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let's get  back to  space complexity um so what  uh just to review what we've been doing  um  from last lecture which  feels like a long time ago but it was  only two days ago  um we  were looking at um  those three theorems  which all had this basically the same  proof  one was the latter about the latter  problem of uh where you're trying to see  if you can get from one  string to another string  changing one symbol at a time  um and all of the strings were in the  language of some finite automaton or you  had some reasonable rule for  saying which of the valid strings which  are not  and then uh we built on that idea we  showed savage's theorem that going from  any non-deterministic machine to a  deterministic machine only squares the  amount of space  and then um  we finally  uh proved that tqbf uh this problem  about quantified boolean formulas  testing whether they are true  uh that that problem is complete  for p space um so we're going to build  off of that last last theorem today  and talk about  uh  one other  complete problem um and also  show a connection between  p space  and testing uh which side uh  can win in certain kinds of games  um  so um and then the at the end of the  lecture we will the second half of the  lecture we'll talk about  diving deeper into space complexity  we're going to talk about a different  part of the parameter space where  instead of looking at polynomial space  we're going to talk about what happens  if you have logarithmic space which is  another natural  point to look at for from  for space complexity but we'll get to  that after the after the break  okay um  uh talk about game complexity and games  um  so when i was little uh we used to play  me and my sisters would play a game  called geography i don't know if how  many of you have heard it or maybe it go  goes under a different name um  but uh  it was a game about basically about  words and places and it has a very  simple um  set of rules um let's say there were two  people you take turns picking  names of places  and then each person  has to pick a place  which starts with the letter that the  previous person's place ended with  for example  you know you have two people playing um  perhaps you have you know you agree on  some starting place like a major city  nearby like boston  um so one you know one player the first  player takes boston  and then after that the next player has  to pick a place that starts with n  because boston ends with n  and so maybe nebraska  would be a possible response to boston  and then the first player would have to  respond to that with a place that starts  with an a because nebraska ends with an  a it's like maybe alaska  uh which also  starts with a ends with a  so  then arkansas maybe would be a nat  reasonable response to that and then  that ends with s um so uh san francisco  and then so on and so on um  and uh  of course you want to  forbid people from  reusing  names because then you'll just get into  a loop you know saying alaska alaska  alaska won't make the game very  interesting so you have to forbid  that possibility  names cannot be reused they can be used  at most once  and then um so eventually  one side or the other is going to run  out of names uh and not have a either  because of the lack of their  geographical knowledge or maybe you've  just exhausted all the possibilities  and i will will not have a response and  that person will be stuck and that  person is the loser um the other person  has won the game  so the objective is to try to avoid  getting stuck um so let's just take a  look um i'm going to think about that  games in a somewhat  slightly more abstracted formalized way  um so you write down all of the  legitimate places um they become nodes  of a graph such as i've shown here  and then you draw an edge from one node  to another if um that  if that corresponds to a legal move in  the game a legal step of the game so you  know boston connects to places that  start within like new york and nebraska  um so the first person might pick boston  the second person might pick either of  these two um and then back to the first  person if they pick nebraska they could  pick arkansas they can pick alaska and  so on just as i described  believe it or not  we actually played this game i can  remember playing this game when i was a  kid that was of course before we had um  uh  you know league of legends and other fun  fun stuff but this is what kids used to  do uh  so  anyway um  um  so just to  get it down on the slide the rules are  we'll assume we have two players we'll  call them player one and player two  player one goes first  um and they take turns picking places  that start with the letter which ended  the previous place  not no repeats allowed and the first  person players the first player stuck  loses  you know the person who doesn't have a  move to make  so what we're going to look at  um today is a generalized version of  that where we're kind of um just going  to take away the names but just leave  the underlying graph  and we're going to call that generalized  geography so here um that can be played  on any directed graph you take away  the names of the edges the names of the  nodes now you just have a some arbitrary  graph  um you're going to designate a  particular node as the starting node and  then the players take turns following  those edges  um and uh what uh you're  because you're gonna forbid having any  loops reusing any um any nodes uh what  you end up  doing is working together you're going  to be constructing a simple path in that  graph  simple path just follows those edges and  never intersects itself  and the first person the first player to  be stuck is the loser so the other one  is the winner okay so that's we're going  to call that generalized geography and  we're going to look at the computational  complexity of deciding for a given graph  and starting point  which side would win if both  uh players  played  optimally as best uh the best possible  okay  and we'll name that problem gg or look  at its associated language so here is a  graph and the starting node  and you want to say um  that pair is in  the language gigi  if the first player player one  who must play at that node a um to start  off with has a forced win  um  uh  in that  general geography game on that graph  starting at a okay where again what i  mean by a force when i'm not going to  define this formally that we could do  that it's not hard to do but a forced  wind also it's also called a winning  strategy  um that just means that if both sides  play optimal they pay the best possible  not up their capability i mean the  absolutely the best possible play  then that  player one  would still win there's nothing that  player two can do to prevent player one  from winning um if player one has a  winning strategy or a forced win um you  may have seen for example uh  there are examples of this that  i mean i don't even know if they still  exist but it used to be in the newspaper  they used to be examples of chess boards  and they would say you know if they  would start from a certain position and  they would say white to win in three  and so that means white has a forced win  no matter what black does white is going  to end up winning but you have to think  of what the strategy is it may not be so  obvious to think through what are the  right moves that what makes but the  point is that no matter what black would  do white would end up winning um and  what you can show we're not going to do  that but for  a class of games that includes this  generalized geography  either one side or the other is going to  be guaranteed to have a forced win  that's not necessarily true for all  possible games obviously  but for a large class of games um  in one side or the other  is guaranteed to have a winning strategy  or forced win  okay  so let's just review that because this  is going to be essential for  you know the first half of the lecture  to understand what we mean by a game and  what we mean by one side or the other to  have a force to win okay so let's let's  just uh oh yeah and what we're going to  show is that gg is piece based complete  that the problem of giving one of these  graphs and a starting point figuring out  does player 1 have a force win or is it  player 2 that has a force win  um  which side is guaranteed to win under  optimal play that's a piece based  complete problem  um  and so let's do a little checking on  that and so i'm going to give you a very  small example  of the generalized geography game  you have to figure out  whether it's player one or player two  that has the forced win the winning  strategy or maybe neither or both those  are the four options okay and you  understand player one has to play here  because that's the starting place of  this general geography game so then it's  up to player two to continue on  all right so let's um launch that poll  uh you'll have to think about it a  little bit um  and uh  let me know what you think which side  has a forced win  which player makes the first move it's  player one makes the first move and  player one plays as i showed you here  this is player one playing at a so  player one picks a  picks and then player two has to pick  one of these two  okay all right i think we're gonna end  the poll  everybody's in  pick something  um  you you guys did not do well on this  bowl  uh at least not too many of you picked  d  that's that's true that's a that's a  little reassuring because you can't have  both players having a forced win  i did say that in games of this kind one  side or the other is going to have a  force win so c is not a very good choice  either um  so you're  maybe you should spend less time reading  your email and more time listening to  the lecture uh  um  the actual correct answer is b  player two  um which is obviously a is a minority  position here player two has the forced  win let's understand why um  so  player one plays here  as i said that's the b that's the first  move player two can either take the  upper  node or the lower note  okay  player two takes the upper note  then player one has no choice but to  take the rightmost node  and now there's no move for player two  uh so player two will lose  um if player two takes the upper note  the upper choice there because  player two will go here player one will  go there and it'll be game over  and player one will have one because  player two was stuck  however  player two had another choice  um  uh player two had another choice um  player two could have also gone down  here  um so player two uh if it played down  here um  things are looking a lot better for  player two because player two goes here  player one's move is forced goes here  but now there's another move left uh  that player two could make so play  player so two goes here one goes there  player two goes up there  and now player one is stuck  so at player two's choice  um player two can make a move which will  end up  causing player one to get stuck  and so because  it you know that's the nature of what we  mean by uh having a winning strategy  there is some way to guarantee if you're  playing optimally and you're certainly  playing optimally mean you'll take that  lower move  uh  then you will  take that so  under  uh optimal play  player two will win this game  and so player two has a winning strategy  it has a forced win  okay  um so i i see so  i'm not going to define playing  optimally i'm going to you know there's  some questions here about that about  that that's um  you know you can think about  i'm going to i'm going to leave it  um  you know  somewhat informal um but we could make  that all precise i i don't think it  would be clarifying to to go through a  precise definition of that um  uh  you know you you can just think you we  don't even have to think about it in  terms of optimal play you you can say  that there is um  a uh forever  uh a  um  you say a player has a force win  if  uh  they have moves that they could make in  response to every possible move the  opponent could make  uh that'll guarantee that they will end  up winning  so it doesn't say we don't even have to  talk about optimality here just talk  about every possible opponent  every possible opponent is going to lose  to somebody who um  uh  you know  has a has a winning strategy and pl and  follows that strategy  okay um  okay i can see there was some confusion  about who was going first and so on i'm  glad we that was part of the reason i  gave this check and just so hopefully to  clear up some of those points um so  player one plays here then player two uh  goes and i got some comments that some  people were confused about what does it  mean for the first uh  where does the first person move okay  hopefully that's  that's uh  clear all right so let's move on  because we're gonna make we're gonna be  spending a while  um  proving something about generalized  geography and these graphs  um okay so let's let's continue  so in order to make a connection  between  um games  and complexity  we're going to need to  talk about a problem one that we've  actually related to one that we've  already seen where we can reformulate  that problem as a game and that's going  to be a problem on quantified boolean  formulas called the formula game  okay so this is a different game now um  not one that you're likely to end up  playing but in in in in a sense in which  you'll see this is a still a reasonable  game where two players can play against  one another and one of them is going to  end up winning and one of them is going  to end up having  uh  one or the other will have a winning  strategy so let's understand what the  game is  uh the game is played  the game is played on a formula  so you so you write down a quantified  boolean formula remember we talked about  that you know for several lectures now  uh you know it's a bunch of quantum  variables with quantifiers  and um  then there's the part that has no  quantifiers  um which  you know uh you can always put into  conjunctive normal form if you want but  for the purposes of this discussion so  far it doesn't matter later on we'll  actually want it to be in quant in  in cnf but okay but but for now we just  have some formula  [Music]  and  associated to that formula there is a  game in a very natural sense so first of  all let me they're going to be two  players  but now the names of those players are  going to be player exist and player for  all  okay and the way the way this is how the  game goes  the uh  the player exists  but both players are going to be picking  values for variables  they're going to say  you know variable x1 is true variable x2  is false variable x3 is false variable  x4 is true  and and they're going to be that that's  how the moves of the game go  they're just assigning values to  variables  so that in the end  all of the variables have been assigned  a value a boolean value and so we end up  with an assignment  but  before we get to that stage  which players get to pick  which variables and the way it works is  the exist player is going to assign  values to the exist variables  so exist we'll pick x1 and x3 and assign  them val values  the for all player is going to pick the  very ver assigned values to the  variables that have referral quantifier  attached  so in this case would be x2 would be um  the the the for all player will pick the  value for x the x2 variable  um fading out here  okay  now um so  and the order of play is  according to the order of the appearance  of the quantifiers in the formula  so the way i have it written out in this  particular formula it's exist x1 so  exist will pick the value for x1  then it's for all's turn for all pick  the value of x2 then it's going to be  exist picking the value for x3  and and so on until they get to the end  and all of the variables have been  assigned a value by one side or the  other  and now the game is over  okay  so  what's left is to understand who has won  at that point  um  and the way we  say who has won is  exist  wins  if in the end  this part psi  which is the part without quantifiers  ended up being satisfied by those  um by that assignment that the two  players together ended up picking  and then for all  will  win  if that  part of the formula is not satisfied  so let me just write that down here  after the variables have been assigned  values we determine the winner  exist wins if the assignment that they  co-op together built they built  cooperatively together uh satisfied psy  and otherwise for all ones if it doesn't  satisfy psi okay so you kind of think  about it this way  you know exist is picking the values of  those variables  he's trying to make  this  uh part of the formula satisfied trying  to satisfy this part of the formula for  all  is picking values but she's trying to  make this part of the formula  unsatisfied so they're  you know they're uh in opposition to one  another  and  um in the end one of them is going to  have succeeded and the other one will  have failed uh  and now thinking about it as a game  one side depending upon the formula one  side or the other is going to have the  ability to dominate the other one and  always win  um  and so  the question is  which side has that um ability to always  win which side has the winning strategy  of course it'll depend upon the formula  some formulas it'll be the exist player  who has the winning strategy and other  formulas it'll be the for all player who  has the winning strategy and  computationally we'd like to make that  determination which side is going to win  um  you know for a given formula  and there's a very very nice  and actually very easy  um and simple way to understand that  because we've already run into that  problem before  the problem of determining which side  has a force force win is exactly the  same thing as determining whether this  quantified boolean formula is true  because the exist player has a force win  when that formula game on that um on  that formula  um  let me say it again the exist player has  a force win when that formula is true  um  so if this quantified boolean formula  were a true quantified bullying formula  then exist is guaranteed to be able to  win that game if it if it plays  its hand right  if this is a false formula for all the  for all player will always be able to  win um if it plays its hand right  okay so um why is that  and it really follows kind of almost  you know without doing any work at all  this is kind of a you know  this proof even though it looks  superficially like it might be not so  easy to prove  it follows for a very simple reason  because  the meaning of what it means for exist  to have a winning strategy is exactly  captured by the semantics of the  quantifiers  let's see what that means what does it  mean that exists has a force to win  let's say up here  that means that well exist has a move  that it could make  that  exists can  pick some value for x1 so there was some  move for x for there was some way to  assign x1  so that no matter what the opponent does  no matter what for all does  there's going to be a response that it  exists can make um some some assignment  to it x3  um  uh so that no matter what for all does  there's going to be an assignment to the  next variable  um and so on and so on so that this  thing ends up being true  so let me just say that again somehow i  didn't feel it came out very clear  uh so  exist has a winning strategy exactly  when there was some assignment to x1  that exists can make such that no matter  what assignment that for all makes to x2  there is some assignment to x3  um such that no matter what assignment  that um for all makes to x4  and so on  makes this thing true  that's what it means for exist to have a  winning strategy  that's exactly what the quantifiers are  saying in the first place  so even really without doing anything  you can see that exists has a winning  strategy  means exactly the same thing as the  formula the quantified being formula  being true  so making the test of whether  uh which side has a winning strategy is  the same as testing whether the formula  is true  okay um  [Music]  so let's let me try to turn to the chat  now um before before we move on to  anything else  um  all right uh whoops  uh here we go  [Music]  okay so i'm getting some questions about  the order of play and about the  alternation of the quantifiers here  so the way i'm showing it  is that the quantifiers always alternate  exist for all exists for all that  doesn't really matter  you know you could have several exists  in a row and that would correspond to  exist making several moves in a row  before turning it over to parole  but  alternatively  we could just add extra variables which  don't play any role in psi  and um they're just kind of dummy  variables that serve as spacers so if  you have two exists in a row and you  don't like that you can just add add a  for all of some variable that you never  see again in between  so you can always get it to be  alternating if you want  so  um  um  also question the difference between pha  fee  uh and psi so psi is the part here that  does not have any quantifiers  so the way we always write down our qbfs  is their leading quantifiers with the  variables which  uh you know and all variables have to be  within the scope of one of the  quantifiers of quantified variables um  so you know there are no free variables  that are unquantified in our qbfs  um and so there's a you know all of  these qbfs have a run of quantifiers and  then a part without quantifiers where  there's this sort of boolean logic part  i'm psi here is the boolean logic part  and phi is the whole thing together with  the quantifiers  um  so  i'm not sure i understand some of these  ques why not have the for all player  have a force to win the for all player  might have a full force win no see  one side or the other is guaranteed to  have a forced win so if exist does not  have a force to win then then parole  will have a forced win that you know i'm  not going to prove that but that's a  fairly simple proof by induction which  i'm not going to go through just take it  as a fact  somebody's asking why does for all want  to not satisfy the uh this expression  the the psy part well i mean that's the  way we set up the game um  you know in order to make this  correspond to tqbf  um  just the truth  of the of this  of this expression we want to make exist  try to satisfy the the this part and for  all try to  unsatisfy make it not satisfied um  and so i'm not sure  because that's what works i'm not sure  what else whatever whether why i can  answer there okay somebody  i'm going to answer some of the you know  most basic questions here somebody's  asking how do we know how many variables  to use  so the  the variables are the variables of the  formula so somebody's going to hand you  a formula  that's going to have x1 to x10 whatever  that formula has some number of  variables and so that game is going to  have 10 moves  you know if the quantifiers alternate  then the moves will alternate but  whatever the pattern of the quantifiers  is going to be the pattern of the moves  so exist is going to follow the  um  you know the places where the exist  quantifier occurs and fraud is going to  follow the places where the for all  quantifier occurs so if you have you  know exist for all exist for all exist  will move first then for all move then  exist will move then for all will move  together they're picking values to their  respective variables with with their  opposing goals  existed trying to make the  um the psi part satisfied for all is  trying to avoid making it satisfied to  make to to make make the assignment not  satisfy that part  and if you think about what that means  just in terms of the meaning of that  that it's going to mean exactly the same  thing that the formula is true  um  so  anyway if you don't  let's see i don't know if there's any  more questions that i can try to answer  here  um  well  so it's still about the numbers of  variables the  um  each formula is going to define a  different game you see the formula  so the formula however big it may have  50 variables that may have two variables  but we'll do an example maybe that'll  help  um  the next check-in which is coming up for  pretty soon might actually be now uh  we'll give you an example  um so then we'll  see who's who's understanding and who's  not okay so let's continue here  um  so  so therefore the problem of determining  does exist have a force win is exactly  the tqbf problem because exist has a  force when exactly when the formula is  true  and what we're going to show is that t  qbf  is  a polynomial time reducible to  generalized geography  but it's conceptually we're going to  think about tqbf now as a game because  that's how geography is generally  geography is set up  so we're going to given a formula like  this we're going to construct an  instance of generalized geography we're  going to construct a graph  where play on that graph is going to  mimic  play in the formula game  so making the moves in that graph are  going to correspond to setting variables  true and false because the graph is  going to be specially designed to to  force that behavior  okay so i think we have a check-in  coming in yeah so let's just see how  um i i suggest you pay attention to this  check-in and really try to think about  it and solve it because i think that'll  help you make the connection between the  truth  of the formula and  exist  having a winning strategy  okay so  so if you take this formula here here's  you know this is some particular formula  exist x for all y blah blah blah  um this is going to be associated to  that formula a formula game  where first exist moves  assigns a value for x and depending upon  that value for all is going to move and  they're going to assign a value for y  and after that's done this part here is  either going to be true or false it's  either going to be satisfied or  unsatisfied  and i want to know can exist always find  a way can it does  to to guarantee that this part is going  to be satisfied or can for all always  get find a way to guarantee that this  part is not satisfied  okay so it's you have to stare you have  to look at this  uh formula to understand  which side is going to end up succeeding  um  so you understand you have to if you're  not clear on the rules look back here  exist is trying to make this part  satisfied for all is trying to make this  part unsatisfied but they neither of  them has the no the totally upper hand  because existence depicting one of the  variables for all is picking the other  variable  but they're doing it in a certain  order okay first exist is going to pick  then a for all is going to pick  okay so that's the way the game works  exist picks a value then for all picks a  value and then you see who wins  so who wins  who's guaranteed to win  um you know can can we make sure exist  always wins or can we make sure that it  for all always wins for this particular  formula so there's just two variables  here  and you can think about this in either  of two ways strictly speaking purely as  a game  or you can look at understand whether  that formula is true or not  um  they're equivalent ways of looking at at  the problem  in fact in a certain sense it's the same  that's what i'm trying to  trying to get across  thinking about this as a game or  thinking about in terms of quantifiers  doesn't make any difference um  okay  almost done here  so  all right  last call  closing the poll  okay i think this one you got  you you you did pretty well on so yeah  the correct answer is the for all player  has a uh winning strategy has a forced  win  uh  and um also similarly the  expression um is  fee is is false  because you know if you try to find some  x  such that no matter what y you pick  um this is going to be true  this is going to be satisfied you're out  of luck because you know if you make x  true  well um  then  if you make x  true  now i'm confused uh  what is going on if you make x true then  uh  y  oh if you make x true  then uh x bar is false  um so y bar has to be true  uh  um  now i'm completely confused oh no  if make x true it has to work for all y  so this thing is clearly false if you  make x true  um  uh then then  then this x bar is going to be false and  it's not the case that for every y  of the setting for y because y is forced  here y is going to have to be true  well y bar is going to have to be true  so y is going to have to be false  somebody if you make  it very clearly but if you make if you  and now if you if you try again  to make x false  um  maybe that's a one to think about first  if you make x false  so  you're going to be stuck on the first  clause because you have it has to work  for both settings of y  but maybe thinking about it as a game is  better you know if you  no matter what x does a y is going to  find have a way of  making one of those two clauses  unsatisfied so if x is true why can uh  uh can set itself like why we can set y  to make the second clause  uh unsatisfied and if x is false we can  find y can be assigned to make the first  clause unsatisfied  um  so anyway um  the right answer is uh the for all  player uh has a winning strategy here in  the end of the day it's not critical  that you understand this correspondence  but  but you have to then if you don't quite  understand that you're going to have to  take on faith  that the formula game is the same as  tqbf  because that's we're going to use and by  the way i would like to also mention  here that um  that this correspondence between  uh games and quantifiers is some things  that mathematicians use all the time  um  because if you have a if you have some  expression which comes up often in  mathematics where there are you know a  whole run of quantifiers you know in  front of some  uh some statement  it's really kind of hard for anybody uh  to really uh  get a feeling for what that means  if you have like six alternating  quantifiers which happens often in  mathematics you're going to have  statements that have a lot of  alternations with quantifiers very hard  to get a feeling for what that means but  if you think about it as a game it's  much more intuitive and it's completely  equivalent to think about quantifiers go  back and forth between quantifiers and  games  anyway um so let's  look at the at the reduction the the  construction that shows uh generalized  geography is uh complete took a little  longer than i thought maybe this is i  just want to make sure everybody's with  me on this so why don't we call the  break now  and then we'll come back  um and we'll look at the construction um  after that because the construction  itself is going to take about 10 minutes  to to work through to figure out how to  reduce  tqbf  and build a graph that simulates the  formula  okay so i'm going to move  right on to uh  the um  and we'll come back  okay so feel free to ask some questions  um  and uh  get ready for us diving in to a  construction  uh for the um  uh for the reduction of tqbf to  generalize geography  so  there's kind of an interesting question  about  you know  what's a relationship  you know between a formula and when you  swap the order of the quantifiers um  uh you know the the the the question is  asking does that somehow relate to the  negating the um  the you know the part the quantifier  free part the part without the  quantifiers and i um i don't think that  that's the right correspondence there it  actually is  uh some relationship when you say um  you know there exists for all is  actually a stronger statement than than  saying that for all there exists in  general the um  that actually implies you know exist x  for a why is this  implies uh  for a why there exists an x um because  what you're saying is that there is  the the choice of y  if  there exists an x  uh for all y there was one x that works  for every one of the y's if you're  saying for y there exists an x the  choice of that x can depend on the y  um  so there  there is a connection but um you have to  think through what it means in order to  understand that connection  not i think you won't need to know that  in order to process what we're going to  be doing but maybe it just helps you  think about it  um  other questions here  okay i think we're  um  we're out of time here  so let's  return to  our  our lecture  um  so i'm gonna go back  hopefully that won't  crash everything  [Music]  here we go  okay so now we're going to be reducing  tqbf  to generalize geography are we all  together now  um  uh  okay  so i'm gonna  illustrate this construction which is  very nice construction by the way i'm  gonna illustrate this construction just  by doing an example or a partial  example okay but i think it'll give you  the idea  so you understand so what i'm trying to  do here is i'm starting off with a  quantified boolean formula  so here it is  um i'm going to assume it's in  conjunctive normal form  which i can always  convert it into that form uh maybe by  adding some additional variables but  without  doing anything  too drastic to it  and so now what i'm gonna uh  starting from that formula  i'm gonna now build a graph  we're playing the geography game on that  graph which is remember taking turns  picking nodes which may form a path  is going to correspond to um  playing the formula game which is  picking the variables of that formula  and then you want uh the  if the exist player wins in the formula  then player one is supposed to win  in the geography game  okay  um  so here is how the graph is going to  look okay so good to follow try to  follow this um  so it's going to be  kind of there's going to be kind of two  parts one that's going to  correspond to the variables and the  other part that's going to correspond to  the clauses  um  and so for each variable i'm going to  have like a diamond structure here so  there's a little starting thing that's  going to be unique to the to the first  variable but then  every variable is going to have like a  little diamond structure here um and  they're going to be attached one to the  next  and we'll to understand how this what  this means  when we play  geography  on this piece of the graph  so but let's just understand the  structure first  okay  so this is the start node  and now um  here i'm player one is going to play the  role of exist player two is going to  play the role of parole and in fact i'm  going to identify that so i'm going to  call because it's going to be helpful  just to have  uh  helpful just to  think about player 1 is being the exist  player so the exist player is going to  be playing on this graph  the exist and the for all players are  going to be playing on this graph it's  really just players one and two  so exist the exist player player one  has to  b  by the rules has to pick the start note  and now with player two's move these the  four players move  now  if you're with me  on this graph  the for all for the for all players move  um is now not very interesting because  it's forced  it has to go to here because again  they're just picking  the  um  the nodes of a simple path  in this graph so  for the fraud  player to the for all player has to go  here is if exist started over there  now um now it's now it's the exist  player's turn  and now now something interesting  happens the exist player has a choice  you can either go left or right  there are two possibilities  and then  after that  the four player's turn  who's again just forced so so far for  all has not had much interesting stuff  to do in this hasn't has not had to have  has not had to think hard  um  so for all players  no matter whether exists went left or  right the for all players move is forced  and ends up over here  and now let's see if this player's turn  again this is kind of an easy one  oh so before i do that let's just look  at how that play could go  so the first two moves that's kind of i  was suggesting exist goes here and then  for all goes there  i'm gonna i'm gonna illustrate  kind of possible plays through this  graph by tracing them out in green  um  so now exists goes here for all goes  there those those were forced but now  the exist player goes could either go  this way  or  could go that way  so those are two  possible different  ways of playing this the game that we  said we're building  um  and now  uh now whose turn is it here  this was the for all player pick this  one so the the the next turn is the  exist player and that's forced  but now  notice now for the first time the for  all player has to think because the for  all player can either go left or right  so could either go left or it could go  right  um  and so  then there's going to be a sequence of  these diamond structures where  kind of they're constructed so that  alternately either the exist has a  choice or the for all player has a  choice  until you get all the way down to the  bottom  okay  that is the graph we've built so far now  if we stopped here  um  then  whoever ended up at this point whether  it was existing for all would be the  winner because there's nowhere for the  other for the opponent to go of course  that wouldn't be very interesting  so there's going to be more stuff we're  going to add on which are going to  correspond to the clauses  now  how to think about what's happened so  far  now so maybe as some of you might be  guessing  is that um when the exist player over  here has a choice could have either gone  left or right that's going to correspond  because this is supposed to be  mimicking the formula game that's going  to correspond to the  uh  exist player  in the formula picking the variable  either true or false  so that the left or right is going to  correspond to true or false  and so um let's just arbitrarily say  left is going to correspond to picking  it true  and right is going to correspond to  picking false  okay so that  so the way i've said it so far is  exist ended up picking the first  variable false then for all pick the  second variable false  and the empty variable also got set  folds everything got set for so far who  knows what happened over here of course  um  uh  okay so just to understand what we've  done so far  now i'm going to show you how to build  the rest of the graph so let's take away  the green part um and now we're just  back to the construction  the green part is actually how you use  that construction so back to the  construction here now what do we want to  achieve  now by the time the  the players on the graph have got down  to here they've effectively made an  assignment to the variables by going  either left or right at each one of  these diamonds so the assignment is done  from the game from the perspective of  the formula game the game is over and  now one side or the other has one  here  we want to build some extra structure  here as a kind of an end game  to make sure  that  the for all player gets stuck  if the exist player has made an  assignment which satisfies the formula  satisfies this part of the formula  and  the exist player should get stuck  if the for all player if the assignment  that they made  does not satisfy the form  okay so let's see how we're going to  achieve that with some additional  structure  so  there's going to be some extra node over  here  let me just tell you what the structure  is and then we'll argue why it works  so there's going to be  going from this bottom node we're going  to start a kind of  you know  the second part of the graph there's  going to be a node which  fans out to a node for each one of the  clauses  and each one of those clauses in turn  fans out to a node for each one of its  literals  so you see we have clause c1  and it has the literals x1 x2 bar x3 x1  x2 bar x3  so there's no note for each of the  literals and clause c1 same for clause  c2  and so on  now we're almost done  now i'm going to tell you how to connect  up these literal nodes  so x1 each node is going to get  correspond  back to  its own diamond so now we're going to  tie it back to the the first part where  we made the assignment the assignment  part of the graph  x1 is going to connect back to the true  part of the x1 diamond  and x2 because it's negated is going to  tie back to the false part of the x2  diamond  because it's an x2 variable  similarly x1 bar  now in  you know in  because x1 bar  you know here's x1 bar and clause 2.  x1 bar  is going to connect up now to the full  side of the x1 diamond  and x2 bar is going to connect up to the  full side of the x2 diamond  and so on so if we had i don't have the  other diamonds here but the x4 would  connect up to the true side of the x4  diamond  for example  that's the whole construction let's  understand why it works  okay so the end game here is we want  exist to win  um  if the assignment satisfied all the  clauses  and for all should win if there was some  unsatisfied clause  okay so why why does this happen so  let's put back an assignment here that  was made so here's here for the in the  one i'm i'm putting back is  you know the assignment that they  cooperatively built  had x1 being true  x2 being false  and some other stuff  now why does this work so what we want  to have happen now now so now the  move proceeds over to here and you need  to arrange it so that exists  is the one who picks that note  if you have a um if you know if you you  know  if this would have been for all just add  an extra node to switch who's  whose turn it is  but you want exist to be up here and  what you want is for all to be picking  the clause  the clause note  now here is  here's the part to understand what's  going on  um  we want  for all to win  if this is not  satisfied so the  for all is going to make a claim  um  if this is not satisfied there is some  clause which is not satisfied there's  some clause which ended up being false  in the assignment so  the for all player gonna say i think i  won  and i won because clause number one is  unsatisfied  so it's going to pick clause number one  so here's the four players more going  his exist player  and so  the four player picks the clause claimed  unsatisfied so over here  full player says claude c1 is  unsatisfied i'm going to move here  if this player says  i don't agree with you the assignment  you know your your line um  uh  the  assignment actually makes one of the  literals true in clause c1  let's say it made x1 was true  as it as in fact it is here x1 in fact  is true so  exist is now  its turn it's going to pick the the  literal that was true  in the assignment in that clause  that the fraud player is claiming is  unsatisfied  one of them is lying at this point  this is either now true in here which  means a for all player did not pick an  unsatisfied clause or the exist player  picked a false literal which is the the  the exist player picked the false letter  over to the exist players lie  so now we're gonna now  the moment of truth  because  let's see if exist gold picked here  now it's um the for all's players turn  it's connected back to the true side of  the diamond  so that node was already used that's the  only place that for all could possibly  go for all is stuck  and an exist has won the game which is  what you want  because  um  for all claims are false and the and the  exist was was was was correct in saying  that x1 satisfied clause c1  so x1 is true  so um there's nowhere for for all to go  but compare that with the situation if  the assignment on this part of the graph  had gone through the false  had assigned x1 to be false  so the path had gone through the full  side of the diamond  now this node would still be unoccupied  would still be um  available  so now the for role play so if if exist  was claiming that x1 was true and it  really was false  the fraud player would be able to move  on to that node and now it's the exist  player's turn and the exist would be  stuck because  this node down here is guaranteed to  have been used  okay so that  is the idea  it's really very beautiful um  and actually not that complicated you  have to you have to stare at it a bit so  let's just see what happens for example  if um  i had one other case maybe it's not  necessary at this point um but you know  if the for all player claimed  well  it's the second clause which is  unsatisfied in the  mutually selected assignment um  then  uh  so this is sort of the the case where  the for all players is correct um you  know in in that if the exist player now  says well i think x1 bar satisfied that  uh second clause  but you know this  assignment made x1 true so x1 bar is  false and so the exist player is lying  this time  and so now the for all player can take  this last edge and go here  and has one one last move to make and  then the exist player stuck  so the for all will win in that case  so let's turn now shifting gears  entirely  to a different um  uh  part of the spate you know  of space complexity instead of looking  at polynomial space which is like very  powerful we're going to look at log  space which is comparatively speaking  much much weaker  so log space are the things that you can  do when you only have  enough space to write down essentially a  pointer or some fixed number of pointers  into the input that's what you get when  you have logarithmic space  and  you know because log space is enough to  write down an index of something um so  you can just this is what you can do  with a bunch of pointers  and  in order to make sense of this we have  to change the model of computation  because if we use the ordinary one tape  turing machine  just scanning across the input  will  just reading the input with the way  we've defined it would cost space n  and so you can't even read the input  if you have less than n space available  like log space  so we're going to introduce a different  model just  to allow us to define this  and that's a model where it's going to  be have two tapes  where the part that contains the input  doesn't count  and that's in order to make sure it  doesn't you know it's not being cheated  the the input tape is going to be read  only  um but it doesn't count toward the space  used only the work tape  which is now has can read can be read or  or written or red  is going to count toward the space bound  okay  um  so now what we're going to define is  you know  using this definition for  our space complexity  um  we'll define  space log n  the things that you can do if you have  only a log amount of space here  on the work tape  so the length of the input is n  the length of the um  of the work tape is lo order log n  so we have uh the deterministic and  non-deterministic associated classes uh  space log n and n space log n we're  calling that l and n l  log space and not deterministic log  space  okay um and as i said that's just enough  space to write down some pointers into  the input and let's do some quick  examples  um  so if you take the language of  palindromes essentially or ww reverse um  that's  in log space uh  so here is a string i don't know i was  here's a string that's in www reverse  and  i mean the ordinary way you might use to  test whether a string is of this form in  ww reverse might be to cross off  corresponding locations here but you  can't do that it's a read-only input  tape so you have to somehow avoid  writing on the input tape but still  testing whether you're in this language  it's not hard to do you can use the work  tape just to keep track of where your  pointers would be  um  and  in so doing you can just  make sure you're  matching off corresponding locations  with each other in the input  so log space because of its ability to  store some fixed number of pointers  it gives you enough to test membership  in this language  so this is in this is solvable in log  space  uh  the path problem which we've seen before  you know you're given a graph and a  start node and  an ending node or a start and a target  you're given a graph of an s and t  it's a directed graph and you know can i  know can i get from s to t  um so that problem is solvable in  non-deterministic log space  uh because as  um the way we would do that i'm not not  writing this down in in any detail but  uh what you would do in your  um not you know non-deterministic log  space machine you have your input graph  written here in the input  and you're just going to guess the  sequence of  nodes  one by one which takes you from the  start node to the target node  you're not going to write down that  whole sequence because that in advance  because that would be  cost you way too much space to write  down the only space that you're going to  use is to remember the current location  we are sitting  so you start out you know you write down  on the work tape the start node  then you're going to  non-deterministically pick one of the  outgoing edges from the start node and  look at its associated node and replace  that on your work tape and keep keep  repeating that if you ever get to the  node t then you can accept  and you also have to be a little careful  i don't have this on on the slide you  also have to make sure that if the graph  has a loop in it  or because that's not allowed in space  complexity uh so you're going to need  also a counter to make sure that if you  count up to a um you've you you you  you've gone through a number of nodes  which exceeds the total number of nodes  in the graph  uh then you can shut down that branch of  the non-determinism uh because  you know it's  you know just it's just going in  a loop  um  if there's any path that connects s to t  there's certainly going to be a path  that has at most the number of nodes of  the graph in it  okay um so path is in nl this language  here is an in l um  what's the relationship between l and n  l well certainly the deterministic class  is contained in the non-deterministic  class  that's all that's known  whether these two collapse down to be  the same  is unsolved and so we're going to spend  a little time next lecture exploring  this  um but but let's let's first  look at some of the basic facts about  log space  um sort of setting ourselves up for next  lecture  so  first of all anything that you can do in  log space you can do in polynomial time  in a sense kind of trivially and in a  sense we almost kind of really proved  this already because if you remember  we said that anything that you can do in  a certain amount of space you can  do  in that much  uh you can do  in time  in  that's exponential in the amount of  space  in the corresponding amount of space so  going from space to time blows you up  exponentially  and  order log the exponential of order log n  is polynomial  so the way we're going to prove this  and again we kind of proved this already  but let's just go through it again  specific to  log space  uh we'll say and this is going to be a  useful definition for us anyway um  a configuration for m on w so we would  also we already talked about a  configuration of turing machine m which  is just a snapshot which is the tape  the state and the head position  when we have  an  input which is sort of an uh read-only  input which is not being counted toward  the space we don't include that in the  configuration um we just say  it's a configuration of m on that  particular input but we're going to be  counting configurations and i don't want  to count all the different inputs as  well i'm going to fix the input and  count all of the configurations relative  to that input  and  so the number of such configurations is  just simply going to be  the number of states times the number of  head positions for the two  heads now  times the number of possible tape  contents and which as i mentioned here  is the number of different possible tape  contents  it's d which is the tape alphabet to the  order log n and that's just going to be  n to the k for some k so it's going to  be polynomial so that tells us  that because the total number of  configurations  um  for m on w is polynomial  this machine  can only be running for a polynomial  number of steps otherwise it will be  looping people are repeating a  configuration and so therefore  that machine has to run itself in  polynomial time so you don't really in a  sense have to do any work if a machine  is deterministically designing a  language in log space  it's also deterministically deciding the  language in polynomial time because  that's all the time you can use when you  have log space unless you're looping  which is not allowed okay so therefore  the same machine shows that you're also  in pain  oh  i'll get to that in a second one thing i  forgot to mention on the previous slide  when i'm talking about the model is by  the way this model is not so  unreasonable um  where you have kind of a imagine having  a very large read-only input  and and your local storage is much  smaller it's much too small to pull in  the entire input  the way i used to explain this  years ago was like you have a your input  is a cd-rom which you guys probably  barely even  know what it is anymore but used to  distribute software that way so the rom  was like on a dvd uh or cd  which contain your  whatever you're trying to like yourself  whether you're trying to distribute  you know it was some large thing uh  relatively and so  you would imagine having a smaller  amount of  memory relative to that so you didn't  want to necessarily copy that whole  thing into your storage  maybe even a better example now is like  you think of the input as being the  entire internet  um obviously you can't download unless  you're google you can't download  the whole internet into your own uh  local memory  but you're going to have references  pointers into the input into different  places that's perhaps more analogous to  this sort of read-only input touring  machine model that i'm describing um  okay  uh  so  um  and it's another fact i want to mention  is that i'm not going to  savage just the anything that you can do  non-deterministic log space you can do  in deterministic  uh space but now with a squaring  and that's using the same proof that's  using savage's theorem which you have to  check also works down to log space  same proof  um  so anything that you do not  deterministically in log n space you can  use deterministically log squared n  space  okay so let me just see if there's a  couple of questions i want to answer  here  [Music]  the relationship between  l and nl is not known to be strict  nobody knows of an example no one knows  whether they're equal  and  and have people looked at  um  sublinear time classes  yes generally for  when you have non-deterministic or  probabilistic which we're not haven't  defined yet but we will um people have  looked at um those sub-linear time  classes as well deterministically  doesn't make so much sense because then  you can't even talk about  either the whole input um  okay uh let me move on um  okay so this this is my last slide let  me see if we can do this before we break  um not only is l contained within p but  this much stronger statement is that nl  is contained within p and that we you'll  that for that you'll have to do some  work because converting your  non-deterministic log space machine to a  deterministic  machine  um obviously you're going to have to  change the machine um  and so we'll introduce a new method here  maybe we'll  quickly go over this at the beginning of  the next lecture  i don't like rushing through things at  the end  but for this  if i if i'm given  um  a non-deterministic machine  that runs in log space  i want to make a new machine that runs  in polynomial time deterministically for  the same language  and i'm going to define something called  the configuration graph for m on an  input w  and that's just you take  m  and it's and w  its input and you look at all of the  configurations  um  for m w actually configuration graph  actually that's  should be called the computation graph  that's what it's called in the book but  okay this is a typo yeah i'll fix that  next time  uh it doesn't matter computation graph  configuration will graph uh all the same  basically you're going to take all of  the configurations of m on w of which we  already observed  only number or only polynomial in number  and may they become the nodes of a graph  and the edges connect two configurations  if one follows another according to the  rules of the machine  okay so here's a picture  these are  ins this is some graph the nodes of this  graph are the configurations of m on w  right so each node here corresponds to  the machine a snapshot of the machine  a  tape contents head position and state  so writing down all those different  configurations i connect one to the  other if i can get to cj cj from ci in  one step legally on the machine  and then  uh  the non-deterministic machine m accepts  w  exactly when there's a path from the  start configuration to the to to to an  accept configuration let's assume we  have just a single accept configuration  as we argue we can do  one or two lectures back  um if we clean up the tapes  so  testing whether you can get from the  start to the accept um  uh is the same as testing whether the  machine accepts its input  and so  because we can test  whether there's a path in a graph  connecting two nodes in polynomial time  we can solve this problem on this  computation configuration graph in  polynomial time and so we can figure out  whether the non-deterministic machine  accepts its input  sorry that came a little faster than i  like to do so we'll we'll we'll see it  again um so here's the polynomial time  algorithm  you construct the graph  you accept if there's a path from the  star to the accept and you reject it  does not  um  and so that tells us that  not only is l contained with an nl but  nl itself is also contained within p so  here's that here's a kind of a nice  hierarchy of languages  we don't not only do we not know whether  l equals n l we don't know whether l  equals p  it's possible that anything you can do  in polynomial time you can do  deterministically in log space shocking  as that might be because it's a pretty  weak class  but we don't know how to prove that uh  there's anything different  anything in p that's not in here  um last check in  uh so we showed that path is in nl  what's the best thing we can do about  the deterministic space complexity of  path so deterministic so this is  non-deterministic log space what can we  say deterministically  about path um  hint  this should not be this should not be  hard  uh if you think back to what we've  shown very recently  okay  get your check in points closing up  closing stop here  all set one two three  yes so the correct answer is log squared  space because this is just savages there  we can do it in log space  non-deterministically so you can do it  in log squared space deterministically  okay so that's um this is what we did  today  um  and i as i mentioned i will uh do this  again  on at tuesday's lecture just to recap  that  all right so i'll stick around a little  bit for questions and um  okay someone says to be about the  nomenclature  why is it l in l space  um  i you know  because there's not really people don't  usually talk about l time so l is sort  of  everybody knows it's the only reasonable  option is space so people just say l and  then l  um  i mean some of these things have had  you know  you know these things these names have a  little bit evolved over time and even  now some people say talk about uh  um  you know i call time classes some call  some people call it d time classes  you know it's just you know it  uh you can make different choices there  um  okay uh let's see  good  um  why does savage's theorem work for log n  you have to you have to look back and  see what you needed  um  and all you needed to be able to do  was to um  write down  uh the configurations  um  and uh if you look back at how savage's  theorem works you're just needing to  write down the configuration so the  deterministic machine can write down the  configurations for the non-deterministic  machine they take exactly the same  size  and then you look at the recursion  and the the depth of the recursion is  going to be exponential  in um the size of the configurations and  so you're going to end up with a  squaring again you have to go back and  just  rethink that proof and you'll see  nowhere did it need a linear amount of  space it works down to a lot it actually  does not work for less than log n  log n is sort of the lower threshold  there  and the reason for that is because you  also need to store the  input location  and that already takes log space  uh the you know not the  the the the head tape heads the tape  heads already kind of  um  have kind of a log space aspect to them  and so  if you're going to you use less than log  space then funky things happen with the  tape head storing the tape heads and so  less than log space usually turns out  not to be interesting  very specific to turing machines and not  sort of general general models  yeah so somebody's asking  suppose in the reduction um  to generalize geography from tqbf if the  formula had two exists in a row  um then you would do kind of the natural  thing in the graph instead of having  that spacer edge between the two  diamonds you could just have one diamond  connecting directly to the other diamond  without a spacer edge  and that would be give you the effect  of  not switching whose turn it is  uh somebody's asking me just a general  question  are people thinking about  these open problems  um  i don't know  people don't don't don't say um  there was a lot of work  on problems related that seemed to be  related to those  those many open questions like p versus  np  l versus nl or l versus p and so on p  versus p space  we'll say we'll talk a little bit more  about some of that there's some very  interesting things that have been  developed i think there's a sense within  the community  that people are stuck  and it's not something you're going to  need some sort of major new  uh idea in order to push the thing  forward so i don't know how many people  are still thinking about them  i hope people are because i would like  to see the answer at some point but  or get the answer  i think about them sometimes myself but  one has to acknowledge  chances of success are not high  uh  okay um  so we're a little after past the end of  the hour here um unless there's any  other questions if i didn't answer your  question i may have missed it so you can  ask again  um  but otherwise i'm going to  close this close the session here  okay bye bye all have a good weekend see  you next week  take care  you

