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hi everybody  can you hear me  yes good  um  welcome back uh  to um  to the course and um  so we are now at lecture 10 and um uh  i  let's see what have we been doing we've  been talking about um  undecidability  um so we  introduced last time  uh reducibilities and mapping  reducibilities for proving  various problems are undecidable  and  this time we're going to and today we're  going to introduce a a  more sophisticated method for proving  undecidability uh using reduceabilities  and that's called the computation  history method and and that's pretty  much  what's  used in all of the  you know more  serious cases where people have proved  problems undecidable it's it's pretty  much a widespread uh  uh method for uh especially you know  it's not an active area of research  these days but at times when people were  proving problems undecidable  this was frequently a method that was  used  um so we'll go over that today and we'll  prove um  a few examples of problems undecidable  using that method  all right so why don't we get started um  uh so first of all  most importantly  you should remember how we you how we  prove um problems are undecidable in the  first place  and that is by showing  um some other undecidable problem that  we already know is undecidable is  reducible to the problem of interest  so typically you might be reducing atm  um which is a problem that we started  with and showed to be undecidable by the  diagonalization method  um or might be some other problem that  we've subsequently shown to be  undecidable and you take that problem  and you reduce it to  the problem you're trying to show  undecidable  okay oops  let me fix that  sorry  um  so um  so this is the thing to keep in mind to  prove a language is undecidable show  that atm or some other known undecidable  language is reducible to the problem b  that you have at hand that you're trying  to show undecidable  all right so we are now going to  um  um  [Music]  first i'm going to start off by  remembering um hilbert's 10th problem  which we discussed briefly a few  lectures back um  and as you may remember  so that's a problem where uh you're  given  say some polynomial  you know over several variables  you know like three x squared y minus  15z  plus  2xz  equals 5.  and you want to solve that problem  you want to find a solution  that solves that equation  but you require that the solution  involves only integers  so those are so called diaphantine  problems because the requirement is that  the solution being integers  and  hilbert's problem was to  ask for a  an algorithm  not the language that he used but  doesn't matter he essentially he asked  for an algorithm to test whether a  polynomial has a solution in integers or  the polynomial equation has a solution  in integers  and um  as  uh  um  uh  um as we now know that problem was posed  back in 1900  it took 71 years to find the solution  but the answer is no  there is no there is no such algorithm  we talked about this when we were  discussing the church turing thesis  um  and um  the method  showing  that there is no algorithm is by a  reduction from atm to that problem  so one can use exactly the method that  we're using today to prove that problem  is undecidable  about polynomials the only thing is is  that the reduction that single reduction  would take the entire semester  um  and i know that's  in fact correct because when i was a  graduate student um there was a woman in  the math department at berkeley where i  where i studied and she uh the whole the  whole course  was to go through the proof of hilbert's  tenth problem  um of the solution to hillary problem  the the proof of the undecidability of  the uh of the solution  uh of the undecidability of testing  whether a polynomial has an integral  solution um  so  um  the thing is that involves some  fairly hairy number theory in order to  um  to come up with the right polynomials to  basically simulate the turing machine  which so that's kind of getting ahead of  ourselves that's what we're going to be  doing today um but we're going to  instead of looking at that problem  because we don't have a whole semester  to spend on it um we're going to look at  a toy problem instead  where there's no number theory but it  still has the same basic underlying idea  um to that that was employed in um the  solution to hilbert's tenth problem  um so that toy problem is called the  post correspondence problem  and it's going to have some other  utility for us  well  the main reason we're studying it is  just to illustrate the method um  uh  so  and the method is going to be as i've  been saying is this computation history  method  okay so one we will first talk about uh  this post-correspondence problem because  that's very easy to understand  and then we'll spend a little time  introducing that method  um  all right  so um  the  uh the post the post correspondence  problem is  as follows you're given  uh  a bunch of pairs of strings  um so here is  one pair t1 and b1  and i'm writing them as  dominoes i'm calling them dominoes  because i'm writing one pair one one of  the strings on top of the other string  in fact i'm  using t and b for top and bottom here so  this is t1 is the top string b1 is the  bottom string in the first domino  and then in the second domino we have  two other strings to a top and a bottom  and so on  um  so we have these here  uh this collection of dominoes  um  as  sort of the legal dominoes um and what  the goal is to find a sequence of those  dominoes um  uh  so you want to pick from these dominoes  here you're allowed to repeat dominoes  so you want to pick a sequence of those  dominoes  line them up next to each other  and so that the string that you get by  reading along the top all of the t's  together  is going to be exactly the same as the  string you get by reading along all the  bottoms  i'll give you an example  um so here to write it down a little bit  more formally  so uh what i'm calling a match  is a sequence of these dominoes so it's  t i1 ti2  as well it's it's i1 is the first domino  i2 is the second domino and so on  and then you what you're going to have  is  all the top strings concatenated  concatenated together is exactly the  same as what you get by concatenating  together all the bottom strings  um so here's an example and i think  it'll become clear if you didn't  understand it so far but it'll be  completely clear from the example i hope  so here is a bunch of dominoes  um four  um  and um  what i wanna know is is there some  selection of these dominoes again you  can repeat dominoes otherwise it would  be not an interesting problem so you can  repeat these as many times as you like  and i want to pick up pick these  dominoes and line them up so that what  you get by reading along the top is the  same as what you get by reading along  the bottom  so first of all if you just stare at  this  and you're trying to make a match  you'll see that there's only one  possibility possible way to start this  um you know  for example if you started with the  third domino here b a over a a  it would fail immediately because  um you know the b and the a are  different  um and and if you you use the second  domino as your starting point that would  fail the match as well  because a a has to would have to e um be  the the the top string would start with  a a the bottom string would start with a  b and they could never be equal  um so by looking at this you can see  that the only possible choice that you  have is to start with the first domino  okay so i'm going to start to build the  domino for you just so you can see it so  here's the very the the match as um so  you can see um the process so the match  starts with the very first domino and  the way i'm going to write it is i'm  going to take the dominoes and kind of  skew them  so that you can see how the top and  bottom strings are lining up  um  but they're just mouth the same dominoes  they've just been  i changed the shape  so this is the first domino a b over aba  now uh the second domino in my and my  match it's gotta start with an a  as the leftmost symbol on the top string  so that would rule out this one for  example um but there might be several  choices and so it's not exactly obvious  which one you should take  what i'm going to suggest is we take  this one here  so we take a a over a b a you see it a a  over a b a it's just  you know written i'm just skewing uh  skewing it so that um i can line up the  a over a the b over b the a over a and  so on so i'm starting to build  this concatenation of the top being  equal to the concatenation of the bottom  following me so far i hope  so now um what's what's next so i need  to have something where  the top string is going to start with a  ba  in fact there's only one choice for that  so it's clearly going to have to be this  domino is going to be next so it's going  to put a ba up here but then it's going  to a force an a a down below because  that's what this domino does um  so this ba matches with that now we have  the aaa  to deal with um  so  we're going to reuse this domino because  that's the only one that starts with an  aa so we're going to reuse that one  now we have a a and aba now we need  something that starts with aba  i see two choices  it could be this one because this is  consistent with a it's at least it  captures the a b we could we could try  putting this one over here but a better  choice would it be would be to use this  one because if we use this one we finish  the match  and therefore we have found what we're  looking for  um and this  this  this collection of  dominoes here has a mat has a match  now it's not always obvious  some some collections of dominoes may  not have a match  and so the computational problem is  is there a match  um  and  the theorem that we're going to prove  today is that this problem is  undecidable  simple as it is simple little  combinatorial problem of you know uh  trying to put together different um  uh different tiles like that to form  form a match  uh there was no algorithm for solving  that problem  um and and i think this is kind of  interesting because it's the first  example of a problem that we have where  the underlying  question  does not really seem to have anything to  do with computation  you know you might argue that atm being  undecidable  is perhaps  a little unsurprising after the fact  because you know it's a problem about  touring machines so you can imagine  turing machines are going to have a  tough time answering it um but here's a  problem just about strings  um and we're going to show it's  undecidable  so  just to formulate this as a language i'm  going to call the pcp language  is the collection of these um  uh  pcp  uh problems pcp instances which are just  collect themselves each one is a  collection of dominoes so it's the  language of all  uh collections of these dominoes where  there is a match  so it's all of the pcp problems where  you can solve them with a match  and  uh we're going to prove that it's  undecidable by showing that atm is  reducible to that pcp language  um now uh before we do that i'm going to  have to i'm going to explain to you what  the computation history method is in the  first place that we're going to be using  uh so before we do that here's our first  check-in for the day  just to make sure you're following me  about  uh what the um  what this pcp problem is  i am  going to give you  a question to  uh  test  is there a match  with these three dominoes  so what you think about that  i mean the main thing i'm i want to make  sure you understand is what a match is  in the first place this is not a super  hard problem  uh to tell whether or not there's a  match  okay  um  five seconds  already  okay  closing it out um  okay um  so  uh  for those of you who said there is a  match  i want you to  exhibit exhibit the match because in  fact there is no such map there is no  match  um  and uh i mean you could get by fiddling  around with it i think if you try to  find a match you'll pretty quickly uh  see that it's you're gonna get um i  think what happens is you kind of get  into if you try to build build a match  with this particular setup  you're just going to get stuck  sort of repeating yourself and  um  one point here that you may have missed  is that a match has to be a finite  sequence  so if you were thinking about there  there is there is a kind of an infinite  match uh that you can build with this uh  this this set of dominoes but that's not  allowed we're only allowing finite  matches  and so that may they might change your  answer as a result but too late um  so  um  and one way to see that you can't have a  uh a match in this problem is that you  know you're going to have to start  you you  you know neither of these two are  possible starting points so this one is  going to be clearly the only chance that  you're going to have as a match uh the  first domino here but then once you put  this one down  um the bottom one the bottom string the  the bottom b's that you're going to get  the bottom string is going to be longer  than the top string  and then the other two are going to be  have the same length um so you're never  going to have the bottom uh the top be  the same length as the bottom the bottom  is always going to be longer no matter  how many uh  more dominoes you lay out so this  there's no chance of there being a match  here um  okay  so now we're gonna  um  uh  work our way up to introducing this  computation history method  [Music]  um um  first  let me  define something for a turing machine  that i'm going to call a configuration  um  configuration is just a snapshot of the  turing machine in the middle of its  computation  so  if you're running the turing machine and  you just  stop it at some moment  it's going to be in a certain state  the head is going to be on a certain  position  and the tape is going to have a certain  contents  and with that information  you can then um continue the copy the  computation of the turing machine that's  a full set of the information that you  need about the turing machine that tells  you everything about its cut its  computation at that moment in time  the state head position and tape  contents and so that we're calling that  that information together  uh state head position tape contents  we're calling that a configuration it's  a  fairly  um  basic notion i mean if you if you  program if you have something if you  have like the states of all the  variables and where the current  execution of the program is at  a moment in time same idea  um  okay so in terms of a picture here here  is the turing machine  imagine it's in state q3  um the head position is in the sixth  position on the tape so  p would be six  um because it's in the sixth position  and the tape contents is going to be  this bunch of a's followed by this bunch  of b's so i would write it down like  this  states in q3  head position number six  this is the contents of the tape  um now what we're going to often do  for convenience in using this notion in  proofs  is that we're going to want to represent  a con  a configuration as a string  um  in a particular way that's going to be  that's going to  this is this concept is going to come up  kind of  again and again during the during during  uh the course and so it's going to be  handy to have a particular way of  writing down configurations  um  you know for doing um  proofs about them and so the way we're  going to write them down is  i think it's just maybe i can just put  it right up here like this  um  we're going to write down  the  the symbols of the tape  but what we're going to do is stick in  the middle of those symbols the the  state  of the machine  immediately to the left  of the  position that the machine is currently  reading  so you can imagine  you know the the head here which is  coming you know imagine the state is  kind of where the head is it's pointing  at the symbol immediately to its right  so this is just another way of writing  down a configuration here's the tape  contents which i'm i'm sort of writing  formally here i'm breaking the the the  the tape contents into two parts which  i'm calling t1 and t2  um this is the t1 part the t2 part  and i'm  putting the uh  the  the state  in between t1 and t2  where i have in mind that the machine's  head position is right at the beginning  of t2  but maybe this picture just says it all  and it's clear enough  um  so just keep in mind that when when  we're going to be writing down  configurations of machines we're  typically going to write it down this  way  um  okay so i think this is a good moment to  take uh to take a moment uh for  questions  oh i see there's a lot of questions  already um  um so one's question is how does the  encoding differentiate between the  string and the state  um  i if i'm understanding you correctly i'm  going to be assuming  that um  the state the symbols that represent  states and the symbols that appear on  the tape  are distinct from one another so you can  always tell just like usually the way we  write things down when you have a state  symbol or whether you have a tape symbol  and so in a configuration you're going  to have a bunch  of uh tape symbols and a single state  symbol represent you know in the  position where the head is  um  somebody says  6 here is just you know this is one two  three four five six we're in head the  head is in position six that's all i had  in mind for six and that's why that six  is appearing over there because that's  the position of the head  um  good  all right  so why don't we continue um  now we're all ready to start defining  computation histories  um which themselves is not very  difficult concept  computation history for a turing machine  m on an input w  is just the sequence of configurations  you go through  when you start the machine  at the beginning  um with uh m on the um  on the tape  with w on the tape i'm sorry um and then  and the  the st the head at the beginning is in  position one  and the uh state in q zero or whatever  the start state is of the machine  so  that's going to be the very the starting  configuration here and these are the  sequence of configuration that mean  machines go through that the machine  goes through step by step  every step of the machine is getting  represented here until it ends up at an  accept  that's what we mean by a computation  history or sometimes we're going to  emphasize that by calling and accepting  computation history  represent meaning that the machine has  accepted its input and these are the  sequence of configurations that the  machine goes through  that's what i mean by a computation  history you know i'm sure in i mean i  think the terminology changes over the  years but the there's a notion similar  to that  for any kind of um you know if you're  you know running a program  and you just want to keep track of how  the variables are changing um as you're  you know in this particular run of the  machine and you're writing them all down  um it's like a log of the uh  history of the settings of the internal  states of the machine and variables and  so on  um  so this is just all of the  configurations the machine goes through  are on the way to accepting its input  if the machine does not accept its input  there is no computation history  okay so there is no accepting  computation history at least to  emphasize that point um  because that can only occur obviously if  the machine has accepted its input  okay so just as we had with  configurations  we're going to might want to be able to  write down computation histories as  strings  and it'll be convenient to have a  particular format for doing that as well  and it's kind of the obvious thing  we're just going to take  [Music]  the sequence of configurations in the  computation history and write them down  as a string where each  configuration is going to be the string  for that configuration  i'll show you kind of a little example  in a second but just to  you know focus on the concept here we're  just going to write down the string for  this configuration the starting  configuration and then the next  configuration and so on until you get to  the accepting configuration and separate  each of those strings by some pound sign  okay so here is a computation history  for m on w um  [Music]  uh  where w is the string w on w2 dot wn  so here is the uh  here is the first  uh configuration  so that this part here is that is the  computation history encoded as a string  i'm just i'm giving you this extra  um  side information just to make it clearer  what's what's going on there so this is  the first configuration this is the  second configuration and so on and i'm  i'm trying to even give you a little bit  more you know um detail of how it might  look uh by kind of  you know making the example a little bit  more concrete so let's say that the  turing machine  when it's in the starting state q0  looking at the very first symbol of of  the end of the the input tape in this  particular input say w1  um it goes from q0 to q7 and writes an a  on the tape and moves its head to the  right  so that's why the second configuration  reflects that  see here it went from q0 to q7  now the head is in the second position  so that's why that state symbol has  moved over one  and  um  the very first position now has become  an a  because the machine has changed whatever  was on that  input tape there on the first position  to an a  and then the next step it goes from q7  reading w2 which is where the head is uh  now it's looking at the w2  and goes to q8 writing a c and again  moves right  okay so that's why i drew these in the  way i did and so you go through a  sequence of those you get to cue accept  and that's  the encoded form of the computation  history  all right  um  i think that's is that all i wanted to  say yeah um  so you can uh feel free  to ask another question if you want um  i'm just gonna relaunch this  yeah good  so if you for all to for all together on  configurations  computation histories  and how we're going to be writing them  down as strings  that's what we've done so far  no questions so i'm going to move on  all right  so let me define a new  um automaton  that we're going to  mainly use  as uh just  to  provide an example  for us  today  i'm going to call this a linearly  bounded automaton  and all it is is a turing machine  [Music]  where the turing machine is going to be  restricted  in where it can uh  the tape is not going to be infinite  anymore the tape is just going to be big  enough to hold the input  so the machine no longer has the ability  to move into the portion of the tape to  the right of the input because there is  no tape out there it just has the tape  sitting here um that contains the input  which can the tape itself can vary in  size however big the input is that's how  big the tape is  so the tape adjusts to the length of the  input  but once you've started the machine with  some particular input that's as big as  the tape is there's no more  um the reason why it's called linearly  bounded is because the amount of memory  is a linear function of the of the size  of the input because you can effectively  get  somewhat more memory by enlarging the  tape alphabet  but that's going to be fixed for any  given machine so that's  where the linear comes from if that's  helpful but if you don't get that  that's  sort of a side remark  uh  but um  what's important to me is that your  understand what i mean by an uh a  linearly bounded automaton or an lba  it's just like a turing machine but this  that portion of the tape that originally  had blanks is just not there  the machine tries to move its head off  the off the right end of the input it  just sticks there just as if it tried to  move its head off the left end of the  input  doesn't go anywhere  okay so now i'm gonna we're gonna ask  the same kinds of questions about lbas  that we asked for other automata  so the acceptance problem  if i give you a  an input  and some particular  lba  and i want to know  does the lba accept that input  well you know and now the question is is  that decidable um or not  uh so  it's  um at first glance you might think well  um an lba is like a turing machine  and the atm problem is undecidable  so uh  you know that that might be a good first  guess  um  [Music]  and also if you try to uh  to simulate them if you try to figure  out how you would go about simulating  the machine um  you know if you if a given b and w if  you actually try to simulate the machine  to get the answer so you run b on w well  of course if you run it for a while and  it eventually halts either accepting or  rejecting then you know the answer and  you're finished but you know this  machine might get into a loop  um  you know nothing to prevent the machine  from looping on that finite amount of on  that limited amount of tape that it has  um and then  you know  you might be in trouble  but in fact that's not the case  because  when you start out with a limited amount  of tape  if you run the machine for a long time  and it's not holding it's going and  going and going  inevitably it's going to have to repeat  um  get into exactly the same configuration  that it did before because there's only  a limited number of configurations that  the machine has  and once it repeats a configuration it's  going to be repeating that configuration  forever and it's going to be in a loop  so  this problem in fact is decidable um  because the idea is if bmw runs for for  a very long time and  an amount that you can calculate then  you know it's got to be you know cycling  you know more than just looping it's got  to be repeating itself  um  and so therefore once it starts  repeating itself it's going to be going  forever  so um and here's the actual calculation  which is something i'm sure you could do  on your own but just to spell it out  so if you have an input of length n  um that you're providing to uh b so if w  is of length n  um the the lba can only go  uh for  uh this number of different it can only  have this number of different  configurations the number of states  times the number of head positions which  is n the number of head positions on the  tape  times the number of different tape  contents  you know if if the tape was only one  long you know this is the this is the  size of the tape alphabet so if the tape  were too long it would you know  had the tape had two  uh  two cells on it the number of possible  tape contents is would be um the square  of the of the alphabet and then and if  if the tape is going to be n  uh symbols long it's going to be  the tape alphabet size to the nth power  uh so therefore if a turning machine  runs for longer it's got to repeat  some configuration and it'll never hold  so the decider is going to be  hopefully uh clear at this point  you're given b and w  says that this is the decider for alba  it's going to run bmw for this number of  steps  if it's accepted by then then you accept  and if it hasn't if it's rejected or  it's still running  then you can reject and you know if it's  still running at this point it's never  going to accept  all right  any questions on this  okay let's move on  um  all right so now um  but what's different now or what's it  perhaps interesting now is that even  though the  acceptance problem for lbas is the is is  decidable  the emptiness problem is undecidable  and that's where the computation history  method is going to come in so this is  where we're going to be starting to do  something new in terms of a proof  technique  um because  so we're going to reduce atm to elba the  emptiness problem for lbas  um  so given an lba does it accept anything  or not  and this is going to use the computation  history method so  this will be a chance to illustrate that  method for the first time  so this the setup initially is just like  before we're going to assume we have a  turing machine that decides the cb elba  problem that of interest  and we're going to use that to construct  turing machine deciding atm for our  contradiction  okay so here's s  supposedly deciding atm  what it's going to do and this is going  to be the tricky part  s  is going to use m and w to design an  llba  that lba  is going to be built  with the knowledge of m w in fact it's  going to have m and w built into it  so  that's why i'm calling that lba  b sub mw  because it depends on mw  um  as i'll describe and what that lba does  it takes its input  let's call it the x the input to the the  lba  and  examines that input to see  if that input is an accepting  computation history for m on w  then that lba is just looking for  computation histories for mw  and that's the only thing it's going to  ever accept  if you feed it something which is not  a computation history you know that  sequence of configurations  uh for mmw leading to an accept if you  feed it something else  the lba is just going to reject it  it only accepts  the accepting computation history for m  w  and now  if you can build such a thing  then  you can  use that machine  in your emptiness tester to see if it  accepts any strings at all  because the only thing that could  possibly be accepting is an accepting  computation history you don't even know  if there exists one because you don't  know if m accepts w  so you're going to build this lba  um  and uh  which is looking for accepting  computation histories  and then see if its language is empty or  not  if its language is not empty  the only thing it could be accepting is  this computation history so you know m  except w  whereas if the language is empty you  know that there is no computation  history for m w and so m does not accept  the leaf  so that's the whole idea the trick is  how do you build  this uh this lba um  so first you're going to make this lba  which tests its input to see if it's an  accepting computation history for m on w  and you have to build this  um  uh lba without even knowing whether  there is a computation history  um it's not like you can take that  string and just build the string into  the lba because they don't even know if  that string exists  but what you can do  is you can make the lba  um  follow the rules of m it knows w  and knows m so it can test take the its  input and using w and the rules of m see  if that input is a computation history  for m and w  um and that's what it's going to do  so here is  um  this machine i'm going to construct i'll  i'll give it to you written down and  with a little bit more um details of its  procedure but just to kind of illustrate  it so here is a proposed input to b sub  mw this would be the x here this would  be an x that would be accepted  but anything else that would be provided  would not be accepted  so this is the sequence of  configurations written down as a string  that would be the x that i'm providing  to the b c b b sub mw and it's supposed  to be checking this to make sure it's  legit  um so  so on on input x the way it's gonna the  way it's going to proceed  it's going to first check to see whether  x begins in the right way  because you know  this lba it knows m and it knows w  so the very first thing is it takes a  look at the first part of the string up  to the pound sign you know there's no  pound sign if you're going to just feed  junk in it's going to be easily  identifiable as junk  um  so uh  you know if you're going to feed  something so it's going to the lba is  going to take everything up to the first  pound sign and just confirm that that  thing is  the first configuration the