36:13,670 --> 00:36:16,630

 

1121
00:36:16,630 --> 00:36:18,470

 

1122
00:36:18,470 --> 00:36:20,470

 

1123
00:36:20,470 --> 00:36:23,270

 

1124
00:36:23,270 --> 00:36:28,710

 

1125
00:36:28,710 --> 00:36:32,470

 

1126
00:36:32,470 --> 00:36:34,950

 

1127
00:36:34,950 --> 00:36:37,270

 

1128
00:36:37,270 --> 00:36:40,710

 

1129
00:36:40,710 --> 00:36:43,109

 

1130
00:36:43,109 --> 00:36:44,550

 

1131
00:36:44,550 --> 00:36:47,910

 

1132
00:36:47,910 --> 00:36:50,230

 

1133
00:36:50,230 --> 00:36:52,069

 

1134
00:36:52,069 --> 00:36:53,109

 

1135
00:36:53,109 --> 00:36:56,150

 

1136
00:36:56,150 --> 00:36:58,310

 

1137
00:36:58,310 --> 00:37:00,150

 

1138
00:37:00,150 --> 00:37:02,069

 

1139
00:37:02,069 --> 00:37:04,230

 

1140
00:37:04,230 --> 00:37:06,470

 

1141
00:37:06,470 --> 00:37:09,190

 

1142
00:37:09,190 --> 00:37:12,069

 

1143
00:37:12,069 --> 00:37:13,990

 

1144
00:37:13,990 --> 00:37:16,310

 

1145
00:37:16,310 --> 00:37:16,320

 

1146
00:37:16,320 --> 00:37:20,550


1147
00:37:20,550 --> 00:37:20,560

 

1148
00:37:20,560 --> 00:37:21,349


1149
00:37:21,349 --> 00:37:23,589

 

1150
00:37:23,589 --> 00:37:23,599

 

1151
00:37:23,599 --> 00:37:24,710


1152
00:37:24,710 --> 00:37:26,630

 

1153
00:37:26,630 --> 00:37:28,630

 

1154
00:37:28,630 --> 00:37:31,750

 

1155
00:37:31,750 --> 00:37:33,030

 

1156
00:37:33,030 --> 00:37:35,109

 

1157
00:37:35,109 --> 00:37:36,630

 

1158
00:37:36,630 --> 00:37:36,640

 

1159
00:37:36,640 --> 00:37:37,589


1160
00:37:37,589 --> 00:37:39,910

 

1161
00:37:39,910 --> 00:37:42,710

 

1162
00:37:42,710 --> 00:37:45,109

 

1163
00:37:45,109 --> 00:37:46,390

 

1164
00:37:46,390 --> 00:37:48,150

 

1165
00:37:48,150 --> 00:37:50,230

 

1166
00:37:50,230 --> 00:37:52,710

 

1167
00:37:52,710 --> 00:37:52,720

 

1168
00:37:52,720 --> 00:37:55,030


1169
00:37:55,030 --> 00:37:57,030

 

1170
00:37:57,030 --> 00:37:57,040

 

1171
00:37:57,040 --> 00:37:57,829


1172
00:37:57,829 --> 00:37:59,589

 

1173
00:37:59,589 --> 00:38:01,589

 

1174
00:38:01,589 --> 00:38:04,390

 

1175
00:38:04,390 --> 00:38:05,990

 

1176
00:38:05,990 --> 00:38:08,390

 

1177
00:38:08,390 --> 00:38:08,400

 

1178
00:38:08,400 --> 00:38:09,430


1179
00:38:09,430 --> 00:38:11,109

 

1180
00:38:11,109 --> 00:38:13,589

 

1181
00:38:13,589 --> 00:38:15,430

 

1182
00:38:15,430 --> 00:38:17,910

 

1183
00:38:17,910 --> 00:38:21,829

 

1184
00:38:21,829 --> 00:38:24,390

 

1185
00:38:24,390 --> 00:38:27,430

 

1186
00:38:27,430 --> 00:38:27,440

 

1187
00:38:27,440 --> 00:38:28,630


1188
00:38:28,630 --> 00:38:28,640

 

1189
00:38:28,640 --> 00:38:30,710


1190
00:38:30,710 --> 00:38:32,870

 

1191
00:38:32,870 --> 00:38:33,990

 

1192
00:38:33,990 --> 00:38:36,550

 

1193
00:38:36,550 --> 00:38:38,230

 

1194
00:38:38,230 --> 00:38:40,230

 

1195
00:38:40,230 --> 00:38:42,550

 

1196
00:38:42,550 --> 00:38:44,310

 

1197
00:38:44,310 --> 00:38:47,510

 

1198
00:38:47,510 --> 00:38:49,670

 

1199
00:38:49,670 --> 00:38:52,870

 

1200
00:38:52,870 --> 00:38:52,880

 

1201
00:38:52,880 --> 00:38:53,990


1202
00:38:53,990 --> 00:38:55,349

 

1203
00:38:55,349 --> 00:38:57,349

 

1204
00:38:57,349 --> 00:38:59,990

 

1205
00:38:59,990 --> 00:39:00,000

 

1206
00:39:00,000 --> 00:39:01,270


1207
00:39:01,270 --> 00:39:03,109

 

1208
00:39:03,109 --> 00:39:04,870

 

1209
00:39:04,870 --> 00:39:07,589

 

1210
00:39:07,589 --> 00:39:10,710

 

1211
00:39:10,710 --> 00:39:12,870

 

1212
00:39:12,870 --> 00:39:15,349

 

1213
00:39:15,349 --> 00:39:17,829

 

1214
00:39:17,829 --> 00:39:19,430

 

1215
00:39:19,430 --> 00:39:22,069

 

1216
00:39:22,069 --> 00:39:24,310

 

1217
00:39:24,310 --> 00:39:26,870

 

1218
00:39:26,870 --> 00:39:29,430

 

1219
00:39:29,430 --> 00:39:31,190

 

1220
00:39:31,190 --> 00:39:33,430

 

1221
00:39:33,430 --> 00:39:35,750

 

1222
00:39:35,750 --> 00:39:37,349

 

1223
00:39:37,349 --> 00:39:39,430

 

1224
00:39:39,430 --> 00:39:40,950

 

1225
00:39:40,950 --> 00:39:43,589

 

1226
00:39:43,589 --> 00:39:43,599

 

1227
00:39:43,599 --> 00:39:44,790


1228
00:39:44,790 --> 00:39:46,310

 

1229
00:39:46,310 --> 00:39:47,510

 

1230
00:39:47,510 --> 00:39:50,790

 

1231
00:39:50,790 --> 00:39:53,750

 

1232
00:39:53,750 --> 00:39:55,829

 

1233
00:39:55,829 --> 00:39:58,069

 

1234
00:39:58,069 --> 00:39:58,079

 

1235
00:39:58,079 --> 00:40:00,630


1236
00:40:00,630 --> 00:40:03,270

 

1237
00:40:03,270 --> 00:40:04,790

 

1238
00:40:04,790 --> 00:40:04,800

 

1239
00:40:04,800 --> 00:40:05,829


1240
00:40:05,829 --> 00:40:07,670

 

1241
00:40:07,670 --> 00:40:09,510

 

1242
00:40:09,510 --> 00:40:11,510

 

1243
00:40:11,510 --> 00:40:11,520

 

1244
00:40:11,520 --> 00:40:12,390


1245
00:40:12,390 --> 00:40:12,400

 

1246
00:40:12,400 --> 00:40:14,069


1247
00:40:14,069 --> 00:40:16,470

 

1248
00:40:16,470 --> 00:40:16,480

 

1249
00:40:16,480 --> 00:40:17,910


1250
00:40:17,910 --> 00:40:20,950

 

1251
00:40:20,950 --> 00:40:22,950

 

1252
00:40:22,950 --> 00:40:24,230

 

1253
00:40:24,230 --> 00:40:25,910

 

1254
00:40:25,910 --> 00:40:26,950

 

1255
00:40:26,950 --> 00:40:26,960

 

1256
00:40:26,960 --> 00:40:27,910


1257
00:40:27,910 --> 00:40:30,309

 

1258
00:40:30,309 --> 00:40:30,319

 

1259
00:40:30,319 --> 00:40:31,430


1260
00:40:31,430 --> 00:40:34,710

 

1261
00:40:34,710 --> 00:40:37,109

 

1262
00:40:37,109 --> 00:40:40,390

 

1263
00:40:40,390 --> 00:40:40,400

 

1264
00:40:40,400 --> 00:40:41,910


1265
00:40:41,910 --> 00:40:45,270

 

1266
00:40:45,270 --> 00:40:47,030

 

1267
00:40:47,030 --> 00:40:49,510

 

1268
00:40:49,510 --> 00:40:51,750

 

1269
00:40:51,750 --> 00:40:53,910

 

1270
00:40:53,910 --> 00:40:55,270

 

1271
00:40:55,270 --> 00:40:57,270

 

1272
00:40:57,270 --> 00:40:58,870

 

1273
00:40:58,870 --> 00:41:00,630

 

1274
00:41:00,630 --> 00:41:02,470

 

1275
00:41:02,470 --> 00:41:05,349

 

1276
00:41:05,349 --> 00:41:05,359

 

1277
00:41:05,359 --> 00:41:06,710


1278
00:41:06,710 --> 00:41:07,750

 