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let's get  back to  space complexity um so what  uh just to review what we've been doing  um  from last lecture which  feels like a long time ago but it was  only two days ago  um we  were looking at um  those three theorems  which all had this basically the same  proof  one was the latter about the latter  problem of uh where you're trying to see  if you can get from one  string to another string  changing one symbol at a time  um and all of the strings were in the  language of some finite automaton or you  had some reasonable rule for  saying which of the valid strings which  are not  and then uh we built on that idea we  showed savage's theorem that going from  any non-deterministic machine to a  deterministic machine only squares the  amount of space  and then um  we finally  uh proved that tqbf uh this problem  about quantified boolean formulas  testing whether they are true  uh that that problem is complete  for p space um so we're going to build  off of that last last theorem today  and talk about  uh  one other  complete problem um and also  show a connection between  p space  and testing uh which side uh  can win in certain kinds of games  um  so um and then the at the end of the  lecture we will the second half of the  lecture we'll talk about  diving deeper into space complexity  we're going to talk about a different  part of the parameter space where  instead of looking at polynomial space  we're going to talk about what happens  if you have logarithmic space which is  another natural  point to look at for from  for space complexity but we'll get to  that after the after the break  okay um  uh talk about game complexity and games  um  so when i was little uh we used to play  me and my sisters would play a game  called geography i don't know if how  many of you have heard it or maybe it go  goes under a different name um  but uh  it was a game about basically about  words and places and it has a very  simple um  set of rules um let's say there were two  people you take turns picking  names of places  and then each person  has to pick a place  which starts with the letter that the  previous person's place ended with  for example  you know you have two people playing um  perhaps you have you know you agree on  some starting place like a major city  nearby like boston  um so one you know one player the first  player takes boston  and then after that the next player has  to pick a place that starts with n  because boston ends with n  and so maybe nebraska  would be a possible response to boston  and then the first player would have to  respond to that with a place that starts  with an a because nebraska ends with an  a it's like maybe alaska  uh which also  starts with a ends with a  so  then arkansas maybe would be a nat  reasonable response to that and then  that ends with s um so uh san francisco  and then so on and so on um  and uh  of course you want to  forbid people from  reusing  names because then you'll just get into  a loop you know saying alaska alaska  alaska won't make the game very  interesting so you have to forbid  that possibility  names cannot be reused they can be used  at most once  and then um so eventually  one side or the other is going to run  out of names uh and not have a either  because of the lack of their  geographical knowledge or maybe you've  just exhausted all the possibilities  and i will will not have a response and  that person will be stuck and that  person is the loser um the other person  has won the game  so the objective is to try to avoid  getting stuck um so let's just take a  look um i'm going to think about that  games in a somewhat  slightly more abstracted formalized way  um so you write down all of the  legitimate places um they become nodes  of a graph such as i've shown here  and then you draw an edge from one node  to another if um that  if that corresponds to a legal move in  the game a legal step of the game so you  know boston connects to places that  start within like new york and nebraska  um so the first person might pick boston  the second person might pick either of  these two um and then back to the first  person if they pick nebraska they could  pick arkansas they can pick alaska and  so on just as i described  believe it or not  we actually played this game i can  remember playing this game when i was a  kid that was of course before we had um  uh  you know league of legends and other fun  fun stuff but this is what kids used to  do uh  so  anyway um  um  so just to  get it down on the slide the rules are  we'll assume we have two players we'll  call them player one and player two  player one goes first  um and they take turns picking places  that start with the letter which ended  the previous place  not no repeats allowed and the first  person players the first player stuck  loses  you know the person who doesn't have a  move to make  so what we're going to look at  um today is a generalized version of  that where we're kind of um just going  to take away the names but just leave  the underlying graph  and we're going to call that generalized  geography so here um that can be played  on any directed graph you take away  the names of the edges the names of the  nodes now you just have a some arbitrary  graph  um you're going to designate a  particular node as the starting node and  then the players take turns following  those edges  um and uh what uh you're  because you're gonna forbid having any  loops reusing any um any nodes uh what  you end up  doing is working together you're going  to be constructing a simple path in that  graph  simple path just follows those edges and  never intersects itself  and the first person the first player to  be stuck is the loser so the other one  is the winner okay so that's we're going  to call that generalized geography and  we're going to look at the computational  complexity of deciding for a given graph  and starting point  which side would win if both  uh players  played  optimally as best uh the best possible  okay  and we'll name that problem gg or look  at its associated language so here is a  graph and the starting node  and you want to say um  that pair is in  the language gigi  if the first player player one  who must play at that node a um to start  off with has a forced win  um  uh  in that  general geography game on that graph  starting at a okay where again what i  mean by a force when i'm not going to  define this formally that we could do  that it's not hard to do but a forced  wind also it's also called a winning  strategy  um that just means that if both sides  play optimal they pay the best possible  not up their capability i mean the  absolutely the best possible play  then that  player one  would still win there's nothing that  player two can do to prevent player one  from winning um if player one has a  winning strategy or a forced win um you  may have seen for example uh  there are examples of this that  i mean i don't even know if they still  exist but it used to be in the newspaper  they used to be examples of chess boards  and they would say you know if they  would start from a certain position and  they would say white to win in three  and so that means white has a forced win  no matter what black does white is going  to end up winning but you have to think  of what the strategy is it may not be so  obvious to think through what are the  right moves that what makes but the  point is that no matter what black would  do white would end up winning um and  what you can show we're not going to do  that but for  a class of games that includes this  generalized geography  either one side or the other is going to  be guaranteed to have a forced win  that's not necessarily true for all  possible games obviously  but for a large class of games um  in one side or the other  is guaranteed to have a winning strategy  or forced win  okay  so let's just review that because this  is going to be essential for  you know the first half of the lecture  to understand what we mean by a game and  what we mean by one side or the other to  have a force to win okay so let's let's  just uh oh yeah and what we're going to  show is that gg is piece based complete  that the problem of giving one of these  graphs and a starting point figuring out  does player 1 have a force win or is it  player 2 that has a force win  um  which side is guaranteed to win under  optimal play that's a piece based  complete problem  um  and so let's do a little checking on  that and so i'm going to give you a very  small example  of the generalized geography game  you have to figure out  whether it's player one or player two  that has the forced win the winning  strategy or maybe neither or both those  are the four options okay and you  understand player one has to play here  because that's the starting place of  this general geography game so then it's  up to player two to continue on  all right so let's um launch that poll  uh you'll have to think about it a  little bit um  and uh  let me know what you think which side  has a forced win  which player makes the first move it's  player one makes the first move and  player one plays as i showed you here  this is player one playing at a so  player one picks a  picks and then player two has to pick  one of these two  okay all right i think we're gonna end  the poll  everybody's in  pick something  um  you you guys did not do well on this  bowl  uh at least not too many of you picked  d  that's that's true that's a that's a  little reassuring because you can't have  both players having a forced win  i did say that in games of this kind one  side or the other is going to have a  force win so c is not a very good choice  either um  so you're  maybe you should spend less time reading  your email and more time listening to  the lecture uh  um  the actual correct answer is b  player two  um which is obviously a is a minority  position here player two has the forced  win let's understand why um  so  player one plays here  as i said that's the b that's the first  move player two can either take the  upper  node or the lower note  okay  player two takes the upper note  then player one has no choice but to  take the rightmost node  and now there's no move for player two  uh so player two will lose  um if player two takes the upper note  the upper choice there because  player two will go here player one will  go there and it'll be game over  and player one will have one because  player two was stuck  however  player two had another choice  um  uh player two had another choice um  player two could have also gone down  here  um so player two uh if it played down  here um  things are looking a lot better for  player two because player two goes here  player one's move is forced goes here  but now there's another move left uh  that player two could make so play  player so two goes here one goes there  player two goes up there  and now player one is stuck  so at player two's choice  um player two can make a move which will  end up  causing player one to get stuck  and so because  it you know that's the nature of what we  mean by uh having a winning strategy  there is some way to guarantee if you're  playing optimally and you're certainly  playing optimally mean you'll take that  lower move  uh  then you will  take that so  under  uh optimal play  player two will win this game  and so player two has a winning strategy  it has a forced win  okay  um so i i see so  i'm not going to define playing  optimally i'm going to you know there's  some questions here about that about  that that's um  you know you can think about  i'm going to i'm going to leave it  um  you know  somewhat informal um but we could make  that all precise i i don't think it  would be clarifying to to go through a  precise definition of that um  uh  you know you you can just think you we  don't even have to think about it in  terms of optimal play you you can say  that there is um  a uh forever  uh a  um  you say a player has a force win  if  uh  they have moves that they could make in  response to every possible move the  opponent could make  uh that'll guarantee that they will end  up winning  so it doesn't say we don't even have to  talk about optimality here just talk  about every possible opponent  every possible opponent is going to lose  to somebody who um  uh  you know  has a has a winning strategy and pl and  follows that strategy  okay um  okay i can see there was some confusion  about who was going first and so on i'm  glad we that was part of the reason i  gave this check and just so hopefully to  clear up some of those points um so  player one plays here then player two uh  goes and i got some comments that some  people were confused about what does it  mean for the first uh  where does the first person move okay  hopefully that's  that's uh  clear all right so let's move on  because we're gonna make we're gonna be  spending a while  um  proving something about generalized  geography and these graphs  um okay so let's let's continue  so in order to make a connection  between  um games  and complexity  we're going to need to  talk about a problem one that we've  actually related to one that we've  already seen where we can reformulate  that problem as a game and that's going  to be a problem on quantified boolean  formulas called the formula game  okay so this is a different game now um  not one that you're likely to end up  playing but in in in in a sense in which  you'll see this is a still a reasonable  game where two players can play against  one another and one of them is going to  end up winning and one of them is going  to end up having  uh  one or the other will have a winning  strategy so let's understand what the  game is  uh the game is played  the game is played on a formula  so you so you write down a quantified  boolean formula remember we talked about  that you know for several lectures now  uh you know it's a bunch of quantum  variables with quantifiers  and um  then there's the part that has no  quantifiers  um which  you know uh you can always put into  conjunctive normal form if you want but  for the purposes of this discussion so  far it doesn't matter later on we'll  actually want it to be in quant in  in cnf but okay but but for now we just  have some formula  [Music]  and  associated to that formula there is a  game in a very natural sense so first of  all let me they're going to be two  players  but now the names of those players are  going to be player exist and player for  all  okay and the way the way this is how the  game goes  the uh  the player exists  but both players are going to be picking  values for variables  they're going to say  you know variable x1 is true variable x2  is false variable x3 is false variable  x4 is true  and and they're going to be that that's  how the moves of the game go  they're just assigning values to  variables  so that in the end  all of the variables have been assigned  a value a boolean value and so we end up  with an assignment  but  before we get to that stage  which players get to pick  which variables and the way it works is  the exist player is going to assign  values to the exist variables  so exist we'll pick x1 and x3 and assign  them val values  the for all player is going to pick the  very ver assigned values to the  variables that have referral quantifier  attached  so in this case would be x2 would be um  the the the for all player will pick the  value for x the x2 variable  um fading out here  okay  now um so  and the order of play is  according to the order of the appearance  of the quantifiers in the formula  so the way i have it written out in this  particular formula it's exist x1 so  exist will pick the value for x1  then it's for all's turn for all pick  the value of x2 then it's going to be  exist picking the value for x3  and and so on until they get to the end  and all of the variables have been  assigned a value by one side or the  other  and now the game is over  okay  so  what's left is to understand who has won  at that point  um  and the way we  say who has won is  exist  wins  if in the end  this part psi  which is the part without quantifiers  ended up being satisfied by those  um by that assignment that the two  players together ended up picking  and then for all  will  win  if that  part of the formula is not satisfied  so let me just write that down here  after the variables have been assigned  values we determine the winner  exist wins if the assignment that they  co-op together built they built  cooperatively together uh satisfied psy  and otherwise for all ones if it doesn't  satisfy psi okay so you kind of think  about it this way  you know exist is picking the values of  those variables  he's trying to make  this  uh part of the formula satisfied trying  to satisfy this part of the formula for  all  is picking values but she's trying to  make this part of the formula  unsatisfied so they're  you know they're uh in opposition to one  another  and  um in the end one of them is going to  have succeeded and the other one will  have failed uh  and now thinking about it as a game  one side depending upon the formula one  side or the other is going to have the  ability to dominate the other one and  always win  um  and so  the question is  which side has that um ability to always  win which side has the winning strategy  of course it'll depend upon the formula  some formulas it'll be the exist player  who has the winning strategy and other  formulas it'll be the for all player who  has the winning strategy and  computationally we'd like to make that  determination which side is going to win  um  you know for a given formula  and there's a very very nice  and actually very easy  um and simple way to understand that  because we've already run into that  problem before  the problem of determining which side  has a force force win is exactly the  same thing as determining whether this  quantified boolean formula is true  because the exist player has a force win  when that formula game on that um on  that formula  um  let me say it again the exist player has  a force win when that formula is true  um  so if this quantified boolean formula  were a true quantified bullying formula  then exist is guaranteed to be able to  win that game if it if it plays  its hand right  if this is a false formula for all the  for all player will always be able to  win um if it plays its hand right  okay so um why is that  and it really follows kind of almost  you know without doing any work at all  this is kind of a you know  this proof even though it looks  superficially like it might be not so  easy to prove  it follows for a very simple reason  because  the meaning of what it means for exist  to have a winning strategy is exactly  captured by the semantics of the  quantifiers  let's see what that means what does it  mean that exists has a force to win  let's say up here  that means that well exist has a move  that it could make  that  exists can  pick some value for x1 so there was some  move for x for there was some way to  assign x1  so that no matter what the opponent does  no matter what for all does  there's going to be a response that it  exists can make um some some assignment  to it x3  um  uh so that no matter what for all does  there's going to be an assignment to the  next variable  um and so on and so on so that this  thing ends up being true  so let me just say that again somehow i  didn't feel it came out very clear  uh so  exist has a winning strategy exactly  when there was some assignment to x1  that exists can make such that no matter  what assignment that for all makes to x2  there is some assignment to x3  um such that no matter what assignment  that um for all makes to x4  and so on  makes this thing true  that's what it means for exist to have a  winning strategy  that's exactly what the quantifiers are  saying in the first place  so even really without doing anything  you can see that exists has a winning  strategy  means exactly the same thing as the  formula the quantified being formula  being true  so making the test of whether  uh which side has a winning strategy is  the same as testing whether the formula  is true  okay um  [Music]  so let's let me try to turn to the chat  now um before before we move on to  anything else  um  all right uh whoops  uh here we go  [Music]  okay so i'm getting some questions about  the order of play and about the  alternation of the quantifiers here  so the way i'm showing it  is that the quantifiers always alternate  exist for all exists for all that  doesn't really matter  you know you could have several exists  in a row and that would correspond to  exist making several moves in a row  before turning it over to parole  but  alternatively  we could just add extra variables which  don't play any role in psi  and um they're just kind of dummy  variables that serve as spacers so if  you have two exists in a row and you  don't like that you can just add add a  for all of some variable that you never  see again in between  so you can always get it to be  alternating if you want  so  um  um  also question the difference between pha  fee  uh and psi so psi is the part here that  does not have any quantifiers  so the way we always write down our qbfs  is their leading quantifiers with the  variables which  uh you know and all variables have to be  within the scope of one of the  quantifiers of quantified variables um  so you know there are no free variables  that are unquantified in our qbfs  um and so there's a you know all of  these qbfs have a run of quantifiers and  then a part without quantifiers where  there's this sort of boolean logic part  i'm psi here is the boolean logic part  and phi is the whole thing together with  the quantifiers  um  so  i'm not sure i understand some of these  ques why not have the for all player  have a force to win the for all player  might have a full force win no see  one side or the other is guaranteed to  have a forced win so if exist does not  have a force to win then then parole  will have a forced win that you know i'm  not going to prove that but that's a  fairly simple proof by induction which  i'm not going to go through just take it  as a fact  somebody's asking why does for all want  to not satisfy the uh this expression  the the psy part well i mean that's the  way we set up the game um  you know in order to make this  correspond to tqbf  um  just the truth  of the of this  of this expression we want to make exist  try to satisfy the the this part and for  all try to  unsatisfy make it not satisfied um  and so i'm not sure  because that's what works i'm not sure  what else whatever whether why i can  answer there okay somebody  i'm going to answer some of the you know  most basic questions here somebody's  asking how do we know how many variables  to use  so the  the variables are the variables of the  formula so somebody's going to hand you  a formula  that's going to have x1 to x10 whatever  that formula has some number of  variables and so that game is going to  have 10 moves  you know if the quantifiers alternate  then the moves will alternate but  whatever the pattern of the quantifiers  is going to be the pattern of the moves  so exist is going to follow the  um  you know the places where the exist  quantifier occurs and fraud is going to  follow the places where the for all  quantifier occurs so if you have you  know exist for all exist for all exist  will move first then for all move then  exist will move then for all will move  together they're picking values to their  respective variables with with their  opposing goals  existed trying to make the  um the psi part satisfied for all is  trying to avoid making it satisfied to  make to to make make the assignment not  satisfy that part  and if you think about what that means  just in terms of the meaning of that  that it's going to mean exactly the same  thing that the formula is true  um  so  anyway if you don't  let's see i don't know if there's any  more questions that i can try to answer  here  um  well  so it's still about the numbers of  variables the  um  each formula is going to define a  different game you see the formula  so the formula however big it may have  50 variables that may have two variables  but we'll do an example maybe that'll  help  um  the next check-in which is coming up for  pretty soon might actually be now uh  we'll give you an example  um so then we'll  see who's who's understanding and who's  not okay so let's continue here  um  so  so therefore the problem of determining  does exist have a force win is exactly  the tqbf problem because exist has a  force when exactly when the formula is  true  and what we're going to show is that t  qbf  is  a polynomial time reducible to  generalized geography  but it's conceptually we're going to  think about tqbf now as a game because  that's how geography is generally  geography is set up  so we're going to given a formula like  this we're going to construct an  instance of generalized geography we're  going to construct a graph  where play on that graph is going to  mimic  play in the formula game  so making the moves in that graph are  going to correspond to setting variables  true and false because the graph is  going to be specially designed to to  force that behavior  okay so i think we have a check-in  coming in yeah so let's just see how  um i i suggest you pay attention to this  check-in and really try to think about  it and solve it because i think that'll  help you make the connection between the  truth  of the formula and  exist  having a winning strategy  okay so  so if you take this formula here here's  you know this is some particular formula  exist x for all y blah blah blah  um this is going to be associated to  that formula a formula game  where first exist moves  assigns a value for x and depending upon  that value for all is going to move and  they're going to assign a value for y  and after that's done this part here is  either going to be true or false it's  either going to be satisfied or  unsatisfied  and i want to know can exist always find  a way can it does  to to guarantee that this part is going  to be satisfied or can for all always  get find a way to guarantee that this  part is not satisfied  okay so it's you have to stare you have  to look at this  uh formula to understand  which side is going to end up succeeding  um  so you understand you have to if you're  not clear on the rules look back here  exist is trying to make this part  satisfied for all is trying to make this  part unsatisfied but they neither of  them has the no the totally upper hand  because existence depicting one of the  variables for all is picking the other  variable  but they're doing it in a certain  order okay first exist is going to pick  then a for all is going to pick  okay so that's the way the game works  exist picks a value then for all picks a  value and then you see who wins  so who wins  who's guaranteed to win  um you know can can we make sure exist  always wins or can we make sure that it  for all always wins for this particular  formula so there's just two variables  here  and you can think about this in either  of two ways strictly speaking purely as  a game  or you can look at understand whether  that formula is true or not  um  they're equivalent ways of looking at at  the problem  in fact in a certain sense it's the same  that's what i'm trying to  trying to get across  thinking about this as a game or  thinking about in terms of quantifiers  doesn't make any difference um  okay  almost done here  so  all right  last call  closing the poll  okay i think this one you got  you you you did pretty well on so yeah  the correct answer is the for all player  has a uh winning strategy has a forced  win  uh  and um also similarly the  expression um is  fee is is false  because you know if you try to find some  x  such that no matter what y you pick  um this is going to be true  this is going to be satisfied you're out  of luck because you know if you make x  true  well um  then  if you make x  true  now i'm confused uh  what is going on if you make x true then  uh  y  oh if you make x true  then uh x bar is false  um so y bar has to be true  uh  um  now i'm completely confused oh no  if make x true it has to work for all y  so this thing is clearly false if you  make x true  um  uh then then  then this x bar is going to be false and  it's not the case that for every y  of the setting for y because y is forced  here y is going to have to be true  well y bar is going to have to be true  so y is going to have to be false  somebody if you make  it very clearly but if you make if you  and now if you if you try again  to make x false  um  maybe that's a one to think about first  if you make x false  so  you're going to be stuck on the first  clause because you have it has to work  for both settings of y  but maybe thinking about it as a game is  better you know if you  no matter what x does a y is going to  find have a way of  making one of those two clauses  unsatisfied so if x is true why can uh  uh can set itself like why we can set y  to make the second clause  uh unsatisfied and if x is false we can  find y can be assigned to make the first  clause unsatisfied  um  so anyway um  the right answer is uh the for all  player uh has a winning strategy here in  the end of the day it's not critical  that you understand this correspondence  but  but you have to then if you don't quite  understand that you're going to have to  take on faith  that the formula game is the same as  tqbf  because that's we're going to use and by  the way i would like to also mention  here that um  that this correspondence between  uh games and quantifiers is some things  that mathematicians use all the time  um  because if you have a if you have some  expression which comes up often in  mathematics where there are you know a  whole run of quantifiers you know in  front of some  uh some statement  it's really kind of hard for anybody uh  to really uh  get a feeling for what that means  if you have like six alternating  quantifiers which happens often in  mathematics you're going to have  statements that have a lot of  alternations with quantifiers very hard  to get a feeling for what that means but  if you think about it as a game it's  much more intuitive and it's completely  equivalent to think about quantifiers go  back and forth between quantifiers and  games  anyway um so let's  look at the at the reduction the the  construction that shows uh generalized  geography is uh complete took a little  longer than i thought maybe this is i  just want to make sure everybody's with  me on this so why don't we call the  break now  and then we'll come back  um and we'll look at the construction um  after that because the construction  itself is going to take about 10 minutes  to to work through to figure out how to  reduce  tqbf  and build a graph that simulates the  formula  okay so i'm going to move  right on to uh  the um  and we'll come back  okay so feel free to ask some questions  um  and uh  get ready for us diving in to a  construction  uh for the um  uh for the reduction of tqbf to  generalize geography  so  there's kind of an interesting question  about  you know  what's a relationship  you know between a formula and when you  swap the order of the quantifiers um  uh you know the the the the question