starting  configuration of um  of m on w which means it has to start  with the start state and here is w  so just going to check that  um  then it's going to check that each  each one of these guys follows legally  from the previous one according to the  rules of f it's going to first checks  that this one is correct and then this  one leads to that one  according to the rules of m then this  one leads to that one according to the  rules of m  all of that you know the knowledge of m  and w is built in so it can do that  um  and then it just  follows along checking this computation  you know it's easy to check that a  computation is correct that's all that's  really going on here  um  it's going to check that the computation  is correct until it gets to the end and  then make sure that the last uh  configuration that it's been given as  input is an accepting configuration  that there's an accept state in it  um and if everything that passes it it  accepts otherwise it uh rejects  okay  um  and the point is that this is an lba you  don't oh i just kind of jumped ahead of  myself you don't need any additional  information so how does it actually do  this  so you know how do you actually do this  on the tape so i claim that the lba  doesn't need any extra space to do this  um to do this check you know it's just  going to be zigzagging back and forth  here on the on on the  on the input  you know checking that the corresponding  symbols match up except around the the  head position where it gets updated or  correctly  and it seems it may need to mark it will  need to mark on the tape it's allowed to  write on the tape just to make sure it  keeps track of where it is  but this is a very simple  i mean i'm not trying to uh if you're  not following i'm not trying to uh alarm  you but i'm just trying to help you  understand that  this is not a complicated procedure here  to do this check that the that the  actual computation of the that's written  down of the turing machine is valid  um but you know we can deal with a  question here  uh  oh good  now we got some questions  oops  you can be thinking about while i'm  thinking about this you can think about  that uh check-in over there  so here's a good question  going from each configuration to the  next configuration is it a unique  uh next  step  um well i'm assuming that the turing  machine we're starting with is  deterministic so this should be only one  way to go so the answer to that  question is yes  this is going to be a unique  um string here there's really going to  be only one  um  computation history not that it really  matters in a sense um the answer to that  question um  as long as that accepting computation  history corresponds to the machine  actually accepting  uh oh boy we're all  all over the place here uh okay  are you ready to end this um  in this um  in this bowl  uh  all right  well this is where we are  uh  everybody who said they'd rather be in  six or four fig six or four six i know  who you are  you're all  going to be expelled  no i'm joking uh  i don't know who you are so um  uh  yeah good  so um  uh  wha what so we are here at the at the  break  um i and i made this poll on purpose uh  at this moment so we can spend a little  time let me just start our clock going  and i can try to help you those of you  who are who answered see that you're  baffled um  i'll i can try to uh help you understand  what's going on  all right so this is a this is a good  question here um  uh  okay well okay there's several good  questions here so um  somebody asks why don't we just test all  possible strings uh for b remember if  we're trying to test emptiness  for b's language  um  that's the e-lba problem  nor do we just try all possible strings  um  and uh  that would work if we had enough time  but there's infinitely many strings and  so that's not going to be good if we're  trying to be a decider  so  so that is  that's why we don't just try all strings  so but here's this other question here  this is a  okay  so  um  the question is how do we find the input  x  so this is an important question um  because  uh  we don't find the input x  there's no  you know we  the input x um at least the accepted  input x would be the accepting  computation history  um  for m on w  we're never going to find it  you know we're never going to find an x  we're not we're building this um this  lba  not we're never going to run that lba  we're designing that lba not for the  purposes of running it  we're building that lba only for the  purpose  of  uh using the lba language emptiness  tester we're going to build that end lba  and feed it into the machine turing  machine r  which is going to tell us whether that  lba's language is empty we ourselves is  not never going to run that machine  we're never going to come up with an x  we're just having we're we're we are  designing a computation  checker  and then  um  the emptiness tester for that  that we assumed to exist for for  elba is going to say yes there is some  computation  which this machine accepts or no there  is no computation which this machine  accepts and that's going to be a  computation for m on w  and so that's going to tell us whether  or not m accepts w  so we're never going to actually be  finding an x we're never going to be  running that lba  we're just feeding using that lba  as  as an input  to r  um  does the computation history method  always use lbas no  as you'll see uh right after the break  because that's the lbas is just  kind of an easy place to get started but  we're going to use the computation  history method and computation histories  next on the on the post correspondence  problem  yes the the computation histories and  the input x are always going to be  finite  so that that was an answer to a question  about whether these histories or the  inputs are going to be finite or not so  the the they're all those strings are  always going to be finite and the  computation history is going to be  finite  um  so we are at the end of our five minutes  um let me just see if there was another  question i wanted to answer  um  let me let's move on  okay now  getting coming back to the  undecidability of this  post-correspondence problem so remember  the post course upon you know this this  problem you're given those dominoes you  want to know if there's a match here is  a little mini version of the the diagram  if that helps you remember it um  and we're going to prove  that this language is undecidable and so  it's undecidable to test whether  you have  a match given a set of dominoes  whether a match is possible  and we're going to use we're going to  reduce atm to to this language using the  computation history method so how in the  world are we going to do that  um  first of all there's a little detail  here i want to mention  uh just  uh um  don't focus on this but  i'm going to assume the matches always  start  with the very first domino on the list  um the very first domino in the  collection  so this is going to be like a starting  domino this is going to make my proof a  little simpler to do it that way and  then you can fix that assumption uh  later and if we have time at the end  i'll do that or maybe after the lecture  is over if there are questions i can  show you how to fix that assumption but  for now  um to simplify the proof we're going to  assume that there is a starting  domino that the matches always have to  start with that particular domino  um  if you didn't follow that point don't  worry you can just ignore it you'll see  where it comes up later  so um now we're going to reduce atm to  the pcp  so assuming we have a machine that  decides the pcp we're going to use that  to make a machine that decides the atm  as before so here's the turing machine s  which is going to decide atm  um  and the way we're going to do it is like  this  we're given m and w we want to know  as always  does m accept w  what we're going to do is we're going to  build an instance  of the pcp problem so we're going to  build a bun a collection of dominoes  [Music]  uh which are going to dip  and that collection is going to be built  knowing m w so that's going to affect  the dominoes we're going to create so  this collection of dominoes is going to  be called p sub mw  depends on an m w  and  finding a match for this set of w  is is going to force you to simulate  m on w because the match is going to  correspond to a computation history for  m on w  so uh we're going to use  once we do that  um once we build  uh the set of dominoes who's where a  match corresponds to a computation  history  we're going to use r to determine  whether or not there is a match  or in other words whether or not there  is a computation history which is  whether or not m accepts w  um so if there is a match we're going to  accept because we know m accepts w  and if and if there is no uh match we're  going to reject  because we know m does not accept w  okay um  this is the plan i haven't told you how  to do this yet  i mean i'm worried about the  significant number of you who are  feeling um  confused by the method i mean you should  you guys should be uh texting me in the  tas  um to try to at least get a sense of how  this is working i mean that we're going  to be going  we're going to be doing the method again  um it's just going to be a little more  complicated because we have to also deal  with these dominoes and all this all  that stuff and that's why i presented it  to the fir to you the first time  in the setting of the lbas which where  in a sense the method comes out  perhaps more simply  so a match is going to correspond to an  accepting computation history  um sometimes i  if i don't use the word accepting  um  you know i'm just  being a little sloppy  computation history and accepting  computation history for us right now  they're the same  i mean later on we may actually talk  about rejecting computation histories  but  let's not  get ourselves confused  all the except computation histories  always have to end with the machine  accepting  okay uh somebody's asking what does it  mean for a match to correspond to a  computation history you'll see that's  going to be on my next slide  oh so this is a good question um  is my step two trying to use r to  determine whether m accepts w  well yes but i'm doing that by testing  whether whether this these dominoes have  a match  because r  is besides pcp i can only use r to test  whether things have a match  because someone asked  should step two be to use r to determine  whether m accepts w  well in effect that's what it's doing  but it's doing indirectly through  um testing whether this  um pcp whether these dominoes have a  match because those dominoes are going  to force you to simulate mlw  okay i think i'm repeating myself here  so let's uh avoid getting into a loop on  the lecture and um  uh see how we actually build piece of mw  so now you understand what you have to  under have the plan in your mind  uh  um  we are given m w  we're trying to make a set of dominoes  where a match is going to be  a computation history for m w so the  string you're going to get in that match  the top string and the bottom string  which are going to be you have to have  to have to match  um there they're going to be computation  histories  but they're going to be a computation  history of m w so i want to figure out  how to make my dominoes force that  okay  so my starting domino as i told you  there's going to be a special starting  domino in my collection so which is  going to require the match to start with  that domino it's going to be this one  here  it's going to have these two strings in  it  and you can see already it's starting to  look like a  a computation history  the dominoes going to have that  feature  so  uh it's the pair of strings and i've  written it kind of to help you see  what's kind of going on it's a pound  sign on the top  and the  the starting configuration for mmw on  the bottom  and what i'm going to do for you here  is at the bottom of the slide  i'm going to take the dominoes i've  written down so far and try to be  building a match and you'll see how that  match is forcing a simulation of the  machine  okay  so let's take this as an illustration  let's assume the input to m  so i'm running m on w now i'm trying to  see does m accept w  um w is a string two two three  so the  start configuration for m on w  is  q zero that's the starting state for m  and then w following is 2 2 3 that's  what it's going to appear on the on the  tape of the turing machine to start off  so this is the start configuration  for on w  and  assuming that i had this particular  string is input string to w  and so uh given the match  the dominoes that i've given you so far  just one  um this is how the match is going to  start  uh now there's going to be more dominoes  coming  so for every possible tape symbol and  state  don't get confused by the language here  this is the important thing if if in the  turing machine  um and i'll just read this in english to  you if the turing machine when it's in  state q  and the head is reading an a  it moves to state r  writes a b at that point  and its head moves to the right  i'm focusing on right rightward moving  uh  turing machine steps right now  but for every possible  uh  state and and tape symbol  and looking at where what happens  i'm going to have a  domino that's going to capture this  information  and it's going to be this domino here  q a  on top  br on the bottom so just the string qa  on the top  br on the bottom  so let's see why with that sort of  arcane looking  why is that a good thing to why is that  a useful domino to put in here  well if you take a look  um  let's assume from my turing machine  when it's in state q0 which is the  starting state and the head is reading a  two which it will happen to be reading  because the  uh this the string w starts with a two  if it goes into state q4 no q7 and  writes a four and then moves right  i'm going to have in this in this domino  i'm going to have  q02 on top  and 4  q7 on the bottom  because 4 is what i wrote right here  that's you know that's the b  and this and the new state that i'm  going into is q7  so that's going to be this  the domino that i'm going to get  and here it is here's that domino  appearing in the match  so um  it's going to take  match up with q q 0 2  um  which you know  you know don't forget the the the top  string has to equal the bottom string so  and this is going to be the only choice  that i have for extending the match so  q02 um is going to be on the top when i  put that there but that's going to force  um 427 to appear at the bottom because  that's what's going to be the bottom  string corresponding to the q02 on the  top  okay  so  if you're looking down here this is the  the beginning of the second  configuration of the machine that's what  i want to be happening  i want to be  for  that match should look like a  computation history  so now um so this is going to be all  possible right moves are going to be  handled in this way it's going to be a  similar process for the left moves but  let me not you know i'll leave that to  your imagination  now  how do i continue on from there what  does the rest of this configuration look  like  well that should just be a copying over  of the remaining symbols that were on  the tape because they don't change  things only change around the head  the rest of those symbols which is the  two and the three those should just get  copied over here so i'm gonna have two  additional  i'm gonna have additional uh symbols in  my dominoes in my uh  in my collection so for every  um  tape symbol i'm gonna have a a  be a  domino so and for every type symbol a  i'm going to have a a be a domino in my  collection  and so that says  uh  i can have  uh so that there's going to be a 2 2  domino so i can match up this 2 over  here but that forces me to put a 2 down  over there  there's also going to be a 3 3 domino  which is going to match up with that 3  but it's going to force me to push it  put a 3 down over there that's the only  way i can extend this  um  match that i've built so far  i have no choice  those are the only  dominoes that i'm going to be given  and by and so doing i'm forcing you to  basically simulate the machine  now what i want to have happen next is a  pound sign to appear here and that will  conclude my second configuration  so there's going to be a pound sign  pound sign domino as well because  there's a pound sign here  it's going to get matched with a pound  sign up top forces a pound sign to come  down um on the bottom  and if you look at where we are right  now we're exactly like where we were at  the beginning when we had just this  first domino appearing but now we're one  configuration later  so if you understood that  and i admit it it's it's  you know this is a there's just one idea  here um  [Music]  and  once you get the idea  it's all trivial it's all very simple  but there's just you have to get that  idea i'm trying to figure out how to get  that idea into your head once you get  the idea you can write all this stuff  yourself  um  so uh  so following the this uh the this  description of how uh the you know the  the transition function of the machine  works and copying over uh the symbols  from the tape to the next configuration  i'm going to be able to get  configuration after configuration going  until i get to a point when there's an  accept  and now from the machine's perspective  we're done  the machine has accepted its input this  is our computation history but is it a  match  well  you know up until this this bottom thing  um is the computation history but it's  not a match because the top doesn't  equal the bottom there's still extra  stuff at the bottom which the top is not  uh  you know  it doesn't have  so what i'm going to need to do now is  is add some additional kind of  pseudo steps of the machine  where i'm going to allow  the top to catch up to the bottom  and the way i'm going to be thinking  about that and this is not real for a  turning machine as much as the turn  machine's real anyway but you're going  to imagine i'm going to add  a new kind of uh move to the machine  which is going to allow the head to eat  the symbols off the tape  like pac-man  okay and the way i'm gonna get that  effect um  on the top if i have any tape symbol  next to a cue except on either side  on the top  what i'm going to get you to write on  the bottom is just to accept with that  taste with that tape symbol gone  and by repeating that  you know move after move the the the  actual tape is going to be shrinking the  symbols on the tape are going to be  going down one by one move by move until  there's nothing left except just the um  you accept all by itself  okay so i have this sort of pac-man idea  of eating the symbols on the tape um  and then  finally i get to a point when there's  just the cue except  alone on the tape there's no symbols  left to consume  and then i'm going to add one last  domino here which is q accept pound  pound matching with just pound which  just conveniently  finishes off the um  uh the  uh the match  and so the match is completed  there's actually a little detail here i  i hesitate even to bring it up because i  i think it's a kind of thing where if  you understand everything up to this  point you could fill it in yourself but  just for completeness sake  uh you have to deal with the situation  when the the  uh the head  um of the turing machine  might move into the blank portion of the  tape  which is not taken into account here and  the way we get that effect is by having  another domino here which allows me to  add blank symbols at the bottom as  needed  that's just a detail  you know i'm more worried that you  understand the underlying concept um  so here's going to be um  uh  a check-in um  trying to remember  um  okay but you know  so okay so  what else can we conclude from this  information  we know so at this point we know that  pcp is undecidable that's what we just  finished proving  what else do we know  if anything  or do we even know that  uh see my picture is a little bit  covering the check in but it's still  readable  so i mean this is not an easy concept to  get the to get the idea but once you get  it  you know it's um  you know it's you'll see it's not that  bad  okay  another 15 seconds  so this portion doesn't this question is  not really relying so much on the  computation history method it's just  relying on the fact about  pcp being undecidable  um  [Music]  okay everybody  almost done  five seconds  all right  um  so i can see that there is uh  some level of confusion um  uh so the reason why b is correct  is that  we know pcp is undecidable  but  we also know that it's recognizable  because you could try all possible  uh ways of combining dominoes  and  one after the  next um and you accept if you ever find  a match  so they might go forever but if there is  a match  uh possible you'll find it eventually so  pcp is a recognizable language  undecidable and so therefore we know  it's complement it has to be  unrecognizable because you know that's  something we've shown before if a  language and its complement are both  recognizable then it's decidable but we  know pcp is not decidable so they both  both sides can't be recognizable  um okay so let's  let me prove to you one last theorem  that's going to be useful for your  homework  uh involving the computation history  method so you'll have one last chance  to get the  way we're going to use this though this  one is going to be a little bit more  similar in spirit  to the lba version  uh than to the pcp version which has  this extra kind of complication about  coding the computations of the machine  into dominoes here um you know we're  going to operate with another automaton  which where it's going to be more  straightforward but anyway okay getting  ahead of myself  so if you remember for the for  context-free grammars we had the  problem the emptiness problem  we showed that was decidable  so testing whether a context-free  grammar's language is empty  is decidable  however testing whether a context-free  grammar's language is everything whether  it's equal to sigma star  that turns out to be um  uh that turns out to be undecidable  okay so emptiness testing for  context-free grammars decidable  sigma-star testing  undecidable  um  okay so  we'll show that atm is reducible to all  pda via the computation history method  um  and uh  so uh assume we have same patterns  assuming we have a decider for all pda  and make a decider for ulti  decider for atm  here's the atm decider  and now  similar to what we did for the  uh  lba case but with a with a twist  we're going to make a push down  automaton  that's going to check its input to see  whether it's an accepting computation  history of m w just like the lba did if  you think back to how that worked  remember remember you know we um the lba  tested its input to see whether it's uh  an accepting computation history accept  it if it did and then we test the lba's  language for emptiness to see if there  are any computation histories so we're  doing the same kind of thing  except now  uh the pda is going to test its input to  see whether it's a an accepting  computation history and if it is it's  going to reject  it's going to do  the reverse of what the lba  did  and that's going to turn out to be  necessary but for the for the moment  let's just go with it and maybe we'll  see in the proof why the proof needs it  to be this way  so um this p pushdown automaton is going  to accept its input if it's not  an accepting computation history for m  on w otherwise it will accept so you can  think of this pda  as  accepting all the junk  it just  doesn't accept  um the good stuff the accepting  computation history  okay it's it's just it it's uh accepting  all the things which fail  to be an accepting computation history  okay and then once we have that we we  test whether  ours language is everything but whether  bmw's languages everything using r  because it if if that pda's language is  everything  then we know um  uh  there could not be an accepting  computation history because that's the  one thing that gets um  not accepted that's the one thing that  gets rejected  so if it's accepting everything there is  not going to be an accepting computation  history  so if there is no  uh if this is equal to sigma star  then there couldn't be an accepting  computation history and so we're going  to accept  okay so how is this going to work so  what's different now  about uh this case  uh from the lba case is remember the lba  uh got you know in  the the computation history on its input  and it used its ability to write on the  tape to check that each  configuration follows the next one  we don't have the ability to write on  the tape and push on automaton and by  the way i hope you're all comfortable  with my using pdas instead of grammars  because we can interchange one to the  other um i should have mentioned that  when when i did it but  um  so the the pda is going to be using its  stack to compare each configuration with  the next one  okay  so the way that's going to happen is  it's going to non-deterministically  take one of these so the very first  thing it does  is as before it checks to make sure that  the beginning  of the input is the start configuration  but then once that's that's the easy  part but once we get going with that we  can we push each configuration well we  first of all we now deterministically  choose which configuration might be the  bet the one that fails  um  where  where one fails to go to the next one  because those are the ones we're trying  to accept when there's a failure  so we now deterministically look for the  place that there's a failure we push  that onto the stack  um and then we pop it off the stack and  compare it with the next configuration  so that's how we're using the stack  instead of being able to mark on the  input  now there's a so i'm kind of going to  illustrate that here  so  you know we're going as we're going to  read this thing here and we're going to  put it onto the stack so this thing  moves over to here  and  this input here got put onto the stack  you know q 0 w 1 w 2 it's q 0 w 1 w 2  you see that that first configuration is  now sitting on the stack now as we're  going to pop it off we're going to match  it with the second configuration  now if you're following me  you'll realize that there's a there's a  difficulty here because it's coming out  in reverse  you know it comes out in the reverse  order that we put it in and that's not  what we need to do so what we're going  to do here and here's a little sort of a  twist we're going to change the way  we're writing down computation histories  by making them by writing them uh  reversing the even numbered ones so this  one here is going to be written down in  reverse  and that's perfectly okay we can write  down configuration computation histories  in any way we want  to um to meet our needs so we're going  to reverse we're going to reverse the  alternate ones  and now when we push one it's going to  come off in the right order to compare  with the next one  and so that's how the the procedure  works  okay  we're running a little short on time um  you need to be let's see so let let me  just go to um  uh  quick recap the computation history  method is useful for showing the  undecidability of problems when you're  testing for the existence of an object  each of those four cases that we showed  today involving testing whether  something exists  so is there an integer solution to the  polynomial  is there some string in the language is  there a string that's not in the  language  um or is there a match  so that's a typical case  when um  this computation history method comes up  and so as a quick review  uh these are the things we showed today  okay so why don't we uh uh we're we're  just out of time um and so i'm i'm gonna  let you go um  but i will stick around uh for another  five minutes or ten minutes uh happy to  answer any questions if you if you have  them um but that's that's officially the  end of the lecture  um okay  so let's let me try to get to there's a  lot of questions in the chat don't  forget you know write to the tas too  so  if m does not accept w  what will happen to the computation  history so if m does not accept w  um  uh then there is no computation there is  no accepting computation history that's  the only kind of computation histories  we're considering  so if m does not accept w there is no  accepting computation history there's no  computation history  um so i don't know what it means what  will happen to it it just doesn't  doesn't exist  um  so uh  i hope that's  helpful um  don't we need to be able to add states  to the end of the tape  i don't know what that means  it's taped to the end of the tape  i i don't know you'll have to repeat  that one sorry or explain that better  um  ah  so this is a good this is a very good  question here where does the previous  proof fail  when trying to reduce atm to  ecfg  it's got to fail  because ecfg is  decidable um  so that that's a very good question  maybe we can just go back to that  last slide here  because i was running a little short on  time i didn't really focus on why we  have to accept  um  the non-computation history strings we  why are we accepting all the junk  strings  and  only rejecting  um  the strings which are the uh the  computation histories so you know we're  looking for strings that fail  you know if the string could fail  because it starts wrong  string it doesn't have the start  configurat configuration or it doesn't  maybe doesn't end with the accepting  configuration or maybe one configuration  doesn't lead properly to the next one  any of those cases are a failure  and there are going to be reason to  accept  the reason why we can do do it do that  um is because the we're taking advantage  of the push downs non-determinism  we don't know where the  um the failure  might occur  so we're going to non-deterministically  guess where that failure occurs  if we were going to flip this around and  say  let's accept only the good ones and  reject everything else we would have to  test that each configuration led to the  next one  so we have to check them all  so  if something fails it only has to fail  in one place  and then you know we can accept if it um  if it's a good computation history  we have to  it has to be good everywhere  and so the push down automaton really if  you're going to get down to the  nitty-gritty the push on automaton can  check that this configuration leads to  that configuration  using pushing on the stack and then  popping it  but now by the time you get to here you  want to check that this second  configuration leads to the third one  you would need to push the second  configuration onto the stack but at this  point you're already  at after the second configuration so we  cannot back up and push the second one  on the stack  so the the push on tom dot is not able  to check um in the positive sense that  you have a computation history it's only  able to and accept them it's only to  check in the negative sense that you  don't have a con  a computation history  and uh accept all the ones that fail  and that's just because it only has to  fail in one place  i hope that helps um  so that's why we couldn't flip this  around and make this proof work for the  emptiness problem and only have to work  for the the everything problem  the old problem  all right um  okay so bye bye everybody and i will see  you on tuesday