1279
00:41:07,750 --> 00:41:07,760

 

1280
00:41:07,760 --> 00:41:09,030


1281
00:41:09,030 --> 00:41:11,990

 

1282
00:41:11,990 --> 00:41:14,069

 

1283
00:41:14,069 --> 00:41:14,079

 

1284
00:41:14,079 --> 00:41:15,430


1285
00:41:15,430 --> 00:41:15,440

 

1286
00:41:15,440 --> 00:41:16,710


1287
00:41:16,710 --> 00:41:18,470

 

1288
00:41:18,470 --> 00:41:20,069

 

1289
00:41:20,069 --> 00:41:21,430

 

1290
00:41:21,430 --> 00:41:23,270

 

1291
00:41:23,270 --> 00:41:24,870

 

1292
00:41:24,870 --> 00:41:26,390

 

1293
00:41:26,390 --> 00:41:26,400

 

1294
00:41:26,400 --> 00:41:27,190


1295
00:41:27,190 --> 00:41:29,670

 

1296
00:41:29,670 --> 00:41:31,109

 

1297
00:41:31,109 --> 00:41:32,069

 

1298
00:41:32,069 --> 00:41:34,870

 

1299
00:41:34,870 --> 00:41:36,390

 

1300
00:41:36,390 --> 00:41:38,630

 

1301
00:41:38,630 --> 00:41:40,390

 

1302
00:41:40,390 --> 00:41:41,670

 

1303
00:41:41,670 --> 00:41:44,069

 

1304
00:41:44,069 --> 00:41:45,270

 

1305
00:41:45,270 --> 00:41:47,510

 

1306
00:41:47,510 --> 00:41:49,190

 

1307
00:41:49,190 --> 00:41:52,150

 

1308
00:41:52,150 --> 00:41:54,870

 

1309
00:41:54,870 --> 00:41:56,470

 

1310
00:41:56,470 --> 00:41:58,630

 

1311
00:41:58,630 --> 00:42:03,270

 

1312
00:42:03,270 --> 00:42:05,910

 

1313
00:42:05,910 --> 00:42:05,920

 

1314
00:42:05,920 --> 00:42:08,390


1315
00:42:08,390 --> 00:42:08,400

 

1316
00:42:08,400 --> 00:42:09,829


1317
00:42:09,829 --> 00:42:12,309

 

1318
00:42:12,309 --> 00:42:13,910

 

1319
00:42:13,910 --> 00:42:13,920

 

1320
00:42:13,920 --> 00:42:15,030


1321
00:42:15,030 --> 00:42:18,470

 

1322
00:42:18,470 --> 00:42:20,470

 

1323
00:42:20,470 --> 00:42:23,829

 

1324
00:42:23,829 --> 00:42:23,839

 

1325
00:42:23,839 --> 00:42:24,870


1326
00:42:24,870 --> 00:42:28,630

 

1327
00:42:28,630 --> 00:42:29,910

 

1328
00:42:29,910 --> 00:42:31,670

 

1329
00:42:31,670 --> 00:42:35,349

 

1330
00:42:35,349 --> 00:42:39,030

 

1331
00:42:39,030 --> 00:42:42,390

 

1332
00:42:42,390 --> 00:42:43,829

 

1333
00:42:43,829 --> 00:42:45,510

 

1334
00:42:45,510 --> 00:42:47,510

 

1335
00:42:47,510 --> 00:42:51,750

 

1336
00:42:51,750 --> 00:42:53,750

 

1337
00:42:53,750 --> 00:42:53,760

 

1338
00:42:53,760 --> 00:42:54,470


1339
00:42:54,470 --> 00:42:56,069

 

1340
00:42:56,069 --> 00:42:59,670

 

1341
00:42:59,670 --> 00:43:05,030

 

1342
00:43:05,030 --> 00:43:07,829

 

1343
00:43:07,829 --> 00:43:11,030

 

1344
00:43:11,030 --> 00:43:13,750

 

1345
00:43:13,750 --> 00:43:15,349

 

1346
00:43:15,349 --> 00:43:15,359

 

1347
00:43:15,359 --> 00:43:17,190


1348
00:43:17,190 --> 00:43:17,200

 

1349
00:43:17,200 --> 00:43:20,309


1350
00:43:20,309 --> 00:43:23,270

 

1351
00:43:23,270 --> 00:43:25,190

 

1352
00:43:25,190 --> 00:43:25,200

 

1353
00:43:25,200 --> 00:43:27,270


1354
00:43:27,270 --> 00:43:29,270

 

1355
00:43:29,270 --> 00:43:32,069

 

1356
00:43:32,069 --> 00:43:34,390

 

1357
00:43:34,390 --> 00:43:35,670

 

1358
00:43:35,670 --> 00:43:37,349

 

1359
00:43:37,349 --> 00:43:37,359

 

1360
00:43:37,359 --> 00:43:38,069


1361
00:43:38,069 --> 00:43:41,589

 

1362
00:43:41,589 --> 00:43:43,270

 

1363
00:43:43,270 --> 00:43:45,829

 

1364
00:43:45,829 --> 00:43:48,390

 

1365
00:43:48,390 --> 00:43:49,750

 

1366
00:43:49,750 --> 00:43:51,589

 

1367
00:43:51,589 --> 00:43:51,599

 

1368
00:43:51,599 --> 00:43:53,030


1369
00:43:53,030 --> 00:43:55,510

 

1370
00:43:55,510 --> 00:43:58,470

 

1371
00:43:58,470 --> 00:44:00,069

 

1372
00:44:00,069 --> 00:44:02,790

 

1373
00:44:02,790 --> 00:44:04,309

 

1374
00:44:04,309 --> 00:44:04,319

 

1375
00:44:04,319 --> 00:44:05,430


1376
00:44:05,430 --> 00:44:08,150

 

1377
00:44:08,150 --> 00:44:11,109

 

1378
00:44:11,109 --> 00:44:12,950

 

1379
00:44:12,950 --> 00:44:14,390

 

1380
00:44:14,390 --> 00:44:16,309

 

1381
00:44:16,309 --> 00:44:17,270

 

1382
00:44:17,270 --> 00:44:20,630

 

1383
00:44:20,630 --> 00:44:22,390

 

1384
00:44:22,390 --> 00:44:25,990

 

1385
00:44:25,990 --> 00:44:28,069

 

1386
00:44:28,069 --> 00:44:38,309

 

1387
00:44:38,309 --> 00:44:38,319

 

1388
00:44:38,319 --> 00:44:39,510


1389
00:44:39,510 --> 00:44:39,520

 

1390
00:44:39,520 --> 00:44:43,750


1391
00:44:43,750 --> 00:44:46,950

 

1392
00:44:46,950 --> 00:44:50,390

 

1393
00:44:50,390 --> 00:44:53,190

 

1394
00:44:53,190 --> 00:44:54,710

 

1395
00:44:54,710 --> 00:44:54,720

 

1396
00:44:54,720 --> 00:44:55,910


1397
00:44:55,910 --> 00:44:59,270

 

1398
00:44:59,270 --> 00:44:59,280

 

1399
00:44:59,280 --> 00:45:00,470


1400
00:45:00,470 --> 00:45:03,349

 

1401
00:45:03,349 --> 00:45:04,950

 

1402
00:45:04,950 --> 00:45:07,750

 

1403
00:45:07,750 --> 00:45:11,430

 

1404
00:45:11,430 --> 00:45:14,309

 

1405
00:45:14,309 --> 00:45:14,319

 

1406
00:45:14,319 --> 00:45:17,349


1407
00:45:17,349 --> 00:45:19,510

 

1408
00:45:19,510 --> 00:45:22,710

 

1409
00:45:22,710 --> 00:45:23,910

 

1410
00:45:23,910 --> 00:45:26,150

 

1411
00:45:26,150 --> 00:45:26,160

 

1412
00:45:26,160 --> 00:45:27,589


1413
00:45:27,589 --> 00:45:32,710

 

1414
00:45:32,710 --> 00:45:35,349

 

1415
00:45:35,349 --> 00:45:37,670

 

1416
00:45:37,670 --> 00:45:41,510

 

1417
00:45:41,510 --> 00:45:44,710

 

1418
00:45:44,710 --> 00:45:47,670

 

1419
00:45:47,670 --> 00:45:49,670

 

1420
00:45:49,670 --> 00:45:52,950

 

1421
00:45:52,950 --> 00:45:55,430

 

1422
00:45:55,430 --> 00:45:57,270

 

1423
00:45:57,270 --> 00:45:59,829

 

1424
00:45:59,829 --> 00:46:00,790

 

1425
00:46:00,790 --> 00:46:02,870

 

1426
00:46:02,870 --> 00:46:05,910

 

1427
00:46:05,910 --> 00:46:05,920

 

1428
00:46:05,920 --> 00:46:07,270


1429
00:46:07,270 --> 00:46:08,710

 

1430
00:46:08,710 --> 00:46:11,430

 

1431
00:46:11,430 --> 00:46:12,470

 

1432
00:46:12,470 --> 00:46:15,510

 

1433
00:46:15,510 --> 00:46:17,270

 

1434
00:46:17,270 --> 00:46:17,280

 

1435
00:46:17,280 --> 00:46:18,470


1436
00:46:18,470 --> 00:46:18,480

 