is  asking does that somehow relate to the  negating the um  the you know the part the quantifier  free part the part without the  quantifiers and i um i don't think that  that's the right correspondence there it  actually is  uh some relationship when you say um  you know there exists for all is  actually a stronger statement than than  saying that for all there exists in  general the um  that actually implies you know exist x  for a why is this  implies uh  for a why there exists an x um because  what you're saying is that there is  the the choice of y  if  there exists an x  uh for all y there was one x that works  for every one of the y's if you're  saying for y there exists an x the  choice of that x can depend on the y  um  so there  there is a connection but um you have to  think through what it means in order to  understand that connection  not i think you won't need to know that  in order to process what we're going to  be doing but maybe it just helps you  think about it  um  other questions here  okay i think we're  um  we're out of time here  so let's  return to  our  our lecture  um  so i'm gonna go back  hopefully that won't  crash everything  [Music]  here we go  okay so now we're going to be reducing  tqbf  to generalize geography are we all  together now  um  uh  okay  so i'm gonna  illustrate this construction which is  very nice construction by the way i'm  gonna illustrate this construction just  by doing an example or a partial  example okay but i think it'll give you  the idea  so you understand so what i'm trying to  do here is i'm starting off with a  quantified boolean formula  so here it is  um i'm going to assume it's in  conjunctive normal form  which i can always  convert it into that form uh maybe by  adding some additional variables but  without  doing anything  too drastic to it  and so now what i'm gonna uh  starting from that formula  i'm gonna now build a graph  we're playing the geography game on that  graph which is remember taking turns  picking nodes which may form a path  is going to correspond to um  playing the formula game which is  picking the variables of that formula  and then you want uh the  if the exist player wins in the formula  then player one is supposed to win  in the geography game  okay  um  so here is how the graph is going to  look okay so good to follow try to  follow this um  so it's going to be  kind of there's going to be kind of two  parts one that's going to  correspond to the variables and the  other part that's going to correspond to  the clauses  um  and so for each variable i'm going to  have like a diamond structure here so  there's a little starting thing that's  going to be unique to the to the first  variable but then  every variable is going to have like a  little diamond structure here um and  they're going to be attached one to the  next  and we'll to understand how this what  this means  when we play  geography  on this piece of the graph  so but let's just understand the  structure first  okay  so this is the start node  and now um  here i'm player one is going to play the  role of exist player two is going to  play the role of parole and in fact i'm  going to identify that so i'm going to  call because it's going to be helpful  just to have  uh  helpful just to  think about player 1 is being the exist  player so the exist player is going to  be playing on this graph  the exist and the for all players are  going to be playing on this graph it's  really just players one and two  so exist the exist player player one  has to  b  by the rules has to pick the start note  and now with player two's move these the  four players move  now  if you're with me  on this graph  the for all for the for all players move  um is now not very interesting because  it's forced  it has to go to here because again  they're just picking  the  um  the nodes of a simple path  in this graph so  for the fraud  player to the for all player has to go  here is if exist started over there  now um now it's now it's the exist  player's turn  and now now something interesting  happens the exist player has a choice  you can either go left or right  there are two possibilities  and then  after that  the four player's turn  who's again just forced so so far for  all has not had much interesting stuff  to do in this hasn't has not had to have  has not had to think hard  um  so for all players  no matter whether exists went left or  right the for all players move is forced  and ends up over here  and now let's see if this player's turn  again this is kind of an easy one  oh so before i do that let's just look  at how that play could go  so the first two moves that's kind of i  was suggesting exist goes here and then  for all goes there  i'm gonna i'm gonna illustrate  kind of possible plays through this  graph by tracing them out in green  um  so now exists goes here for all goes  there those those were forced but now  the exist player goes could either go  this way  or  could go that way  so those are two  possible different  ways of playing this the game that we  said we're building  um  and now  uh now whose turn is it here  this was the for all player pick this  one so the the the next turn is the  exist player and that's forced  but now  notice now for the first time the for  all player has to think because the for  all player can either go left or right  so could either go left or it could go  right  um  and so  then there's going to be a sequence of  these diamond structures where  kind of they're constructed so that  alternately either the exist has a  choice or the for all player has a  choice  until you get all the way down to the  bottom  okay  that is the graph we've built so far now  if we stopped here  um  then  whoever ended up at this point whether  it was existing for all would be the  winner because there's nowhere for the  other for the opponent to go of course  that wouldn't be very interesting  so there's going to be more stuff we're  going to add on which are going to  correspond to the clauses  now  how to think about what's happened so  far  now so maybe as some of you might be  guessing  is that um when the exist player over  here has a choice could have either gone  left or right that's going to correspond  because this is supposed to be  mimicking the formula game that's going  to correspond to the  uh  exist player  in the formula picking the variable  either true or false  so that the left or right is going to  correspond to true or false  and so um let's just arbitrarily say  left is going to correspond to picking  it true  and right is going to correspond to  picking false  okay so that  so the way i've said it so far is  exist ended up picking the first  variable false then for all pick the  second variable false  and the empty variable also got set  folds everything got set for so far who  knows what happened over here of course  um  uh  okay so just to understand what we've  done so far  now i'm going to show you how to build  the rest of the graph so let's take away  the green part um and now we're just  back to the construction  the green part is actually how you use  that construction so back to the  construction here now what do we want to  achieve  now by the time the  the players on the graph have got down  to here they've effectively made an  assignment to the variables by going  either left or right at each one of  these diamonds so the assignment is done  from the game from the perspective of  the formula game the game is over and  now one side or the other has one  here  we want to build some extra structure  here as a kind of an end game  to make sure  that  the for all player gets stuck  if the exist player has made an  assignment which satisfies the formula  satisfies this part of the formula  and  the exist player should get stuck  if the for all player if the assignment  that they made  does not satisfy the form  okay so let's see how we're going to  achieve that with some additional  structure  so  there's going to be some extra node over  here  let me just tell you what the structure  is and then we'll argue why it works  so there's going to be  going from this bottom node we're going  to start a kind of  you know  the second part of the graph there's  going to be a node which  fans out to a node for each one of the  clauses  and each one of those clauses in turn  fans out to a node for each one of its  literals  so you see we have clause c1  and it has the literals x1 x2 bar x3 x1  x2 bar x3  so there's no note for each of the  literals and clause c1 same for clause  c2  and so on  now we're almost done  now i'm going to tell you how to connect  up these literal nodes  so x1 each node is going to get  correspond  back to  its own diamond so now we're going to  tie it back to the the first part where  we made the assignment the assignment  part of the graph  x1 is going to connect back to the true  part of the x1 diamond  and x2 because it's negated is going to  tie back to the false part of the x2  diamond  because it's an x2 variable  similarly x1 bar  now in  you know in  because x1 bar  you know here's x1 bar and clause 2.  x1 bar  is going to connect up now to the full  side of the x1 diamond  and x2 bar is going to connect up to the  full side of the x2 diamond  and so on so if we had i don't have the  other diamonds here but the x4 would  connect up to the true side of the x4  diamond  for example  that's the whole construction let's  understand why it works  okay so the end game here is we want  exist to win  um  if the assignment satisfied all the  clauses  and for all should win if there was some  unsatisfied clause  okay so why why does this happen so  let's put back an assignment here that  was made so here's here for the in the  one i'm i'm putting back is  you know the assignment that they  cooperatively built  had x1 being true  x2 being false  and some other stuff  now why does this work so what we want  to have happen now now so now the  move proceeds over to here and you need  to arrange it so that exists  is the one who picks that note  if you have a um if you know if you you  know  if this would have been for all just add  an extra node to switch who's  whose turn it is  but you want exist to be up here and  what you want is for all to be picking  the clause  the clause note  now here is  here's the part to understand what's  going on  um  we want  for all to win  if this is not  satisfied so the  for all is going to make a claim  um  if this is not satisfied there is some  clause which is not satisfied there's  some clause which ended up being false  in the assignment so  the for all player gonna say i think i  won  and i won because clause number one is  unsatisfied  so it's going to pick clause number one  so here's the four players more going  his exist player  and so  the four player picks the clause claimed  unsatisfied so over here  full player says claude c1 is  unsatisfied i'm going to move here  if this player says  i don't agree with you the assignment  you know your your line um  uh  the  assignment actually makes one of the  literals true in clause c1  let's say it made x1 was true  as it as in fact it is here x1 in fact  is true so  exist is now  its turn it's going to pick the the  literal that was true  in the assignment in that clause  that the fraud player is claiming is  unsatisfied  one of them is lying at this point  this is either now true in here which  means a for all player did not pick an  unsatisfied clause or the exist player  picked a false literal which is the the  the exist player picked the false letter  over to the exist players lie  so now we're gonna now  the moment of truth  because  let's see if exist gold picked here  now it's um the for all's players turn  it's connected back to the true side of  the diamond  so that node was already used that's the  only place that for all could possibly  go for all is stuck  and an exist has won the game which is  what you want  because  um  for all claims are false and the and the  exist was was was was correct in saying  that x1 satisfied clause c1  so x1 is true  so um there's nowhere for for all to go  but compare that with the situation if  the assignment on this part of the graph  had gone through the false  had assigned x1 to be false  so the path had gone through the full  side of the diamond  now this node would still be unoccupied  would still be um  available  so now the for role play so if if exist  was claiming that x1 was true and it  really was false  the fraud player would be able to move  on to that node and now it's the exist  player's turn and the exist would be  stuck because  this node down here is guaranteed to  have been used  okay so that  is the idea  it's really very beautiful um  and actually not that complicated you  have to you have to stare at it a bit so  let's just see what happens for example  if um  i had one other case maybe it's not  necessary at this point um but you know  if the for all player claimed  well  it's the second clause which is  unsatisfied in the  mutually selected assignment um  then  uh  so this is sort of the the case where  the for all players is correct um you  know in in that if the exist player now  says well i think x1 bar satisfied that  uh second clause  but you know this  assignment made x1 true so x1 bar is  false and so the exist player is lying  this time  and so now the for all player can take  this last edge and go here  and has one one last move to make and  then the exist player stuck  so the for all will win in that case  so let's turn now shifting gears  entirely  to a different um  uh  part of the spate you know  of space complexity instead of looking  at polynomial space which is like very  powerful we're going to look at log  space which is comparatively speaking  much much weaker  so log space are the things that you can  do when you only have  enough space to write down essentially a  pointer or some fixed number of pointers  into the input that's what you get when  you have logarithmic space  and  you know because log space is enough to  write down an index of something um so  you can just this is what you can do  with a bunch of pointers  and  in order to make sense of this we have  to change the model of computation  because if we use the ordinary one tape  turing machine  just scanning across the input  will  just reading the input with the way  we've defined it would cost space n  and so you can't even read the input  if you have less than n space available  like log space  so we're going to introduce a different  model just  to allow us to define this  and that's a model where it's going to  be have two tapes  where the part that contains the input  doesn't count  and that's in order to make sure it  doesn't you know it's not being cheated  the the input tape is going to be read  only  um but it doesn't count toward the space  used only the work tape  which is now has can read can be read or  or written or red  is going to count toward the space bound  okay  um  so now what we're going to define is  you know  using this definition for  our space complexity  um  we'll define  space log n  the things that you can do if you have  only a log amount of space here  on the work tape  so the length of the input is n  the length of the um  of the work tape is lo order log n  so we have uh the deterministic and  non-deterministic associated classes uh  space log n and n space log n we're  calling that l and n l  log space and not deterministic log  space  okay um and as i said that's just enough  space to write down some pointers into  the input and let's do some quick  examples  um  so if you take the language of  palindromes essentially or ww reverse um  that's  in log space uh  so here is a string i don't know i was  here's a string that's in www reverse  and  i mean the ordinary way you might use to  test whether a string is of this form in  ww reverse might be to cross off  corresponding locations here but you  can't do that it's a read-only input  tape so you have to somehow avoid  writing on the input tape but still  testing whether you're in this language  it's not hard to do you can use the work  tape just to keep track of where your  pointers would be  um  and  in so doing you can just  make sure you're  matching off corresponding locations  with each other in the input  so log space because of its ability to  store some fixed number of pointers  it gives you enough to test membership  in this language  so this is in this is solvable in log  space  uh  the path problem which we've seen before  you know you're given a graph and a  start node and  an ending node or a start and a target  you're given a graph of an s and t  it's a directed graph and you know can i  know can i get from s to t  um so that problem is solvable in  non-deterministic log space  uh because as  um the way we would do that i'm not not  writing this down in in any detail but  uh what you would do in your  um not you know non-deterministic log  space machine you have your input graph  written here in the input  and you're just going to guess the  sequence of  nodes  one by one which takes you from the  start node to the target node  you're not going to write down that  whole sequence because that in advance  because that would be  cost you way too much space to write  down the only space that you're going to  use is to remember the current location  we are sitting  so you start out you know you write down  on the work tape the start node  then you're going to  non-deterministically pick one of the  outgoing edges from the start node and  look at its associated node and replace  that on your work tape and keep keep  repeating that if you ever get to the  node t then you can accept  and you also have to be a little careful  i don't have this on on the slide you  also have to make sure that if the graph  has a loop in it  or because that's not allowed in space  complexity uh so you're going to need  also a counter to make sure that if you  count up to a um you've you you you  you've gone through a number of nodes  which exceeds the total number of nodes  in the graph  uh then you can shut down that branch of  the non-determinism uh because  you know it's  you know just it's just going in  a loop  um  if there's any path that connects s to t  there's certainly going to be a path  that has at most the number of nodes of  the graph in it  okay um so path is in nl this language  here is an in l um  what's the relationship between l and n  l well certainly the deterministic class  is contained in the non-deterministic  class  that's all that's known  whether these two collapse down to be  the same  is unsolved and so we're going to spend  a little time next lecture exploring  this  um but but let's let's first  look at some of the basic facts about  log space  um sort of setting ourselves up for next  lecture  so  first of all anything that you can do in  log space you can do in polynomial time  in a sense kind of trivially and in a  sense we almost kind of really proved  this already because if you remember  we said that anything that you can do in  a certain amount of space you can  do  in that much  uh you can do  in time  in  that's exponential in the amount of  space  in the corresponding amount of space so  going from space to time blows you up  exponentially  and  order log the exponential of order log n  is polynomial  so the way we're going to prove this  and again we kind of proved this already  but let's just go through it again  specific to  log space  uh we'll say and this is going to be a  useful definition for us anyway um  a configuration for m on w so we would  also we already talked about a  configuration of turing machine m which  is just a snapshot which is the tape  the state and the head position  when we have  an  input which is sort of an uh read-only  input which is not being counted toward  the space we don't include that in the  configuration um we just say  it's a configuration of m on that  particular input but we're going to be  counting configurations and i don't want  to count all the different inputs as  well i'm going to fix the input and  count all of the configurations relative  to that input  and  so the number of such configurations is  just simply going to be  the number of states times the number of  head positions for the two  heads now  times the number of possible tape  contents and which as i mentioned here  is the number of different possible tape  contents  it's d which is the tape alphabet to the  order log n and that's just going to be  n to the k for some k so it's going to  be polynomial so that tells us  that because the total number of  configurations  um  for m on w is polynomial  this machine  can only be running for a polynomial  number of steps otherwise it will be  looping people are repeating a  configuration and so therefore  that machine has to run itself in  polynomial time so you don't really in a  sense have to do any work if a machine  is deterministically designing a  language in log space  it's also deterministically deciding the  language in polynomial time because  that's all the time you can use when you  have log space unless you're looping  which is not allowed okay so therefore  the same machine shows that you're also  in pain  oh  i'll get to that in a second one thing i  forgot to mention on the previous slide  when i'm talking about the model is by  the way this model is not so  unreasonable um  where you have kind of a imagine having  a very large read-only input  and and your local storage is much  smaller it's much too small to pull in  the entire input  the way i used to explain this  years ago was like you have a your input  is a cd-rom which you guys probably  barely even  know what it is anymore but used to  distribute software that way so the rom  was like on a dvd uh or cd  which contain your  whatever you're trying to like yourself  whether you're trying to distribute  you know it was some large thing uh  relatively and so  you would imagine having a smaller  amount of  memory relative to that so you didn't  want to necessarily copy that whole  thing into your storage  maybe even a better example now is like  you think of the input as being the  entire internet  um obviously you can't download unless  you're google you can't download  the whole internet into your own uh  local memory  but you're going to have references  pointers into the input into different  places that's perhaps more analogous to  this sort of read-only input touring  machine model that i'm describing um  okay  uh  so  um  and it's another fact i want to mention  is that i'm not going to  savage just the anything that you can do  non-deterministic log space you can do  in deterministic  uh space but now with a squaring  and that's using the same proof that's  using savage's theorem which you have to  check also works down to log space  same proof  um  so anything that you do not  deterministically in log n space you can  use deterministically log squared n  space  okay so let me just see if there's a  couple of questions i want to answer  here  [Music]  the relationship between  l and nl is not known to be strict  nobody knows of an example no one knows  whether they're equal  and  and have people looked at  um  sublinear time classes  yes generally for  when you have non-deterministic or  probabilistic which we're not haven't  defined yet but we will um people have  looked at um those sub-linear time  classes as well deterministically  doesn't make so much sense because then  you can't even talk about  either the whole input um  okay uh let me move on um  okay so this this is my last slide let  me see if we can do this before we break  um not only is l contained within p but  this much stronger statement is that nl  is contained within p and that we you'll  that for that you'll have to do some  work because converting your  non-deterministic log space machine to a  deterministic  machine  um obviously you're going to have to  change the machine um  and so we'll introduce a new method here  maybe we'll  quickly go over this at the beginning of  the next lecture  i don't like rushing through things at  the end  but for this  if i if i'm given  um  a non-deterministic machine  that runs in log space  i want to make a new machine that runs  in polynomial time deterministically for  the same language  and i'm going to define something called  the configuration graph for m on an  input w  and that's just you take  m  and it's and w  its input and you look at all of the  configurations  um  for m w actually configuration graph  actually that's  should be called the computation graph  that's what it's called in the book but  okay this is a typo yeah i'll fix that  next time  uh it doesn't matter computation graph  configuration will graph uh all the same  basically you're going to take all of  the configurations of m on w of which we  already observed  only number or only polynomial in number  and may they become the nodes of a graph  and the edges connect two configurations  if one follows another according to the  rules of the machine  okay so here's a picture  these are  ins this is some graph the nodes of this  graph are the configurations of m on w  right so each node here corresponds to  the machine a snapshot of the machine  a  tape contents head position and state  so writing down all those different  configurations i connect one to the  other if i can get to cj cj from ci in  one step legally on the machine  and then  uh  the non-deterministic machine m accepts  w  exactly when there's a path from the  start configuration to the to to to an  accept configuration let's assume we  have just a single accept configuration  as we argue we can do  one or two lectures back  um if we clean up the tapes  so  testing whether you can get from the  start to the accept um  uh is the same as testing whether the  machine accepts its input  and so  because we can test  whether there's a path in a graph  connecting two nodes in polynomial time  we can solve this problem on this  computation configuration graph in  polynomial time and so we can figure out  whether the non-deterministic machine  accepts its input  sorry that came a little faster than i  like to do so we'll we'll we'll see it  again um so here's the polynomial time  algorithm  you construct the graph  you accept if there's a path from the  star to the accept and you reject it  does not  um  and so that tells us that  not only is l contained with an nl but  nl itself is also contained within p so  here's that here's a kind of a nice  hierarchy of languages  we don't not only do we not know whether  l equals n l we don't know whether l  equals p  it's possible that anything you can do  in polynomial time you can do  deterministically in log space shocking  as that might be because it's a pretty  weak class  but we don't know how to prove that uh  there's anything different  anything in p that's not in here  um last check in  uh so we showed that path is in nl  what's the best thing we can do about  the deterministic space complexity of  path so deterministic so this is  non-deterministic log space what can we  say deterministically  about path um  hint  this should not be this should not be  hard  uh if you think back to what we've  shown very recently  okay  get your check in points closing up  closing stop here  all set one two three  yes so the correct answer is log squared  space because this is just savages there  we can do it in log space  non-deterministically so you can do it  in log squared space deterministically  okay so that's um this is what we did  today  um  and i as i mentioned i will uh do this  again  on at tuesday's lecture just to recap  that  all right so i'll stick around a little  bit for questions and um  okay someone says to be about the  nomenclature  why is it l in l space  um  i you know  because there's not really people don't  usually talk about l time so l is sort  of  everybody knows it's the only reasonable  option is space so people just say l and  then l  um  i mean some of these things have had  you know  you know these things these names have a  little bit evolved over time and even  now some people say talk about uh  um  you know i call time classes some call  some people call it d time classes  you know it's just you know it  uh you can make different choices there  um  okay uh let's see  good  um  why does savage's theorem work for log n  you have to you have to look back and  see what you needed  um  and all you needed to be able to do  was to um  write down  uh the configurations  um  and uh if you look back at how savage's  theorem works you're just needing to  write down the configuration so the  deterministic machine can write down the  configurations for the non-deterministic  machine they take exactly the same  size  and then you look at the recursion  and the the depth of the recursion is  going to be exponential  in um the size of the configurations and  so you're going to end up with a  squaring again you have to go back and  just  rethink that proof and you'll see  nowhere did it need a linear amount of  space it works down to a lot it actually  does not work for less than log n  log n is sort of the lower threshold  there  and the reason for that is because you  also need to store the  input location  and that already takes log space  uh the you know not the  the the the head tape heads the tape  heads already kind of  um  have kind of a log space aspect to them  and so  if you're going to you use less than log  space then funky things happen with the  tape head storing the tape heads and so  less than log space usually turns out  not to be interesting  very specific to turing machines and not  sort of general general models  yeah so somebody's asking  suppose in the reduction um  to generalize geography from tqbf if the  formula had two exists in a row  um then you would do kind of the natural  thing in the graph instead of having  that spacer edge between the two  diamonds you could just have one diamond  connecting directly to the other diamond  without a spacer edge  and that would be give you the effect  of  not switching whose turn it is  uh somebody's asking me just a general  question  are people thinking about  these open problems  um  i don't know  people don't don't don't say um  there was a lot of work  on problems related that seemed to be  related to those  those many open questions like p versus  np  l versus nl or l versus p and so on p  versus p space  we'll say we'll talk a little bit more  about some of that there's some very  interesting things that have been  developed i think there's a sense within  the community  that people are stuck  and it's not something you're going to  need some sort of major new  uh idea in order to push the thing  forward so i don't know how many people  are still thinking about them  i hope people are because i would like  to see the answer at some point but  or get the answer  i think about them sometimes myself but  one has to acknowledge  chances of success are not high  uh  okay um  so we're a little after past the end of  the hour here um unless there's any  other questions if i didn't answer your  question i may have missed it so you can  ask again  um  but otherwise i'm going to  close this close the session here  okay bye bye all have a good weekend see  you next week  take care  you
 