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hi everybody  can you hear me  yes good  um  welcome back uh  to um  to the course and um  so we are now at lecture 10 and um uh  i  let's see what have we been doing we've  been talking about um  undecidability  um so we  introduced last time  uh reducibilities and mapping  reducibilities for proving  various problems are undecidable  and  this time we're going to and today we're  going to introduce a a  more sophisticated method for proving  undecidability uh using reduceabilities  and that's called the computation  history method and and that's pretty  much  what's  used in all of the  you know more  serious cases where people have proved  problems undecidable it's it's pretty  much a widespread uh  uh method for uh especially you know  it's not an active area of research  these days but at times when people were  proving problems undecidable  this was frequently a method that was  used  um so we'll go over that today and we'll  prove um  a few examples of problems undecidable  using that method  all right so why don't we get started um  uh so first of all  most importantly  you should remember how we you how we  prove um problems are undecidable in the  first place  and that is by showing  um some other undecidable problem that  we already know is undecidable is  reducible to the problem of interest  so typically you might be reducing atm  um which is a problem that we started  with and showed to be undecidable by the  diagonalization method  um or might be some other problem that  we've subsequently shown to be  undecidable and you take that problem  and you reduce it to  the problem you're trying to show  undecidable  okay oops  let me fix that  sorry  um  so um  so this is the thing to keep in mind to  prove a language is undecidable show  that atm or some other known undecidable  language is reducible to the problem b  that you have at hand that you're trying  to show undecidable  all right so we are now going to  um  um  [Music]  first i'm going to start off by  remembering um hilbert's 10th problem  which we discussed briefly a few  lectures back um  and as you may remember  so that's a problem where uh you're  given  say some polynomial  you know over several variables  you know like three x squared y minus  15z  plus  2xz  equals 5.  and you want to solve that problem  you want to find a solution  that solves that equation  but you require that the solution  involves only integers  so those are so called diaphantine  problems because the requirement is that  the solution being integers  and  hilbert's problem was to  ask for a  an algorithm  not the language that he used but  doesn't matter he essentially he asked  for an algorithm to test whether a  polynomial has a solution in integers or  the polynomial equation has a solution  in integers  and um  as  uh  um  uh  um as we now know that problem was posed  back in 1900  it took 71 years to find the solution  but the answer is no  there is no there is no such algorithm  we talked about this when we were  discussing the church turing thesis  um  and um  the method  showing  that there is no algorithm is by a  reduction from atm to that problem  so one can use exactly the method that  we're using today to prove that problem  is undecidable  about polynomials the only thing is is  that the reduction that single reduction  would take the entire semester  um  and i know that's  in fact correct because when i was a  graduate student um there was a woman in  the math department at berkeley where i  where i studied and she uh the whole the  whole course  was to go through the proof of hilbert's  tenth problem  um of the solution to hillary problem  the the proof of the undecidability of  the uh of the solution  uh of the undecidability of testing  whether a polynomial has an integral  solution um  so  um  the thing is that involves some  fairly hairy number theory in order to  um  to come up with the right polynomials to  basically simulate the turing machine  which so that's kind of getting ahead of  ourselves that's what we're going to be  doing today um but we're going to  instead of looking at that problem  because we don't have a whole semester  to spend on it um we're going to look at  a toy problem instead  where there's no number theory but it  still has the same basic underlying idea  um to that that was employed in um the  solution to hilbert's tenth problem  um so that toy problem is called the  post correspondence problem  and it's going to have some other  utility for us  well  the main reason we're studying it is  just to illustrate the method um  uh  so  and the method is going to be as i've  been saying is this computation history  method  okay so one we will first talk about uh  this post-correspondence problem because  that's very easy to understand  and then we'll spend a little time  introducing that method  um  all right  so um  the  uh the post the post correspondence  problem is  as follows you're given  uh  a bunch of pairs of strings  um so here is  one pair t1 and b1  and i'm writing them as  dominoes i'm calling them dominoes  because i'm writing one pair one one of  the strings on top of the other string  in fact i'm  using t and b for top and bottom here so  this is t1 is the top string b1 is the  bottom string in the first domino  and then in the second domino we have  two other strings to a top and a bottom  and so on  um  so we have these here  uh this collection of dominoes  um  as  sort of the legal dominoes um and what  the goal is to find a sequence of those  dominoes um  uh  so you want to pick from these dominoes  here you're allowed to repeat dominoes  so you want to pick a sequence of those  dominoes  line them up next to each other  and so that the string that you get by  reading along the top all of the t's  together  is going to be exactly the same as the  string you get by reading along all the  bottoms  i'll give you an example  um so here to write it down a little bit  more formally  so uh what i'm calling a match  is a sequence of these dominoes so it's  t i1 ti2  as well it's it's i1 is the first domino  i2 is the second domino and so on  and then you what you're going to have  is  all the top strings concatenated  concatenated together is exactly the  same as what you get by concatenating  together all the bottom strings  um so here's an example and i think  it'll become clear if you didn't  understand it so far but it'll be  completely clear from the example i hope  so here is a bunch of dominoes  um four  um  and um  what i wanna know is is there some  selection of these dominoes again you  can repeat dominoes otherwise it would  be not an interesting problem so you can  repeat these as many times as you like  and i want to pick up pick these  dominoes and line them up so that what  you get by reading along the top is the  same as what you get by reading along  the bottom  so first of all if you just stare at  this  and you're trying to make a match  you'll see that there's only one  possibility possible way to start this  um you know  for example if you started with the  third domino here b a over a a  it would fail immediately because  um you know the b and the a are  different  um and and if you you use the second  domino as your starting point that would  fail the match as well  because a a has to would have to e um be  the the the top string would start with  a a the bottom string would start with a  b and they could never be equal  um so by looking at this you can see  that the only possible choice that you  have is to start with the first domino  okay so i'm going to start to build the  domino for you just so you can see it so  here's the very the the match as um so  you can see um the process so the match  starts with the very first domino and  the way i'm going to write it is i'm  going to take the dominoes and kind of  skew them  so that you can see how the top and  bottom strings are lining up  um  but they're just mouth the same dominoes  they've just been  i changed the shape  so this is the first domino a b over aba  now uh the second domino in my and my  match it's gotta start with an a  as the leftmost symbol on the top string  so that would rule out this one for  example um but there might be several  choices and so it's not exactly obvious  which one you should take  what i'm going to suggest is we take  this one here  so we take a a over a b a you see it a a  over a b a it's just  you know written i'm just skewing uh  skewing it so that um i can line up the  a over a the b over b the a over a and  so on so i'm starting to build  this concatenation of the top being  equal to the concatenation of the bottom  following me so far i hope  so now um what's what's next so i need  to have something where  the top string is going to start with a  ba  in fact there's only one choice for that  so it's clearly going to have to be this  domino is going to be next so it's going  to put a ba up here but then it's going  to a force an a a down below because  that's what this domino does um  so this ba matches with that now we have  the aaa  to deal with um  so  we're going to reuse this domino because  that's the only one that starts with an  aa so we're going to reuse that one  now we have a a and aba now we need  something that starts with aba  i see two choices  it could be this one because this is  consistent with a it's at least it  captures the a b we could we could try  putting this one over here but a better  choice would it be would be to use this  one because if we use this one we finish  the match  and therefore we have found what we're  looking for  um and this  this  this collection of  dominoes here has a mat has a match  now it's not always obvious  some some collections of dominoes may  not have a match  and so the computational problem is  is there a match  um  and  the theorem that we're going to prove  today is that this problem is  undecidable  simple as it is simple little  combinatorial problem of you know uh  trying to put together different um  uh different tiles like that to form  form a match  uh there was no algorithm for solving  that problem  um and and i think this is kind of  interesting because it's the first  example of a problem that we have where  the underlying  question  does not really seem to have anything to  do with computation  you know you might argue that atm being  undecidable  is perhaps  a little unsurprising after the fact  because you know it's a problem about  touring machines so you can imagine  turing machines are going to have a  tough time answering it um but here's a  problem just about strings  um and we're going to show it's  undecidable  so  just to formulate this as a language i'm  going to call the pcp language  is the collection of these um  uh  pcp  uh problems pcp instances which are just  collect themselves each one is a  collection of dominoes so it's the  language of all  uh collections of these dominoes where  there is a match  so it's all of the pcp problems where  you can solve them with a match  and  uh we're going to prove that it's  undecidable by showing that atm is  reducible to that pcp language  um now uh before we do that i'm going to  have to i'm going to explain to you what  the computation history method is in the  first place that we're going to be using  uh so before we do that here's our first  check-in for the day  just to make sure you're following me  about  uh what the um  what this pcp problem is  i am  going to give you  a question to  uh  test  is there a match  with these three dominoes  so what you think about that  i mean the main thing i'm i want to make  sure you understand is what a match is  in the first place this is not a super  hard problem  uh to tell whether or not there's a  match  okay  um  five seconds  already  okay  closing it out um  okay um  so  uh  for those of you who said there is a  match  i want you to  exhibit exhibit the match because in  fact there is no such map there is no  match  um  and uh i mean you could get by fiddling  around with it i think if you try to  find a match you'll pretty quickly uh  see that it's you're gonna get um i  think what happens is you kind of get  into if you try to build build a match  with this particular setup  you're just going to get stuck  sort of repeating yourself and  um  one point here that you may have missed  is that a match has to be a finite  sequence  so if you were thinking about there  there is there is a kind of an infinite  match uh that you can build with this uh  this this set of dominoes but that's not  allowed we're only allowing finite  matches  and so that may they might change your  answer as a result but too late um  so  um  and one way to see that you can't have a  uh a match in this problem is that you  know you're going to have to start  you you  you know neither of these two are  possible starting points so this one is  going to be clearly the only chance that  you're going to have as a match uh the  first domino here but then once you put  this one down  um the bottom one the bottom string the  the bottom b's that you're going to get  the bottom string is going to be longer  than the top string  and then the other two are going to be  have the same length um so you're never  going to have the bottom uh the top be  the same length as the bottom the bottom  is always going to be longer no matter  how many uh  more dominoes you lay out so this  there's no chance of there being a match  here um  okay  so now we're gonna  um  uh  work our way up to introducing this  computation history method  [Music]  um um  first  let me  define something for a turing machine  that i'm going to call a configuration  um  configuration is just a snapshot of the  turing machine in the middle of its  computation  so  if you're running the turing machine and  you just  stop it at some moment  it's going to be in a certain state  the head is going to be on a certain  position  and the tape is going to have a certain  contents  and with that information  you can then um continue the copy the  computation of the turing machine that's  a full set of the information that you  need about the turing machine that tells  you everything about its cut its  computation at that moment in time  the state head position and tape  contents and so that we're calling that  that information together  uh state head position tape contents  we're calling that a configuration it's  a  fairly  um  basic notion i mean if you if you  program if you have something if you  have like the states of all the  variables and where the current  execution of the program is at  a moment in time same idea  um  okay so in terms of a picture here here  is the turing machine  imagine it's in state q3  um the head position is in the sixth  position on the tape so  p would be six  um because it's in the sixth position  and the tape contents is going to be  this bunch of a's followed by this bunch  of b's so i would write it down like  this  states in q3  head position number six  this is the contents of the tape  um now what we're going to often do  for convenience in using this notion in  proofs  is that we're going to want to represent  a con  a configuration as a string  um  in a particular way that's going to be  that's going to  this is this concept is going to come up  kind of  again and again during the during during  uh the course and so it's going to be  handy to have a particular way of  writing down configurations  um  you know for doing um  proofs about them and so the way we're  going to write them down is  i think it's just maybe i can just put  it right up here like this  um  we're going to write down  the  the symbols of the tape  but what we're going to do is stick in  the middle of those symbols the the  state  of the machine  immediately to the left  of the  position that the machine is currently  reading  so you can imagine  you know the the head here which is  coming you know imagine the state is  kind of where the head is it's pointing  at the symbol immediately to its right  so this is just another way of writing  down a configuration here's the tape  contents which i'm i'm sort of writing  formally here i'm breaking the the the  the tape contents into two parts which  i'm calling t1 and t2  um this is the t1 part the t2 part  and i'm  putting the uh  the  the state  in between t1 and t2  where i have in mind that the machine's  head position is right at the beginning  of t2  but maybe this picture just says it all  and it's clear enough  um  so just keep in mind that when when  we're going to be writing down  configurations of machines we're  typically going to write it down this  way  um  okay so i think this is a good moment to  take uh to take a moment uh for  questions  oh i see there's a lot of questions  already um  um so one's question is how does the  encoding differentiate between the  string and the state  um  i if i'm understanding you correctly i'm  going to be assuming  that um  the state the symbols that represent  states and the symbols that appear on  the tape  are distinct from one another so you can  always tell just like usually the way we  write things down when you have a state  symbol or whether you have a tape symbol  and so in a configuration you're going  to have a bunch  of uh tape symbols and a single state  symbol represent you know in the  position where the head is  um  somebody says  6 here is just you know this is one two  three four five six we're in head the  head is in position six that's all i had  in mind for six and that's why that six  is appearing over there because that's  the position of the head  um  good  all right  so why don't we continue um  now we're all ready to start defining  computation histories  um which themselves is not very  difficult concept  computation history for a turing machine  m on an input w  is just the sequence of configurations  you go through  when you start the machine  at the beginning  um with uh m on the um  on the tape  with w on the tape i'm sorry um and then  and the  the st the head at the beginning is in  position one  and the uh state in q zero or whatever  the start state is of the machine  so  that's going to be the very the starting  configuration here and these are the  sequence of configuration that mean  machines go through that the machine  goes through step by step  every step of the machine is getting  represented here until it ends up at an  accept  that's what we mean by a computation  history or sometimes we're going to  emphasize that by calling and accepting  computation history  represent meaning that the machine has  accepted its input and these are the  sequence of configurations that the  machine goes through  that's what i mean by a computation  history you know i'm sure in i mean i  think the terminology changes over the  years but the there's a notion similar  to that  for any kind of um you know if you're  you know running a program  and you just want to keep track of how  the variables are changing um as you're  you know in this particular run of the  machine and you're writing them all down  um it's like a log of the uh  history of the settings of the internal  states of the machine and variables and  so on  um  so this is just all of the  configurations the machine goes through  are on the way to accepting its input  if the machine does not accept its input  there is no computation history  okay so there is no accepting  computation history at least to  emphasize that point um  because that can only occur obviously if  the machine has accepted its input  okay so just as we had with  configurations  we're going to might want to be able to  write down computation histories as  strings  and it'll be convenient to have a  particular format for doing that as well  and it's kind of the obvious thing  we're just going to take  [Music]  the sequence of configurations in the  computation history and write them down  as a string where each  configuration is going to be the string  for that configuration  i'll show you kind of a little example  in a second but just to  you know focus on the concept here we're  just going to write down the string for  this configuration the starting  configuration and then the next  configuration and so on until you get to  the accepting configuration and separate  each of those strings by some pound sign  okay so here is a computation history  for m on w um  [Music]  uh  where w is the string w on w2 dot wn  so here is the uh  here is the first  uh configuration  so that this part here is that is the  computation history encoded as a string  i'm just i'm giving you this extra  um  side information just to make it clearer  what's what's going on there so this is  the first configuration this is the  second configuration and so on and i'm  i'm trying to even give you a little bit  more you know um detail of how it might  look uh by kind of  you know making the example a little bit  more concrete so let's say that the  turing machine  when it's in the starting state q0  looking at the very first symbol of of  the end of the the input tape in this  