1437
00:46:18,480 --> 00:46:19,430


1438
00:46:19,430 --> 00:46:21,589

 

1439
00:46:21,589 --> 00:46:21,599

 

1440
00:46:21,599 --> 00:46:22,710


1441
00:46:22,710 --> 00:46:24,630

 

1442
00:46:24,630 --> 00:46:27,030

 

1443
00:46:27,030 --> 00:46:30,309

 

1444
00:46:30,309 --> 00:46:32,550

 

1445
00:46:32,550 --> 00:46:33,750

 

1446
00:46:33,750 --> 00:46:35,510

 

1447
00:46:35,510 --> 00:46:37,270

 

1448
00:46:37,270 --> 00:46:38,950

 

1449
00:46:38,950 --> 00:46:38,960

 

1450
00:46:38,960 --> 00:46:39,910


1451
00:46:39,910 --> 00:46:41,910

 

1452
00:46:41,910 --> 00:46:45,190

 

1453
00:46:45,190 --> 00:46:46,870

 

1454
00:46:46,870 --> 00:46:49,990

 

1455
00:46:49,990 --> 00:46:52,710

 

1456
00:46:52,710 --> 00:46:55,030

 

1457
00:46:55,030 --> 00:46:58,390

 

1458
00:46:58,390 --> 00:47:00,069

 

1459
00:47:00,069 --> 00:47:01,829

 

1460
00:47:01,829 --> 00:47:05,190

 

1461
00:47:05,190 --> 00:47:07,750

 

1462
00:47:07,750 --> 00:47:09,510

 

1463
00:47:09,510 --> 00:47:12,069

 

1464
00:47:12,069 --> 00:47:13,910

 

1465
00:47:13,910 --> 00:47:15,750

 

1466
00:47:15,750 --> 00:47:17,430

 

1467
00:47:17,430 --> 00:47:17,440

 

1468
00:47:17,440 --> 00:47:20,549


1469
00:47:20,549 --> 00:47:24,230

 

1470
00:47:24,230 --> 00:47:26,150

 

1471
00:47:26,150 --> 00:47:27,430

 

1472
00:47:27,430 --> 00:47:29,349

 

1473
00:47:29,349 --> 00:47:31,750

 

1474
00:47:31,750 --> 00:47:33,030

 

1475
00:47:33,030 --> 00:47:38,069

 

1476
00:47:38,069 --> 00:47:40,069

 

1477
00:47:40,069 --> 00:47:43,030

 

1478
00:47:43,030 --> 00:47:46,630

 

1479
00:47:46,630 --> 00:47:46,640

 

1480
00:47:46,640 --> 00:47:47,829


1481
00:47:47,829 --> 00:47:47,839

 

1482
00:47:47,839 --> 00:47:51,109


1483
00:47:51,109 --> 00:47:52,870

 

1484
00:47:52,870 --> 00:47:55,349

 

1485
00:47:55,349 --> 00:47:55,359

 

1486
00:47:55,359 --> 00:47:55,840


1487
00:47:55,840 --> 00:47:55,850

 

1488
00:47:55,850 --> 00:47:57,750


1489
00:47:57,750 --> 00:48:00,309

 

1490
00:48:00,309 --> 00:48:02,150

 

1491
00:48:02,150 --> 00:48:06,069

 

1492
00:48:06,069 --> 00:48:11,589

 

1493
00:48:11,589 --> 00:48:13,750

 

1494
00:48:13,750 --> 00:48:18,829

 

1495
00:48:18,829 --> 00:48:21,349

 

1496
00:48:21,349 --> 00:48:24,470

 

1497
00:48:24,470 --> 00:48:26,069

 

1498
00:48:26,069 --> 00:48:27,349

 

1499
00:48:27,349 --> 00:48:29,030

 

1500
00:48:29,030 --> 00:48:30,950

 

1501
00:48:30,950 --> 00:48:33,589

 

1502
00:48:33,589 --> 00:48:35,829

 

1503
00:48:35,829 --> 00:48:35,839

 

1504
00:48:35,839 --> 00:48:36,870


1505
00:48:36,870 --> 00:48:39,270

 

1506
00:48:39,270 --> 00:48:41,349

 

1507
00:48:41,349 --> 00:48:43,589

 

1508
00:48:43,589 --> 00:48:45,349

 

1509
00:48:45,349 --> 00:48:47,430

 

1510
00:48:47,430 --> 00:48:49,589

 

1511
00:48:49,589 --> 00:48:51,990

 

1512
00:48:51,990 --> 00:48:53,109

 

1513
00:48:53,109 --> 00:48:55,510

 

1514
00:48:55,510 --> 00:48:57,430

 

1515
00:48:57,430 --> 00:48:59,670

 

1516
00:48:59,670 --> 00:49:01,430

 

1517
00:49:01,430 --> 00:49:04,950

 

1518
00:49:04,950 --> 00:49:07,510

 

1519
00:49:07,510 --> 00:49:11,910

 

1520
00:49:11,910 --> 00:49:14,390

 

1521
00:49:14,390 --> 00:49:16,309

 

1522
00:49:16,309 --> 00:49:16,319

 

1523
00:49:16,319 --> 00:49:17,750


1524
00:49:17,750 --> 00:49:19,829

 

1525
00:49:19,829 --> 00:49:21,030

 

1526
00:49:21,030 --> 00:49:23,589

 

1527
00:49:23,589 --> 00:49:25,670

 

1528
00:49:25,670 --> 00:49:27,190

 

1529
00:49:27,190 --> 00:49:29,750

 

1530
00:49:29,750 --> 00:49:31,270

 

1531
00:49:31,270 --> 00:49:33,589

 

1532
00:49:33,589 --> 00:49:36,790

 

1533
00:49:36,790 --> 00:49:36,800

 

1534
00:49:36,800 --> 00:49:37,589


1535
00:49:37,589 --> 00:49:39,990

 

1536
00:49:39,990 --> 00:49:42,549

 

1537
00:49:42,549 --> 00:49:45,270

 

1538
00:49:45,270 --> 00:49:47,190

 

1539
00:49:47,190 --> 00:49:50,390

 

1540
00:49:50,390 --> 00:49:52,069

 

1541
00:49:52,069 --> 00:49:53,750

 

1542
00:49:53,750 --> 00:49:55,430

 

1543
00:49:55,430 --> 00:49:56,549

 

1544
00:49:56,549 --> 00:49:59,670

 

1545
00:49:59,670 --> 00:50:01,510

 

1546
00:50:01,510 --> 00:50:03,829

 

1547
00:50:03,829 --> 00:50:05,990

 

1548
00:50:05,990 --> 00:50:08,069

 

1549
00:50:08,069 --> 00:50:10,710

 

1550
00:50:10,710 --> 00:50:13,270

 

1551
00:50:13,270 --> 00:50:17,270

 

1552
00:50:17,270 --> 00:50:17,280

 

1553
00:50:17,280 --> 00:50:18,390


1554
00:50:18,390 --> 00:50:20,870

 

1555
00:50:20,870 --> 00:50:25,510

 

1556
00:50:25,510 --> 00:50:27,030

 

1557
00:50:27,030 --> 00:50:28,470

 

1558
00:50:28,470 --> 00:50:34,630

 

1559
00:50:34,630 --> 00:50:37,349

 

1560
00:50:37,349 --> 00:50:40,790

 

1561
00:50:40,790 --> 00:50:42,950

 

1562
00:50:42,950 --> 00:50:45,990

 

1563
00:50:45,990 --> 00:50:48,309

 

1564
00:50:48,309 --> 00:50:50,790

 

1565
00:50:50,790 --> 00:50:53,750

 

1566
00:50:53,750 --> 00:50:56,150

 

1567
00:50:56,150 --> 00:50:58,710

 

1568
00:50:58,710 --> 00:51:01,589

 

1569
00:51:01,589 --> 00:51:05,109

 

1570
00:51:05,109 --> 00:51:05,119

 

1571
00:51:05,119 --> 00:51:06,150


1572
00:51:06,150 --> 00:51:09,990

 

1573
00:51:09,990 --> 00:51:10,000

 

1574
00:51:10,000 --> 00:51:11,349


1575
00:51:11,349 --> 00:51:11,359

 

1576
00:51:11,359 --> 00:51:12,150


1577
00:51:12,150 --> 00:51:12,160

 

1578
00:51:12,160 --> 00:51:12,950


1579
00:51:12,950 --> 00:51:15,349

 

1580
00:51:15,349 --> 00:51:18,230

 

1581
00:51:18,230 --> 00:51:19,190

 

1582
00:51:19,190 --> 00:51:21,349

 

1583
00:51:21,349 --> 00:51:23,910

 

1584
00:51:23,910 --> 00:51:26,950

 

1585
00:51:26,950 --> 00:51:26,960

 

1586
00:51:26,960 --> 00:51:29,270


1587
00:51:29,270 --> 00:51:31,589

 

1588
00:51:31,589 --> 00:51:33,430

 

1589
00:51:33,430 --> 00:51:37,109

 

1590
00:51:37,109 --> 00:51:38,870

 

1591
00:51:38,870 --> 00:51:41,430

 

1592
00:51:41,430 --> 00:51:43,910

 

1593
00:51:43,910 --> 00:51:46,150

 

1594
00:51:46,150 --> 00:51:46,160

 

1595
00:51:46,160 --> 00:51:47,589


1596
00:51:47,589 --> 00:51:47,599

 