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267
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280
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281
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282
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283
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284
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285
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287
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289
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290
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291
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292
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293
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295
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296
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297
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298
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299
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300
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301
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302
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303
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304
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305
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306
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307
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308
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309
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310
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311
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312
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313
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314
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315
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316
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317
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318
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319
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320
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321
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322
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323
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324
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325
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326
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327
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328
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329
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330
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331
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332
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333
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334
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335
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336
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337
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338
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339
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340
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341
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342
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343
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344
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345
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346
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347
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348
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349
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350
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351
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352
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353
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354
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355
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356
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357
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358
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359
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360
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361
00:12:39,829 --> 00:12:42,069

 

362
00:12:42,069 --> 00:12:42,079

 

363
00:12:42,079 --> 00:12:43,910


364
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365
00:12:46,230 --> 00:12:48,790

 

366
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367
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368
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369
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370
00:13:00,310 --> 00:13:03,190

 

371
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372
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373
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374
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375
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376
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377
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378
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379
00:13:16,000 --> 00:13:18,710


380
00:13:18,710 --> 00:13:21,670

 

381
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382
00:13:23,269 --> 00:13:23,279

 

383
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384
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385
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386
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387
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388
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389
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390
00:13:37,910 --> 00:13:39,829

 

391
00:13:39,829 --> 00:13:42,550

 

392
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393
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394
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395
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396
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397
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398
00:13:56,550 --> 00:13:57,829

 

399
00:13:57,829 --> 00:13:59,910

 

400
00:13:59,910 --> 00:14:01,750

 

401
00:14:01,750 --> 00:14:03,990

 

402
00:14:03,990 --> 00:14:05,750

 

403
00:14:05,750 --> 00:14:07,670

 

404
00:14:07,670 --> 00:14:08,870

 

405
00:14:08,870 --> 00:14:08,880

 

406
00:14:08,880 --> 00:14:10,230


407
00:14:10,230 --> 00:14:12,470

 

408
00:14:12,470 --> 00:14:14,470

 

409
00:14:14,470 --> 00:14:14,480

 

410
00:14:14,480 --> 00:14:15,350


411
00:14:15,350 --> 00:14:17,269

 

412
00:14:17,269 --> 00:14:19,750

 

413
00:14:19,750 --> 00:14:21,910

 

414
00:14:21,910 --> 00:14:23,430

 

415
00:14:23,430 --> 00:14:23,440

 

416
00:14:23,440 --> 00:14:24,230


417
00:14:24,230 --> 00:14:27,509

 

418
00:14:27,509 --> 00:14:28,790

 

419
00:14:28,790 --> 00:14:31,509

 

420
00:14:31,509 --> 00:14:33,509

 

421
00:14:33,509 --> 00:14:36,949

 

422
00:14:36,949 --> 00:14:38,949

 

423
00:14:38,949 --> 00:14:40,790

 

424
00:14:40,790 --> 00:14:40,800

 

425
00:14:40,800 --> 00:14:43,269


426
00:14:43,269 --> 00:14:44,710

 

427
00:14:44,710 --> 00:14:47,110

 

428
00:14:47,110 --> 00:14:49,350

 

429
00:14:49,350 --> 00:14:52,150

 

430
00:14:52,150 --> 00:14:55,269

 

431
00:14:55,269 --> 00:14:55,279

 

432
00:14:55,279 --> 00:14:57,110


433
00:14:57,110 --> 00:14:59,030

 

434
00:14:59,030 --> 00:15:00,389

 

435
00:15:00,389 --> 00:15:02,949

 

436
00:15:02,949 --> 00:15:05,030

 

437
00:15:05,030 --> 00:15:08,069

 

438
00:15:08,069 --> 00:15:09,990

 

439
00:15:09,990 --> 00:15:10,000

 

440
00:15:10,000 --> 00:15:13,350


441
00:15:13,350 --> 00:15:15,670

 

442
00:15:15,670 --> 00:15:15,680

 

443
00:15:15,680 --> 00:15:16,470


444
00:15:16,470 --> 00:15:16,480

 

445
00:15:16,480 --> 00:15:17,509


446
00:15:17,509 --> 00:15:19,990

 

447
00:15:19,990 --> 00:15:21,990

 

448
00:15:21,990 --> 00:15:23,910

 

449
00:15:23,910 --> 00:15:25,509

 

450
00:15:25,509 --> 00:15:27,269

 

451
00:15:27,269 --> 00:15:28,870

 

452
00:15:28,870 --> 00:15:30,550

 

453
00:15:30,550 --> 00:15:33,350

 

454
00:15:33,350 --> 00:15:35,749

 

455
00:15:35,749 --> 00:15:38,389

 

456
00:15:38,389 --> 00:15:38,399

 

457
00:15:38,399 --> 00:15:39,670


458
00:15:39,670 --> 00:15:41,509

 

459
00:15:41,509 --> 00:15:44,389

 

460
00:15:44,389 --> 00:15:47,430

 

461
00:15:47,430 --> 00:15:48,870

 

462
00:15:48,870 --> 00:15:50,470

 

463
00:15:50,470 --> 00:15:52,550

 

464
00:15:52,550 --> 00:15:53,910

 

465
00:15:53,910 --> 00:15:55,590

 

466
00:15:55,590 --> 00:15:58,150

 

467
00:15:58,150 --> 00:16:01,350

 

468
00:16:01,350 --> 00:16:03,509

 

469
00:16:03,509 --> 00:16:05,509

 

470
00:16:05,509 --> 00:16:08,710

 

471
00:16:08,710 --> 00:16:10,870

 

472
00:16:10,870 --> 00:16:12,470

 

473
00:16:12,470 --> 00:16:14,069

 

474
00:16:14,069 --> 00:16:16,389

 

475
00:16:16,389 --> 00:16:18,069

 

476
00:16:18,069 --> 00:16:20,230

 

477
00:16:20,230 --> 00:16:20,240

 

478
00:16:20,240 --> 00:16:21,350


479
00:16:21,350 --> 00:16:23,189

 

480
00:16:23,189 --> 00:16:27,590

 

481
00:16:27,590 --> 00:16:31,030

 

482
00:16:31,030 --> 00:16:33,749

 

483
00:16:33,749 --> 00:16:33,759

 

484
00:16:33,759 --> 00:16:34,829


485
00:16:34,829 --> 00:16:37,350

 

486
00:16:37,350 --> 00:16:39,350

 

487
00:16:39,350 --> 00:16:42,710

 

488
00:16:42,710 --> 00:16:44,710

 

489
00:16:44,710 --> 00:16:46,150

 

490
00:16:46,150 --> 00:16:48,310

 

491
00:16:48,310 --> 00:16:50,710

 

492
00:16:50,710 --> 00:16:52,710

 

493
00:16:52,710 --> 00:16:56,389

 

494
00:16:56,389 --> 00:17:00,870

 

495
00:17:00,870 --> 00:17:02,470

 

496
00:17:02,470 --> 00:17:05,350

 

497
00:17:05,350 --> 00:17:07,750

 

498
00:17:07,750 --> 00:17:09,590

 

499
00:17:09,590 --> 00:17:11,429

 

500
00:17:11,429 --> 00:17:13,189

 

501
00:17:13,189 --> 00:17:14,549

 

502
00:17:14,549 --> 00:17:14,559

 

503
00:17:14,559 --> 00:17:15,350


504
00:17:15,350 --> 00:17:17,029

 

505
00:17:17,029 --> 00:17:19,029

 

506
00:17:19,029 --> 00:17:20,309

 

507
00:17:20,309 --> 00:17:26,470

 

508
00:17:26,470 --> 00:17:29,029

 

509
00:17:29,029 --> 00:17:30,710

 

510
00:17:30,710 --> 00:17:33,190

 

511
00:17:33,190 --> 00:17:35,430

 

512
00:17:35,430 --> 00:17:37,590

 

513
00:17:37,590 --> 00:17:40,390

 

514
00:17:40,390 --> 00:17:41,909

 

515
00:17:41,909 --> 00:17:43,510

 

516
00:17:43,510 --> 00:17:43,520

 

517
00:17:43,520 --> 00:17:44,830


518
00:17:44,830 --> 00:17:46,710

 

519
00:17:46,710 --> 00:17:48,950

 

520
00:17:48,950 --> 00:17:51,190

 

521
00:17:51,190 --> 00:17:53,029

 

522
00:17:53,029 --> 00:17:54,870

 

523
00:17:54,870 --> 00:17:56,950

 

524
00:17:56,950 --> 00:18:00,630

 

525
00:18:00,630 --> 00:18:01,670

 

526
00:18:01,670 --> 00:18:01,680

 

527
00:18:01,680 --> 00:18:03,830


528
00:18:03,830 --> 00:18:03,840

 

529
00:18:03,840 --> 00:18:05,029


530
00:18:05,029 --> 00:18:07,190

 

531
00:18:07,190 --> 00:18:09,510

 

532
00:18:09,510 --> 00:18:11,270

 

533
00:18:11,270 --> 00:18:11,280

 

534
00:18:11,280 --> 00:18:12,310


535
00:18:12,310 --> 00:18:13,990

 

536
00:18:13,990 --> 00:18:15,909

 

537
00:18:15,909 --> 00:18:15,919

 

538
00:18:15,919 --> 00:18:20,150


539
00:18:20,150 --> 00:18:22,150

 

540
00:18:22,150 --> 00:18:23,669

 

541
00:18:23,669 --> 00:18:25,270

 

542
00:18:25,270 --> 00:18:28,230

 

543
00:18:28,230 --> 00:18:30,070

 

544
00:18:30,070 --> 00:18:32,150

 

545
00:18:32,150 --> 00:18:33,430

 

546
00:18:33,430 --> 00:18:36,789

 

547
00:18:36,789 --> 00:18:39,669

 

548
00:18:39,669 --> 00:18:41,110

 

549
00:18:41,110 --> 00:18:43,029

 

550
00:18:43,029 --> 00:18:45,750

 

551
00:18:45,750 --> 00:18:47,110

 

552
00:18:47,110 --> 00:18:47,120

 

553
00:18:47,120 --> 00:18:48,230


554
00:18:48,230 --> 00:18:49,669

 

555
00:18:49,669 --> 00:18:51,669

 

556
00:18:51,669 --> 00:18:53,669

 

557
00:18:53,669 --> 00:18:55,029

 

558
00:18:55,029 --> 00:18:55,039

 

559
00:18:55,039 --> 00:18:56,950


560
00:18:56,950 --> 00:18:59,510

 

561
00:18:59,510 --> 00:19:01,830

 

562
00:19:01,830 --> 00:19:04,070

 

563
00:19:04,070 --> 00:19:06,150

 

564
00:19:06,150 --> 00:19:08,710

 

565
00:19:08,710 --> 00:19:12,549

 

566
00:19:12,549 --> 00:19:14,230

 

567
00:19:14,230 --> 00:19:16,390

 

568
00:19:16,390 --> 00:19:18,390

 

569
00:19:18,390 --> 00:19:20,310

 

570
00:19:20,310 --> 00:19:20,320

 

571
00:19:20,320 --> 00:19:21,270


572
00:19:21,270 --> 00:19:26,070

 

573
00:19:26,070 --> 00:19:28,789

 

574
00:19:28,789 --> 00:19:31,750

 

575
00:19:31,750 --> 00:19:34,710

 

576
00:19:34,710 --> 00:19:34,720

 

577
00:19:34,720 --> 00:19:36,789


578
00:19:36,789 --> 00:19:39,190

 

579
00:19:39,190 --> 00:19:41,270

 

580
00:19:41,270 --> 00:19:43,990

 

581
00:19:43,990 --> 00:19:47,350

 

582
00:19:47,350 --> 00:19:49,190

 

583
00:19:49,190 --> 00:19:51,990

 

584
00:19:51,990 --> 00:19:54,549

 

585
00:19:54,549 --> 00:19:56,549

 

586
00:19:56,549 --> 00:19:58,789

 

587
00:19:58,789 --> 00:20:03,029

 

588
00:20:03,029 --> 00:20:06,149

 

589
00:20:06,149 --> 00:20:08,549

 

590
00:20:08,549 --> 00:20:10,710

 

591
00:20:10,710 --> 00:20:10,720

 

592
00:20:10,720 --> 00:20:11,669


593
00:20:11,669 --> 00:20:14,149

 

594
00:20:14,149 --> 00:20:14,159

 

595
00:20:14,159 --> 00:20:15,350


596
00:20:15,350 --> 00:20:15,360

 

597
00:20:15,360 --> 00:20:19,669


598
00:20:19,669 --> 00:20:23,190

 

599
00:20:23,190 --> 00:20:25,029

 

600
00:20:25,029 --> 00:20:25,039

 

601
00:20:25,039 --> 00:20:26,870


602
00:20:26,870 --> 00:20:29,110

 

603
00:20:29,110 --> 00:20:31,270

 

604
00:20:31,270 --> 00:20:31,280

 

605
00:20:31,280 --> 00:20:32,870


606
00:20:32,870 --> 00:20:32,880

 

607
00:20:32,880 --> 00:20:34,390


608
00:20:34,390 --> 00:20:37,029

 

609
00:20:37,029 --> 00:20:39,110

 

610
00:20:39,110 --> 00:20:41,430

 

611
00:20:41,430 --> 00:20:44,789

 

612
00:20:44,789 --> 00:20:47,750

 

613
00:20:47,750 --> 00:20:50,149

 

614
00:20:50,149 --> 00:20:51,590

 

615
00:20:51,590 --> 00:20:51,600

 

616
00:20:51,600 --> 00:20:53,190


617
00:20:53,190 --> 00:20:53,200

 

618
00:20:53,200 --> 00:20:54,230


619
00:20:54,230 --> 00:20:55,830

 

620
00:20:55,830 --> 00:20:58,470

 

621
00:20:58,470 --> 00:21:00,149

 

622
00:21:00,149 --> 00:21:01,750

 

623
00:21:01,750 --> 00:21:03,830

 

624
00:21:03,830 --> 00:21:05,990

 

625
00:21:05,990 --> 00:21:08,070

 

626
00:21:08,070 --> 00:21:11,430

 

627
00:21:11,430 --> 00:21:14,070

 

628
00:21:14,070 --> 00:21:17,830

 

629
00:21:17,830 --> 00:21:19,270

 

630
00:21:19,270 --> 00:21:21,110

 

631
00:21:21,110 --> 00:21:22,310

 

632
00:21:22,310 --> 00:21:24,390

 

633
00:21:24,390 --> 00:21:24,400

 

634
00:21:24,400 --> 00:21:25,909


635
00:21:25,909 --> 00:21:28,310

 

636
00:21:28,310 --> 00:21:30,470

 

637
00:21:30,470 --> 00:21:30,480

 

638
00:21:30,480 --> 00:21:31,510


639
00:21:31,510 --> 00:21:33,830

 