particular input say w1  um it goes from q0 to q7 and writes an a  on the tape and moves its head to the  right  so that's why the second configuration  reflects that  see here it went from q0 to q7  now the head is in the second position  so that's why that state symbol has  moved over one  and  um  the very first position now has become  an a  because the machine has changed whatever  was on that  input tape there on the first position  to an a  and then the next step it goes from q7  reading w2 which is where the head is uh  now it's looking at the w2  and goes to q8 writing a c and again  moves right  okay so that's why i drew these in the  way i did and so you go through a  sequence of those you get to cue accept  and that's  the encoded form of the computation  history  all right  um  i think that's is that all i wanted to  say yeah um  so you can uh feel free  to ask another question if you want um  i'm just gonna relaunch this  yeah good  so if you for all to for all together on  configurations  computation histories  and how we're going to be writing them  down as strings  that's what we've done so far  no questions so i'm going to move on  all right  so let me define a new  um automaton  that we're going to  mainly use  as uh just  to  provide an example  for us  today  i'm going to call this a linearly  bounded automaton  and all it is is a turing machine  [Music]  where the turing machine is going to be  restricted  in where it can uh  the tape is not going to be infinite  anymore the tape is just going to be big  enough to hold the input  so the machine no longer has the ability  to move into the portion of the tape to  the right of the input because there is  no tape out there it just has the tape  sitting here um that contains the input  which can the tape itself can vary in  size however big the input is that's how  big the tape is  so the tape adjusts to the length of the  input  but once you've started the machine with  some particular input that's as big as  the tape is there's no more  um the reason why it's called linearly  bounded is because the amount of memory  is a linear function of the of the size  of the input because you can effectively  get  somewhat more memory by enlarging the  tape alphabet  but that's going to be fixed for any  given machine so that's  where the linear comes from if that's  helpful but if you don't get that  that's  sort of a side remark  uh  but um  what's important to me is that your  understand what i mean by an uh a  linearly bounded automaton or an lba  it's just like a turing machine but this  that portion of the tape that originally  had blanks is just not there  the machine tries to move its head off  the off the right end of the input it  just sticks there just as if it tried to  move its head off the left end of the  input  doesn't go anywhere  okay so now i'm gonna we're gonna ask  the same kinds of questions about lbas  that we asked for other automata  so the acceptance problem  if i give you a  an input  and some particular  lba  and i want to know  does the lba accept that input  well you know and now the question is is  that decidable um or not  uh so  it's  um at first glance you might think well  um an lba is like a turing machine  and the atm problem is undecidable  so uh  you know that that might be a good first  guess  um  [Music]  and also if you try to uh  to simulate them if you try to figure  out how you would go about simulating  the machine um  you know if you if a given b and w if  you actually try to simulate the machine  to get the answer so you run b on w well  of course if you run it for a while and  it eventually halts either accepting or  rejecting then you know the answer and  you're finished but you know this  machine might get into a loop  um  you know nothing to prevent the machine  from looping on that finite amount of on  that limited amount of tape that it has  um and then  you know  you might be in trouble  but in fact that's not the case  because  when you start out with a limited amount  of tape  if you run the machine for a long time  and it's not holding it's going and  going and going  inevitably it's going to have to repeat  um  get into exactly the same configuration  that it did before because there's only  a limited number of configurations that  the machine has  and once it repeats a configuration it's  going to be repeating that configuration  forever and it's going to be in a loop  so  this problem in fact is decidable um  because the idea is if bmw runs for for  a very long time and  an amount that you can calculate then  you know it's got to be you know cycling  you know more than just looping it's got  to be repeating itself  um  and so therefore once it starts  repeating itself it's going to be going  forever  so um and here's the actual calculation  which is something i'm sure you could do  on your own but just to spell it out  so if you have an input of length n  um that you're providing to uh b so if w  is of length n  um the the lba can only go  uh for  uh this number of different it can only  have this number of different  configurations the number of states  times the number of head positions which  is n the number of head positions on the  tape  times the number of different tape  contents  you know if if the tape was only one  long you know this is the this is the  size of the tape alphabet so if the tape  were too long it would you know  had the tape had two  uh  two cells on it the number of possible  tape contents is would be um the square  of the of the alphabet and then and if  if the tape is going to be n  uh symbols long it's going to be  the tape alphabet size to the nth power  uh so therefore if a turning machine  runs for longer it's got to repeat  some configuration and it'll never hold  so the decider is going to be  hopefully uh clear at this point  you're given b and w  says that this is the decider for alba  it's going to run bmw for this number of  steps  if it's accepted by then then you accept  and if it hasn't if it's rejected or  it's still running  then you can reject and you know if it's  still running at this point it's never  going to accept  all right  any questions on this  okay let's move on  um  all right so now um  but what's different now or what's it  perhaps interesting now is that even  though the  acceptance problem for lbas is the is is  decidable  the emptiness problem is undecidable  and that's where the computation history  method is going to come in so this is  where we're going to be starting to do  something new in terms of a proof  technique  um because  so we're going to reduce atm to elba the  emptiness problem for lbas  um  so given an lba does it accept anything  or not  and this is going to use the computation  history method so  this will be a chance to illustrate that  method for the first time  so this the setup initially is just like  before we're going to assume we have a  turing machine that decides the cb elba  problem that of interest  and we're going to use that to construct  turing machine deciding atm for our  contradiction  okay so here's s  supposedly deciding atm  what it's going to do and this is going  to be the tricky part  s  is going to use m and w to design an  llba  that lba  is going to be built  with the knowledge of m w in fact it's  going to have m and w built into it  so  that's why i'm calling that lba  b sub mw  because it depends on mw  um  as i'll describe and what that lba does  it takes its input  let's call it the x the input to the the  lba  and  examines that input to see  if that input is an accepting  computation history for m on w  then that lba is just looking for  computation histories for mw  and that's the only thing it's going to  ever accept  if you feed it something which is not  a computation history you know that  sequence of configurations  uh for mmw leading to an accept if you  feed it something else  the lba is just going to reject it  it only accepts  the accepting computation history for m  w  and now  if you can build such a thing  then  you can  use that machine  in your emptiness tester to see if it  accepts any strings at all  because the only thing that could  possibly be accepting is an accepting  computation history you don't even know  if there exists one because you don't  know if m accepts w  so you're going to build this lba  um  and uh  which is looking for accepting  computation histories  and then see if its language is empty or  not  if its language is not empty  the only thing it could be accepting is  this computation history so you know m  except w  whereas if the language is empty you  know that there is no computation  history for m w and so m does not accept  the leaf  so that's the whole idea the trick is  how do you build  this uh this lba um  so first you're going to make this lba  which tests its input to see if it's an  accepting computation history for m on w  and you have to build this  um  uh lba without even knowing whether  there is a computation history  um it's not like you can take that  string and just build the string into  the lba because they don't even know if  that string exists  but what you can do  is you can make the lba  um  follow the rules of m it knows w  and knows m so it can test take the its  input and using w and the rules of m see  if that input is a computation history  for m and w  um and that's what it's going to do  so here is  um  this machine i'm going to construct i'll  i'll give it to you written down and  with a little bit more um details of its  procedure but just to kind of illustrate  it so here is a proposed input to b sub  mw this would be the x here this would  be an x that would be accepted  but anything else that would be provided  would not be accepted  so this is the sequence of  configurations written down as a string  that would be the x that i'm providing  to the b c b b sub mw and it's supposed  to be checking this to make sure it's  legit  um so  so on on input x the way it's gonna the  way it's going to proceed  it's going to first check to see whether  x begins in the right way  because you know  this lba it knows m and it knows w  so the very first thing is it takes a  look at the first part of the string up  to the pound sign you know there's no  pound sign if you're going to just feed  junk in it's going to be easily  identifiable as junk  um  so uh  you know if you're going to feed  something so it's going to the lba is  going to take everything up to the first  pound sign and just confirm that that  thing is  the first configuration the starting  configuration of um  of m on w which means it has to start  with the start state and here is w  so just going to check that  um  then it's going to check that each  each one of these guys follows legally  from the previous one according to the  rules of f it's going to first checks  that this one is correct and then this  one leads to that one  according to the rules of m then this  one leads to that one according to the  rules of m  all of that you know the knowledge of m  and w is built in so it can do that  um  and then it just  follows along checking this computation  you know it's easy to check that a  computation is correct that's all that's  really going on here  um  it's going to check that the computation  is correct until it gets to the end and  then make sure that the last uh  configuration that it's been given as  input is an accepting configuration  that there's an accept state in it  um and if everything that passes it it  accepts otherwise it uh rejects  okay  um  and the point is that this is an lba you  don't oh i just kind of jumped ahead of  myself you don't need any additional  information so how does it actually do  this  so you know how do you actually do this  on the tape so i claim that the lba  doesn't need any extra space to do this  um to do this check you know it's just  going to be zigzagging back and forth  here on the on on the  on the input  you know checking that the corresponding  symbols match up except around the the  head position where it gets updated or  correctly  and it seems it may need to mark it will  need to mark on the tape it's allowed to  write on the tape just to make sure it  keeps track of where it is  but this is a very simple  i mean i'm not trying to uh if you're  not following i'm not trying to uh alarm  you but i'm just trying to help you  understand that  this is not a complicated procedure here  to do this check that the that the  actual computation of the that's written  down of the turing machine is valid  um but you know we can deal with a  question here  uh  oh good  now we got some questions  oops  you can be thinking about while i'm  thinking about this you can think about  that uh check-in over there  so here's a good question  going from each configuration to the  next configuration is it a unique  uh next  step  um well i'm assuming that the turing  machine we're starting with is  deterministic so this should be only one  way to go so the answer to that  question is yes  this is going to be a unique  um string here there's really going to  be only one  um  computation history not that it really  matters in a sense um the answer to that  question um  as long as that accepting computation  history corresponds to the machine  actually accepting  uh oh boy we're all  all over the place here uh okay  are you ready to end this um  in this um  in this bowl  uh  all right  well this is where we are  uh  everybody who said they'd rather be in  six or four fig six or four six i know  who you are  you're all  going to be expelled  no i'm joking uh  i don't know who you are so um  uh  yeah good  so um  uh  wha what so we are here at the at the  break  um i and i made this poll on purpose uh  at this moment so we can spend a little  time let me just start our clock going  and i can try to help you those of you  who are who answered see that you're  baffled um  i'll i can try to uh help you understand  what's going on  all right so this is a this is a good  question here um  uh  okay well okay there's several good  questions here so um  somebody asks why don't we just test all  possible strings uh for b remember if  we're trying to test emptiness  for b's language  um  that's the e-lba problem  nor do we just try all possible strings  um  and uh  that would work if we had enough time  but there's infinitely many strings and  so that's not going to be good if we're  trying to be a decider  so  so that is  that's why we don't just try all strings  so but here's this other question here  this is a  okay  so  um  the question is how do we find the input  x  so this is an important question um  because  uh  we don't find the input x  there's no  you know we  the input x um at least the accepted  input x would be the accepting  computation history  um  for m on w  we're never going to find it  you know we're never going to find an x  we're not we're building this um this  lba  not we're never going to run that lba  we're designing that lba not for the  purposes of running it  we're building that lba only for the  purpose  of  uh using the lba language emptiness  tester we're going to build that end lba  and feed it into the machine turing  machine r  which is going to tell us whether that  lba's language is empty we ourselves is  not never going to run that machine  we're never going to come up with an x  we're just having we're we're we are  designing a computation  checker  and then  um  the emptiness tester for that  that we assumed to exist for for  elba is going to say yes there is some  computation  which this machine accepts or no there  is no computation which this machine  accepts and that's going to be a  computation for m on w  and so that's going to tell us whether  or not m accepts w  so we're never going to actually be  finding an x we're never going to be  running that lba  we're just feeding using that lba  as  as an input  to r  um  does the computation history method  always use lbas no  as you'll see uh right after the break  because that's the lbas is just  kind of an easy place to get started but  we're going to use the computation  history method and computation histories  next on the on the post correspondence  problem  yes the the computation histories and  the input x are always going to be  finite  so that that was an answer to a question  about whether these histories or the  inputs are going to be finite or not so  the the they're all those strings are  always going to be finite and the  computation history is going to be  finite  um  so we are at the end of our five minutes  um let me just see if there was another  question i wanted to answer  um  let me let's move on  okay now  getting coming back to the  undecidability of this  post-correspondence problem so remember  the post course upon you know this this  problem you're given those dominoes you  want to know if there's a match here is  a little mini version of the the diagram  if that helps you remember it um  and we're going to prove  that this language is undecidable and so  it's undecidable to test whether  you have  a match given a set of dominoes  whether a match is possible  and we're going to use we're going to  reduce atm to to this language using the  computation history method so how in the  world are we going to do that  um  first of all there's a little detail  here i want to mention  uh just  uh um  don't focus on this but  i'm going to assume the matches always  start  with the very first domino on the list  um the very first domino in the  collection  so this is going to be like a starting  domino this is going to make my proof a  little simpler to do it that way and  then you can fix that assumption uh  later and if we have time at the end  i'll do that or maybe after the lecture  is over if there are questions i can  show you how to fix that assumption but  for now  um to simplify the proof we're going to  assume that there is a starting  domino that the matches always have to  start with that particular domino  um  if you didn't follow that point don't  worry you can just ignore it you'll see  where it comes up later  so um now we're going to reduce atm to  the pcp  so assuming we have a machine that  decides the pcp we're going to use that  to make a machine that decides the atm  as before so here's the turing machine s  which is going to decide atm  um  and the way we're going to do it is like  this  we're given m and w we want to know  as always  does m accept w  what we're going to do is we're going to  build an instance  of the pcp problem so we're going to  build a bun a collection of dominoes  [Music]  uh which are going to dip  and that collection is going to be built  knowing m w so that's going to affect  the dominoes we're going to create so  this collection of dominoes is going to  be called p sub mw  depends on an m w  and  finding a match for this set of w  is is going to force you to simulate  m on w because the match is going to  correspond to a computation history for  m on w  so uh we're going to use  once we do that  um once we build  uh the set of dominoes who's where a  match corresponds to a computation  history  we're going to use r to determine  whether or not there is a match  or in other words whether or not there  is a computation history which is  whether or not m accepts w  um so if there is a match we're going to  accept because we know m accepts w  and if and if there is no uh match we're  going to reject  because we know m does not accept w  okay um  this is the plan i haven't told you how  to do this yet  i mean i'm worried about the  significant number of you who are  feeling um  confused by the method i mean you should  you guys should be uh texting me in the  tas  um to try to at least get a sense of how  this is working i mean that we're going  to be going  we're going to be doing the method again  um it's just going to be a little more  complicated because we have to also deal  with these dominoes and all this all  that stuff and that's why i presented it  to the fir to you the first time  in the setting of the lbas which where  in a sense the method comes out  perhaps more simply  so a match is going to correspond to an  accepting computation history  um sometimes i  if i don't use the word accepting  um  you know i'm just  being a little sloppy  computation history and accepting  computation history for us right now  they're the same  i mean later on we may actually talk  about rejecting computation histories  but  let's not  get ourselves confused  all the except computation histories  always have to end with the machine  accepting  okay uh somebody's asking what does it  mean for a match to correspond to a  computation history you'll see that's  going to be on my next slide  oh so this is a good question um  is my step two trying to use r to  determine whether m accepts w  well yes but i'm doing that by testing  whether whether this these dominoes have  a match  because r  is besides pcp i can only use r to test  whether things have a match  because someone asked  should step two be to use r to determine  whether m accepts w  well in effect