1597
00:51:47,599 --> 00:51:49,670


1598
00:51:49,670 --> 00:51:49,680

 

1599
00:51:49,680 --> 00:51:50,549


1600
00:51:50,549 --> 00:51:51,910

 

1601
00:51:51,910 --> 00:51:54,710

 

1602
00:51:54,710 --> 00:51:56,870

 

1603
00:51:56,870 --> 00:51:59,510

 

1604
00:51:59,510 --> 00:52:04,390

 

1605
00:52:04,390 --> 00:52:08,790

 

1606
00:52:08,790 --> 00:52:12,069

 

1607
00:52:12,069 --> 00:52:15,190

 

1608
00:52:15,190 --> 00:52:17,589

 

1609
00:52:17,589 --> 00:52:20,549

 

1610
00:52:20,549 --> 00:52:23,829

 

1611
00:52:23,829 --> 00:52:27,349

 

1612
00:52:27,349 --> 00:52:30,390

 

1613
00:52:30,390 --> 00:52:34,230

 

1614
00:52:34,230 --> 00:52:36,150

 

1615
00:52:36,150 --> 00:52:38,710

 

1616
00:52:38,710 --> 00:52:40,950

 

1617
00:52:40,950 --> 00:52:44,150

 

1618
00:52:44,150 --> 00:52:46,549

 

1619
00:52:46,549 --> 00:52:48,710

 

1620
00:52:48,710 --> 00:52:50,630

 

1621
00:52:50,630 --> 00:52:52,390

 

1622
00:52:52,390 --> 00:52:54,710

 

1623
00:52:54,710 --> 00:52:56,230

 

1624
00:52:56,230 --> 00:52:56,240

 

1625
00:52:56,240 --> 00:52:56,950


1626
00:52:56,950 --> 00:53:00,790

 

1627
00:53:00,790 --> 00:53:03,829

 

1628
00:53:03,829 --> 00:53:03,839

 

1629
00:53:03,839 --> 00:53:05,910


1630
00:53:05,910 --> 00:53:08,549

 

1631
00:53:08,549 --> 00:53:11,670

 

1632
00:53:11,670 --> 00:53:13,589

 

1633
00:53:13,589 --> 00:53:15,750

 

1634
00:53:15,750 --> 00:53:18,390

 

1635
00:53:18,390 --> 00:53:20,630

 

1636
00:53:20,630 --> 00:53:22,069

 

1637
00:53:22,069 --> 00:53:23,750

 

1638
00:53:23,750 --> 00:53:23,760

 

1639
00:53:23,760 --> 00:53:27,270


1640
00:53:27,270 --> 00:53:28,710

 

1641
00:53:28,710 --> 00:53:32,470

 

1642
00:53:32,470 --> 00:53:32,480

 

1643
00:53:32,480 --> 00:53:35,670


1644
00:53:35,670 --> 00:53:35,680

 

1645
00:53:35,680 --> 00:53:36,950


1646
00:53:36,950 --> 00:53:40,309

 

1647
00:53:40,309 --> 00:53:42,549

 

1648
00:53:42,549 --> 00:53:45,430

 

1649
00:53:45,430 --> 00:53:47,430

 

1650
00:53:47,430 --> 00:53:49,030

 

1651
00:53:49,030 --> 00:53:49,040

 

1652
00:53:49,040 --> 00:53:50,470


1653
00:53:50,470 --> 00:53:53,430

 

1654
00:53:53,430 --> 00:53:55,109

 

1655
00:53:55,109 --> 00:53:56,470

 

1656
00:53:56,470 --> 00:53:58,470

 

1657
00:53:58,470 --> 00:54:01,030

 

1658
00:54:01,030 --> 00:54:04,309

 

1659
00:54:04,309 --> 00:54:04,319

 

1660
00:54:04,319 --> 00:54:05,430


1661
00:54:05,430 --> 00:54:07,510

 

1662
00:54:07,510 --> 00:54:09,030

 

1663
00:54:09,030 --> 00:54:11,750

 

1664
00:54:11,750 --> 00:54:15,109

 

1665
00:54:15,109 --> 00:54:16,870

 

1666
00:54:16,870 --> 00:54:19,109

 

1667
00:54:19,109 --> 00:54:21,430

 

1668
00:54:21,430 --> 00:54:22,790

 

1669
00:54:22,790 --> 00:54:25,109

 

1670
00:54:25,109 --> 00:54:26,950

 

1671
00:54:26,950 --> 00:54:28,470

 

1672
00:54:28,470 --> 00:54:28,480

 

1673
00:54:28,480 --> 00:54:31,750


1674
00:54:31,750 --> 00:54:33,829

 

1675
00:54:33,829 --> 00:54:36,069

 

1676
00:54:36,069 --> 00:54:41,190

 

1677
00:54:41,190 --> 00:54:43,430

 

1678
00:54:43,430 --> 00:54:43,440

 

1679
00:54:43,440 --> 00:54:44,230


1680
00:54:44,230 --> 00:54:45,750

 

1681
00:54:45,750 --> 00:54:45,760

 

1682
00:54:45,760 --> 00:54:47,910


1683
00:54:47,910 --> 00:54:47,920

 

1684
00:54:47,920 --> 00:54:49,510


1685
00:54:49,510 --> 00:54:50,870

 

1686
00:54:50,870 --> 00:54:51,910

 

1687
00:54:51,910 --> 00:54:53,750

 

1688
00:54:53,750 --> 00:54:56,870

 

1689
00:54:56,870 --> 00:54:58,549

 

1690
00:54:58,549 --> 00:55:00,630

 

1691
00:55:00,630 --> 00:55:03,349

 

1692
00:55:03,349 --> 00:55:05,030

 

1693
00:55:05,030 --> 00:55:07,910

 

1694
00:55:07,910 --> 00:55:09,829

 

1695
00:55:09,829 --> 00:55:11,910

 

1696
00:55:11,910 --> 00:55:13,430

 

1697
00:55:13,430 --> 00:55:16,150

 

1698
00:55:16,150 --> 00:55:17,670

 

1699
00:55:17,670 --> 00:55:18,630

 

1700
00:55:18,630 --> 00:55:19,829

 

1701
00:55:19,829 --> 00:55:21,430

 

1702
00:55:21,430 --> 00:55:24,069

 

1703
00:55:24,069 --> 00:55:26,390

 

1704
00:55:26,390 --> 00:55:26,400

 

1705
00:55:26,400 --> 00:55:29,109


1706
00:55:29,109 --> 00:55:30,789

 

1707
00:55:30,789 --> 00:55:33,270

 

1708
00:55:33,270 --> 00:55:33,280

 

1709
00:55:33,280 --> 00:55:35,190


1710
00:55:35,190 --> 00:55:35,200

 

1711
00:55:35,200 --> 00:55:36,390


1712
00:55:36,390 --> 00:55:38,470

 

1713
00:55:38,470 --> 00:55:38,480

 

1714
00:55:38,480 --> 00:55:39,270


1715
00:55:39,270 --> 00:55:43,670

 

1716
00:55:43,670 --> 00:55:45,750

 

1717
00:55:45,750 --> 00:55:46,870

 

1718
00:55:46,870 --> 00:55:49,510

 

1719
00:55:49,510 --> 00:55:51,829

 

1720
00:55:51,829 --> 00:55:54,069

 

1721
00:55:54,069 --> 00:55:56,069

 

1722
00:55:56,069 --> 00:55:57,750

 

1723
00:55:57,750 --> 00:55:59,349

 

1724
00:55:59,349 --> 00:56:01,349

 

1725
00:56:01,349 --> 00:56:03,670

 

1726
00:56:03,670 --> 00:56:03,680

 

1727
00:56:03,680 --> 00:56:04,549


1728
00:56:04,549 --> 00:56:06,150

 

1729
00:56:06,150 --> 00:56:08,390

 

1730
00:56:08,390 --> 00:56:11,510

 

1731
00:56:11,510 --> 00:56:14,950

 

1732
00:56:14,950 --> 00:56:16,789

 

1733
00:56:16,789 --> 00:56:16,799

 

1734
00:56:16,799 --> 00:56:20,470


1735
00:56:20,470 --> 00:56:21,990

 

1736
00:56:21,990 --> 00:56:24,390

 

1737
00:56:24,390 --> 00:56:26,630

 

1738
00:56:26,630 --> 00:56:29,190

 

1739
00:56:29,190 --> 00:56:31,270

 

1740
00:56:31,270 --> 00:56:31,280

 

1741
00:56:31,280 --> 00:56:32,789


1742
00:56:32,789 --> 00:56:34,230

 

1743
00:56:34,230 --> 00:56:37,349

 

1744
00:56:37,349 --> 00:56:40,789

 

1745
00:56:40,789 --> 00:56:43,030

 

1746
00:56:43,030 --> 00:56:46,470

 

1747
00:56:46,470 --> 00:56:48,789

 

1748
00:56:48,789 --> 00:56:51,190

 

1749
00:56:51,190 --> 00:56:53,190

 

1750
00:56:53,190 --> 00:56:55,109

 

1751
00:56:55,109 --> 00:56:57,270

 

1752
00:56:57,270 --> 00:56:59,510

 

1753
00:56:59,510 --> 00:57:01,190

 

1754
00:57:01,190 --> 00:57:01,200

 