640
00:21:33,830 --> 00:21:35,669

 

641
00:21:35,669 --> 00:21:37,750

 

642
00:21:37,750 --> 00:21:40,230

 

643
00:21:40,230 --> 00:21:40,240

 

644
00:21:40,240 --> 00:21:41,990


645
00:21:41,990 --> 00:21:42,000

 

646
00:21:42,000 --> 00:21:42,830


647
00:21:42,830 --> 00:21:45,750

 

648
00:21:45,750 --> 00:21:47,190

 

649
00:21:47,190 --> 00:21:49,270

 

650
00:21:49,270 --> 00:21:51,590

 

651
00:21:51,590 --> 00:21:54,230

 

652
00:21:54,230 --> 00:21:56,070

 

653
00:21:56,070 --> 00:21:58,230

 

654
00:21:58,230 --> 00:22:00,789

 

655
00:22:00,789 --> 00:22:00,799

 

656
00:22:00,799 --> 00:22:01,909


657
00:22:01,909 --> 00:22:03,590

 

658
00:22:03,590 --> 00:22:05,350

 

659
00:22:05,350 --> 00:22:09,029

 

660
00:22:09,029 --> 00:22:11,909

 

661
00:22:11,909 --> 00:22:13,510

 

662
00:22:13,510 --> 00:22:15,270

 

663
00:22:15,270 --> 00:22:16,950

 

664
00:22:16,950 --> 00:22:18,549

 

665
00:22:18,549 --> 00:22:20,149

 

666
00:22:20,149 --> 00:22:22,070

 

667
00:22:22,070 --> 00:22:24,630

 

668
00:22:24,630 --> 00:22:24,640

 

669
00:22:24,640 --> 00:22:26,789


670
00:22:26,789 --> 00:22:29,029

 

671
00:22:29,029 --> 00:22:31,750

 

672
00:22:31,750 --> 00:22:33,830

 

673
00:22:33,830 --> 00:22:36,950

 

674
00:22:36,950 --> 00:22:38,310

 

675
00:22:38,310 --> 00:22:40,549

 

676
00:22:40,549 --> 00:22:42,950

 

677
00:22:42,950 --> 00:22:45,270

 

678
00:22:45,270 --> 00:22:47,190

 

679
00:22:47,190 --> 00:22:50,710

 

680
00:22:50,710 --> 00:22:54,390

 

681
00:22:54,390 --> 00:22:57,669

 

682
00:22:57,669 --> 00:22:59,110

 

683
00:22:59,110 --> 00:22:59,120

 

684
00:22:59,120 --> 00:23:01,190


685
00:23:01,190 --> 00:23:03,029

 

686
00:23:03,029 --> 00:23:07,029

 

687
00:23:07,029 --> 00:23:07,039

 

688
00:23:07,039 --> 00:23:08,470


689
00:23:08,470 --> 00:23:10,870

 

690
00:23:10,870 --> 00:23:13,669

 

691
00:23:13,669 --> 00:23:15,909

 

692
00:23:15,909 --> 00:23:18,390

 

693
00:23:18,390 --> 00:23:21,430

 

694
00:23:21,430 --> 00:23:24,070

 

695
00:23:24,070 --> 00:23:26,230

 

696
00:23:26,230 --> 00:23:30,230

 

697
00:23:30,230 --> 00:23:34,830

 

698
00:23:34,830 --> 00:23:38,950

 

699
00:23:38,950 --> 00:23:40,710

 

700
00:23:40,710 --> 00:23:43,269

 

701
00:23:43,269 --> 00:23:44,789

 

702
00:23:44,789 --> 00:23:47,029

 

703
00:23:47,029 --> 00:23:48,630

 

704
00:23:48,630 --> 00:23:50,710

 

705
00:23:50,710 --> 00:23:50,720

 

706
00:23:50,720 --> 00:23:52,230


707
00:23:52,230 --> 00:23:55,669

 

708
00:23:55,669 --> 00:23:58,470

 

709
00:23:58,470 --> 00:24:00,549

 

710
00:24:00,549 --> 00:24:00,559

 

711
00:24:00,559 --> 00:24:02,070


712
00:24:02,070 --> 00:24:03,909

 

713
00:24:03,909 --> 00:24:06,870

 

714
00:24:06,870 --> 00:24:10,230

 

715
00:24:10,230 --> 00:24:12,470

 

716
00:24:12,470 --> 00:24:14,149

 

717
00:24:14,149 --> 00:24:14,159

 

718
00:24:14,159 --> 00:24:14,870


719
00:24:14,870 --> 00:24:17,190

 

720
00:24:17,190 --> 00:24:19,350

 

721
00:24:19,350 --> 00:24:21,909

 

722
00:24:21,909 --> 00:24:23,430

 

723
00:24:23,430 --> 00:24:26,470

 

724
00:24:26,470 --> 00:24:29,029

 

725
00:24:29,029 --> 00:24:30,710

 

726
00:24:30,710 --> 00:24:33,669

 

727
00:24:33,669 --> 00:24:35,590

 

728
00:24:35,590 --> 00:24:35,600

 

729
00:24:35,600 --> 00:24:37,990


730
00:24:37,990 --> 00:24:40,470

 

731
00:24:40,470 --> 00:24:42,310

 

732
00:24:42,310 --> 00:24:43,990

 

733
00:24:43,990 --> 00:24:46,789

 

734
00:24:46,789 --> 00:24:48,630

 

735
00:24:48,630 --> 00:24:50,070

 

736
00:24:50,070 --> 00:24:51,990

 

737
00:24:51,990 --> 00:24:53,430

 

738
00:24:53,430 --> 00:24:55,909

 

739
00:24:55,909 --> 00:24:58,310

 

740
00:24:58,310 --> 00:25:00,549

 

741
00:25:00,549 --> 00:25:03,510

 

742
00:25:03,510 --> 00:25:06,470

 

743
00:25:06,470 --> 00:25:08,149

 

744
00:25:08,149 --> 00:25:11,029

 

745
00:25:11,029 --> 00:25:12,549

 

746
00:25:12,549 --> 00:25:14,630

 

747
00:25:14,630 --> 00:25:17,510

 

748
00:25:17,510 --> 00:25:18,870

 

749
00:25:18,870 --> 00:25:21,029

 

750
00:25:21,029 --> 00:25:22,630

 

751
00:25:22,630 --> 00:25:25,110

 

752
00:25:25,110 --> 00:25:27,750

 

753
00:25:27,750 --> 00:25:27,760

 

754
00:25:27,760 --> 00:25:28,870


755
00:25:28,870 --> 00:25:30,789

 

756
00:25:30,789 --> 00:25:32,710

 

757
00:25:32,710 --> 00:25:33,830

 

758
00:25:33,830 --> 00:25:36,390

 

759
00:25:36,390 --> 00:25:39,029

 

760
00:25:39,029 --> 00:25:40,789

 

761
00:25:40,789 --> 00:25:43,110

 

762
00:25:43,110 --> 00:25:43,960

 

763
00:25:43,960 --> 00:25:43,970

 

764
00:25:43,970 --> 00:25:45,350


765
00:25:45,350 --> 00:25:47,510

 

766
00:25:47,510 --> 00:25:49,590

 

767
00:25:49,590 --> 00:25:50,630

 

768
00:25:50,630 --> 00:25:50,640

 

769
00:25:50,640 --> 00:25:52,470


770
00:25:52,470 --> 00:25:57,269

 

771
00:25:57,269 --> 00:26:00,880

 

772
00:26:00,880 --> 00:26:00,890

 

773
00:26:00,890 --> 00:26:08,830


774
00:26:08,830 --> 00:26:11,590

 

775
00:26:11,590 --> 00:26:13,269

 

776
00:26:13,269 --> 00:26:15,510

 

777
00:26:15,510 --> 00:26:17,350

 

778
00:26:17,350 --> 00:26:19,750

 

779
00:26:19,750 --> 00:26:21,909

 

780
00:26:21,909 --> 00:26:24,710

 

781
00:26:24,710 --> 00:26:26,390

 

782
00:26:26,390 --> 00:26:28,149

 

783
00:26:28,149 --> 00:26:30,710

 

784
00:26:30,710 --> 00:26:33,269

 

785
00:26:33,269 --> 00:26:33,279

 

786
00:26:33,279 --> 00:26:34,950


787
00:26:34,950 --> 00:26:34,960

 

788
00:26:34,960 --> 00:26:36,789


789
00:26:36,789 --> 00:26:39,430

 

790
00:26:39,430 --> 00:26:42,870

 

791
00:26:42,870 --> 00:26:44,950

 

792
00:26:44,950 --> 00:26:47,110

 

793
00:26:47,110 --> 00:26:48,950

 

794
00:26:48,950 --> 00:26:50,950

 

795
00:26:50,950 --> 00:26:52,870

 

796
00:26:52,870 --> 00:26:55,269

 

797
00:26:55,269 --> 00:26:56,470

 

798
00:26:56,470 --> 00:27:00,149

 

799
00:27:00,149 --> 00:27:00,159

 

800
00:27:00,159 --> 00:27:01,350


801
00:27:01,350 --> 00:27:01,360

 

802
00:27:01,360 --> 00:27:06,470


803
00:27:06,470 --> 00:27:06,480

 

804
00:27:06,480 --> 00:27:07,510


805
00:27:07,510 --> 00:27:10,230

 

806
00:27:10,230 --> 00:27:10,240

 

807
00:27:10,240 --> 00:27:11,190


808
00:27:11,190 --> 00:27:14,149

 

809
00:27:14,149 --> 00:27:16,310

 

810
00:27:16,310 --> 00:27:19,350

 

811
00:27:19,350 --> 00:27:21,269

 

812
00:27:21,269 --> 00:27:23,029

 

813
00:27:23,029 --> 00:27:24,950

 

814
00:27:24,950 --> 00:27:26,310

 

815
00:27:26,310 --> 00:27:28,870

 

816
00:27:28,870 --> 00:27:31,269

 

817
00:27:31,269 --> 00:27:35,830

 

818
00:27:35,830 --> 00:27:38,470

 

819
00:27:38,470 --> 00:27:40,789

 

820
00:27:40,789 --> 00:27:42,310

 

821
00:27:42,310 --> 00:27:44,710

 

822
00:27:44,710 --> 00:27:47,669

 

823
00:27:47,669 --> 00:27:49,590

 

824
00:27:49,590 --> 00:27:51,269

 

825
00:27:51,269 --> 00:27:51,279

 

826
00:27:51,279 --> 00:27:53,590


827
00:27:53,590 --> 00:27:53,600

 

828
00:27:53,600 --> 00:27:59,110


829
00:27:59,110 --> 00:28:00,470

 

830
00:28:00,470 --> 00:28:02,230

 

831
00:28:02,230 --> 00:28:03,990

 

832
00:28:03,990 --> 00:28:07,029

 

833
00:28:07,029 --> 00:28:08,870

 

834
00:28:08,870 --> 00:28:11,029

 

835
00:28:11,029 --> 00:28:13,029

 

836
00:28:13,029 --> 00:28:14,630

 

837
00:28:14,630 --> 00:28:17,350

 

838
00:28:17,350 --> 00:28:19,510

 

839
00:28:19,510 --> 00:28:20,710

 

840
00:28:20,710 --> 00:28:23,110

 

841
00:28:23,110 --> 00:28:26,389

 

842
00:28:26,389 --> 00:28:30,710

 

843
00:28:30,710 --> 00:28:32,870

 

844
00:28:32,870 --> 00:28:35,029

 

845
00:28:35,029 --> 00:28:36,470

 

846
00:28:36,470 --> 00:28:38,710

 

847
00:28:38,710 --> 00:28:38,720

 

848
00:28:38,720 --> 00:28:39,909


849
00:28:39,909 --> 00:28:41,350

 

850
00:28:41,350 --> 00:28:42,630

 

851
00:28:42,630 --> 00:28:45,590

 

852
00:28:45,590 --> 00:28:48,950

 

853
00:28:48,950 --> 00:28:50,230

 

854
00:28:50,230 --> 00:28:53,750

 

855
00:28:53,750 --> 00:28:55,909

 

856
00:28:55,909 --> 00:28:57,430

 

857
00:28:57,430 --> 00:28:59,430

 

858
00:28:59,430 --> 00:29:02,149

 

859
00:29:02,149 --> 00:29:03,750

 

860
00:29:03,750 --> 00:29:05,350

 

861
00:29:05,350 --> 00:29:07,350

 

862
00:29:07,350 --> 00:29:08,789

 

863
00:29:08,789 --> 00:29:09,830

 

864
00:29:09,830 --> 00:29:12,310

 

865
00:29:12,310 --> 00:29:14,149

 

866
00:29:14,149 --> 00:29:15,350

 

867
00:29:15,350 --> 00:29:17,990

 

868
00:29:17,990 --> 00:29:19,350

 

869
00:29:19,350 --> 00:29:21,990

 

870
00:29:21,990 --> 00:29:23,830

 

871
00:29:23,830 --> 00:29:26,630

 

872
00:29:26,630 --> 00:29:28,549

 

873
00:29:28,549 --> 00:29:30,389

 

874
00:29:30,389 --> 00:29:32,230

 

875
00:29:32,230 --> 00:29:34,789

 

876
00:29:34,789 --> 00:29:34,799

 

877
00:29:34,799 --> 00:29:35,830


878
00:29:35,830 --> 00:29:37,830

 

879
00:29:37,830 --> 00:29:40,310

 

880
00:29:40,310 --> 00:29:41,430

 

881
00:29:41,430 --> 00:29:43,830

 

882
00:29:43,830 --> 00:29:45,909

 

883
00:29:45,909 --> 00:29:47,590

 

884
00:29:47,590 --> 00:29:49,350

 

885
00:29:49,350 --> 00:29:51,029

 

886
00:29:51,029 --> 00:29:53,190

 

887
00:29:53,190 --> 00:29:54,549

 

888
00:29:54,549 --> 00:29:56,310

 

889
00:29:56,310 --> 00:30:00,310

 

890
00:30:00,310 --> 00:30:02,149

 

891
00:30:02,149 --> 00:30:04,630

 

892
00:30:04,630 --> 00:30:07,190

 

893
00:30:07,190 --> 00:30:10,230

 

894
00:30:10,230 --> 00:30:12,870

 

895
00:30:12,870 --> 00:30:14,870

 

896
00:30:14,870 --> 00:30:18,710

 

897
00:30:18,710 --> 00:30:18,720

 

898
00:30:18,720 --> 00:30:20,389


899
00:30:20,389 --> 00:30:20,399

 

900
00:30:20,399 --> 00:30:22,230


901
00:30:22,230 --> 00:30:24,710

 

902
00:30:24,710 --> 00:30:26,230

 

903
00:30:26,230 --> 00:30:28,230

 

904
00:30:28,230 --> 00:30:28,240

 

905
00:30:28,240 --> 00:30:29,269


906
00:30:29,269 --> 00:30:29,279

 

907
00:30:29,279 --> 00:30:35,269


908
00:30:35,269 --> 00:30:35,279

 

909
00:30:35,279 --> 00:30:37,430


910
00:30:37,430 --> 00:30:38,870

 

911
00:30:38,870 --> 00:30:40,710

 

912
00:30:40,710 --> 00:30:40,720

 

913
00:30:40,720 --> 00:30:43,110


914
00:30:43,110 --> 00:30:45,750

 

915
00:30:45,750 --> 00:30:49,190

 

916
00:30:49,190 --> 00:30:51,430

 

917
00:30:51,430 --> 00:30:53,510

 

918
00:30:53,510 --> 00:30:55,110

 

919
00:30:55,110 --> 00:30:55,120

 

920
00:30:55,120 --> 00:30:55,909


921
00:30:55,909 --> 00:30:55,919

 

922
00:30:55,919 --> 00:30:57,509


923
00:30:57,509 --> 00:30:59,269

 

924
00:30:59,269 --> 00:31:01,269

 

925
00:31:01,269 --> 00:31:02,789

 

926
00:31:02,789 --> 00:31:04,710

 

927
00:31:04,710 --> 00:31:06,950

 

928
00:31:06,950 --> 00:31:09,509

 

929
00:31:09,509 --> 00:31:09,519

 

930
00:31:09,519 --> 00:31:11,909


931
00:31:11,909 --> 00:31:11,919

 

932
00:31:11,919 --> 00:31:13,830


933
00:31:13,830 --> 00:31:15,509

 

934
00:31:15,509 --> 00:31:17,750

 

935
00:31:17,750 --> 00:31:20,149

 

936
00:31:20,149 --> 00:31:22,149

 

937
00:31:22,149 --> 00:31:22,159

 

938
00:31:22,159 --> 00:31:24,630


939
00:31:24,630 --> 00:31:26,549

 

940
00:31:26,549 --> 00:31:26,559

 

941
00:31:26,559 --> 00:31:27,830


942
00:31:27,830 --> 00:31:27,840

 

943
00:31:27,840 --> 00:31:28,789


944
00:31:28,789 --> 00:31:31,190

 

945
00:31:31,190 --> 00:31:33,029

 

946
00:31:33,029 --> 00:31:34,630

 

947
00:31:34,630 --> 00:31:37,269

 

948
00:31:37,269 --> 00:31:39,029

 

949
00:31:39,029 --> 00:31:40,549

 

950
00:31:40,549 --> 00:31:42,549

 

951
00:31:42,549 --> 00:31:44,789

 

952
00:31:44,789 --> 00:31:46,310

 

953
00:31:46,310 --> 00:31:48,710

 

954
00:31:48,710 --> 00:31:51,110

 

955
00:31:51,110 --> 00:31:51,120

 

956
00:31:51,120 --> 00:31:52,710


957
00:31:52,710 --> 00:31:54,950

 

958
00:31:54,950 --> 00:31:56,710

 

959
00:31:56,710 --> 00:31:59,350

 

960
00:31:59,350 --> 00:32:01,190

 

961
00:32:01,190 --> 00:32:03,430

 

962
00:32:03,430 --> 00:32:06,230

 

963
00:32:06,230 --> 00:32:08,230

 

964
00:32:08,230 --> 00:32:10,310

 

965
00:32:10,310 --> 00:32:13,029

 

966
00:32:13,029 --> 00:32:14,389

 

967
00:32:14,389 --> 00:32:16,070

 

968
00:32:16,070 --> 00:32:18,310

 

969
00:32:18,310 --> 00:32:18,320

 

970
00:32:18,320 --> 00:32:20,230


971
00:32:20,230 --> 00:32:22,230

 

972
00:32:22,230 --> 00:32:22,240

 

973
00:32:22,240 --> 00:32:23,190


974
00:32:23,190 --> 00:32:25,350

 

975
00:32:25,350 --> 00:32:28,630

 

976
00:32:28,630 --> 00:32:31,190

 

977
00:32:31,190 --> 00:32:32,870

 

978
00:32:32,870 --> 00:32:35,590

 

979
00:32:35,590 --> 00:32:37,669

 

980
00:32:37,669 --> 00:32:39,669

 

981
00:32:39,669 --> 00:32:41,509

 

982
00:32:41,509 --> 00:32:44,070

 

983
00:32:44,070 --> 00:32:45,909

 

984
00:32:45,909 --> 00:32:47,990

 

985
00:32:47,990 --> 00:32:50,470

 

986
00:32:50,470 --> 00:32:51,750

 

987
00:32:51,750 --> 00:32:52,870

 

988
00:32:52,870 --> 00:32:52,880

 

989
00:32:52,880 --> 00:32:54,710


990
00:32:54,710 --> 00:32:57,350

 

991
00:32:57,350 --> 00:32:58,870

 

992
00:32:58,870 --> 00:33:00,710

 

993
00:33:00,710 --> 00:33:02,950

 

994
00:33:02,950 --> 00:33:04,950

 

995
00:33:04,950 --> 00:33:07,190

 

996
00:33:07,190 --> 00:33:09,190

 

997
00:33:09,190 --> 00:33:11,750

 

998
00:33:11,750 --> 00:33:14,149

 

999
00:33:14,149 --> 00:33:16,549

 

1000
00:33:16,549 --> 00:33:16,559

 

1001
00:33:16,559 --> 00:33:18,389


1002
00:33:18,389 --> 00:33:19,750

 

1003
00:33:19,750 --> 00:33:21,830

 

1004
00:33:21,830 --> 00:33:23,509

 

1005
00:33:23,509 --> 00:33:25,350

 

1006
00:33:25,350 --> 00:33:27,750

 

1007
00:33:27,750 --> 00:33:29,669

 

1008
00:33:29,669 --> 00:33:31,830

 

1009
00:33:31,830 --> 00:33:33,269

 

1010
00:33:33,269 --> 00:33:33,279

 

1011
00:33:33,279 --> 00:33:34,230


1012
00:33:34,230 --> 00:33:36,830

 

1013
00:33:36,830 --> 00:33:39,590

 

1014
00:33:39,590 --> 00:33:43,669

 

1015
00:33:43,669 --> 00:33:45,990

 

1016
00:33:45,990 --> 00:33:48,070

 

1017
00:33:48,070 --> 00:33:51,190

 

1018
00:33:51,190 --> 00:33:52,789

 

1019
00:33:52,789 --> 00:33:54,630

 

1020
00:33:54,630 --> 00:33:57,509

 

1021
00:33:57,509 --> 00:33:59,350

 

1022
00:33:59,350 --> 00:34:01,430

 

1023
00:34:01,430 --> 00:34:03,430

 

1024
00:34:03,430 --> 00:34:03,440

 

1025
00:34:03,440 --> 00:34:05,190


1026
00:34:05,190 --> 00:34:06,389

 

1027
00:34:06,389 --> 00:34:08,310

 

1028
00:34:08,310 --> 00:34:09,349

 

1029
00:34:09,349 --> 00:34:11,829

 

1030
00:34:11,829 --> 00:34:14,550

 

1031
00:34:14,550 --> 00:34:14,560

 

1032
00:34:14,560 --> 00:34:19,990


1033
00:34:19,990 --> 00:34:21,909

 

1034
00:34:21,909 --> 00:34:23,589

 

1035
00:34:23,589 --> 00:34:25,990

 

1036
00:34:25,990 --> 00:34:27,349

 

1037
00:34:27,349 --> 00:34:29,190

 

1038
00:34:29,190 --> 00:34:30,310

 

1039
00:34:30,310 --> 00:34:31,990

 

1040
00:34:31,990 --> 00:34:35,030

 

1041
00:34:35,030 --> 00:34:35,040

 

1042
00:34:35,040 --> 00:34:37,589


1043
00:34:37,589 --> 00:34:39,349

 

1044
00:34:39,349 --> 00:34:39,359

 

1045
00:34:39,359 --> 00:34:40,869


1046
00:34:40,869 --> 00:34:41,750

 

1047
00:34:41,750 --> 00:34:44,310

 

1048
00:34:44,310 --> 00:34:48,550

 

1049
00:34:48,550 --> 00:34:50,869

 

1050
00:34:50,869 --> 00:34:52,869

 

1051
00:34:52,869 --> 00:34:55,589

 

1052
00:34:55,589 --> 00:34:58,150

 

1053
00:34:58,150 --> 00:34:58,160

 

1054
00:34:58,160 --> 00:34:59,109


1055
00:34:59,109 --> 00:34:59,119

 

1056
00:34:59,119 --> 00:35:00,950


1057
00:35:00,950 --> 00:35:04,390

 

1058
00:35:04,390 --> 00:35:06,630

 

1059
00:35:06,630 --> 00:35:09,030

 

1060
00:35:09,030 --> 00:35:11,349

 

1061
00:35:11,349 --> 00:35:11,359

 

1062
00:35:11,359 --> 00:35:12,310


1063
00:35:12,310 --> 00:35:15,030

 

1064
00:35:15,030 --> 00:35:17,109

 

1065
00:35:17,109 --> 00:35:18,550

 

1066
00:35:18,550 --> 00:35:20,470

 

1067
00:35:20,470 --> 00:35:20,480

 

1068
00:35:20,480 --> 00:35:21,510


1069
00:35:21,510 --> 00:35:23,109

 

1070
00:35:23,109 --> 00:35:23,119

 

1071
00:35:23,119 --> 00:35:28,790


1072
00:35:28,790 --> 00:35:30,470

 

1073
00:35:30,470 --> 00:35:30,480

 

1074
00:35:30,480 --> 00:35:32,230


1075
00:35:32,230 --> 00:35:37,829

 

1076
00:35:37,829 --> 00:35:40,710

 

1077
00:35:40,710 --> 00:35:40,720

 

1078
00:35:40,720 --> 00:35:41,670


1079
00:35:41,670 --> 00:35:41,680

 

1080
00:35:41,680 --> 00:35:42,550


1081
00:35:42,550 --> 00:35:44,550

 

1082
00:35:44,550 --> 00:35:48,150

 

1083
00:35:48,150 --> 00:35:50,790

 

1084
00:35:50,790 --> 00:35:50,800

 

1085
00:35:50,800 --> 00:35:53,270


1086
00:35:53,270 --> 00:35:53,280

 

1087
00:35:53,280 --> 00:35:55,589


1088
00:35:55,589 --> 00:35:59,510

 

1089
00:35:59,510 --> 00:36:01,829

 

1090
00:36:01,829 --> 00:36:03,270

 

1091
00:36:03,270 --> 00:36:04,790

 

1092
00:36:04,790 --> 00:36:04,800

 

1093
00:36:04,800 --> 00:36:05,829


1094
00:36:05,829 --> 00:36:08,069

 

1095
00:36:08,069 --> 00:36:10,390

 

1096
00:36:10,390 --> 00:36:12,870

 

1097
00:36:12,870 --> 00:36:15,829

 

1098
00:36:15,829 --> 00:36:17,589

 

1099
00:36:17,589 --> 00:36:18,950

 

1100
00:36:18,950 --> 00:36:20,310

 

1101
00:36:20,310 --> 00:36:22,550

 

1102
00:36:22,550 --> 00:36:24,470

 

1103
00:36:24,470 --> 00:36:26,390

 

1104
00:36:26,390 --> 00:36:28,069

 

1105
00:36:28,069 --> 00:36:28,079

 

1106
00:36:28,079 --> 00:36:29,109


1107
00:36:29,109 --> 00:36:31,430

 

1108
00:36:31,430 --> 00:36:33,030

 

1109
00:36:33,030 --> 00:36:33,040

 

1110
00:36:33,040 --> 00:36:34,390


1111
00:36:34,390 --> 00:36:35,510

 