that's what it's doing  but it's doing indirectly through  um testing whether this  um pcp whether these dominoes have a  match because those dominoes are going  to force you to simulate mlw  okay i think i'm repeating myself here  so let's uh avoid getting into a loop on  the lecture and um  uh see how we actually build piece of mw  so now you understand what you have to  under have the plan in your mind  uh  um  we are given m w  we're trying to make a set of dominoes  where a match is going to be  a computation history for m w so the  string you're going to get in that match  the top string and the bottom string  which are going to be you have to have  to have to match  um there they're going to be computation  histories  but they're going to be a computation  history of m w so i want to figure out  how to make my dominoes force that  okay  so my starting domino as i told you  there's going to be a special starting  domino in my collection so which is  going to require the match to start with  that domino it's going to be this one  here  it's going to have these two strings in  it  and you can see already it's starting to  look like a  a computation history  the dominoes going to have that  feature  so  uh it's the pair of strings and i've  written it kind of to help you see  what's kind of going on it's a pound  sign on the top  and the  the starting configuration for mmw on  the bottom  and what i'm going to do for you here  is at the bottom of the slide  i'm going to take the dominoes i've  written down so far and try to be  building a match and you'll see how that  match is forcing a simulation of the  machine  okay  so let's take this as an illustration  let's assume the input to m  so i'm running m on w now i'm trying to  see does m accept w  um w is a string two two three  so the  start configuration for m on w  is  q zero that's the starting state for m  and then w following is 2 2 3 that's  what it's going to appear on the on the  tape of the turing machine to start off  so this is the start configuration  for on w  and  assuming that i had this particular  string is input string to w  and so uh given the match  the dominoes that i've given you so far  just one  um this is how the match is going to  start  uh now there's going to be more dominoes  coming  so for every possible tape symbol and  state  don't get confused by the language here  this is the important thing if if in the  turing machine  um and i'll just read this in english to  you if the turing machine when it's in  state q  and the head is reading an a  it moves to state r  writes a b at that point  and its head moves to the right  i'm focusing on right rightward moving  uh  turing machine steps right now  but for every possible  uh  state and and tape symbol  and looking at where what happens  i'm going to have a  domino that's going to capture this  information  and it's going to be this domino here  q a  on top  br on the bottom so just the string qa  on the top  br on the bottom  so let's see why with that sort of  arcane looking  why is that a good thing to why is that  a useful domino to put in here  well if you take a look  um  let's assume from my turing machine  when it's in state q0 which is the  starting state and the head is reading a  two which it will happen to be reading  because the  uh this the string w starts with a two  if it goes into state q4 no q7 and  writes a four and then moves right  i'm going to have in this in this domino  i'm going to have  q02 on top  and 4  q7 on the bottom  because 4 is what i wrote right here  that's you know that's the b  and this and the new state that i'm  going into is q7  so that's going to be this  the domino that i'm going to get  and here it is here's that domino  appearing in the match  so um  it's going to take  match up with q q 0 2  um  which you know  you know don't forget the the the top  string has to equal the bottom string so  and this is going to be the only choice  that i have for extending the match so  q02 um is going to be on the top when i  put that there but that's going to force  um 427 to appear at the bottom because  that's what's going to be the bottom  string corresponding to the q02 on the  top  okay  so  if you're looking down here this is the  the beginning of the second  configuration of the machine that's what  i want to be happening  i want to be  for  that match should look like a  computation history  so now um so this is going to be all  possible right moves are going to be  handled in this way it's going to be a  similar process for the left moves but  let me not you know i'll leave that to  your imagination  now  how do i continue on from there what  does the rest of this configuration look  like  well that should just be a copying over  of the remaining symbols that were on  the tape because they don't change  things only change around the head  the rest of those symbols which is the  two and the three those should just get  copied over here so i'm gonna have two  additional  i'm gonna have additional uh symbols in  my dominoes in my uh  in my collection so for every  um  tape symbol i'm gonna have a a  be a  domino so and for every type symbol a  i'm going to have a a be a domino in my  collection  and so that says  uh  i can have  uh so that there's going to be a 2 2  domino so i can match up this 2 over  here but that forces me to put a 2 down  over there  there's also going to be a 3 3 domino  which is going to match up with that 3  but it's going to force me to push it  put a 3 down over there that's the only  way i can extend this  um  match that i've built so far  i have no choice  those are the only  dominoes that i'm going to be given  and by and so doing i'm forcing you to  basically simulate the machine  now what i want to have happen next is a  pound sign to appear here and that will  conclude my second configuration  so there's going to be a pound sign  pound sign domino as well because  there's a pound sign here  it's going to get matched with a pound  sign up top forces a pound sign to come  down um on the bottom  and if you look at where we are right  now we're exactly like where we were at  the beginning when we had just this  first domino appearing but now we're one  configuration later  so if you understood that  and i admit it it's it's  you know this is a there's just one idea  here um  [Music]  and  once you get the idea  it's all trivial it's all very simple  but there's just you have to get that  idea i'm trying to figure out how to get  that idea into your head once you get  the idea you can write all this stuff  yourself  um  so uh  so following the this uh the this  description of how uh the you know the  the transition function of the machine  works and copying over uh the symbols  from the tape to the next configuration  i'm going to be able to get  configuration after configuration going  until i get to a point when there's an  accept  and now from the machine's perspective  we're done  the machine has accepted its input this  is our computation history but is it a  match  well  you know up until this this bottom thing  um is the computation history but it's  not a match because the top doesn't  equal the bottom there's still extra  stuff at the bottom which the top is not  uh  you know  it doesn't have  so what i'm going to need to do now is  is add some additional kind of  pseudo steps of the machine  where i'm going to allow  the top to catch up to the bottom  and the way i'm going to be thinking  about that and this is not real for a  turning machine as much as the turn  machine's real anyway but you're going  to imagine i'm going to add  a new kind of uh move to the machine  which is going to allow the head to eat  the symbols off the tape  like pac-man  okay and the way i'm gonna get that  effect um  on the top if i have any tape symbol  next to a cue except on either side  on the top  what i'm going to get you to write on  the bottom is just to accept with that  taste with that tape symbol gone  and by repeating that  you know move after move the the the  actual tape is going to be shrinking the  symbols on the tape are going to be  going down one by one move by move until  there's nothing left except just the um  you accept all by itself  okay so i have this sort of pac-man idea  of eating the symbols on the tape um  and then  finally i get to a point when there's  just the cue except  alone on the tape there's no symbols  left to consume  and then i'm going to add one last  domino here which is q accept pound  pound matching with just pound which  just conveniently  finishes off the um  uh the  uh the match  and so the match is completed  there's actually a little detail here i  i hesitate even to bring it up because i  i think it's a kind of thing where if  you understand everything up to this  point you could fill it in yourself but  just for completeness sake  uh you have to deal with the situation  when the the  uh the head  um of the turing machine  might move into the blank portion of the  tape  which is not taken into account here and  the way we get that effect is by having  another domino here which allows me to  add blank symbols at the bottom as  needed  that's just a detail  you know i'm more worried that you  understand the underlying concept um  so here's going to be um  uh  a check-in um  trying to remember  um  okay but you know  so okay so  what else can we conclude from this  information  we know so at this point we know that  pcp is undecidable that's what we just  finished proving  what else do we know  if anything  or do we even know that  uh see my picture is a little bit  covering the check in but it's still  readable  so i mean this is not an easy concept to  get the to get the idea but once you get  it  you know it's um  you know it's you'll see it's not that  bad  okay  another 15 seconds  so this portion doesn't this question is  not really relying so much on the  computation history method it's just  relying on the fact about  pcp being undecidable  um  [Music]  okay everybody  almost done  five seconds  all right  um  so i can see that there is uh  some level of confusion um  uh so the reason why b is correct  is that  we know pcp is undecidable  but  we also know that it's recognizable  because you could try all possible  uh ways of combining dominoes  and  one after the  next um and you accept if you ever find  a match  so they might go forever but if there is  a match  uh possible you'll find it eventually so  pcp is a recognizable language  undecidable and so therefore we know  it's complement it has to be  unrecognizable because you know that's  something we've shown before if a  language and its complement are both  recognizable then it's decidable but we  know pcp is not decidable so they both  both sides can't be recognizable  um okay so let's  let me prove to you one last theorem  that's going to be useful for your  homework  uh involving the computation history  method so you'll have one last chance  to get the  way we're going to use this though this  one is going to be a little bit more  similar in spirit  to the lba version  uh than to the pcp version which has  this extra kind of complication about  coding the computations of the machine  into dominoes here um you know we're  going to operate with another automaton  which where it's going to be more  straightforward but anyway okay getting  ahead of myself  so if you remember for the for  context-free grammars we had the  problem the emptiness problem  we showed that was decidable  so testing whether a context-free  grammar's language is empty  is decidable  however testing whether a context-free  grammar's language is everything whether  it's equal to sigma star  that turns out to be um  uh that turns out to be undecidable  okay so emptiness testing for  context-free grammars decidable  sigma-star testing  undecidable  um  okay so  we'll show that atm is reducible to all  pda via the computation history method  um  and uh  so uh assume we have same patterns  assuming we have a decider for all pda  and make a decider for ulti  decider for atm  here's the atm decider  and now  similar to what we did for the  uh  lba case but with a with a twist  we're going to make a push down  automaton  that's going to check its input to see  whether it's an accepting computation  history of m w just like the lba did if  you think back to how that worked  remember remember you know we um the lba  tested its input to see whether it's uh  an accepting computation history accept  it if it did and then we test the lba's  language for emptiness to see if there  are any computation histories so we're  doing the same kind of thing  except now  uh the pda is going to test its input to  see whether it's a an accepting  computation history and if it is it's  going to reject  it's going to do  the reverse of what the lba  did  and that's going to turn out to be  necessary but for the for the moment  let's just go with it and maybe we'll  see in the proof why the proof needs it  to be this way  so um this p pushdown automaton is going  to accept its input if it's not  an accepting computation history for m  on w otherwise it will accept so you can  think of this pda  as  accepting all the junk  it just  doesn't accept  um the good stuff the accepting  computation history  okay it's it's just it it's uh accepting  all the things which fail  to be an accepting computation history  okay and then once we have that we we  test whether  ours language is everything but whether  bmw's languages everything using r  because it if if that pda's language is  everything  then we know um  uh  there could not be an accepting  computation history because that's the  one thing that gets um  not accepted that's the one thing that  gets rejected  so if it's accepting everything there is  not going to be an accepting computation  history  so if there is no  uh if this is equal to sigma star  then there couldn't be an accepting  computation history and so we're going  to accept  okay so how is this going to work so  what's different now  about uh this case  uh from the lba case is remember the lba  uh got you know in  the the computation history on its input  and it used its ability to write on the  tape to check that each  configuration follows the next one  we don't have the ability to write on  the tape and push on automaton and by  the way i hope you're all comfortable  with my using pdas instead of grammars  because we can interchange one to the  other um i should have mentioned that  when when i did it but  um  so the the pda is going to be using its  stack to compare each configuration with  the next one  okay  so the way that's going to happen is  it's going to non-deterministically  take one of these so the very first  thing it does  is as before it checks to make sure that  the beginning  of the input is the start configuration  but then once that's that's the easy  part but once we get going with that we  can we push each configuration well we  first of all we now deterministically  choose which configuration might be the  bet the one that fails  um  where  where one fails to go to the next one  because those are the ones we're trying  to accept when there's a failure  so we now deterministically look for the  place that there's a failure we push  that onto the stack  um and then we pop it off the stack and  compare it with the next configuration  so that's how we're using the stack  instead of being able to mark on the  input  now there's a so i'm kind of going to  illustrate that here  so  you know we're going as we're going to  read this thing here and we're going to  put it onto the stack so this thing  moves over to here  and  this input here got put onto the stack  you know q 0 w 1 w 2 it's q 0 w 1 w 2  you see that that first configuration is  now sitting on the stack now as we're  going to pop it off we're going to match  it with the second configuration  now if you're following me  you'll realize that there's a there's a  difficulty here because it's coming out  in reverse  you know it comes out in the reverse  order that we put it in and that's not  what we need to do so what we're going  to do here and here's a little sort of a  twist we're going to change the way  we're writing down computation histories  by making them by writing them uh  reversing the even numbered ones so this  one here is going to be written down in  reverse  and that's perfectly okay we can write  down configuration computation histories  in any way we want  to um to meet our needs so we're going  to reverse we're going to reverse the  alternate ones  and now when we push one it's going to  come off in the right order to compare  with the next one  and so that's how the the procedure  works  okay  we're running a little short on time um  you need to be let's see so let let me  just go to um  uh  quick recap the computation history  method is useful for showing the  undecidability of problems when you're  testing for the existence of an object  each of those four cases that we showed  today involving testing whether  something exists  so is there an integer solution to the  polynomial  is there some string in the language is  there a string that's not in the  language  um or is there a match  so that's a typical case  when um  this computation history method comes up  and so as a quick review  uh these are the things we showed today  okay so why don't we uh uh we're we're  just out of time um and so i'm i'm gonna  let you go um  but i will stick around uh for another  five minutes or ten minutes uh happy to  answer any questions if you if you have  them um but that's that's officially the  end of the lecture  um okay  so let's let me try to get to there's a  lot of questions in the chat don't  forget you know write to the tas too  so  if m does not accept w  what will happen to the computation  history so if m does not accept w  um  uh then there is no computation there is  no accepting computation history that's  the only kind of computation histories  we're considering  so if m does not accept w there is no  accepting computation history there's no  computation history  um so i don't know what it means what  will happen to it it just doesn't  doesn't exist  um  so uh  i hope that's  helpful um  don't we need to be able to add states  to the end of the tape  i don't know what that means  it's taped to the end of the tape  i i don't know you'll have to repeat  that one sorry or explain that better  um  ah  so this is a good this is a very good  question here where does the previous  proof fail  when trying to reduce atm to  ecfg  it's got to fail  because ecfg is  decidable um  so that that's a very good question  maybe we can just go back to that  last slide here  because i was running a little short on  time i didn't really focus on why we  have to accept  um  the non-computation history strings we  why are we accepting all the junk  strings  and  only rejecting  um  the strings which are the uh the  computation histories so you know we're  looking for strings that fail  you know if the string could fail  because it starts wrong  string it doesn't have the start  configurat configuration or it doesn't  maybe doesn't end with the accepting  configuration or maybe one configuration  doesn't lead properly to the next one  any of those cases are a failure  and there are going to be reason to  accept  the reason why we can do do it do that  um is because the we're taking advantage  of the push downs non-determinism  we don't know where the  um the failure  might occur  so we're going to non-deterministically  guess where that failure occurs  if we were going to flip this around and  say  let's accept only the good ones and  reject everything else we would have to  test that each configuration led to the  next one  so we have to check them all  so  if something fails it only has to fail  in one place  and then you know we can accept if it um  if it's a good computation history  we have to  it has to be good everywhere  and so the push down automaton really if  you're going to get down to the  nitty-gritty the push on automaton can  check that this configuration leads to  that configuration  using pushing on the stack and then  popping it  but now by the time you get to here you  want to check that this second  configuration leads to the third one  you would need to push the second  configuration onto the stack but at this  point you're already  at after the second configuration so we  cannot back up and push the second one  on the stack  so the the push on tom dot is not able  to check um in the positive sense that  you have a computation history it's only  able to and accept them it's only to  check in the negative sense that you  don't have a con  a computation history  and uh accept all the ones that fail  and that's just because it only has to  fail in one place  i hope that helps um  so that's why we couldn't flip this  around and make this proof work for the  emptiness problem and only have to work  for the the everything problem  the old problem  all right um  okay so bye bye everybody and i will see  you on tuesday
 