1755
00:57:01,200 --> 00:57:03,030


1756
00:57:03,030 --> 00:57:04,710

 

1757
00:57:04,710 --> 00:57:07,109

 

1758
00:57:07,109 --> 00:57:09,109

 

1759
00:57:09,109 --> 00:57:11,190

 

1760
00:57:11,190 --> 00:57:13,030

 

1761
00:57:13,030 --> 00:57:18,069

 

1762
00:57:18,069 --> 00:57:20,230

 

1763
00:57:20,230 --> 00:57:21,750

 

1764
00:57:21,750 --> 00:57:24,069

 

1765
00:57:24,069 --> 00:57:30,789

 

1766
00:57:30,789 --> 00:57:32,390

 

1767
00:57:32,390 --> 00:57:36,069

 

1768
00:57:36,069 --> 00:57:37,109

 

1769
00:57:37,109 --> 00:57:39,589

 

1770
00:57:39,589 --> 00:57:41,910

 

1771
00:57:41,910 --> 00:57:43,829

 

1772
00:57:43,829 --> 00:57:45,510

 

1773
00:57:45,510 --> 00:57:46,390

 

1774
00:57:46,390 --> 00:57:48,789

 

1775
00:57:48,789 --> 00:57:50,549

 

1776
00:57:50,549 --> 00:57:53,510

 

1777
00:57:53,510 --> 00:57:56,549

 

1778
00:57:56,549 --> 00:57:58,150

 

1779
00:57:58,150 --> 00:57:59,750

 

1780
00:57:59,750 --> 00:58:02,470

 

1781
00:58:02,470 --> 00:58:05,030

 

1782
00:58:05,030 --> 00:58:06,630

 

1783
00:58:06,630 --> 00:58:09,190

 

1784
00:58:09,190 --> 00:58:09,200

 

1785
00:58:09,200 --> 00:58:11,109


1786
00:58:11,109 --> 00:58:14,630

 

1787
00:58:14,630 --> 00:58:19,190

 

1788
00:58:19,190 --> 00:58:21,670

 

1789
00:58:21,670 --> 00:58:21,680

 

1790
00:58:21,680 --> 00:58:22,470


1791
00:58:22,470 --> 00:58:25,349

 

1792
00:58:25,349 --> 00:58:25,359

 

1793
00:58:25,359 --> 00:58:26,150


1794
00:58:26,150 --> 00:58:30,069

 

1795
00:58:30,069 --> 00:58:30,079

 

1796
00:58:30,079 --> 00:58:31,190


1797
00:58:31,190 --> 00:58:34,390

 

1798
00:58:34,390 --> 00:58:37,109

 

1799
00:58:37,109 --> 00:58:39,030

 

1800
00:58:39,030 --> 00:58:41,349

 

1801
00:58:41,349 --> 00:58:42,870

 

1802
00:58:42,870 --> 00:58:44,549

 

1803
00:58:44,549 --> 00:58:46,950

 

1804
00:58:46,950 --> 00:58:48,390

 

1805
00:58:48,390 --> 00:58:52,069

 

1806
00:58:52,069 --> 00:58:53,910

 

1807
00:58:53,910 --> 00:58:56,069

 

1808
00:58:56,069 --> 00:58:57,990

 

1809
00:58:57,990 --> 00:58:58,000

 

1810
00:58:58,000 --> 00:59:00,950


1811
00:59:00,950 --> 00:59:02,950

 

1812
00:59:02,950 --> 00:59:05,430

 

1813
00:59:05,430 --> 00:59:08,950

 

1814
00:59:08,950 --> 00:59:10,789

 

1815
00:59:10,789 --> 00:59:13,030

 

1816
00:59:13,030 --> 00:59:15,109

 

1817
00:59:15,109 --> 00:59:16,470

 

1818
00:59:16,470 --> 00:59:19,349

 

1819
00:59:19,349 --> 00:59:20,950

 

1820
00:59:20,950 --> 00:59:25,990

 

1821
00:59:25,990 --> 00:59:26,000

 

1822
00:59:26,000 --> 00:59:27,030


1823
00:59:27,030 --> 00:59:28,069

 

1824
00:59:28,069 --> 00:59:30,150

 

1825
00:59:30,150 --> 00:59:32,230

 

1826
00:59:32,230 --> 00:59:35,349

 

1827
00:59:35,349 --> 00:59:38,309

 

1828
00:59:38,309 --> 00:59:41,910

 

1829
00:59:41,910 --> 00:59:43,829

 

1830
00:59:43,829 --> 00:59:43,839

 

1831
00:59:43,839 --> 00:59:45,109


1832
00:59:45,109 --> 00:59:48,549

 

1833
00:59:48,549 --> 00:59:48,559

 

1834
00:59:48,559 --> 00:59:49,829


1835
00:59:49,829 --> 00:59:54,549

 

1836
00:59:54,549 --> 00:59:54,559

 

1837
00:59:54,559 --> 00:59:55,829


1838
00:59:55,829 --> 00:59:58,549

 

1839
00:59:58,549 --> 01:00:00,470

 

1840
01:00:00,470 --> 01:00:02,069

 

1841
01:00:02,069 --> 01:00:03,670

 

1842
01:00:03,670 --> 01:00:05,510

 

1843
01:00:05,510 --> 01:00:08,069

 

1844
01:00:08,069 --> 01:00:10,630

 

1845
01:00:10,630 --> 01:00:11,750

 

1846
01:00:11,750 --> 01:00:14,630

 

1847
01:00:14,630 --> 01:00:16,309

 

1848
01:00:16,309 --> 01:00:19,190

 

1849
01:00:19,190 --> 01:00:20,710

 

1850
01:00:20,710 --> 01:00:22,069

 

1851
01:00:22,069 --> 01:00:24,549

 

1852
01:00:24,549 --> 01:00:26,710

 

1853
01:00:26,710 --> 01:00:28,950

 

1854
01:00:28,950 --> 01:00:31,270

 

1855
01:00:31,270 --> 01:00:33,190

 

1856
01:00:33,190 --> 01:00:35,910

 

1857
01:00:35,910 --> 01:00:38,069

 

1858
01:00:38,069 --> 01:00:40,230

 

1859
01:00:40,230 --> 01:00:45,030

 

1860
01:00:45,030 --> 01:00:48,069

 

1861
01:00:48,069 --> 01:00:49,990

 

1862
01:00:49,990 --> 01:00:51,990

 

1863
01:00:51,990 --> 01:00:54,150

 

1864
01:00:54,150 --> 01:00:56,470

 

1865
01:00:56,470 --> 01:00:58,470

 

1866
01:00:58,470 --> 01:01:01,670

 

1867
01:01:01,670 --> 01:01:05,270

 

1868
01:01:05,270 --> 01:01:08,230

 

1869
01:01:08,230 --> 01:01:09,670

 

1870
01:01:09,670 --> 01:01:11,589

 

1871
01:01:11,589 --> 01:01:11,599

 

1872
01:01:11,599 --> 01:01:12,390


1873
01:01:12,390 --> 01:01:12,400

 

1874
01:01:12,400 --> 01:01:13,670


1875
01:01:13,670 --> 01:01:15,829

 

1876
01:01:15,829 --> 01:01:17,430

 

1877
01:01:17,430 --> 01:01:18,950

 

1878
01:01:18,950 --> 01:01:20,470

 

1879
01:01:20,470 --> 01:01:22,390

 

1880
01:01:22,390 --> 01:01:24,789

 

1881
01:01:24,789 --> 01:01:24,799

 

1882
01:01:24,799 --> 01:01:26,950


1883
01:01:26,950 --> 01:01:29,190

 

1884
01:01:29,190 --> 01:01:31,030

 

1885
01:01:31,030 --> 01:01:34,390

 

1886
01:01:34,390 --> 01:01:36,150

 

1887
01:01:36,150 --> 01:01:38,069

 

1888
01:01:38,069 --> 01:01:39,829

 

1889
01:01:39,829 --> 01:01:40,950

 

1890
01:01:40,950 --> 01:01:42,870

 

1891
01:01:42,870 --> 01:01:44,789

 

1892
01:01:44,789 --> 01:01:46,950

 

1893
01:01:46,950 --> 01:01:46,960

 

1894
01:01:46,960 --> 01:01:48,150


1895
01:01:48,150 --> 01:01:49,670

 

1896
01:01:49,670 --> 01:01:52,549

 

1897
01:01:52,549 --> 01:01:54,230

 

1898
01:01:54,230 --> 01:01:56,950

 

1899
01:01:56,950 --> 01:02:01,029

 

1900
01:02:01,029 --> 01:02:03,349

 

1901
01:02:03,349 --> 01:02:04,829

 

1902
01:02:04,829 --> 01:02:07,349

 

1903
01:02:07,349 --> 01:02:09,349

 

1904
01:02:09,349 --> 01:02:11,430

 

1905
01:02:11,430 --> 01:02:11,440

 

1906
01:02:11,440 --> 01:02:12,470


1907
01:02:12,470 --> 01:02:14,950

 

1908
01:02:14,950 --> 01:02:18,069

 

1909
01:02:18,069 --> 01:02:20,230

 

1910
01:02:20,230 --> 01:02:22,549

 

1911
01:02:22,549 --> 01:02:22,559

 