1112
00:36:35,510 --> 00:36:37,829

 

1113
00:36:37,829 --> 00:36:39,589

 

1114
00:36:39,589 --> 00:36:41,670

 

1115
00:36:41,670 --> 00:36:44,069

 

1116
00:36:44,069 --> 00:36:47,109

 

1117
00:36:47,109 --> 00:36:49,510

 

1118
00:36:49,510 --> 00:36:51,190

 

1119
00:36:51,190 --> 00:36:56,550

 

1120
00:36:56,550 --> 00:37:00,069

 

1121
00:37:00,069 --> 00:37:01,750

 

1122
00:37:01,750 --> 00:37:04,710

 

1123
00:37:04,710 --> 00:37:07,030

 

1124
00:37:07,030 --> 00:37:08,550

 

1125
00:37:08,550 --> 00:37:08,560

 

1126
00:37:08,560 --> 00:37:09,430


1127
00:37:09,430 --> 00:37:12,069

 

1128
00:37:12,069 --> 00:37:14,069

 

1129
00:37:14,069 --> 00:37:16,630

 

1130
00:37:16,630 --> 00:37:18,710

 

1131
00:37:18,710 --> 00:37:20,550

 

1132
00:37:20,550 --> 00:37:20,560

 

1133
00:37:20,560 --> 00:37:21,349


1134
00:37:21,349 --> 00:37:23,430

 

1135
00:37:23,430 --> 00:37:24,950

 

1136
00:37:24,950 --> 00:37:26,470

 

1137
00:37:26,470 --> 00:37:28,790

 

1138
00:37:28,790 --> 00:37:28,800

 

1139
00:37:28,800 --> 00:37:30,150


1140
00:37:30,150 --> 00:37:31,670

 

1141
00:37:31,670 --> 00:37:33,190

 

1142
00:37:33,190 --> 00:37:35,109

 

1143
00:37:35,109 --> 00:37:37,750

 

1144
00:37:37,750 --> 00:37:40,870

 

1145
00:37:40,870 --> 00:37:43,270

 

1146
00:37:43,270 --> 00:37:43,280

 

1147
00:37:43,280 --> 00:37:44,230


1148
00:37:44,230 --> 00:37:46,470

 

1149
00:37:46,470 --> 00:37:47,910

 

1150
00:37:47,910 --> 00:37:50,470

 

1151
00:37:50,470 --> 00:37:53,030

 

1152
00:37:53,030 --> 00:37:55,510

 

1153
00:37:55,510 --> 00:37:57,109

 

1154
00:37:57,109 --> 00:38:00,310

 

1155
00:38:00,310 --> 00:38:02,230

 

1156
00:38:02,230 --> 00:38:04,390

 

1157
00:38:04,390 --> 00:38:06,870

 

1158
00:38:06,870 --> 00:38:08,630

 

1159
00:38:08,630 --> 00:38:09,670

 

1160
00:38:09,670 --> 00:38:10,950

 

1161
00:38:10,950 --> 00:38:12,870

 

1162
00:38:12,870 --> 00:38:14,710

 

1163
00:38:14,710 --> 00:38:16,390

 

1164
00:38:16,390 --> 00:38:18,710

 

1165
00:38:18,710 --> 00:38:21,190

 

1166
00:38:21,190 --> 00:38:22,790

 

1167
00:38:22,790 --> 00:38:22,800

 

1168
00:38:22,800 --> 00:38:24,870


1169
00:38:24,870 --> 00:38:28,470

 

1170
00:38:28,470 --> 00:38:30,230

 

1171
00:38:30,230 --> 00:38:32,069

 

1172
00:38:32,069 --> 00:38:34,390

 

1173
00:38:34,390 --> 00:38:36,150

 

1174
00:38:36,150 --> 00:38:37,589

 

1175
00:38:37,589 --> 00:38:39,670

 

1176
00:38:39,670 --> 00:38:40,950

 

1177
00:38:40,950 --> 00:38:42,710

 

1178
00:38:42,710 --> 00:38:45,670

 

1179
00:38:45,670 --> 00:38:46,790

 

1180
00:38:46,790 --> 00:38:48,710

 

1181
00:38:48,710 --> 00:38:51,190

 

1182
00:38:51,190 --> 00:38:51,200

 

1183
00:38:51,200 --> 00:38:52,950


1184
00:38:52,950 --> 00:38:52,960

 

1185
00:38:52,960 --> 00:38:54,550


1186
00:38:54,550 --> 00:38:56,630

 

1187
00:38:56,630 --> 00:38:56,640

 

1188
00:38:56,640 --> 00:38:57,990


1189
00:38:57,990 --> 00:38:59,670

 

1190
00:38:59,670 --> 00:39:01,910

 

1191
00:39:01,910 --> 00:39:05,430

 

1192
00:39:05,430 --> 00:39:06,950

 

1193
00:39:06,950 --> 00:39:09,589

 

1194
00:39:09,589 --> 00:39:09,599

 

1195
00:39:09,599 --> 00:39:10,790


1196
00:39:10,790 --> 00:39:12,790

 

1197
00:39:12,790 --> 00:39:15,430

 

1198
00:39:15,430 --> 00:39:15,440

 

1199
00:39:15,440 --> 00:39:16,630


1200
00:39:16,630 --> 00:39:18,630

 

1201
00:39:18,630 --> 00:39:21,270

 

1202
00:39:21,270 --> 00:39:23,349

 

1203
00:39:23,349 --> 00:39:23,359

 

1204
00:39:23,359 --> 00:39:24,230


1205
00:39:24,230 --> 00:39:25,430

 

1206
00:39:25,430 --> 00:39:25,440

 

1207
00:39:25,440 --> 00:39:26,470


1208
00:39:26,470 --> 00:39:27,829

 

1209
00:39:27,829 --> 00:39:29,829

 

1210
00:39:29,829 --> 00:39:32,310

 

1211
00:39:32,310 --> 00:39:36,710

 

1212
00:39:36,710 --> 00:39:39,510

 

1213
00:39:39,510 --> 00:39:41,510

 

1214
00:39:41,510 --> 00:39:43,910

 

1215
00:39:43,910 --> 00:39:45,750

 

1216
00:39:45,750 --> 00:39:46,950

 

1217
00:39:46,950 --> 00:39:49,670

 

1218
00:39:49,670 --> 00:39:51,190

 

1219
00:39:51,190 --> 00:39:52,390

 

1220
00:39:52,390 --> 00:39:57,030

 

1221
00:39:57,030 --> 00:39:59,270

 

1222
00:39:59,270 --> 00:40:01,430

 

1223
00:40:01,430 --> 00:40:03,349

 

1224
00:40:03,349 --> 00:40:05,109

 

1225
00:40:05,109 --> 00:40:07,589

 

1226
00:40:07,589 --> 00:40:09,510

 

1227
00:40:09,510 --> 00:40:11,990

 

1228
00:40:11,990 --> 00:40:14,710

 

1229
00:40:14,710 --> 00:40:16,550

 

1230
00:40:16,550 --> 00:40:18,470

 

1231
00:40:18,470 --> 00:40:18,480

 

1232
00:40:18,480 --> 00:40:19,190


1233
00:40:19,190 --> 00:40:21,270

 

1234
00:40:21,270 --> 00:40:24,069

 

1235
00:40:24,069 --> 00:40:25,670

 

1236
00:40:25,670 --> 00:40:27,990

 

1237
00:40:27,990 --> 00:40:30,790

 

1238
00:40:30,790 --> 00:40:30,800

 

1239
00:40:30,800 --> 00:40:31,910


1240
00:40:31,910 --> 00:40:33,190

 

1241
00:40:33,190 --> 00:40:35,750

 

1242
00:40:35,750 --> 00:40:37,270

 

1243
00:40:37,270 --> 00:40:39,670

 

1244
00:40:39,670 --> 00:40:42,550

 

1245
00:40:42,550 --> 00:40:44,150

 

1246
00:40:44,150 --> 00:40:45,510

 

1247
00:40:45,510 --> 00:40:48,309

 

1248
00:40:48,309 --> 00:40:48,319

 

1249
00:40:48,319 --> 00:40:49,430


1250
00:40:49,430 --> 00:40:59,349

 

1251
00:40:59,349 --> 00:41:01,030

 

1252
00:41:01,030 --> 00:41:01,040

 

1253
00:41:01,040 --> 00:41:02,230


1254
00:41:02,230 --> 00:41:06,230

 

1255
00:41:06,230 --> 00:41:07,190

 

1256
00:41:07,190 --> 00:41:09,430

 

1257
00:41:09,430 --> 00:41:09,440

 

1258
00:41:09,440 --> 00:41:10,630


1259
00:41:10,630 --> 00:41:12,470

 

1260
00:41:12,470 --> 00:41:12,480

 

1261
00:41:12,480 --> 00:41:14,710


1262
00:41:14,710 --> 00:41:17,510

 

1263
00:41:17,510 --> 00:41:19,430

 

1264
00:41:19,430 --> 00:41:20,940

 

1265
00:41:20,940 --> 00:41:20,950

 

1266
00:41:20,950 --> 00:41:23,990


1267
00:41:23,990 --> 00:41:26,230

 

1268
00:41:26,230 --> 00:41:28,230

 

1269
00:41:28,230 --> 00:41:28,240

 

1270
00:41:28,240 --> 00:41:29,750


1271
00:41:29,750 --> 00:41:31,430

 

1272
00:41:31,430 --> 00:41:32,829

 

1273
00:41:32,829 --> 00:41:32,839

 

1274
00:41:32,839 --> 00:41:34,550


1275
00:41:34,550 --> 00:41:34,560

 

1276
00:41:34,560 --> 00:41:36,230


1277
00:41:36,230 --> 00:41:36,240

 

1278
00:41:36,240 --> 00:41:37,589


1279
00:41:37,589 --> 00:41:38,950

 

1280
00:41:38,950 --> 00:41:40,950

 

1281
00:41:40,950 --> 00:41:42,790

 

1282
00:41:42,790 --> 00:41:44,230

 

1283
00:41:44,230 --> 00:41:46,829

 

1284
00:41:46,829 --> 00:41:48,790

 

1285
00:41:48,790 --> 00:41:50,230

 

1286
00:41:50,230 --> 00:41:51,829

 

1287
00:41:51,829 --> 00:41:56,470

 

1288
00:41:56,470 --> 00:41:58,470

 

1289
00:41:58,470 --> 00:42:00,069

 

1290
00:42:00,069 --> 00:42:03,190

 

1291
00:42:03,190 --> 00:42:05,030

 

1292
00:42:05,030 --> 00:42:06,550

 

1293
00:42:06,550 --> 00:42:08,790

 

1294
00:42:08,790 --> 00:42:11,349

 

1295
00:42:11,349 --> 00:42:11,359

 

1296
00:42:11,359 --> 00:42:13,349


1297
00:42:13,349 --> 00:42:14,630

 

1298
00:42:14,630 --> 00:42:17,990

 

1299
00:42:17,990 --> 00:42:21,510

 

1300
00:42:21,510 --> 00:42:23,430

 

1301
00:42:23,430 --> 00:42:25,990

 

1302
00:42:25,990 --> 00:42:27,990

 

1303
00:42:27,990 --> 00:42:30,470

 

1304
00:42:30,470 --> 00:42:33,670

 

1305
00:42:33,670 --> 00:42:37,030

 

1306
00:42:37,030 --> 00:42:38,550

 

1307
00:42:38,550 --> 00:42:41,190

 

1308
00:42:41,190 --> 00:42:44,630

 

1309
00:42:44,630 --> 00:42:47,190

 

1310
00:42:47,190 --> 00:42:50,150

 

1311
00:42:50,150 --> 00:42:51,829

 

1312
00:42:51,829 --> 00:42:51,839

 

1313
00:42:51,839 --> 00:42:53,190


1314
00:42:53,190 --> 00:42:53,200

 

1315
00:42:53,200 --> 00:42:54,150


1316
00:42:54,150 --> 00:42:56,829

 

1317
00:42:56,829 --> 00:42:59,829

 

1318
00:42:59,829 --> 00:43:01,510

 

1319
00:43:01,510 --> 00:43:03,349

 

1320
00:43:03,349 --> 00:43:04,630

 

1321
00:43:04,630 --> 00:43:06,790

 

1322
00:43:06,790 --> 00:43:08,550

 

1323
00:43:08,550 --> 00:43:09,910

 

1324
00:43:09,910 --> 00:43:11,190

 

1325
00:43:11,190 --> 00:43:11,200

 

1326
00:43:11,200 --> 00:43:12,710


1327
00:43:12,710 --> 00:43:14,870

 

1328
00:43:14,870 --> 00:43:16,790

 

1329
00:43:16,790 --> 00:43:18,309

 

1330
00:43:18,309 --> 00:43:20,390

 

1331
00:43:20,390 --> 00:43:21,750

 

1332
00:43:21,750 --> 00:43:23,030

 

1333
00:43:23,030 --> 00:43:26,309

 

1334
00:43:26,309 --> 00:43:27,670

 

1335
00:43:27,670 --> 00:43:27,680

 

1336
00:43:27,680 --> 00:43:29,270


1337
00:43:29,270 --> 00:43:31,510

 

1338
00:43:31,510 --> 00:43:33,589

 

1339
00:43:33,589 --> 00:43:35,190

 

1340
00:43:35,190 --> 00:43:35,200

 

1341
00:43:35,200 --> 00:43:36,710


1342
00:43:36,710 --> 00:43:38,870

 

1343
00:43:38,870 --> 00:43:40,790

 

1344
00:43:40,790 --> 00:43:42,790

 

1345
00:43:42,790 --> 00:43:42,800

 

1346
00:43:42,800 --> 00:43:44,150


1347
00:43:44,150 --> 00:43:46,630

 

1348
00:43:46,630 --> 00:43:48,230

 

1349
00:43:48,230 --> 00:43:50,309

 

1350
00:43:50,309 --> 00:43:52,309

 

1351
00:43:52,309 --> 00:43:54,230

 

1352
00:43:54,230 --> 00:43:56,630

 

1353
00:43:56,630 --> 00:43:58,550

 

1354
00:43:58,550 --> 00:43:59,670

 

1355
00:43:59,670 --> 00:43:59,680

 

1356
00:43:59,680 --> 00:44:00,470


1357
00:44:00,470 --> 00:44:01,750

 

1358
00:44:01,750 --> 00:44:04,390

 

1359
00:44:04,390 --> 00:44:06,069

 

1360
00:44:06,069 --> 00:44:08,390

 

1361
00:44:08,390 --> 00:44:09,829

 

1362
00:44:09,829 --> 00:44:11,030

 

1363
00:44:11,030 --> 00:44:12,790

 

1364
00:44:12,790 --> 00:44:15,750

 

1365
00:44:15,750 --> 00:44:17,990

 

1366
00:44:17,990 --> 00:44:18,000

 

1367
00:44:18,000 --> 00:44:18,790


1368
00:44:18,790 --> 00:44:22,470

 

1369
00:44:22,470 --> 00:44:24,630

 

1370
00:44:24,630 --> 00:44:26,470

 

1371
00:44:26,470 --> 00:44:26,480

 

1372
00:44:26,480 --> 00:44:27,750


1373
00:44:27,750 --> 00:44:29,430

 

1374
00:44:29,430 --> 00:44:32,230

 

1375
00:44:32,230 --> 00:44:34,630

 

1376
00:44:34,630 --> 00:44:36,870

 

1377
00:44:36,870 --> 00:44:38,390

 

1378
00:44:38,390 --> 00:44:40,710

 

1379
00:44:40,710 --> 00:44:42,069

 

1380
00:44:42,069 --> 00:44:42,079

 

1381
00:44:42,079 --> 00:44:44,230


1382
00:44:44,230 --> 00:44:44,240

 

1383
00:44:44,240 --> 00:44:45,589


1384
00:44:45,589 --> 00:44:47,910

 

1385
00:44:47,910 --> 00:44:49,589

 

1386
00:44:49,589 --> 00:44:51,270

 

1387
00:44:51,270 --> 00:44:53,109

 

1388
00:44:53,109 --> 00:44:56,309

 

1389
00:44:56,309 --> 00:44:59,430

 

1390
00:44:59,430 --> 00:45:00,710

 

1391
00:45:00,710 --> 00:45:03,109

 

1392
00:45:03,109 --> 00:45:05,349

 

1393
00:45:05,349 --> 00:45:08,309

 

1394
00:45:08,309 --> 00:45:11,430

 

1395
00:45:11,430 --> 00:45:12,630

 

1396
00:45:12,630 --> 00:45:13,990

 

1397
00:45:13,990 --> 00:45:15,910

 

1398
00:45:15,910 --> 00:45:18,710

 

1399
00:45:18,710 --> 00:45:20,470

 

1400
00:45:20,470 --> 00:45:22,950

 

1401
00:45:22,950 --> 00:45:24,870

 

1402
00:45:24,870 --> 00:45:24,880

 

1403
00:45:24,880 --> 00:45:26,230


1404
00:45:26,230 --> 00:45:27,910

 

1405
00:45:27,910 --> 00:45:29,910

 

1406
00:45:29,910 --> 00:45:32,150

 

1407
00:45:32,150 --> 00:45:34,470

 

1408
00:45:34,470 --> 00:45:36,309

 

1409
00:45:36,309 --> 00:45:38,390

 

1410
00:45:38,390 --> 00:45:40,870

 

1411
00:45:40,870 --> 00:45:44,470

 

1412
00:45:44,470 --> 00:45:46,230

 

1413
00:45:46,230 --> 00:45:49,270

 

1414
00:45:49,270 --> 00:45:51,910

 

1415
00:45:51,910 --> 00:45:53,910

 

1416
00:45:53,910 --> 00:45:55,910

 

1417
00:45:55,910 --> 00:46:00,550

 

1418
00:46:00,550 --> 00:46:00,560

 

1419
00:46:00,560 --> 00:46:01,430


1420
00:46:01,430 --> 00:46:03,510

 

1421
00:46:03,510 --> 00:46:05,510

 

1422
00:46:05,510 --> 00:46:07,589

 

1423
00:46:07,589 --> 00:46:08,630

 

1424
00:46:08,630 --> 00:46:08,640

 

1425
00:46:08,640 --> 00:46:09,670


1426
00:46:09,670 --> 00:46:14,870

 

1427
00:46:14,870 --> 00:46:16,710

 

1428
00:46:16,710 --> 00:46:18,230

 

1429
00:46:18,230 --> 00:46:20,069

 

1430
00:46:20,069 --> 00:46:22,230

 

1431
00:46:22,230 --> 00:46:22,240

 

1432
00:46:22,240 --> 00:46:23,270


1433
00:46:23,270 --> 00:46:26,829

 

1434
00:46:26,829 --> 00:46:32,390

 

1435
00:46:32,390 --> 00:46:33,990

 

1436
00:46:33,990 --> 00:46:36,390

 

1437
00:46:36,390 --> 00:46:40,309

 

1438
00:46:40,309 --> 00:46:41,430

 

1439
00:46:41,430 --> 00:46:43,990

 

1440
00:46:43,990 --> 00:46:46,230

 

1441
00:46:46,230 --> 00:46:53,829

 

1442
00:46:53,829 --> 00:46:56,790

 

1443
00:46:56,790 --> 00:46:56,800

 

1444
00:46:56,800 --> 00:46:58,309


1445
00:46:58,309 --> 00:46:58,319

 

1446
00:46:58,319 --> 00:47:00,870


1447
00:47:00,870 --> 00:47:03,430

 

1448
00:47:03,430 --> 00:47:05,109

 

1449
00:47:05,109 --> 00:47:07,510

 

1450
00:47:07,510 --> 00:47:09,829

 

1451
00:47:09,829 --> 00:47:11,829

 

1452
00:47:11,829 --> 00:47:14,069

 

1453
00:47:14,069 --> 00:47:14,079

 

1454
00:47:14,079 --> 00:47:15,190


1455
00:47:15,190 --> 00:47:16,470

 

1456
00:47:16,470 --> 00:47:16,480

 

1457
00:47:16,480 --> 00:47:19,349


1458
00:47:19,349 --> 00:47:19,359

 

1459
00:47:19,359 --> 00:47:21,190


1460
00:47:21,190 --> 00:47:23,670

 

1461
00:47:23,670 --> 00:47:25,829

 

1462
00:47:25,829 --> 00:47:25,839

 

1463
00:47:25,839 --> 00:47:26,790


1464
00:47:26,790 --> 00:47:26,800

 

1465
00:47:26,800 --> 00:47:29,750


1466
00:47:29,750 --> 00:47:32,069

 

1467
00:47:32,069 --> 00:47:33,349

 

1468
00:47:33,349 --> 00:47:35,990

 

1469
00:47:35,990 --> 00:47:38,710

 

1470
00:47:38,710 --> 00:47:40,710

 

1471
00:47:40,710 --> 00:47:42,790

 

1472
00:47:42,790 --> 00:47:44,309

 

1473
00:47:44,309 --> 00:47:47,109

 

1474
00:47:47,109 --> 00:47:47,119

 

1475
00:47:47,119 --> 00:47:47,910


1476
00:47:47,910 --> 00:47:49,670

 

1477
00:47:49,670 --> 00:47:49,680

 

1478
00:47:49,680 --> 00:47:50,829


1479
00:47:50,829 --> 00:47:53,270

 

1480
00:47:53,270 --> 00:47:53,280

 

1481
00:47:53,280 --> 00:47:55,349


1482
00:47:55,349 --> 00:47:58,390

 

1483
00:47:58,390 --> 00:47:59,990

 

1484
00:47:59,990 --> 00:48:02,230

 

1485
00:48:02,230 --> 00:48:04,150

 

1486
00:48:04,150 --> 00:48:06,230

 

1487
00:48:06,230 --> 00:48:07,750

 

1488
00:48:07,750 --> 00:48:07,760

 

1489
00:48:07,760 --> 00:48:09,430


1490
00:48:09,430 --> 00:48:10,870

 

1491
00:48:10,870 --> 00:48:13,270

 

1492
00:48:13,270 --> 00:48:15,109

 

1493
00:48:15,109 --> 00:48:16,710

 

1494
00:48:16,710 --> 00:48:19,190

 

1495
00:48:19,190 --> 00:48:22,150

 

1496
00:48:22,150 --> 00:48:23,829

 

1497
00:48:23,829 --> 00:48:24,790

 

1498
00:48:24,790 --> 00:48:26,470

 

1499
00:48:26,470 --> 00:48:29,750

 

1500
00:48:29,750 --> 00:48:32,150

 

1501
00:48:32,150 --> 00:48:34,950

 

1502
00:48:34,950 --> 00:48:36,470

 

1503
00:48:36,470 --> 00:48:39,750

 

1504
00:48:39,750 --> 00:48:41,910

 

1505
00:48:41,910 --> 00:48:43,750

 

1506
00:48:43,750 --> 00:48:45,670

 

1507
00:48:45,670 --> 00:48:47,589

 

1508
00:48:47,589 --> 00:48:47,599

 

1509
00:48:47,599 --> 00:48:48,950


1510
00:48:48,950 --> 00:48:48,960

 

1511
00:48:48,960 --> 00:48:50,870


1512
00:48:50,870 --> 00:48:53,109

 

1513
00:48:53,109 --> 00:48:54,710

 

1514
00:48:54,710 --> 00:48:57,030

 

1515
00:48:57,030 --> 00:48:59,510

 

1516
00:48:59,510 --> 00:49:02,870

 

1517
00:49:02,870 --> 00:49:05,109

 

1518
00:49:05,109 --> 00:49:06,710

 

1519
00:49:06,710 --> 00:49:07,990

 

1520
00:49:07,990 --> 00:49:10,150

 

1521
00:49:10,150 --> 00:49:10,160

 

1522
00:49:10,160 --> 00:49:11,750


1523
00:49:11,750 --> 00:49:15,510

 

1524
00:49:15,510 --> 00:49:17,270

 

1525
00:49:17,270 --> 00:49:19,829

 

1526
00:49:19,829 --> 00:49:21,670

 

1527
00:49:21,670 --> 00:49:23,750

 

1528
00:49:23,750 --> 00:49:26,950

 

1529
00:49:26,950 --> 00:49:28,790

 

1530
00:49:28,790 --> 00:49:31,109

 

1531
00:49:31,109 --> 00:49:34,790

 

1532
00:49:34,790 --> 00:49:34,800

 

1533
00:49:34,800 --> 00:49:36,069


1534
00:49:36,069 --> 00:49:38,150

 

1535
00:49:38,150 --> 00:49:40,230

 

1536
00:49:40,230 --> 00:49:41,589

 

1537
00:49:41,589 --> 00:49:41,599

 

1538
00:49:41,599 --> 00:49:43,270


1539
00:49:43,270 --> 00:49:45,829

 

1540
00:49:45,829 --> 00:49:47,829

 

1541
00:49:47,829 --> 00:49:49,829

 

1542
00:49:49,829 --> 00:49:51,990

 

1543
00:49:51,990 --> 00:49:52,000

 

1544
00:49:52,000 --> 00:49:52,870


1545
00:49:52,870 --> 00:49:55,109

 

1546
00:49:55,109 --> 00:49:57,109

 

1547
00:49:57,109 --> 00:49:58,549

 

1548
00:49:58,549 --> 00:50:01,589

 

1549
00:50:01,589 --> 00:50:03,349

 

1550
00:50:03,349 --> 00:50:04,710

 

1551
00:50:04,710 --> 00:50:04,720

 

1552
00:50:04,720 --> 00:50:06,230


1553
00:50:06,230 --> 00:50:06,240

 

1554
00:50:06,240 --> 00:50:07,030


1555
00:50:07,030 --> 00:50:08,790

 

1556
00:50:08,790 --> 00:50:08,800

 

1557
00:50:08,800 --> 00:50:09,829


1558
00:50:09,829 --> 00:50:11,109

 

1559
00:50:11,109 --> 00:50:13,190

 

1560
00:50:13,190 --> 00:50:14,950

 

1561
00:50:14,950 --> 00:50:16,470

 

1562
00:50:16,470 --> 00:50:18,710

 

1563
00:50:18,710 --> 00:50:19,670

 

1564
00:50:19,670 --> 00:50:21,270

 

1565
00:50:21,270 --> 00:50:23,190

 

1566
00:50:23,190 --> 00:50:25,750

 

1567
00:50:25,750 --> 00:50:25,760

 

1568
00:50:25,760 --> 00:50:29,109


1569
00:50:29,109 --> 00:50:30,950

 