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302
00:09:27,110 --> 00:09:27,120

 

303
00:09:27,120 --> 00:09:28,470


304
00:09:28,470 --> 00:09:31,269

 

305
00:09:31,269 --> 00:09:32,790

 

306
00:09:32,790 --> 00:09:36,790

 

307
00:09:36,790 --> 00:09:38,470

 

308
00:09:38,470 --> 00:09:39,910

 

309
00:09:39,910 --> 00:09:43,110

 

310
00:09:43,110 --> 00:09:45,750

 

311
00:09:45,750 --> 00:09:47,750

 

312
00:09:47,750 --> 00:09:47,760

 

313
00:09:47,760 --> 00:09:49,030


314
00:09:49,030 --> 00:09:51,910

 

315
00:09:51,910 --> 00:09:54,070

 

316
00:09:54,070 --> 00:09:55,750

 

317
00:09:55,750 --> 00:09:59,670

 

318
00:09:59,670 --> 00:10:01,670

 

319
00:10:01,670 --> 00:10:03,670

 

320
00:10:03,670 --> 00:10:06,949

 

321
00:10:06,949 --> 00:10:09,269

 

322
00:10:09,269 --> 00:10:11,110

 

323
00:10:11,110 --> 00:10:14,550

 

324
00:10:14,550 --> 00:10:16,310

 