1912
01:02:22,559 --> 01:02:23,510


1913
01:02:23,510 --> 01:02:23,520

 

1914
01:02:23,520 --> 01:02:24,390


1915
01:02:24,390 --> 01:02:26,789

 

1916
01:02:26,789 --> 01:02:28,549

 

1917
01:02:28,549 --> 01:02:31,029

 

1918
01:02:31,029 --> 01:02:33,029

 

1919
01:02:33,029 --> 01:02:34,470

 

1920
01:02:34,470 --> 01:02:36,230

 

1921
01:02:36,230 --> 01:02:38,390

 

1922
01:02:38,390 --> 01:02:40,230

 

1923
01:02:40,230 --> 01:02:40,240

 

1924
01:02:40,240 --> 01:02:42,390


1925
01:02:42,390 --> 01:02:45,510

 

1926
01:02:45,510 --> 01:02:45,520

 

1927
01:02:45,520 --> 01:02:47,270


1928
01:02:47,270 --> 01:02:47,280

 

1929
01:02:47,280 --> 01:02:48,549


1930
01:02:48,549 --> 01:02:51,349

 

1931
01:02:51,349 --> 01:02:53,510

 

1932
01:02:53,510 --> 01:02:54,870

 

1933
01:02:54,870 --> 01:02:54,880

 

1934
01:02:54,880 --> 01:02:57,910


1935
01:02:57,910 --> 01:02:57,920

 

1936
01:02:57,920 --> 01:02:58,710


1937
01:02:58,710 --> 01:02:58,720

 

1938
01:02:58,720 --> 01:02:59,670


1939
01:02:59,670 --> 01:03:00,950

 

1940
01:03:00,950 --> 01:03:03,990

 

1941
01:03:03,990 --> 01:03:07,750

 

1942
01:03:07,750 --> 01:03:09,990

 

1943
01:03:09,990 --> 01:03:10,000

 

1944
01:03:10,000 --> 01:03:11,270


1945
01:03:11,270 --> 01:03:13,829

 

1946
01:03:13,829 --> 01:03:16,710

 

1947
01:03:16,710 --> 01:03:17,990

 

1948
01:03:17,990 --> 01:03:20,710

 

1949
01:03:20,710 --> 01:03:22,069

 

1950
01:03:22,069 --> 01:03:23,990

 

1951
01:03:23,990 --> 01:03:26,309

 

1952
01:03:26,309 --> 01:03:28,150

 

1953
01:03:28,150 --> 01:03:30,230

 

1954
01:03:30,230 --> 01:03:31,829

 

1955
01:03:31,829 --> 01:03:37,029

 

1956
01:03:37,029 --> 01:03:42,069

 

1957
01:03:42,069 --> 01:03:46,870

 

1958
01:03:46,870 --> 01:03:46,880

 

1959
01:03:46,880 --> 01:03:47,910


1960
01:03:47,910 --> 01:03:47,920

 

1961
01:03:47,920 --> 01:03:49,270


1962
01:03:49,270 --> 01:03:51,670

 

1963
01:03:51,670 --> 01:03:53,430

 

1964
01:03:53,430 --> 01:03:54,950

 

1965
01:03:54,950 --> 01:03:57,349

 

1966
01:03:57,349 --> 01:03:59,029

 

1967
01:03:59,029 --> 01:03:59,039

 

1968
01:03:59,039 --> 01:04:00,230


1969
01:04:00,230 --> 01:04:01,270

 

1970
01:04:01,270 --> 01:04:01,280

 

1971
01:04:01,280 --> 01:04:02,230


1972
01:04:02,230 --> 01:04:04,309

 

1973
01:04:04,309 --> 01:04:07,430

 

1974
01:04:07,430 --> 01:04:07,440

 

1975
01:04:07,440 --> 01:04:08,710


1976
01:04:08,710 --> 01:04:09,990

 

1977
01:04:09,990 --> 01:04:12,470

 

1978
01:04:12,470 --> 01:04:22,390

 

1979
01:04:22,390 --> 01:04:24,069

 

1980
01:04:24,069 --> 01:04:24,079

 

1981
01:04:24,079 --> 01:04:26,470


1982
01:04:26,470 --> 01:04:28,710

 

1983
01:04:28,710 --> 01:04:30,630

 

1984
01:04:30,630 --> 01:04:32,870

 

1985
01:04:32,870 --> 01:04:34,630

 

1986
01:04:34,630 --> 01:04:37,109

 

1987
01:04:37,109 --> 01:04:39,349

 

1988
01:04:39,349 --> 01:04:41,430

 

1989
01:04:41,430 --> 01:04:43,910

 

1990
01:04:43,910 --> 01:04:45,029

 

1991
01:04:45,029 --> 01:04:47,029

 

1992
01:04:47,029 --> 01:04:48,950

 

1993
01:04:48,950 --> 01:04:51,829

 

1994
01:04:51,829 --> 01:04:55,109

 

1995
01:04:55,109 --> 01:04:57,750

 

1996
01:04:57,750 --> 01:05:00,950

 

1997
01:05:00,950 --> 01:05:03,910

 

1998
01:05:03,910 --> 01:05:06,390

 

1999
01:05:06,390 --> 01:05:09,029

 

2000
01:05:09,029 --> 01:05:12,150

 

2001
01:05:12,150 --> 01:05:13,990

 

2002
01:05:13,990 --> 01:05:16,549

 

2003
01:05:16,549 --> 01:05:17,910

 

2004
01:05:17,910 --> 01:05:19,829

 

2005
01:05:19,829 --> 01:05:21,190

 

2006
01:05:21,190 --> 01:05:23,430

 

2007
01:05:23,430 --> 01:05:24,789

 

2008
01:05:24,789 --> 01:05:27,430

 

2009
01:05:27,430 --> 01:05:29,750

 

2010
01:05:29,750 --> 01:05:31,829

 

2011
01:05:31,829 --> 01:05:33,910

 

2012
01:05:33,910 --> 01:05:37,349

 

2013
01:05:37,349 --> 01:05:37,359

 

2014
01:05:37,359 --> 01:05:39,589


2015
01:05:39,589 --> 01:05:43,270

 

2016
01:05:43,270 --> 01:05:45,670

 

2017
01:05:45,670 --> 01:05:47,109

 

2018
01:05:47,109 --> 01:05:48,870

 

2019
01:05:48,870 --> 01:05:52,069

 

2020
01:05:52,069 --> 01:05:53,829

 

2021
01:05:53,829 --> 01:05:55,029

 

2022
01:05:55,029 --> 01:05:57,109

 

2023
01:05:57,109 --> 01:05:59,510

 

2024
01:05:59,510 --> 01:06:03,109

 

2025
01:06:03,109 --> 01:06:03,119

 

2026
01:06:03,119 --> 01:06:07,109


2027
01:06:07,109 --> 01:06:09,190

 

2028
01:06:09,190 --> 01:06:10,710

 

2029
01:06:10,710 --> 01:06:12,789

 

2030
01:06:12,789 --> 01:06:15,430

 

2031
01:06:15,430 --> 01:06:17,349

 

2032
01:06:17,349 --> 01:06:18,950

 

2033
01:06:18,950 --> 01:06:18,960

 

2034
01:06:18,960 --> 01:06:20,150


2035
01:06:20,150 --> 01:06:22,230

 

2036
01:06:22,230 --> 01:06:24,549

 

2037
01:06:24,549 --> 01:06:26,870

 

2038
01:06:26,870 --> 01:06:28,230

 

2039
01:06:28,230 --> 01:06:28,240

 

2040
01:06:28,240 --> 01:06:30,829


2041
01:06:30,829 --> 01:06:30,839

 

2042
01:06:30,839 --> 01:06:32,710


2043
01:06:32,710 --> 01:06:34,870

 

2044
01:06:34,870 --> 01:06:36,950

 

2045
01:06:36,950 --> 01:06:39,430

 

2046
01:06:39,430 --> 01:06:39,440

 

2047
01:06:39,440 --> 01:06:40,789


2048
01:06:40,789 --> 01:06:40,799

 

2049
01:06:40,799 --> 01:06:41,990


2050
01:06:41,990 --> 01:06:43,990

 

2051
01:06:43,990 --> 01:06:45,829

 

2052
01:06:45,829 --> 01:06:47,670

 

2053
01:06:47,670 --> 01:06:50,309

 

2054
01:06:50,309 --> 01:06:54,870

 

2055
01:06:54,870 --> 01:06:56,309

 

2056
01:06:56,309 --> 01:06:57,589

 

2057
01:06:57,589 --> 01:06:59,270

 

2058
01:06:59,270 --> 01:07:02,069

 

2059
01:07:02,069 --> 01:07:03,990

 

2060
01:07:03,990 --> 01:07:06,390

 

2061
01:07:06,390 --> 01:07:07,910

 

2062
01:07:07,910 --> 01:07:09,990

 

2063
01:07:09,990 --> 01:07:11,990

 

2064
01:07:11,990 --> 01:07:14,390

 

2065
01:07:14,390 --> 01:07:16,630

 

2066
01:07:16,630 --> 01:07:16,640

 

2067
01:07:16,640 --> 01:07:19,029


2068
01:07:19,029 --> 01:07:20,710

 

2069
01:07:20,710 --> 01:07:24,870

 

2070
01:07:24,870 --> 01:07:26,470

 

2071
01:07:26,470 --> 01:07:28,789

 