1570
00:50:30,950 --> 00:50:33,270

 

1571
00:50:33,270 --> 00:50:33,280

 

1572
00:50:33,280 --> 00:50:35,829


1573
00:50:35,829 --> 00:50:38,549

 

1574
00:50:38,549 --> 00:50:42,309

 

1575
00:50:42,309 --> 00:50:43,829

 

1576
00:50:43,829 --> 00:50:45,829

 

1577
00:50:45,829 --> 00:50:47,910

 

1578
00:50:47,910 --> 00:50:47,920

 

1579
00:50:47,920 --> 00:50:48,950


1580
00:50:48,950 --> 00:50:52,069

 

1581
00:50:52,069 --> 00:50:53,670

 

1582
00:50:53,670 --> 00:50:55,349

 

1583
00:50:55,349 --> 00:50:58,069

 

1584
00:50:58,069 --> 00:51:01,190

 

1585
00:51:01,190 --> 00:51:01,200

 

1586
00:51:01,200 --> 00:51:02,549


1587
00:51:02,549 --> 00:51:03,990

 

1588
00:51:03,990 --> 00:51:06,069

 

1589
00:51:06,069 --> 00:51:08,069

 

1590
00:51:08,069 --> 00:51:09,910

 

1591
00:51:09,910 --> 00:51:11,349

 

1592
00:51:11,349 --> 00:51:14,470

 

1593
00:51:14,470 --> 00:51:18,150

 

1594
00:51:18,150 --> 00:51:20,630

 

1595
00:51:20,630 --> 00:51:22,790

 

1596
00:51:22,790 --> 00:51:22,800

 

1597
00:51:22,800 --> 00:51:24,710


1598
00:51:24,710 --> 00:51:27,990

 

1599
00:51:27,990 --> 00:51:30,309

 

1600
00:51:30,309 --> 00:51:31,270

 

1601
00:51:31,270 --> 00:51:32,390

 

1602
00:51:32,390 --> 00:51:35,589

 

1603
00:51:35,589 --> 00:51:39,190

 

1604
00:51:39,190 --> 00:51:41,030

 

1605
00:51:41,030 --> 00:51:44,150

 

1606
00:51:44,150 --> 00:51:47,910

 

1607
00:51:47,910 --> 00:51:50,710

 

1608
00:51:50,710 --> 00:51:53,349

 

1609
00:51:53,349 --> 00:51:55,990

 

1610
00:51:55,990 --> 00:51:58,470

 

1611
00:51:58,470 --> 00:52:02,390

 

1612
00:52:02,390 --> 00:52:02,400

 

1613
00:52:02,400 --> 00:52:03,670


1614
00:52:03,670 --> 00:52:05,270

 

1615
00:52:05,270 --> 00:52:06,870

 

1616
00:52:06,870 --> 00:52:08,950

 

1617
00:52:08,950 --> 00:52:11,349

 

1618
00:52:11,349 --> 00:52:12,870

 

1619
00:52:12,870 --> 00:52:12,880

 

1620
00:52:12,880 --> 00:52:14,549


1621
00:52:14,549 --> 00:52:16,069

 

1622
00:52:16,069 --> 00:52:16,079

 

1623
00:52:16,079 --> 00:52:18,230


1624
00:52:18,230 --> 00:52:19,829

 

1625
00:52:19,829 --> 00:52:21,910

 

1626
00:52:21,910 --> 00:52:23,990

 

1627
00:52:23,990 --> 00:52:25,670

 

1628
00:52:25,670 --> 00:52:28,150

 

1629
00:52:28,150 --> 00:52:30,790

 

1630
00:52:30,790 --> 00:52:32,230

 

1631
00:52:32,230 --> 00:52:33,990

 

1632
00:52:33,990 --> 00:52:36,390

 

1633
00:52:36,390 --> 00:52:38,150

 

1634
00:52:38,150 --> 00:52:40,470

 

1635
00:52:40,470 --> 00:52:42,549

 

1636
00:52:42,549 --> 00:52:45,670

 

1637
00:52:45,670 --> 00:52:47,910

 

1638
00:52:47,910 --> 00:52:52,630

 

1639
00:52:52,630 --> 00:52:55,510

 

1640
00:52:55,510 --> 00:52:59,990

 

1641
00:52:59,990 --> 00:53:00,000

 

1642
00:53:00,000 --> 00:53:00,630


1643
00:53:00,630 --> 00:53:02,549

 

1644
00:53:02,549 --> 00:53:04,870

 

1645
00:53:04,870 --> 00:53:06,549

 

1646
00:53:06,549 --> 00:53:08,470

 

1647
00:53:08,470 --> 00:53:11,190

 

1648
00:53:11,190 --> 00:53:12,390

 

1649
00:53:12,390 --> 00:53:13,750

 

1650
00:53:13,750 --> 00:53:15,589

 

1651
00:53:15,589 --> 00:53:17,349

 

1652
00:53:17,349 --> 00:53:19,349

 

1653
00:53:19,349 --> 00:53:19,359

 

1654
00:53:19,359 --> 00:53:22,150


1655
00:53:22,150 --> 00:53:23,510

 

1656
00:53:23,510 --> 00:53:25,589

 

1657
00:53:25,589 --> 00:53:26,829

 

1658
00:53:26,829 --> 00:53:28,710

 

1659
00:53:28,710 --> 00:53:31,030

 

1660
00:53:31,030 --> 00:53:31,040

 

1661
00:53:31,040 --> 00:53:31,910


1662
00:53:31,910 --> 00:53:33,430

 

1663
00:53:33,430 --> 00:53:35,190

 

1664
00:53:35,190 --> 00:53:37,270

 

1665
00:53:37,270 --> 00:53:38,950

 

1666
00:53:38,950 --> 00:53:41,109

 

1667
00:53:41,109 --> 00:53:41,119

 

1668
00:53:41,119 --> 00:53:42,470


1669
00:53:42,470 --> 00:53:45,670

 

1670
00:53:45,670 --> 00:53:45,680

 

1671
00:53:45,680 --> 00:53:47,270


1672
00:53:47,270 --> 00:53:50,549

 

1673
00:53:50,549 --> 00:53:52,309

 

1674
00:53:52,309 --> 00:53:53,910

 

1675
00:53:53,910 --> 00:53:54,950

 

1676
00:53:54,950 --> 00:53:57,990

 

1677
00:53:57,990 --> 00:54:00,710

 

1678
00:54:00,710 --> 00:54:02,390

 

1679
00:54:02,390 --> 00:54:05,990

 

1680
00:54:05,990 --> 00:54:07,910

 

1681
00:54:07,910 --> 00:54:11,430

 

1682
00:54:11,430 --> 00:54:14,549

 

1683
00:54:14,549 --> 00:54:14,559

 

1684
00:54:14,559 --> 00:54:15,589


1685
00:54:15,589 --> 00:54:15,599

 

1686
00:54:15,599 --> 00:54:17,430


1687
00:54:17,430 --> 00:54:19,190

 

1688
00:54:19,190 --> 00:54:23,430

 

1689
00:54:23,430 --> 00:54:25,910

 

1690
00:54:25,910 --> 00:54:28,710

 

1691
00:54:28,710 --> 00:54:30,630

 

1692
00:54:30,630 --> 00:54:32,150

 

1693
00:54:32,150 --> 00:54:35,030

 

1694
00:54:35,030 --> 00:54:37,270

 

1695
00:54:37,270 --> 00:54:39,510

 

1696
00:54:39,510 --> 00:54:41,270

 

1697
00:54:41,270 --> 00:54:41,280

 

1698
00:54:41,280 --> 00:54:43,030


1699
00:54:43,030 --> 00:54:46,470

 

1700
00:54:46,470 --> 00:54:48,470

 

1701
00:54:48,470 --> 00:54:50,150

 

1702
00:54:50,150 --> 00:54:52,789

 

1703
00:54:52,789 --> 00:54:56,150

 

1704
00:54:56,150 --> 00:54:57,750

 

1705
00:54:57,750 --> 00:54:59,750

 

1706
00:54:59,750 --> 00:55:01,829

 

1707
00:55:01,829 --> 00:55:03,670

 

1708
00:55:03,670 --> 00:55:03,680

 

1709
00:55:03,680 --> 00:55:05,990


1710
00:55:05,990 --> 00:55:08,309

 

1711
00:55:08,309 --> 00:55:12,390

 

1712
00:55:12,390 --> 00:55:14,470

 

1713
00:55:14,470 --> 00:55:18,470

 

1714
00:55:18,470 --> 00:55:20,870

 

1715
00:55:20,870 --> 00:55:22,710

 

1716
00:55:22,710 --> 00:55:24,789

 

1717
00:55:24,789 --> 00:55:26,549

 

1718
00:55:26,549 --> 00:55:27,670

 

1719
00:55:27,670 --> 00:55:27,680

 

1720
00:55:27,680 --> 00:55:29,190


1721
00:55:29,190 --> 00:55:29,200

 

1722
00:55:29,200 --> 00:55:30,309


1723
00:55:30,309 --> 00:55:32,870

 

1724
00:55:32,870 --> 00:55:35,910

 

1725
00:55:35,910 --> 00:55:39,349

 

1726
00:55:39,349 --> 00:55:41,109

 

1727
00:55:41,109 --> 00:55:43,349

 

1728
00:55:43,349 --> 00:55:46,390

 

1729
00:55:46,390 --> 00:55:48,950

 

1730
00:55:48,950 --> 00:55:51,190

 

1731
00:55:51,190 --> 00:55:54,069

 

1732
00:55:54,069 --> 00:55:55,910

 

1733
00:55:55,910 --> 00:55:57,829

 

1734
00:55:57,829 --> 00:56:00,549

 

1735
00:56:00,549 --> 00:56:02,789

 

1736
00:56:02,789 --> 00:56:02,799

 

1737
00:56:02,799 --> 00:56:04,630


1738
00:56:04,630 --> 00:56:07,990

 

1739
00:56:07,990 --> 00:56:10,390

 

1740
00:56:10,390 --> 00:56:13,670

 

1741
00:56:13,670 --> 00:56:15,270

 

1742
00:56:15,270 --> 00:56:17,349

 

1743
00:56:17,349 --> 00:56:19,030

 

1744
00:56:19,030 --> 00:56:20,470

 

1745
00:56:20,470 --> 00:56:22,390

 

1746
00:56:22,390 --> 00:56:25,190

 

1747
00:56:25,190 --> 00:56:26,950

 

1748
00:56:26,950 --> 00:56:29,270

 

1749
00:56:29,270 --> 00:56:32,230

 

1750
00:56:32,230 --> 00:56:33,829

 

1751
00:56:33,829 --> 00:56:35,990

 

1752
00:56:35,990 --> 00:56:38,230

 

1753
00:56:38,230 --> 00:56:39,510

 

1754
00:56:39,510 --> 00:56:41,270

 

1755
00:56:41,270 --> 00:56:44,069

 

1756
00:56:44,069 --> 00:56:46,309

 

1757
00:56:46,309 --> 00:56:46,319

 

1758
00:56:46,319 --> 00:56:47,510


1759
00:56:47,510 --> 00:56:49,270

 

1760
00:56:49,270 --> 00:56:50,870

 

1761
00:56:50,870 --> 00:56:53,750

 

1762
00:56:53,750 --> 00:56:53,760

 

1763
00:56:53,760 --> 00:56:55,990


1764
00:56:55,990 --> 00:56:56,000

 

1765
00:56:56,000 --> 00:56:57,750


1766
00:56:57,750 --> 00:56:59,589

 

1767
00:56:59,589 --> 00:57:02,309

 

1768
00:57:02,309 --> 00:57:04,950

 

1769
00:57:04,950 --> 00:57:09,829

 

1770
00:57:09,829 --> 00:57:11,510

 

1771
00:57:11,510 --> 00:57:13,190

 

1772
00:57:13,190 --> 00:57:16,470

 

1773
00:57:16,470 --> 00:57:18,870

 

1774
00:57:18,870 --> 00:57:19,990

 

1775
00:57:19,990 --> 00:57:21,910

 

1776
00:57:21,910 --> 00:57:24,630

 

1777
00:57:24,630 --> 00:57:27,510

 

1778
00:57:27,510 --> 00:57:29,190

 

1779
00:57:29,190 --> 00:57:31,270

 

1780
00:57:31,270 --> 00:57:33,990

 

1781
00:57:33,990 --> 00:57:34,000

 

1782
00:57:34,000 --> 00:57:35,109


1783
00:57:35,109 --> 00:57:37,510

 

1784
00:57:37,510 --> 00:57:37,520

 

1785
00:57:37,520 --> 00:57:38,309


1786
00:57:38,309 --> 00:57:41,030

 

1787
00:57:41,030 --> 00:57:43,030

 

1788
00:57:43,030 --> 00:57:45,270

 

1789
00:57:45,270 --> 00:57:46,870

 

1790
00:57:46,870 --> 00:57:49,270

 

1791
00:57:49,270 --> 00:57:50,630

 

1792
00:57:50,630 --> 00:57:52,789

 

1793
00:57:52,789 --> 00:57:54,950

 

1794
00:57:54,950 --> 00:57:57,589

 

1795
00:57:57,589 --> 00:57:59,510

 

1796
00:57:59,510 --> 00:58:01,510

 

1797
00:58:01,510 --> 00:58:05,910

 

1798
00:58:05,910 --> 00:58:05,920

 

1799
00:58:05,920 --> 00:58:07,430


1800
00:58:07,430 --> 00:58:09,750

 

1801
00:58:09,750 --> 00:58:11,910

 

1802
00:58:11,910 --> 00:58:13,270

 

1803
00:58:13,270 --> 00:58:15,910

 

1804
00:58:15,910 --> 00:58:15,920

 

1805
00:58:15,920 --> 00:58:18,309


1806
00:58:18,309 --> 00:58:20,390

 

1807
00:58:20,390 --> 00:58:22,230

 

1808
00:58:22,230 --> 00:58:23,829

 

1809
00:58:23,829 --> 00:58:25,670

 

1810
00:58:25,670 --> 00:58:29,430

 

1811
00:58:29,430 --> 00:58:29,440

 

1812
00:58:29,440 --> 00:58:30,309


1813
00:58:30,309 --> 00:58:31,750

 

1814
00:58:31,750 --> 00:58:34,549

 

1815
00:58:34,549 --> 00:58:36,549

 

1816
00:58:36,549 --> 00:58:38,309

 

1817
00:58:38,309 --> 00:58:39,990

 

1818
00:58:39,990 --> 00:58:41,349

 

1819
00:58:41,349 --> 00:58:43,030

 

1820
00:58:43,030 --> 00:58:44,870

 

1821
00:58:44,870 --> 00:58:46,710

 

1822
00:58:46,710 --> 00:58:49,430

 

1823
00:58:49,430 --> 00:58:51,589

 

1824
00:58:51,589 --> 00:58:53,109

 

1825
00:58:53,109 --> 00:58:54,789

 

1826
00:58:54,789 --> 00:58:57,109

 

1827
00:58:57,109 --> 00:58:59,109

 

1828
00:58:59,109 --> 00:58:59,119

 

1829
00:58:59,119 --> 00:59:00,230


1830
00:59:00,230 --> 00:59:02,230

 

1831
00:59:02,230 --> 00:59:04,789

 

1832
00:59:04,789 --> 00:59:07,670

 

1833
00:59:07,670 --> 00:59:10,150

 

1834
00:59:10,150 --> 00:59:12,870

 

1835
00:59:12,870 --> 00:59:12,880

 

1836
00:59:12,880 --> 00:59:13,670


1837
00:59:13,670 --> 00:59:13,680

 

1838
00:59:13,680 --> 00:59:14,710


1839
00:59:14,710 --> 00:59:17,670

 

1840
00:59:17,670 --> 00:59:18,549

 

1841
00:59:18,549 --> 00:59:20,870

 

1842
00:59:20,870 --> 00:59:23,750

 

1843
00:59:23,750 --> 00:59:23,760

 

1844
00:59:23,760 --> 00:59:24,549


1845
00:59:24,549 --> 00:59:26,069

 

1846
00:59:26,069 --> 00:59:28,069

 

1847
00:59:28,069 --> 00:59:29,430

 

1848
00:59:29,430 --> 00:59:32,870

 

1849
00:59:32,870 --> 00:59:34,549

 

1850
00:59:34,549 --> 00:59:36,789

 

1851
00:59:36,789 --> 00:59:40,789

 

1852
00:59:40,789 --> 00:59:44,069

 

1853
00:59:44,069 --> 00:59:47,190

 

1854
00:59:47,190 --> 00:59:50,549

 

1855
00:59:50,549 --> 00:59:52,950

 

1856
00:59:52,950 --> 00:59:55,109

 

1857
00:59:55,109 --> 00:59:56,789

 

1858
00:59:56,789 --> 00:59:56,799

 

1859
00:59:56,799 --> 00:59:57,990


1860
00:59:57,990 --> 01:00:00,630

 

1861
01:00:00,630 --> 01:00:02,309

 

1862
01:00:02,309 --> 01:00:04,150

 

1863
01:00:04,150 --> 01:00:04,160

 

1864
01:00:04,160 --> 01:00:05,589


1865
01:00:05,589 --> 01:00:05,599

 

1866
01:00:05,599 --> 01:00:07,750


1867
01:00:07,750 --> 01:00:10,069

 

1868
01:00:10,069 --> 01:00:13,910

 

1869
01:00:13,910 --> 01:00:13,920

 

1870
01:00:13,920 --> 01:00:14,870


1871
01:00:14,870 --> 01:00:17,750

 

1872
01:00:17,750 --> 01:00:19,670

 

1873
01:00:19,670 --> 01:00:23,670

 

1874
01:00:23,670 --> 01:00:23,680

 

1875
01:00:23,680 --> 01:00:25,829


1876
01:00:25,829 --> 01:00:28,870

 

1877
01:00:28,870 --> 01:00:31,190

 

1878
01:00:31,190 --> 01:00:34,069

 

1879
01:00:34,069 --> 01:00:35,910

 

1880
01:00:35,910 --> 01:00:37,510

 

1881
01:00:37,510 --> 01:00:40,309

 

1882
01:00:40,309 --> 01:00:41,750

 

1883
01:00:41,750 --> 01:00:43,990

 

1884
01:00:43,990 --> 01:00:46,230

 

1885
01:00:46,230 --> 01:00:49,270

 

1886
01:00:49,270 --> 01:00:50,789

 

1887
01:00:50,789 --> 01:00:50,799

 

1888
01:00:50,799 --> 01:00:51,750


1889
01:00:51,750 --> 01:00:51,760

 

1890
01:00:51,760 --> 01:00:53,030


1891
01:00:53,030 --> 01:00:55,829

 

1892
01:00:55,829 --> 01:00:57,589

 

1893
01:00:57,589 --> 01:01:00,549

 

1894
01:01:00,549 --> 01:01:03,109

 

1895
01:01:03,109 --> 01:01:05,510

 

1896
01:01:05,510 --> 01:01:07,910

 

1897
01:01:07,910 --> 01:01:10,470

 

1898
01:01:10,470 --> 01:01:11,990

 

1899
01:01:11,990 --> 01:01:13,910

 

1900
01:01:13,910 --> 01:01:13,920

 

1901
01:01:13,920 --> 01:01:15,190


1902
01:01:15,190 --> 01:01:15,200

 

1903
01:01:15,200 --> 01:01:16,710


1904
01:01:16,710 --> 01:01:19,349

 

1905
01:01:19,349 --> 01:01:20,870

 

1906
01:01:20,870 --> 01:01:22,150

 

1907
01:01:22,150 --> 01:01:24,470

 

1908
01:01:24,470 --> 01:01:27,109

 

1909
01:01:27,109 --> 01:01:29,510

 

1910
01:01:29,510 --> 01:01:31,349

 

1911
01:01:31,349 --> 01:01:33,510

 

1912
01:01:33,510 --> 01:01:37,109

 

1913
01:01:37,109 --> 01:01:40,069

 

1914
01:01:40,069 --> 01:01:43,349

 

1915
01:01:43,349 --> 01:01:46,069

 

1916
01:01:46,069 --> 01:01:48,789

 

1917
01:01:48,789 --> 01:01:51,109

 

1918
01:01:51,109 --> 01:01:54,390

 

1919
01:01:54,390 --> 01:01:56,390

 

1920
01:01:56,390 --> 01:01:59,270

 

1921
01:01:59,270 --> 01:02:00,950

 

1922
01:02:00,950 --> 01:02:00,960

 

1923
01:02:00,960 --> 01:02:02,230


1924
01:02:02,230 --> 01:02:04,630

 

1925
01:02:04,630 --> 01:02:09,109

 

1926
01:02:09,109 --> 01:02:10,230

 

1927
01:02:10,230 --> 01:02:11,829

 

1928
01:02:11,829 --> 01:02:13,430

 

1929
01:02:13,430 --> 01:02:15,349

 

1930
01:02:15,349 --> 01:02:17,029

 

1931
01:02:17,029 --> 01:02:19,109

 

1932
01:02:19,109 --> 01:02:20,470

 

1933
01:02:20,470 --> 01:02:22,470

 

1934
01:02:22,470 --> 01:02:24,549

 

1935
01:02:24,549 --> 01:02:25,230

 

1936
01:02:25,230 --> 01:02:27,349

 

1937
01:02:27,349 --> 01:02:30,150

 

1938
01:02:30,150 --> 01:02:32,549

 

1939
01:02:32,549 --> 01:02:35,430

 

1940
01:02:35,430 --> 01:02:38,309

 

1941
01:02:38,309 --> 01:02:42,069

 

1942
01:02:42,069 --> 01:02:43,670

 

1943
01:02:43,670 --> 01:02:46,390

 

1944
01:02:46,390 --> 01:02:48,789

 

1945
01:02:48,789 --> 01:02:52,789

 

1946
01:02:52,789 --> 01:02:54,630

 

1947
01:02:54,630 --> 01:02:56,789

 

1948
01:02:56,789 --> 01:03:00,069

 

1949
01:03:00,069 --> 01:03:03,670

 

1950
01:03:03,670 --> 01:03:05,349

 

1951
01:03:05,349 --> 01:03:07,029

 

1952
01:03:07,029 --> 01:03:08,069

 

1953
01:03:08,069 --> 01:03:09,910

 

1954
01:03:09,910 --> 01:03:12,390

 

1955
01:03:12,390 --> 01:03:14,309

 

1956
01:03:14,309 --> 01:03:17,190

 

1957
01:03:17,190 --> 01:03:18,150

 

1958
01:03:18,150 --> 01:03:18,160

 

1959
01:03:18,160 --> 01:03:19,270


1960
01:03:19,270 --> 01:03:21,829

 

1961
01:03:21,829 --> 01:03:23,430

 

1962
01:03:23,430 --> 01:03:25,190

 

1963
01:03:25,190 --> 01:03:27,430

 

1964
01:03:27,430 --> 01:03:30,950

 

1965
01:03:30,950 --> 01:03:33,750

 

1966
01:03:33,750 --> 01:03:35,910

 

1967
01:03:35,910 --> 01:03:38,549

 

1968
01:03:38,549 --> 01:03:40,069

 

1969
01:03:40,069 --> 01:03:40,079

 

1970
01:03:40,079 --> 01:03:40,950


1971
01:03:40,950 --> 01:03:43,109

 

1972
01:03:43,109 --> 01:03:44,870

 

1973
01:03:44,870 --> 01:03:45,910

 

1974
01:03:45,910 --> 01:03:48,390

 

1975
01:03:48,390 --> 01:03:50,230

 

1976
01:03:50,230 --> 01:03:50,240

 

1977
01:03:50,240 --> 01:03:51,589


1978
01:03:51,589 --> 01:03:55,430

 

1979
01:03:55,430 --> 01:03:58,230

 

1980
01:03:58,230 --> 01:03:59,670

 

1981
01:03:59,670 --> 01:04:02,230

 

1982
01:04:02,230 --> 01:04:02,240

 

1983
01:04:02,240 --> 01:04:06,470


1984
01:04:06,470 --> 01:04:06,480

 

1985
01:04:06,480 --> 01:04:07,270


1986
01:04:07,270 --> 01:04:09,190

 

1987
01:04:09,190 --> 01:04:11,910

 

1988
01:04:11,910 --> 01:04:14,710

 

1989
01:04:14,710 --> 01:04:17,029

 

1990
01:04:17,029 --> 01:04:20,230

 

1991
01:04:20,230 --> 01:04:22,150

 

1992
01:04:22,150 --> 01:04:24,710

 

1993
01:04:24,710 --> 01:04:24,720

 

1994
01:04:24,720 --> 01:04:25,510


1995
01:04:25,510 --> 01:04:27,029

 

1996
01:04:27,029 --> 01:04:28,710

 

1997
01:04:28,710 --> 01:04:31,190

 

1998
01:04:31,190 --> 01:04:31,200

 

1999
01:04:31,200 --> 01:04:32,230


2000
01:04:32,230 --> 01:04:33,829

 

2001
01:04:33,829 --> 01:04:33,839

 

2002
01:04:33,839 --> 01:04:34,870


2003
01:04:34,870 --> 01:04:36,390

 

2004
01:04:36,390 --> 01:04:38,829

 

2005
01:04:38,829 --> 01:04:38,839

 

2006
01:04:38,839 --> 01:04:40,470


2007
01:04:40,470 --> 01:04:40,480

 

2008
01:04:40,480 --> 01:04:41,349


2009
01:04:41,349 --> 01:04:44,230

 

2010
01:04:44,230 --> 01:04:45,990

 

2011
01:04:45,990 --> 01:04:48,309

 

2012
01:04:48,309 --> 01:04:50,470

 

2013
01:04:50,470 --> 01:04:52,309

 

2014
01:04:52,309 --> 01:04:53,670

 

2015
01:04:53,670 --> 01:04:55,430

 

2016
01:04:55,430 --> 01:04:57,190

 

2017
01:04:57,190 --> 01:04:59,990

 

2018
01:04:59,990 --> 01:05:03,190

 

2019
01:05:03,190 --> 01:05:04,950

 

2020
01:05:04,950 --> 01:05:07,510

 

2021
01:05:07,510 --> 01:05:10,630

 

2022
01:05:10,630 --> 01:05:14,309

 

2023
01:05:14,309 --> 01:05:15,829

 

2024
01:05:15,829 --> 01:05:15,839

 