325
00:10:16,310 --> 00:10:18,389

 

326
00:10:18,389 --> 00:10:21,430

 

327
00:10:21,430 --> 00:10:24,790

 

328
00:10:24,790 --> 00:10:27,509

 

329
00:10:27,509 --> 00:10:29,350

 

330
00:10:29,350 --> 00:10:31,269

 

331
00:10:31,269 --> 00:10:32,470

 

332
00:10:32,470 --> 00:10:35,190

 

333
00:10:35,190 --> 00:10:37,110

 

334
00:10:37,110 --> 00:10:37,120

 

335
00:10:37,120 --> 00:10:38,630


336
00:10:38,630 --> 00:10:41,190

 

337
00:10:41,190 --> 00:10:42,949

 

338
00:10:42,949 --> 00:10:45,269

 

339
00:10:45,269 --> 00:10:48,870

 

340
00:10:48,870 --> 00:10:52,389

 

341
00:10:52,389 --> 00:10:55,430

 

342
00:10:55,430 --> 00:10:57,590

 

343
00:10:57,590 --> 00:10:59,030

 

344
00:10:59,030 --> 00:11:01,350

 

345
00:11:01,350 --> 00:11:03,430

 

346
00:11:03,430 --> 00:11:05,030

 

347
00:11:05,030 --> 00:11:06,790

 

348
00:11:06,790 --> 00:11:08,230

 

349
00:11:08,230 --> 00:11:11,269

 

350
00:11:11,269 --> 00:11:13,509

 

351
00:11:13,509 --> 00:11:15,590

 

352
00:11:15,590 --> 00:11:18,710

 

353
00:11:18,710 --> 00:11:21,910

 

354
00:11:21,910 --> 00:11:24,630

 

355
00:11:24,630 --> 00:11:26,790

 

356
00:11:26,790 --> 00:11:31,829

 

357
00:11:31,829 --> 00:11:34,230

 

358
00:11:34,230 --> 00:11:37,190

 

359
00:11:37,190 --> 00:11:39,110

 

360
00:11:39,110 --> 00:11:40,949

 

361
00:11:40,949 --> 00:11:40,959

 

362
00:11:40,959 --> 00:11:42,630


363
00:11:42,630 --> 00:11:44,230

 

364
00:11:44,230 --> 00:11:46,069

 

365
00:11:46,069 --> 00:11:47,990

 

366
00:11:47,990 --> 00:11:50,230

 

367
00:11:50,230 --> 00:11:53,030

 

368
00:11:53,030 --> 00:11:56,710

 

369
00:11:56,710 --> 00:11:59,030

 

370
00:11:59,030 --> 00:12:00,310

 

371
00:12:00,310 --> 00:12:03,030

 

372
00:12:03,030 --> 00:12:03,040

 

373
00:12:03,040 --> 00:12:04,389


374
00:12:04,389 --> 00:12:06,230

 

375
00:12:06,230 --> 00:12:08,310

 

376
00:12:08,310 --> 00:12:11,990

 

377
00:12:11,990 --> 00:12:13,910

 

378
00:12:13,910 --> 00:12:16,069

 

379
00:12:16,069 --> 00:12:18,310

 

380
00:12:18,310 --> 00:12:20,310

 

381
00:12:20,310 --> 00:12:22,310

 

382
00:12:22,310 --> 00:12:24,790

 

383
00:12:24,790 --> 00:12:26,550

 

384
00:12:26,550 --> 00:12:28,389

 

385
00:12:28,389 --> 00:12:30,710

 

386
00:12:30,710 --> 00:12:33,430

 

387
00:12:33,430 --> 00:12:35,430

 

388
00:12:35,430 --> 00:12:36,829

 

389
00:12:36,829 --> 00:12:39,590

 

390
00:12:39,590 --> 00:12:39,600

 

391
00:12:39,600 --> 00:12:40,389


392
00:12:40,389 --> 00:12:41,990

 

393
00:12:41,990 --> 00:12:45,190

 

394
00:12:45,190 --> 00:12:47,430

 

395
00:12:47,430 --> 00:12:49,110

 

396
00:12:49,110 --> 00:12:50,550

 

397
00:12:50,550 --> 00:12:53,030

 

398
00:12:53,030 --> 00:12:55,430

 

399
00:12:55,430 --> 00:12:55,440

 

400
00:12:55,440 --> 00:12:56,470


401
00:12:56,470 --> 00:12:56,480

 

402
00:12:56,480 --> 00:12:57,430


403
00:12:57,430 --> 00:12:58,870

 

404
00:12:58,870 --> 00:13:00,949

 

405
00:13:00,949 --> 00:13:00,959

 

406
00:13:00,959 --> 00:13:02,230


407
00:13:02,230 --> 00:13:04,230

 

408
00:13:04,230 --> 00:13:07,030

 

409
00:13:07,030 --> 00:13:09,590

 

410
00:13:09,590 --> 00:13:12,150

 

411
00:13:12,150 --> 00:13:13,430

 

412
00:13:13,430 --> 00:13:14,949

 

413
00:13:14,949 --> 00:13:16,150

 

414
00:13:16,150 --> 00:13:18,230

 

415
00:13:18,230 --> 00:13:20,550

 

416
00:13:20,550 --> 00:13:22,790

 

417
00:13:22,790 --> 00:13:25,190

 

418
00:13:25,190 --> 00:13:25,200

 

419
00:13:25,200 --> 00:13:27,269


420
00:13:27,269 --> 00:13:28,949

 

421
00:13:28,949 --> 00:13:30,870

 

422
00:13:30,870 --> 00:13:33,190

 

423
00:13:33,190 --> 00:13:33,200

 

424
00:13:33,200 --> 00:13:34,629


425
00:13:34,629 --> 00:13:35,670

 

426
00:13:35,670 --> 00:13:37,750

 

427
00:13:37,750 --> 00:13:39,750

 

428
00:13:39,750 --> 00:13:41,110

 

429
00:13:41,110 --> 00:13:42,470

 

430
00:13:42,470 --> 00:13:45,030

 

431
00:13:45,030 --> 00:13:46,829

 

432
00:13:46,829 --> 00:13:48,870

 

433
00:13:48,870 --> 00:13:48,880

 

434
00:13:48,880 --> 00:13:50,949


435
00:13:50,949 --> 00:13:50,959

 

436
00:13:50,959 --> 00:13:51,750


437
00:13:51,750 --> 00:13:53,910

 

438
00:13:53,910 --> 00:13:56,870

 

439
00:13:56,870 --> 00:14:00,550

 

440
00:14:00,550 --> 00:14:00,560

 

441
00:14:00,560 --> 00:14:02,710


442
00:14:02,710 --> 00:14:02,720

 

443
00:14:02,720 --> 00:14:03,910


444
00:14:03,910 --> 00:14:07,350

 

445
00:14:07,350 --> 00:14:09,269

 

446
00:14:09,269 --> 00:14:11,509

 

447
00:14:11,509 --> 00:14:13,509

 

448
00:14:13,509 --> 00:14:16,310

 

449
00:14:16,310 --> 00:14:17,829

 

450
00:14:17,829 --> 00:14:19,670

 

451
00:14:19,670 --> 00:14:21,750

 

452
00:14:21,750 --> 00:14:21,760

 

453
00:14:21,760 --> 00:14:22,949


454
00:14:22,949 --> 00:14:24,550

 

455
00:14:24,550 --> 00:14:26,710

 

456
00:14:26,710 --> 00:14:30,790

 

457
00:14:30,790 --> 00:14:34,150

 

458
00:14:34,150 --> 00:14:35,829

 

459
00:14:35,829 --> 00:14:37,509

 

460
00:14:37,509 --> 00:14:40,150

 

461
00:14:40,150 --> 00:14:42,389

 

462
00:14:42,389 --> 00:14:48,310

 

463
00:14:48,310 --> 00:14:50,150

 

464
00:14:50,150 --> 00:14:50,160

 

465
00:14:50,160 --> 00:14:51,030


466
00:14:51,030 --> 00:14:54,790

 

467
00:14:54,790 --> 00:14:58,150

 

468
00:14:58,150 --> 00:14:59,750

 

469
00:14:59,750 --> 00:15:01,350

 

470
00:15:01,350 --> 00:15:03,110

 

471
00:15:03,110 --> 00:15:03,120

 

472
00:15:03,120 --> 00:15:04,550


473
00:15:04,550 --> 00:15:04,560

 

474
00:15:04,560 --> 00:15:05,509


475
00:15:05,509 --> 00:15:07,590

 

476
00:15:07,590 --> 00:15:09,990

 

477
00:15:09,990 --> 00:15:14,389

 

478
00:15:14,389 --> 00:15:16,310

 

479
00:15:16,310 --> 00:15:18,069

 

480
00:15:18,069 --> 00:15:19,590

 

481
00:15:19,590 --> 00:15:20,870

 

482
00:15:20,870 --> 00:15:22,389

 

483
00:15:22,389 --> 00:15:22,399

 

484
00:15:22,399 --> 00:15:27,110


485
00:15:27,110 --> 00:15:27,120

 

486
00:15:27,120 --> 00:15:27,910


487
00:15:27,910 --> 00:15:27,920

 

488
00:15:27,920 --> 00:15:30,230


489
00:15:30,230 --> 00:15:33,189

 

490
00:15:33,189 --> 00:15:33,199

 

491
00:15:33,199 --> 00:15:34,550


492
00:15:34,550 --> 00:15:34,560

 

493
00:15:34,560 --> 00:15:35,350


494
00:15:35,350 --> 00:15:38,870

 

495
00:15:38,870 --> 00:15:41,749

 

496
00:15:41,749 --> 00:15:41,759

 

497
00:15:41,759 --> 00:15:42,790


498
00:15:42,790 --> 00:15:42,800

 

499
00:15:42,800 --> 00:15:43,670


500
00:15:43,670 --> 00:15:46,069

 

501
00:15:46,069 --> 00:15:46,079

 

502
00:15:46,079 --> 00:15:47,030


503
00:15:47,030 --> 00:15:49,110

 

504
00:15:49,110 --> 00:15:51,350

 

505
00:15:51,350 --> 00:15:52,710

 

506
00:15:52,710 --> 00:15:52,720

 

507
00:15:52,720 --> 00:15:53,590


508
00:15:53,590 --> 00:15:53,600

 

509
00:15:53,600 --> 00:15:54,790


510
00:15:54,790 --> 00:15:57,749

 

511
00:15:57,749 --> 00:16:00,069

 

512
00:16:00,069 --> 00:16:02,790

 

513
00:16:02,790 --> 00:16:05,430

 

514
00:16:05,430 --> 00:16:06,790

 

515
00:16:06,790 --> 00:16:09,110

 

516
00:16:09,110 --> 00:16:11,189

 

517
00:16:11,189 --> 00:16:13,910

 

518
00:16:13,910 --> 00:16:16,150

 

519
00:16:16,150 --> 00:16:16,160

 

520
00:16:16,160 --> 00:16:18,230


521
00:16:18,230 --> 00:16:21,350

 

522
00:16:21,350 --> 00:16:22,870

 

523
00:16:22,870 --> 00:16:22,880

 

524
00:16:22,880 --> 00:16:25,269


525
00:16:25,269 --> 00:16:27,269

 

526
00:16:27,269 --> 00:16:28,949

 

527
00:16:28,949 --> 00:16:32,389

 

528
00:16:32,389 --> 00:16:34,629

 

529
00:16:34,629 --> 00:16:36,389

 

530
00:16:36,389 --> 00:16:36,399

 

531
00:16:36,399 --> 00:16:38,389


532
00:16:38,389 --> 00:16:39,990

 

533
00:16:39,990 --> 00:16:43,110

 

534
00:16:43,110 --> 00:16:43,120

 

535
00:16:43,120 --> 00:16:43,990


536
00:16:43,990 --> 00:16:44,000

 

537
00:16:44,000 --> 00:16:46,069


538
00:16:46,069 --> 00:16:48,949

 

539
00:16:48,949 --> 00:16:52,470

 

540
00:16:52,470 --> 00:16:54,629

 

541
00:16:54,629 --> 00:16:55,749

 

542
00:16:55,749 --> 00:16:57,110

 

543
00:16:57,110 --> 00:16:59,350

 

544
00:16:59,350 --> 00:17:01,110

 

545
00:17:01,110 --> 00:17:02,790

 

546
00:17:02,790 --> 00:17:05,110

 

547
00:17:05,110 --> 00:17:06,549

 

548
00:17:06,549 --> 00:17:09,829

 

549
00:17:09,829 --> 00:17:11,429

 

550
00:17:11,429 --> 00:17:13,429

 

551
00:17:13,429 --> 00:17:15,110

 

552
00:17:15,110 --> 00:17:16,470

 

553
00:17:16,470 --> 00:17:18,789

 

554
00:17:18,789 --> 00:17:21,510

 

555
00:17:21,510 --> 00:17:23,350

 

556
00:17:23,350 --> 00:17:24,710

 

557
00:17:24,710 --> 00:17:26,710

 

558
00:17:26,710 --> 00:17:29,510

 

559
00:17:29,510 --> 00:17:31,110

 

560
00:17:31,110 --> 00:17:32,870

 

561
00:17:32,870 --> 00:17:32,880

 

562
00:17:32,880 --> 00:17:34,870


563
00:17:34,870 --> 00:17:36,230

 

564
00:17:36,230 --> 00:17:36,240

 

565
00:17:36,240 --> 00:17:38,230


566
00:17:38,230 --> 00:17:38,240

 

567
00:17:38,240 --> 00:17:39,110


568
00:17:39,110 --> 00:17:41,590

 

569
00:17:41,590 --> 00:17:43,620

 

570
00:17:43,620 --> 00:17:43,630

 

571
00:17:43,630 --> 00:17:44,830


572
00:17:44,830 --> 00:17:48,150

 

573
00:17:48,150 --> 00:17:48,160

 

574
00:17:48,160 --> 00:17:50,150


575
00:17:50,150 --> 00:17:51,669

 

576
00:17:51,669 --> 00:17:53,830

 

577
00:17:53,830 --> 00:17:56,070

 

578
00:17:56,070 --> 00:17:56,080

 

579
00:17:56,080 --> 00:17:57,190


580
00:17:57,190 --> 00:17:59,590

 

581
00:17:59,590 --> 00:18:01,029

 

582
00:18:01,029 --> 00:18:01,039

 

583
00:18:01,039 --> 00:18:03,430


584
00:18:03,430 --> 00:18:03,440

 

585
00:18:03,440 --> 00:18:04,230


586
00:18:04,230 --> 00:18:05,750

 

587
00:18:05,750 --> 00:18:06,710

 

588
00:18:06,710 --> 00:18:08,789

 

589
00:18:08,789 --> 00:18:11,110

 

590
00:18:11,110 --> 00:18:12,470

 

591
00:18:12,470 --> 00:18:12,480

 

592
00:18:12,480 --> 00:18:13,510


593
00:18:13,510 --> 00:18:14,950

 

594
00:18:14,950 --> 00:18:14,960

 

595
00:18:14,960 --> 00:18:16,549


596
00:18:16,549 --> 00:18:18,310

 

597
00:18:18,310 --> 00:18:21,830

 

598
00:18:21,830 --> 00:18:23,430

 

599
00:18:23,430 --> 00:18:25,669

 

600
00:18:25,669 --> 00:18:28,070

 

601
00:18:28,070 --> 00:18:29,909

 

602
00:18:29,909 --> 00:18:32,710

 

603
00:18:32,710 --> 00:18:34,390

 

604
00:18:34,390 --> 00:18:36,710

 

605
00:18:36,710 --> 00:18:38,549

 

606
00:18:38,549 --> 00:18:41,029

 

607
00:18:41,029 --> 00:18:42,710

 