2072
01:07:28,789 --> 01:07:29,829

 

2073
01:07:29,829 --> 01:07:32,549

 

2074
01:07:32,549 --> 01:07:34,390

 

2075
01:07:34,390 --> 01:07:36,950

 

2076
01:07:36,950 --> 01:07:39,029

 

2077
01:07:39,029 --> 01:07:39,910

 

2078
01:07:39,910 --> 01:07:41,990

 

2079
01:07:41,990 --> 01:07:47,190

 

2080
01:07:47,190 --> 01:07:48,789

 

2081
01:07:48,789 --> 01:07:50,309

 

2082
01:07:50,309 --> 01:07:53,029

 

2083
01:07:53,029 --> 01:07:53,039

 

2084
01:07:53,039 --> 01:07:54,150


2085
01:07:54,150 --> 01:07:57,829

 

2086
01:07:57,829 --> 01:07:59,029

 

2087
01:07:59,029 --> 01:08:02,390

 

2088
01:08:02,390 --> 01:08:04,789

 

2089
01:08:04,789 --> 01:08:06,549

 

2090
01:08:06,549 --> 01:08:06,559

 

2091
01:08:06,559 --> 01:08:07,750


2092
01:08:07,750 --> 01:08:12,870

 

2093
01:08:12,870 --> 01:08:16,470

 

2094
01:08:16,470 --> 01:08:16,480

 

2095
01:08:16,480 --> 01:08:18,309


2096
01:08:18,309 --> 01:08:19,590

 

2097
01:08:19,590 --> 01:08:21,510

 

2098
01:08:21,510 --> 01:08:22,709

 

2099
01:08:22,709 --> 01:08:25,030

 

2100
01:08:25,030 --> 01:08:25,040

 

2101
01:08:25,040 --> 01:08:26,149


2102
01:08:26,149 --> 01:08:30,149

 

2103
01:08:30,149 --> 01:08:32,470

 

2104
01:08:32,470 --> 01:08:34,789

 

2105
01:08:34,789 --> 01:08:35,829

 

2106
01:08:35,829 --> 01:08:38,470

 

2107
01:08:38,470 --> 01:08:41,669

 

2108
01:08:41,669 --> 01:08:44,309

 

2109
01:08:44,309 --> 01:08:47,669

 

2110
01:08:47,669 --> 01:08:47,679

 

2111
01:08:47,679 --> 01:08:49,349


2112
01:08:49,349 --> 01:08:51,349

 

2113
01:08:51,349 --> 01:08:54,390

 

2114
01:08:54,390 --> 01:08:56,950

 

2115
01:08:56,950 --> 01:08:56,960

 

2116
01:08:56,960 --> 01:08:58,789


2117
01:08:58,789 --> 01:08:58,799

 

2118
01:08:58,799 --> 01:08:59,990


2119
01:08:59,990 --> 01:09:01,430

 

2120
01:09:01,430 --> 01:09:04,309

 

2121
01:09:04,309 --> 01:09:07,189

 

2122
01:09:07,189 --> 01:09:09,189

 

2123
01:09:09,189 --> 01:09:14,550

 

2124
01:09:14,550 --> 01:09:16,630

 

2125
01:09:16,630 --> 01:09:16,640

 

2126
01:09:16,640 --> 01:09:17,510


2127
01:09:17,510 --> 01:09:20,470

 

2128
01:09:20,470 --> 01:09:24,070

 

2129
01:09:24,070 --> 01:09:25,910

 

2130
01:09:25,910 --> 01:09:29,430

 

2131
01:09:29,430 --> 01:09:30,789

 

2132
01:09:30,789 --> 01:09:30,799

 

2133
01:09:30,799 --> 01:09:33,189


2134
01:09:33,189 --> 01:09:34,630

 

2135
01:09:34,630 --> 01:09:34,640

 

2136
01:09:34,640 --> 01:09:35,590


2137
01:09:35,590 --> 01:09:37,749

 

2138
01:09:37,749 --> 01:09:37,759

 

2139
01:09:37,759 --> 01:09:38,630


2140
01:09:38,630 --> 01:09:40,709

 

2141
01:09:40,709 --> 01:09:42,950

 

2142
01:09:42,950 --> 01:09:45,030

 

2143
01:09:45,030 --> 01:09:46,870

 

2144
01:09:46,870 --> 01:09:48,550

 

2145
01:09:48,550 --> 01:09:51,030

 

2146
01:09:51,030 --> 01:09:54,149

 

2147
01:09:54,149 --> 01:09:57,030

 

2148
01:09:57,030 --> 01:09:59,510

 

2149
01:09:59,510 --> 01:10:01,590

 

2150
01:10:01,590 --> 01:10:01,600

 

2151
01:10:01,600 --> 01:10:02,390


2152
01:10:02,390 --> 01:10:02,400

 

2153
01:10:02,400 --> 01:10:03,430


2154
01:10:03,430 --> 01:10:05,510

 

2155
01:10:05,510 --> 01:10:07,110

 

2156
01:10:07,110 --> 01:10:08,070

 

2157
01:10:08,070 --> 01:10:09,910

 

2158
01:10:09,910 --> 01:10:12,470

 

2159
01:10:12,470 --> 01:10:13,990

 

2160
01:10:13,990 --> 01:10:16,310

 

2161
01:10:16,310 --> 01:10:18,470

 

2162
01:10:18,470 --> 01:10:20,390

 

2163
01:10:20,390 --> 01:10:22,630

 

2164
01:10:22,630 --> 01:10:23,990

 

2165
01:10:23,990 --> 01:10:26,950

 

2166
01:10:26,950 --> 01:10:28,870

 

2167
01:10:28,870 --> 01:10:30,390

 

2168
01:10:30,390 --> 01:10:33,270

 

2169
01:10:33,270 --> 01:10:36,149

 

2170
01:10:36,149 --> 01:10:37,430

 

2171
01:10:37,430 --> 01:10:38,950

 

2172
01:10:38,950 --> 01:10:42,550

 

2173
01:10:42,550 --> 01:10:45,270

 

2174
01:10:45,270 --> 01:10:47,910

 

2175
01:10:47,910 --> 01:10:49,830

 

2176
01:10:49,830 --> 01:10:51,510

 

2177
01:10:51,510 --> 01:10:51,520

 

2178
01:10:51,520 --> 01:10:53,669


2179
01:10:53,669 --> 01:10:55,350

 

2180
01:10:55,350 --> 01:10:55,360

 

2181
01:10:55,360 --> 01:10:56,790


2182
01:10:56,790 --> 01:10:59,350

 

2183
01:10:59,350 --> 01:11:02,790

 

2184
01:11:02,790 --> 01:11:04,630

 

2185
01:11:04,630 --> 01:11:05,830

 

2186
01:11:05,830 --> 01:11:08,310

 

2187
01:11:08,310 --> 01:11:10,630

 

2188
01:11:10,630 --> 01:11:15,590

 

2189
01:11:15,590 --> 01:11:15,600

 

2190
01:11:15,600 --> 01:11:19,990


2191
01:11:19,990 --> 01:11:23,669

 

2192
01:11:23,669 --> 01:11:25,110

 

2193
01:11:25,110 --> 01:11:27,189

 

2194
01:11:27,189 --> 01:11:29,350

 

2195
01:11:29,350 --> 01:11:31,830

 

2196
01:11:31,830 --> 01:11:33,750

 

2197
01:11:33,750 --> 01:11:35,350

 

2198
01:11:35,350 --> 01:11:37,030

 

2199
01:11:37,030 --> 01:11:38,950

 

2200
01:11:38,950 --> 01:11:40,790

 

2201
01:11:40,790 --> 01:11:43,110

 

2202
01:11:43,110 --> 01:11:44,950

 

2203
01:11:44,950 --> 01:11:46,950

 

2204
01:11:46,950 --> 01:11:48,470

 

2205
01:11:48,470 --> 01:11:49,910

 

2206
01:11:49,910 --> 01:11:51,830

 

2207
01:11:51,830 --> 01:11:54,470

 

2208
01:11:54,470 --> 01:11:55,590

 

2209
01:11:55,590 --> 01:11:57,990

 

2210
01:11:57,990 --> 01:11:59,189

 

2211
01:11:59,189 --> 01:12:00,790

 

2212
01:12:00,790 --> 01:12:03,270

 

2213
01:12:03,270 --> 01:12:05,270

 

2214
01:12:05,270 --> 01:12:07,750

 

2215
01:12:07,750 --> 01:12:09,270

 

2216
01:12:09,270 --> 01:12:12,550

 

2217
01:12:12,550 --> 01:12:14,630

 

2218
01:12:14,630 --> 01:12:16,870

 

2219
01:12:16,870 --> 01:12:19,270

 

2220
01:12:19,270 --> 01:12:20,830

 

2221
01:12:20,830 --> 01:12:23,990

 

2222
01:12:23,990 --> 01:12:25,590

 

2223
01:12:25,590 --> 01:12:27,270

 

2224
01:12:27,270 --> 01:12:29,270

 

2225
01:12:29,270 --> 01:12:30,790

 

2226
01:12:30,790 --> 01:12:32,870

 

2227
01:12:32,870 --> 01:12:32,880

 

2228
01:12:32,880 --> 01:12:34,070


2229
01:12:34,070 --> 01:12:35,750

 