2025
01:05:15,839 --> 01:05:16,950


2026
01:05:16,950 --> 01:05:19,430

 

2027
01:05:19,430 --> 01:05:21,270

 

2028
01:05:21,270 --> 01:05:24,150

 

2029
01:05:24,150 --> 01:05:27,349

 

2030
01:05:27,349 --> 01:05:29,190

 

2031
01:05:29,190 --> 01:05:30,870

 

2032
01:05:30,870 --> 01:05:33,190

 

2033
01:05:33,190 --> 01:05:34,789

 

2034
01:05:34,789 --> 01:05:36,470

 

2035
01:05:36,470 --> 01:05:38,549

 

2036
01:05:38,549 --> 01:05:42,789

 

2037
01:05:42,789 --> 01:05:42,799

 

2038
01:05:42,799 --> 01:05:43,589


2039
01:05:43,589 --> 01:05:45,990

 

2040
01:05:45,990 --> 01:05:48,309

 

2041
01:05:48,309 --> 01:05:49,990

 

2042
01:05:49,990 --> 01:05:52,390

 

2043
01:05:52,390 --> 01:05:53,589

 

2044
01:05:53,589 --> 01:05:55,270

 

2045
01:05:55,270 --> 01:05:57,589

 

2046
01:05:57,589 --> 01:05:59,510

 

2047
01:05:59,510 --> 01:05:59,520

 

2048
01:05:59,520 --> 01:06:00,630


2049
01:06:00,630 --> 01:06:03,990

 

2050
01:06:03,990 --> 01:06:05,829

 

2051
01:06:05,829 --> 01:06:07,750

 

2052
01:06:07,750 --> 01:06:10,870

 

2053
01:06:10,870 --> 01:06:12,230

 

2054
01:06:12,230 --> 01:06:12,240

 

2055
01:06:12,240 --> 01:06:13,829


2056
01:06:13,829 --> 01:06:13,839

 

2057
01:06:13,839 --> 01:06:14,950


2058
01:06:14,950 --> 01:06:17,589

 

2059
01:06:17,589 --> 01:06:18,950

 

2060
01:06:18,950 --> 01:06:21,029

 

2061
01:06:21,029 --> 01:06:22,950

 

2062
01:06:22,950 --> 01:06:24,470

 

2063
01:06:24,470 --> 01:06:26,630

 

2064
01:06:26,630 --> 01:06:28,309

 

2065
01:06:28,309 --> 01:06:30,150

 

2066
01:06:30,150 --> 01:06:33,270

 

2067
01:06:33,270 --> 01:06:35,109

 

2068
01:06:35,109 --> 01:06:36,789

 

2069
01:06:36,789 --> 01:06:39,670

 

2070
01:06:39,670 --> 01:06:41,029

 

2071
01:06:41,029 --> 01:06:42,870

 

2072
01:06:42,870 --> 01:06:44,630

 

2073
01:06:44,630 --> 01:06:47,750

 

2074
01:06:47,750 --> 01:06:49,910

 

2075
01:06:49,910 --> 01:06:51,349

 

2076
01:06:51,349 --> 01:06:51,359

 

2077
01:06:51,359 --> 01:06:52,470


2078
01:06:52,470 --> 01:06:54,150

 

2079
01:06:54,150 --> 01:06:55,589

 

2080
01:06:55,589 --> 01:06:57,109

 

2081
01:06:57,109 --> 01:06:58,829

 

2082
01:06:58,829 --> 01:07:00,789

 

2083
01:07:00,789 --> 01:07:03,109

 

2084
01:07:03,109 --> 01:07:07,190

 

2085
01:07:07,190 --> 01:07:08,870

 

2086
01:07:08,870 --> 01:07:11,670

 

2087
01:07:11,670 --> 01:07:13,510

 

2088
01:07:13,510 --> 01:07:15,750

 

2089
01:07:15,750 --> 01:07:17,910

 

2090
01:07:17,910 --> 01:07:20,230

 

2091
01:07:20,230 --> 01:07:21,430

 

2092
01:07:21,430 --> 01:07:22,710

 

2093
01:07:22,710 --> 01:07:25,430

 

2094
01:07:25,430 --> 01:07:28,549

 

2095
01:07:28,549 --> 01:07:30,230

 

2096
01:07:30,230 --> 01:07:32,390

 

2097
01:07:32,390 --> 01:07:34,309

 

2098
01:07:34,309 --> 01:07:36,549

 

2099
01:07:36,549 --> 01:07:38,309

 

2100
01:07:38,309 --> 01:07:40,230

 

2101
01:07:40,230 --> 01:07:41,589

 

2102
01:07:41,589 --> 01:07:43,430

 

2103
01:07:43,430 --> 01:07:44,710

 

2104
01:07:44,710 --> 01:07:46,150

 

2105
01:07:46,150 --> 01:07:48,150

 

2106
01:07:48,150 --> 01:07:49,430

 

2107
01:07:49,430 --> 01:07:51,190

 

2108
01:07:51,190 --> 01:07:53,270

 

2109
01:07:53,270 --> 01:07:56,230

 

2110
01:07:56,230 --> 01:07:58,150

 

2111
01:07:58,150 --> 01:08:00,069

 

2112
01:08:00,069 --> 01:08:01,910

 

2113
01:08:01,910 --> 01:08:03,589

 

2114
01:08:03,589 --> 01:08:06,069

 

2115
01:08:06,069 --> 01:08:07,829

 

2116
01:08:07,829 --> 01:08:11,270

 

2117
01:08:11,270 --> 01:08:11,280

 

2118
01:08:11,280 --> 01:08:12,150


2119
01:08:12,150 --> 01:08:12,160

 

2120
01:08:12,160 --> 01:08:13,349


2121
01:08:13,349 --> 01:08:13,359

 

2122
01:08:13,359 --> 01:08:14,390


2123
01:08:14,390 --> 01:08:14,400

 

2124
01:08:14,400 --> 01:08:16,070


2125
01:08:16,070 --> 01:08:17,990

 

2126
01:08:17,990 --> 01:08:21,189

 

2127
01:08:21,189 --> 01:08:23,510

 

2128
01:08:23,510 --> 01:08:25,510

 

2129
01:08:25,510 --> 01:08:26,950

 

2130
01:08:26,950 --> 01:08:30,309

 

2131
01:08:30,309 --> 01:08:32,390

 

2132
01:08:32,390 --> 01:08:34,390

 

2133
01:08:34,390 --> 01:08:37,669

 

2134
01:08:37,669 --> 01:08:39,349

 

2135
01:08:39,349 --> 01:08:39,359

 

2136
01:08:39,359 --> 01:08:41,590


2137
01:08:41,590 --> 01:08:42,950

 

2138
01:08:42,950 --> 01:08:45,110

 

2139
01:08:45,110 --> 01:08:47,910

 

2140
01:08:47,910 --> 01:08:47,920

 

2141
01:08:47,920 --> 01:08:49,030


2142
01:08:49,030 --> 01:08:50,309

 

2143
01:08:50,309 --> 01:08:51,749

 

2144
01:08:51,749 --> 01:08:51,759

 

2145
01:08:51,759 --> 01:08:52,460


2146
01:08:52,460 --> 01:08:52,470

 

2147
01:08:52,470 --> 01:08:55,110


2148
01:08:55,110 --> 01:08:57,430

 

2149
01:08:57,430 --> 01:09:01,829

 

2150
01:09:01,829 --> 01:09:03,749

 

2151
01:09:03,749 --> 01:09:06,789

 

2152
01:09:06,789 --> 01:09:06,799

 

2153
01:09:06,799 --> 01:09:09,430


2154
01:09:09,430 --> 01:09:11,590

 

2155
01:09:11,590 --> 01:09:11,600

 

2156
01:09:11,600 --> 01:09:12,870


2157
01:09:12,870 --> 01:09:16,470

 

2158
01:09:16,470 --> 01:09:18,390

 

2159
01:09:18,390 --> 01:09:19,910

 

2160
01:09:19,910 --> 01:09:21,990

 

2161
01:09:21,990 --> 01:09:24,149

 

2162
01:09:24,149 --> 01:09:26,550

 

2163
01:09:26,550 --> 01:09:28,390

 

2164
01:09:28,390 --> 01:09:30,309

 

2165
01:09:30,309 --> 01:09:32,070

 

2166
01:09:32,070 --> 01:09:38,870

 

2167
01:09:38,870 --> 01:09:42,550

 

2168
01:09:42,550 --> 01:09:44,870

 

2169
01:09:44,870 --> 01:09:47,269

 

2170
01:09:47,269 --> 01:09:50,870

 

2171
01:09:50,870 --> 01:09:53,910

 

2172
01:09:53,910 --> 01:09:56,390

 

2173
01:09:56,390 --> 01:09:57,750

 

2174
01:09:57,750 --> 01:09:59,510

 

2175
01:09:59,510 --> 01:10:01,910

 

2176
01:10:01,910 --> 01:10:01,920

 

2177
01:10:01,920 --> 01:10:03,669


2178
01:10:03,669 --> 01:10:03,679

 

2179
01:10:03,679 --> 01:10:04,790


2180
01:10:04,790 --> 01:10:07,030

 

2181
01:10:07,030 --> 01:10:09,750

 

2182
01:10:09,750 --> 01:10:11,510

 

2183
01:10:11,510 --> 01:10:12,950

 

2184
01:10:12,950 --> 01:10:14,630

 

2185
01:10:14,630 --> 01:10:17,030

 

2186
01:10:17,030 --> 01:10:18,550

 

2187
01:10:18,550 --> 01:10:19,669

 

2188
01:10:19,669 --> 01:10:21,189

 

2189
01:10:21,189 --> 01:10:23,430

 

2190
01:10:23,430 --> 01:10:23,440

 

2191
01:10:23,440 --> 01:10:24,870


2192
01:10:24,870 --> 01:10:27,590

 

2193
01:10:27,590 --> 01:10:29,350

 

2194
01:10:29,350 --> 01:10:31,350

 

2195
01:10:31,350 --> 01:10:33,430

 

2196
01:10:33,430 --> 01:10:34,870

 

2197
01:10:34,870 --> 01:10:36,310

 

2198
01:10:36,310 --> 01:10:39,189

 

2199
01:10:39,189 --> 01:10:41,510

 

2200
01:10:41,510 --> 01:10:44,790

 

2201
01:10:44,790 --> 01:10:44,800

 

2202
01:10:44,800 --> 01:10:45,990


2203
01:10:45,990 --> 01:10:47,669

 

2204
01:10:47,669 --> 01:10:50,390

 

2205
01:10:50,390 --> 01:10:50,400

 

2206
01:10:50,400 --> 01:10:52,390


2207
01:10:52,390 --> 01:10:52,400

 

2208
01:10:52,400 --> 01:10:54,229


2209
01:10:54,229 --> 01:10:56,630

 

2210
01:10:56,630 --> 01:10:58,390

 

2211
01:10:58,390 --> 01:10:59,910

 

2212
01:10:59,910 --> 01:11:01,669

 

2213
01:11:01,669 --> 01:11:03,270

 

2214
01:11:03,270 --> 01:11:04,310

 

2215
01:11:04,310 --> 01:11:06,630

 

2216
01:11:06,630 --> 01:11:10,149

 

2217
01:11:10,149 --> 01:11:11,430

 

2218
01:11:11,430 --> 01:11:13,830

 

2219
01:11:13,830 --> 01:11:15,510

 

2220
01:11:15,510 --> 01:11:19,590

 

2221
01:11:19,590 --> 01:11:23,110

 

2222
01:11:23,110 --> 01:11:26,470

 

2223
01:11:26,470 --> 01:11:28,390

 

2224
01:11:28,390 --> 01:11:30,470

 

2225
01:11:30,470 --> 01:11:32,070

 

2226
01:11:32,070 --> 01:11:33,110

 

2227
01:11:33,110 --> 01:11:35,910

 

2228
01:11:35,910 --> 01:11:40,149

 

2229
01:11:40,149 --> 01:11:43,430

 

2230
01:11:43,430 --> 01:11:47,189

 

2231
01:11:47,189 --> 01:11:47,199

 

2232
01:11:47,199 --> 01:11:48,229


2233
01:11:48,229 --> 01:11:52,630

 

2234
01:11:52,630 --> 01:11:53,990

 

2235
01:11:53,990 --> 01:11:55,750

 

2236
01:11:55,750 --> 01:11:58,950

 

2237
01:11:58,950 --> 01:12:02,550

 

2238
01:12:02,550 --> 01:12:03,669

 

2239
01:12:03,669 --> 01:12:03,679

 

2240
01:12:03,679 --> 01:12:05,669


2241
01:12:05,669 --> 01:12:07,669

 

2242
01:12:07,669 --> 01:12:07,679

 

2243
01:12:07,679 --> 01:12:09,750


2244
01:12:09,750 --> 01:12:11,270

 

2245
01:12:11,270 --> 01:12:13,830

 

2246
01:12:13,830 --> 01:12:15,590

 

2247
01:12:15,590 --> 01:12:18,149

 

2248
01:12:18,149 --> 01:12:20,790

 

2249
01:12:20,790 --> 01:12:22,950

 

2250
01:12:22,950 --> 01:12:24,870

 

2251
01:12:24,870 --> 01:12:24,880

 

2252
01:12:24,880 --> 01:12:26,550


2253
01:12:26,550 --> 01:12:27,910

 

2254
01:12:27,910 --> 01:12:31,030

 

2255
01:12:31,030 --> 01:12:33,750

 

2256
01:12:33,750 --> 01:12:36,630

 

2257
01:12:36,630 --> 01:12:38,390

 

2258
01:12:38,390 --> 01:12:40,229

 

2259
01:12:40,229 --> 01:12:41,990

 

2260
01:12:41,990 --> 01:12:44,950

 

2261
01:12:44,950 --> 01:12:46,630

 

2262
01:12:46,630 --> 01:12:48,709

 

2263
01:12:48,709 --> 01:12:50,870

 

2264
01:12:50,870 --> 01:12:52,550

 

2265
01:12:52,550 --> 01:12:53,990

 

2266
01:12:53,990 --> 01:12:55,590

 

2267
01:12:55,590 --> 01:12:57,669

 

2268
01:12:57,669 --> 01:13:00,149

 

2269
01:13:00,149 --> 01:13:00,159

 

2270
01:13:00,159 --> 01:13:01,110


2271
01:13:01,110 --> 01:13:03,030

 

2272
01:13:03,030 --> 01:13:04,630

 

2273
01:13:04,630 --> 01:13:06,229

 

2274
01:13:06,229 --> 01:13:08,550

 

2275
01:13:08,550 --> 01:13:08,560

 

2276
01:13:08,560 --> 01:13:09,669


2277
01:13:09,669 --> 01:13:12,709

 

2278
01:13:12,709 --> 01:13:14,870

 

2279
01:13:14,870 --> 01:13:17,669

 

2280
01:13:17,669 --> 01:13:19,750

 

2281
01:13:19,750 --> 01:13:22,550

 

2282
01:13:22,550 --> 01:13:24,229

 

2283
01:13:24,229 --> 01:13:27,110

 

2284
01:13:27,110 --> 01:13:28,390

 

2285
01:13:28,390 --> 01:13:30,149

 

2286
01:13:30,149 --> 01:13:31,830

 

2287
01:13:31,830 --> 01:13:34,149

 

2288
01:13:34,149 --> 01:13:35,990

 

2289
01:13:35,990 --> 01:13:37,830

 

2290
01:13:37,830 --> 01:13:39,669

 

2291
01:13:39,669 --> 01:13:41,669

 

2292
01:13:41,669 --> 01:13:44,310

 

2293
01:13:44,310 --> 01:13:46,630

 

2294
01:13:46,630 --> 01:13:49,030

 

2295
01:13:49,030 --> 01:13:50,550

 

2296
01:13:50,550 --> 01:13:52,390

 

2297
01:13:52,390 --> 01:13:54,229

 

2298
01:13:54,229 --> 01:13:56,390

 

2299
01:13:56,390 --> 01:13:58,229

 

2300
01:13:58,229 --> 01:14:00,470

 

2301
01:14:00,470 --> 01:14:00,480

 

2302
01:14:00,480 --> 01:14:01,270


2303
01:14:01,270 --> 01:14:02,950

 

2304
01:14:02,950 --> 01:14:02,960

 

2305
01:14:02,960 --> 01:14:03,830


2306
01:14:03,830 --> 01:14:06,870

 

2307
01:14:06,870 --> 01:14:09,669

 

2308
01:14:09,669 --> 01:14:09,679

 

2309
01:14:09,679 --> 01:14:13,110


2310
01:14:13,110 --> 01:14:15,830

 

2311
01:14:15,830 --> 01:14:17,430

 

2312
01:14:17,430 --> 01:14:21,350

 

2313
01:14:21,350 --> 01:14:23,830

 

2314
01:14:23,830 --> 01:14:26,950

 

2315
01:14:26,950 --> 01:14:28,229

 

2316
01:14:28,229 --> 01:14:29,830

 

2317
01:14:29,830 --> 01:14:32,229

 

2318
01:14:32,229 --> 01:14:34,950

 

2319
01:14:34,950 --> 01:14:34,960

 

2320
01:14:34,960 --> 01:14:36,149


2321
01:14:36,149 --> 01:14:36,159

 

2322
01:14:36,159 --> 01:14:38,709


2323
01:14:38,709 --> 01:14:42,470

 

2324
01:14:42,470 --> 01:14:42,480

 

2325
01:14:42,480 --> 01:14:45,430


2326
01:14:45,430 --> 01:14:47,590

 

2327
01:14:47,590 --> 01:14:47,600

 

2328
01:14:47,600 --> 01:14:48,550


2329
01:14:48,550 --> 01:14:50,630

 

2330
01:14:50,630 --> 01:14:53,669

 

2331
01:14:53,669 --> 01:14:55,030

 

2332
01:14:55,030 --> 01:14:55,040

 

2333
01:14:55,040 --> 01:14:56,390


2334
01:14:56,390 --> 01:14:59,270

 

2335
01:14:59,270 --> 01:14:59,280

 

2336
01:14:59,280 --> 01:15:00,790


2337
01:15:00,790 --> 01:15:02,870

 

2338
01:15:02,870 --> 01:15:04,550

 

2339
01:15:04,550 --> 01:15:06,950

 

2340
01:15:06,950 --> 01:15:06,960

 

2341
01:15:06,960 --> 01:15:07,750


2342
01:15:07,750 --> 01:15:09,990

 

2343
01:15:09,990 --> 01:15:12,149

 

2344
01:15:12,149 --> 01:15:13,189

 

2345
01:15:13,189 --> 01:15:13,199

 

2346
01:15:13,199 --> 01:15:14,390


2347
01:15:14,390 --> 01:15:16,950

 

2348
01:15:16,950 --> 01:15:20,550

 

2349
01:15:20,550 --> 01:15:22,310

 

2350
01:15:22,310 --> 01:15:23,910

 

2351
01:15:23,910 --> 01:15:27,189

 

2352
01:15:27,189 --> 01:15:27,199

 

2353
01:15:27,199 --> 01:15:28,950


2354
01:15:28,950 --> 01:15:30,950

 

2355
01:15:30,950 --> 01:15:33,270

 

2356
01:15:33,270 --> 01:15:36,830

 

2357
01:15:36,830 --> 01:15:39,750

 

2358
01:15:39,750 --> 01:15:39,760

 

2359
01:15:39,760 --> 01:15:40,950


2360
01:15:40,950 --> 01:15:44,870

 

2361
01:15:44,870 --> 01:15:44,880

 

2362
01:15:44,880 --> 01:15:45,990


2363
01:15:45,990 --> 01:15:46,000

 

2364
01:15:46,000 --> 01:15:47,669


2365
01:15:47,669 --> 01:15:50,229

 

2366
01:15:50,229 --> 01:15:51,669

 

2367
01:15:51,669 --> 01:15:53,189

 

2368
01:15:53,189 --> 01:15:53,199

 

2369
01:15:53,199 --> 01:15:54,149


2370
01:15:54,149 --> 01:15:56,390

 

2371
01:15:56,390 --> 01:15:58,310

 

2372
01:15:58,310 --> 01:15:59,830

 

2373
01:15:59,830 --> 01:16:02,310

 

2374
01:16:02,310 --> 01:16:02,320

 

2375
01:16:02,320 --> 01:16:03,669


2376
01:16:03,669 --> 01:16:06,070

 

2377
01:16:06,070 --> 01:16:07,750

 

2378
01:16:07,750 --> 01:16:09,590

 

2379
01:16:09,590 --> 01:16:11,510

 

2380
01:16:11,510 --> 01:16:13,189

 

2381
01:16:13,189 --> 01:16:15,350

 

2382
01:16:15,350 --> 01:16:15,360

 

2383
01:16:15,360 --> 01:16:16,310


2384
01:16:16,310 --> 01:16:18,390

 

2385
01:16:18,390 --> 01:16:20,709

 

2386
01:16:20,709 --> 01:16:22,790

 

2387
01:16:22,790 --> 01:16:25,750

 

2388
01:16:25,750 --> 01:16:26,790

 

2389
01:16:26,790 --> 01:16:28,550

 

2390
01:16:28,550 --> 01:16:28,560

 

2391
01:16:28,560 --> 01:16:29,590


2392
01:16:29,590 --> 01:16:31,830

 

2393
01:16:31,830 --> 01:16:33,910

 

2394
01:16:33,910 --> 01:16:36,070

 

2395
01:16:36,070 --> 01:16:38,149

 

2396
01:16:38,149 --> 01:16:40,149

 

2397
01:16:40,149 --> 01:16:40,159

 

2398
01:16:40,159 --> 01:16:41,510


2399
01:16:41,510 --> 01:16:44,229

 

2400
01:16:44,229 --> 01:16:47,030

 

2401
01:16:47,030 --> 01:16:48,390

 

2402
01:16:48,390 --> 01:16:50,709

 

2403
01:16:50,709 --> 01:16:52,709

 

2404
01:16:52,709 --> 01:16:55,350

 

2405
01:16:55,350 --> 01:16:57,030

 

2406
01:16:57,030 --> 01:16:57,040

 

2407
01:16:57,040 --> 01:16:58,149


2408
01:16:58,149 --> 01:17:01,189

 

2409
01:17:01,189 --> 01:17:02,390

 

2410
01:17:02,390 --> 01:17:04,149

 

2411
01:17:04,149 --> 01:17:06,550

 

2412
01:17:06,550 --> 01:17:09,510

 

2413
01:17:09,510 --> 01:17:11,430

 

2414
01:17:11,430 --> 01:17:14,149

 

2415
01:17:14,149 --> 01:17:15,830

 

2416
01:17:15,830 --> 01:17:21,030

 

2417
01:17:21,030 --> 01:17:23,590

 

2418
01:17:23,590 --> 01:17:26,550

 

2419
01:17:26,550 --> 01:17:29,669

 

2420
01:17:29,669 --> 01:17:32,229

 

2421
01:17:32,229 --> 01:17:33,990

 

2422
01:17:33,990 --> 01:17:35,590

 

2423
01:17:35,590 --> 01:17:37,430

 

2424
01:17:37,430 --> 01:17:39,110

 

2425
01:17:39,110 --> 01:17:41,270

 

2426
01:17:41,270 --> 01:17:43,110

 

2427
01:17:43,110 --> 01:17:46,229

 

2428
01:17:46,229 --> 01:17:46,239

 

2429
01:17:46,239 --> 01:17:47,590


2430
01:17:47,590 --> 01:17:50,229

 

2431
01:17:50,229 --> 01:17:52,870

 

2432
01:17:52,870 --> 01:17:52,880

 

2433
01:17:52,880 --> 01:17:54,149


2434
01:17:54,149 --> 01:17:56,390

 

2435
01:17:56,390 --> 01:17:58,470

 

2436
01:17:58,470 --> 01:17:58,480

 

2437
01:17:58,480 --> 01:18:02,390


2438
01:18:02,390 --> 01:18:05,430

 

2439
01:18:05,430 --> 01:18:09,590

 

2440
01:18:09,590 --> 01:18:11,830

 

2441
01:18:11,830 --> 01:18:14,070

 

2442
01:18:14,070 --> 01:18:15,910

 

2443
01:18:15,910 --> 01:18:18,149

 

2444
01:18:18,149 --> 01:18:18,159

 

2445
01:18:18,159 --> 01:18:19,350


2446
01:18:19,350 --> 01:18:22,229

 

2447
01:18:22,229 --> 01:18:23,669

 

2448
01:18:23,669 --> 01:18:25,030

 

2449
01:18:25,030 --> 01:18:26,709

 

2450
01:18:26,709 --> 01:18:27,590

 

2451
01:18:27,590 --> 01:18:29,590

 

2452
01:18:29,590 --> 01:18:30,790

 

2453
01:18:30,790 --> 01:18:32,550

 

2454
01:18:32,550 --> 01:18:34,149

 

2455
01:18:34,149 --> 01:18:36,390

 

2456
01:18:36,390 --> 01:18:38,630

 

2457
01:18:38,630 --> 01:18:40,310

 

2458
01:18:40,310 --> 01:18:43,990

 

2459
01:18:43,990 --> 01:18:45,510

 

2460
01:18:45,510 --> 01:18:48,390

 

2461
01:18:48,390 --> 01:18:49,910

 

2462
01:18:49,910 --> 01:18:53,430

 

2463
01:18:53,430 --> 01:18:56,149

 

2464
01:18:56,149 --> 01:18:58,550

 

2465
01:18:58,550 --> 01:18:58,560

 

2466
01:18:58,560 --> 01:19:01,669


2467
01:19:01,669 --> 01:19:03,590

 

2468
01:19:03,590 --> 01:19:05,750

 

2469
01:19:05,750 --> 01:19:08,790

 

2470
01:19:08,790 --> 01:19:10,630

 

2471
01:19:10,630 --> 01:19:12,229

 

2472
01:19:12,229 --> 01:19:13,430

 

2473
01:19:13,430 --> 01:19:13,440

 

2474
01:19:13,440 --> 01:19:14,950


2475
01:19:14,950 --> 01:19:17,350

 

2476
01:19:17,350 --> 01:19:19,910

 

2477
01:19:19,910 --> 01:19:22,790

 

2478
01:19:22,790 --> 01:19:24,229

 

2479
01:19:24,229 --> 01:19:37,189

 

2480
01:19:37,189 --> 01:19:37,199

 

2481
01:19:37,199 --> 01:19:39,280