2230
01:12:35,750 --> 01:12:38,790

 

2231
01:12:38,790 --> 01:12:42,390

 

2232
01:12:42,390 --> 01:12:45,510

 

2233
01:12:45,510 --> 01:12:47,110

 

2234
01:12:47,110 --> 01:12:51,350

 

2235
01:12:51,350 --> 01:12:53,510

 

2236
01:12:53,510 --> 01:12:54,950

 

2237
01:12:54,950 --> 01:12:57,110

 

2238
01:12:57,110 --> 01:12:57,120

 

2239
01:12:57,120 --> 01:12:58,870


2240
01:12:58,870 --> 01:13:00,709

 

2241
01:13:00,709 --> 01:13:03,430

 

2242
01:13:03,430 --> 01:13:04,550

 

2243
01:13:04,550 --> 01:13:07,590

 

2244
01:13:07,590 --> 01:13:09,669

 

2245
01:13:09,669 --> 01:13:12,070

 

2246
01:13:12,070 --> 01:13:14,310

 

2247
01:13:14,310 --> 01:13:14,320

 

2248
01:13:14,320 --> 01:13:15,430


2249
01:13:15,430 --> 01:13:17,830

 

2250
01:13:17,830 --> 01:13:20,709

 

2251
01:13:20,709 --> 01:13:23,669

 

2252
01:13:23,669 --> 01:13:25,910

 

2253
01:13:25,910 --> 01:13:28,229

 

2254
01:13:28,229 --> 01:13:30,870

 

2255
01:13:30,870 --> 01:13:33,590

 

2256
01:13:33,590 --> 01:13:33,600

 

2257
01:13:33,600 --> 01:13:34,950


2258
01:13:34,950 --> 01:13:34,960

 

2259
01:13:34,960 --> 01:13:35,750


2260
01:13:35,750 --> 01:13:37,750

 

2261
01:13:37,750 --> 01:13:37,760

 

2262
01:13:37,760 --> 01:13:38,950


2263
01:13:38,950 --> 01:13:41,750

 

2264
01:13:41,750 --> 01:13:44,709

 

2265
01:13:44,709 --> 01:13:46,390

 

2266
01:13:46,390 --> 01:13:48,390

 

2267
01:13:48,390 --> 01:13:48,400

 

2268
01:13:48,400 --> 01:13:49,510


2269
01:13:49,510 --> 01:13:51,830

 

2270
01:13:51,830 --> 01:13:55,350

 

2271
01:13:55,350 --> 01:13:55,360

 

2272
01:13:55,360 --> 01:13:56,229


2273
01:13:56,229 --> 01:13:58,390

 

2274
01:13:58,390 --> 01:14:00,790

 

2275
01:14:00,790 --> 01:14:03,189

 

2276
01:14:03,189 --> 01:14:04,870

 

2277
01:14:04,870 --> 01:14:06,310

 

2278
01:14:06,310 --> 01:14:09,030

 

2279
01:14:09,030 --> 01:14:10,950

 

2280
01:14:10,950 --> 01:14:13,430

 

2281
01:14:13,430 --> 01:14:16,070

 

2282
01:14:16,070 --> 01:14:17,750

 

2283
01:14:17,750 --> 01:14:19,910

 

2284
01:14:19,910 --> 01:14:22,950

 

2285
01:14:22,950 --> 01:14:25,830

 

2286
01:14:25,830 --> 01:14:28,070

 

2287
01:14:28,070 --> 01:14:30,229

 

2288
01:14:30,229 --> 01:14:30,239

 

2289
01:14:30,239 --> 01:14:31,669


2290
01:14:31,669 --> 01:14:34,229

 

2291
01:14:34,229 --> 01:14:34,239

 

2292
01:14:34,239 --> 01:14:35,669


2293
01:14:35,669 --> 01:14:35,679

 

2294
01:14:35,679 --> 01:14:36,630


2295
01:14:36,630 --> 01:14:38,470

 

2296
01:14:38,470 --> 01:14:41,030

 

2297
01:14:41,030 --> 01:14:43,510

 

2298
01:14:43,510 --> 01:14:43,520

 

2299
01:14:43,520 --> 01:14:46,709


2300
01:14:46,709 --> 01:14:48,070

 

2301
01:14:48,070 --> 01:14:48,080

 

2302
01:14:48,080 --> 01:14:49,990


2303
01:14:49,990 --> 01:14:52,790

 

2304
01:14:52,790 --> 01:14:52,800

 

2305
01:14:52,800 --> 01:14:53,750


2306
01:14:53,750 --> 01:14:55,430

 

2307
01:14:55,430 --> 01:14:57,030

 

2308
01:14:57,030 --> 01:14:59,350

 

2309
01:14:59,350 --> 01:15:00,870

 

2310
01:15:00,870 --> 01:15:01,990

 

2311
01:15:01,990 --> 01:15:05,669

 

2312
01:15:05,669 --> 01:15:05,679

 

2313
01:15:05,679 --> 01:15:07,270


2314
01:15:07,270 --> 01:15:10,070

 

2315
01:15:10,070 --> 01:15:10,080

 

2316
01:15:10,080 --> 01:15:12,229


2317
01:15:12,229 --> 01:15:13,750

 

2318
01:15:13,750 --> 01:15:16,790

 

2319
01:15:16,790 --> 01:15:18,709

 

2320
01:15:18,709 --> 01:15:20,470

 

2321
01:15:20,470 --> 01:15:20,480

 

2322
01:15:20,480 --> 01:15:21,350


2323
01:15:21,350 --> 01:15:22,790

 

2324
01:15:22,790 --> 01:15:25,110

 

2325
01:15:25,110 --> 01:15:26,550

 

2326
01:15:26,550 --> 01:15:28,550

 

2327
01:15:28,550 --> 01:15:32,630

 

2328
01:15:32,630 --> 01:15:32,640

 

2329
01:15:32,640 --> 01:15:33,590


2330
01:15:33,590 --> 01:15:36,070

 

2331
01:15:36,070 --> 01:15:37,350

 

2332
01:15:37,350 --> 01:15:39,350

 

2333
01:15:39,350 --> 01:15:41,270

 

2334
01:15:41,270 --> 01:15:43,590

 

2335
01:15:43,590 --> 01:15:45,030

 

2336
01:15:45,030 --> 01:15:47,510

 

2337
01:15:47,510 --> 01:15:49,830

 

2338
01:15:49,830 --> 01:15:53,270

 

2339
01:15:53,270 --> 01:15:55,189

 

2340
01:15:55,189 --> 01:15:56,790

 

2341
01:15:56,790 --> 01:15:59,270

 

2342
01:15:59,270 --> 01:16:00,550

 

2343
01:16:00,550 --> 01:16:07,430

 

2344
01:16:07,430 --> 01:16:07,440

 

2345
01:16:07,440 --> 01:16:08,630


2346
01:16:08,630 --> 01:16:12,070

 

2347
01:16:12,070 --> 01:16:15,350

 

2348
01:16:15,350 --> 01:16:17,430

 

2349
01:16:17,430 --> 01:16:17,440

 

2350
01:16:17,440 --> 01:16:18,830


2351
01:16:18,830 --> 01:16:21,669

 

2352
01:16:21,669 --> 01:16:23,510

 

2353
01:16:23,510 --> 01:16:25,030

 

2354
01:16:25,030 --> 01:16:27,910

 

2355
01:16:27,910 --> 01:16:27,920

 

2356
01:16:27,920 --> 01:16:29,830


2357
01:16:29,830 --> 01:16:29,840

 

2358
01:16:29,840 --> 01:16:33,270


2359
01:16:33,270 --> 01:16:35,189

 

2360
01:16:35,189 --> 01:16:38,149

 

2361
01:16:38,149 --> 01:16:38,159

 

2362
01:16:38,159 --> 01:16:39,110


2363
01:16:39,110 --> 01:16:41,669

 

2364
01:16:41,669 --> 01:16:43,990

 

2365
01:16:43,990 --> 01:16:46,149

 

2366
01:16:46,149 --> 01:16:47,750

 

2367
01:16:47,750 --> 01:16:48,790

 

2368
01:16:48,790 --> 01:16:50,390

 

2369
01:16:50,390 --> 01:16:52,630

 

2370
01:16:52,630 --> 01:16:55,030

 

2371
01:16:55,030 --> 01:16:57,189

 

2372
01:16:57,189 --> 01:16:57,199

 

2373
01:16:57,199 --> 01:16:58,149


2374
01:16:58,149 --> 01:16:59,510

 

2375
01:16:59,510 --> 01:17:01,110

 

2376
01:17:01,110 --> 01:17:05,270

 

2377
01:17:05,270 --> 01:17:07,030

 

2378
01:17:07,030 --> 01:17:12,390

 

2379
01:17:12,390 --> 01:17:12,400

 

2380
01:17:12,400 --> 01:17:13,910


2381
01:17:13,910 --> 01:17:16,149

 

2382
01:17:16,149 --> 01:17:18,070

 

2383
01:17:18,070 --> 01:17:19,430

 

2384
01:17:19,430 --> 01:17:19,440

 

2385
01:17:19,440 --> 01:17:21,669


2386
01:17:21,669 --> 01:17:34,870

 

2387
01:17:34,870 --> 01:17:34,880

 

2388
01:17:34,880 --> 01:17:36,960