608
00:18:42,710 --> 00:18:42,720

 

609
00:18:42,720 --> 00:18:43,350


610
00:18:43,350 --> 00:18:43,360

 

611
00:18:43,360 --> 00:18:44,310


612
00:18:44,310 --> 00:18:44,320

 

613
00:18:44,320 --> 00:18:45,430


614
00:18:45,430 --> 00:18:48,070

 

615
00:18:48,070 --> 00:18:49,990

 

616
00:18:49,990 --> 00:18:51,350

 

617
00:18:51,350 --> 00:18:53,270

 

618
00:18:53,270 --> 00:18:55,270

 

619
00:18:55,270 --> 00:18:57,510

 

620
00:18:57,510 --> 00:18:57,520

 

621
00:18:57,520 --> 00:18:59,909


622
00:18:59,909 --> 00:19:02,310

 

623
00:19:02,310 --> 00:19:04,070

 

624
00:19:04,070 --> 00:19:06,710

 

625
00:19:06,710 --> 00:19:08,950

 

626
00:19:08,950 --> 00:19:10,950

 

627
00:19:10,950 --> 00:19:12,549

 

628
00:19:12,549 --> 00:19:14,710

 

629
00:19:14,710 --> 00:19:16,549

 

630
00:19:16,549 --> 00:19:18,470

 

631
00:19:18,470 --> 00:19:20,230

 

632
00:19:20,230 --> 00:19:20,240

 

633
00:19:20,240 --> 00:19:21,270


634
00:19:21,270 --> 00:19:23,029

 

635
00:19:23,029 --> 00:19:25,029

 

636
00:19:25,029 --> 00:19:26,950

 

637
00:19:26,950 --> 00:19:29,510

 

638
00:19:29,510 --> 00:19:32,310

 

639
00:19:32,310 --> 00:19:32,320

 

640
00:19:32,320 --> 00:19:33,510


641
00:19:33,510 --> 00:19:36,390

 

642
00:19:36,390 --> 00:19:37,669

 

643
00:19:37,669 --> 00:19:40,549

 

644
00:19:40,549 --> 00:19:40,559

 

645
00:19:40,559 --> 00:19:42,710


646
00:19:42,710 --> 00:19:44,470

 

647
00:19:44,470 --> 00:19:45,830

 

648
00:19:45,830 --> 00:19:48,710

 

649
00:19:48,710 --> 00:19:49,510

 

650
00:19:49,510 --> 00:19:51,669

 

651
00:19:51,669 --> 00:19:54,310

 

652
00:19:54,310 --> 00:19:56,230

 

653
00:19:56,230 --> 00:19:58,310

 

654
00:19:58,310 --> 00:19:58,320

 

655
00:19:58,320 --> 00:19:59,350


656
00:19:59,350 --> 00:20:01,110

 

657
00:20:01,110 --> 00:20:03,270

 

658
00:20:03,270 --> 00:20:05,430

 

659
00:20:05,430 --> 00:20:07,110

 

660
00:20:07,110 --> 00:20:08,789

 

661
00:20:08,789 --> 00:20:08,799

 

662
00:20:08,799 --> 00:20:09,669


663
00:20:09,669 --> 00:20:11,350

 

664
00:20:11,350 --> 00:20:11,360

 

665
00:20:11,360 --> 00:20:12,149


666
00:20:12,149 --> 00:20:14,950

 

667
00:20:14,950 --> 00:20:17,350

 

668
00:20:17,350 --> 00:20:19,750

 

669
00:20:19,750 --> 00:20:19,760

 

670
00:20:19,760 --> 00:20:21,110


671
00:20:21,110 --> 00:20:22,870

 

672
00:20:22,870 --> 00:20:25,029

 

673
00:20:25,029 --> 00:20:26,149

 

674
00:20:26,149 --> 00:20:28,310

 

675
00:20:28,310 --> 00:20:28,320

 

676
00:20:28,320 --> 00:20:29,270


677
00:20:29,270 --> 00:20:31,110

 

678
00:20:31,110 --> 00:20:33,510

 

679
00:20:33,510 --> 00:20:35,669

 

680
00:20:35,669 --> 00:20:37,909

 

681
00:20:37,909 --> 00:20:41,990

 

682
00:20:41,990 --> 00:20:43,510

 

683
00:20:43,510 --> 00:20:45,750

 

684
00:20:45,750 --> 00:20:47,990

 

685
00:20:47,990 --> 00:20:50,470

 

686
00:20:50,470 --> 00:20:52,950

 

687
00:20:52,950 --> 00:20:55,190

 

688
00:20:55,190 --> 00:20:58,630

 

689
00:20:58,630 --> 00:21:00,549

 

690
00:21:00,549 --> 00:21:02,950

 

691
00:21:02,950 --> 00:21:02,960

 

692
00:21:02,960 --> 00:21:04,070


693
00:21:04,070 --> 00:21:05,510

 

694
00:21:05,510 --> 00:21:07,750

 

695
00:21:07,750 --> 00:21:11,590

 

696
00:21:11,590 --> 00:21:13,669

 

697
00:21:13,669 --> 00:21:15,270

 

698
00:21:15,270 --> 00:21:17,190

 

699
00:21:17,190 --> 00:21:19,350

 

700
00:21:19,350 --> 00:21:19,360

 

701
00:21:19,360 --> 00:21:20,310


702
00:21:20,310 --> 00:21:21,909

 

703
00:21:21,909 --> 00:21:23,270

 

704
00:21:23,270 --> 00:21:25,110

 

705
00:21:25,110 --> 00:21:26,470

 

706
00:21:26,470 --> 00:21:26,480

 

707
00:21:26,480 --> 00:21:28,870


708
00:21:28,870 --> 00:21:28,880

 

709
00:21:28,880 --> 00:21:31,029


710
00:21:31,029 --> 00:21:33,270

 

711
00:21:33,270 --> 00:21:35,350

 

712
00:21:35,350 --> 00:21:35,360

 

713
00:21:35,360 --> 00:21:36,630


714
00:21:36,630 --> 00:21:38,230

 

715
00:21:38,230 --> 00:21:41,669

 

716
00:21:41,669 --> 00:21:43,990

 

717
00:21:43,990 --> 00:21:45,830

 

718
00:21:45,830 --> 00:21:47,909

 

719
00:21:47,909 --> 00:21:47,919

 

720
00:21:47,919 --> 00:21:48,710


721
00:21:48,710 --> 00:21:51,270

 

722
00:21:51,270 --> 00:21:52,950

 

723
00:21:52,950 --> 00:21:55,270

 

724
00:21:55,270 --> 00:21:57,270

 

725
00:21:57,270 --> 00:21:59,669

 

726
00:21:59,669 --> 00:22:00,710

 

727
00:22:00,710 --> 00:22:03,190

 

728
00:22:03,190 --> 00:22:05,669

 

729
00:22:05,669 --> 00:22:07,590

 

730
00:22:07,590 --> 00:22:11,190

 

731
00:22:11,190 --> 00:22:12,710

 

732
00:22:12,710 --> 00:22:13,909

 

733
00:22:13,909 --> 00:22:16,870

 

734
00:22:16,870 --> 00:22:18,390

 

735
00:22:18,390 --> 00:22:20,870

 

736
00:22:20,870 --> 00:22:20,880

 

737
00:22:20,880 --> 00:22:22,789


738
00:22:22,789 --> 00:22:23,909

 

739
00:22:23,909 --> 00:22:26,870

 

740
00:22:26,870 --> 00:22:29,510

 

741
00:22:29,510 --> 00:22:31,430

 

742
00:22:31,430 --> 00:22:33,590

 

743
00:22:33,590 --> 00:22:34,789

 

744
00:22:34,789 --> 00:22:36,710

 

745
00:22:36,710 --> 00:22:36,720

 

746
00:22:36,720 --> 00:22:38,230


747
00:22:38,230 --> 00:22:38,240

 

748
00:22:38,240 --> 00:22:39,510


749
00:22:39,510 --> 00:22:40,470

 

750
00:22:40,470 --> 00:22:45,510

 

751
00:22:45,510 --> 00:22:47,909

 

752
00:22:47,909 --> 00:22:49,669

 

753
00:22:49,669 --> 00:22:51,830

 

754
00:22:51,830 --> 00:22:54,310

 

755
00:22:54,310 --> 00:22:57,350

 

756
00:22:57,350 --> 00:22:59,669

 

757
00:22:59,669 --> 00:23:01,909

 

758
00:23:01,909 --> 00:23:03,270

 

759
00:23:03,270 --> 00:23:05,510

 

760
00:23:05,510 --> 00:23:06,789

 

761
00:23:06,789 --> 00:23:10,149

 

762
00:23:10,149 --> 00:23:11,350

 

763
00:23:11,350 --> 00:23:14,070

 

764
00:23:14,070 --> 00:23:15,029

 

765
00:23:15,029 --> 00:23:17,350

 

766
00:23:17,350 --> 00:23:18,789

 

767
00:23:18,789 --> 00:23:21,510

 

768
00:23:21,510 --> 00:23:24,149

 

769
00:23:24,149 --> 00:23:24,159

 

770
00:23:24,159 --> 00:23:24,870


771
00:23:24,870 --> 00:23:26,390

 

772
00:23:26,390 --> 00:23:28,390

 

773
00:23:28,390 --> 00:23:30,230

 

774
00:23:30,230 --> 00:23:31,669

 

775
00:23:31,669 --> 00:23:34,230

 

776
00:23:34,230 --> 00:23:36,390

 

777
00:23:36,390 --> 00:23:38,870

 

778
00:23:38,870 --> 00:23:38,880

 

779
00:23:38,880 --> 00:23:40,549


780
00:23:40,549 --> 00:23:42,230

 

781
00:23:42,230 --> 00:23:43,590

 

782
00:23:43,590 --> 00:23:46,070

 

783
00:23:46,070 --> 00:23:48,390

 

784
00:23:48,390 --> 00:23:50,310

 

785
00:23:50,310 --> 00:23:52,390

 

786
00:23:52,390 --> 00:23:54,310

 

787
00:23:54,310 --> 00:23:56,630

 

788
00:23:56,630 --> 00:23:58,070

 

789
00:23:58,070 --> 00:24:00,950

 

790
00:24:00,950 --> 00:24:02,789

 

791
00:24:02,789 --> 00:24:04,870

 

792
00:24:04,870 --> 00:24:05,830

 

793
00:24:05,830 --> 00:24:08,310

 

794
00:24:08,310 --> 00:24:10,230

 

795
00:24:10,230 --> 00:24:11,750

 

796
00:24:11,750 --> 00:24:14,230

 

797
00:24:14,230 --> 00:24:15,669

 

798
00:24:15,669 --> 00:24:17,990

 

799
00:24:17,990 --> 00:24:21,590

 

800
00:24:21,590 --> 00:24:23,750

 

801
00:24:23,750 --> 00:24:25,830

 

802
00:24:25,830 --> 00:24:27,029

 

803
00:24:27,029 --> 00:24:27,039

 

804
00:24:27,039 --> 00:24:28,710


805
00:24:28,710 --> 00:24:30,230

 

806
00:24:30,230 --> 00:24:32,230

 

807
00:24:32,230 --> 00:24:34,789

 

808
00:24:34,789 --> 00:24:36,950

 

809
00:24:36,950 --> 00:24:41,190

 

810
00:24:41,190 --> 00:24:42,789

 

811
00:24:42,789 --> 00:24:44,149

 

812
00:24:44,149 --> 00:24:46,950

 

813
00:24:46,950 --> 00:24:48,710

 

814
00:24:48,710 --> 00:24:53,190

 

815
00:24:53,190 --> 00:24:55,110

 

816
00:24:55,110 --> 00:24:55,120

 

817
00:24:55,120 --> 00:24:56,789


818
00:24:56,789 --> 00:24:58,310

 

819
00:24:58,310 --> 00:25:01,110

 

820
00:25:01,110 --> 00:25:01,120

 

821
00:25:01,120 --> 00:25:02,149


822
00:25:02,149 --> 00:25:03,350

 

823
00:25:03,350 --> 00:25:05,510

 

824
00:25:05,510 --> 00:25:07,350

 

825
00:25:07,350 --> 00:25:09,480

 

826
00:25:09,480 --> 00:25:09,490

 

827
00:25:09,490 --> 00:25:11,750


828
00:25:11,750 --> 00:25:13,669

 

829
00:25:13,669 --> 00:25:16,630

 

830
00:25:16,630 --> 00:25:18,950

 

831
00:25:18,950 --> 00:25:21,190

 

832
00:25:21,190 --> 00:25:23,510

 

833
00:25:23,510 --> 00:25:24,870

 

834
00:25:24,870 --> 00:25:26,870

 

835
00:25:26,870 --> 00:25:29,430

 

836
00:25:29,430 --> 00:25:30,950

 

837
00:25:30,950 --> 00:25:32,310

 

838
00:25:32,310 --> 00:25:34,070

 

839
00:25:34,070 --> 00:25:35,830

 

840
00:25:35,830 --> 00:25:38,070

 

841
00:25:38,070 --> 00:25:41,750

 

842
00:25:41,750 --> 00:25:45,110

 

843
00:25:45,110 --> 00:25:47,100

 

844
00:25:47,100 --> 00:25:47,110

 

845
00:25:47,110 --> 00:25:48,630


846
00:25:48,630 --> 00:25:48,640

 

847
00:25:48,640 --> 00:25:49,430


848
00:25:49,430 --> 00:25:53,350

 

849
00:25:53,350 --> 00:25:57,909

 

850
00:25:57,909 --> 00:26:00,390

 

851
00:26:00,390 --> 00:26:02,630

 

852
00:26:02,630 --> 00:26:05,909

 

853
00:26:05,909 --> 00:26:08,549

 

854
00:26:08,549 --> 00:26:10,950

 

855
00:26:10,950 --> 00:26:10,960

 

856
00:26:10,960 --> 00:26:12,070


857
00:26:12,070 --> 00:26:14,070

 

858
00:26:14,070 --> 00:26:15,909

 

859
00:26:15,909 --> 00:26:17,830

 

860
00:26:17,830 --> 00:26:21,029

 

861
00:26:21,029 --> 00:26:22,830

 

862
00:26:22,830 --> 00:26:26,390

 

863
00:26:26,390 --> 00:26:28,789

 

864
00:26:28,789 --> 00:26:30,630

 

865
00:26:30,630 --> 00:26:33,510

 

866
00:26:33,510 --> 00:26:35,029

 

867
00:26:35,029 --> 00:26:38,310

 

868
00:26:38,310 --> 00:26:40,230

 

869
00:26:40,230 --> 00:26:42,230

 

870
00:26:42,230 --> 00:26:44,870

 

871
00:26:44,870 --> 00:26:48,310

 

872
00:26:48,310 --> 00:26:49,590

 

873
00:26:49,590 --> 00:26:49,600

 

874
00:26:49,600 --> 00:26:50,630


875
00:26:50,630 --> 00:26:52,630

 

876
00:26:52,630 --> 00:26:53,990

 

877
00:26:53,990 --> 00:26:57,430

 

878
00:26:57,430 --> 00:26:59,190

 

879
00:26:59,190 --> 00:27:00,630

 

880
00:27:00,630 --> 00:27:02,149

 

881
00:27:02,149 --> 00:27:02,159

 

882
00:27:02,159 --> 00:27:03,190


883
00:27:03,190 --> 00:27:03,200

 

884
00:27:03,200 --> 00:27:04,549


885
00:27:04,549 --> 00:27:06,789

 

886
00:27:06,789 --> 00:27:07,750

 

887
00:27:07,750 --> 00:27:09,590

 

888
00:27:09,590 --> 00:27:10,710

 

889
00:27:10,710 --> 00:27:13,510

 

890
00:27:13,510 --> 00:27:14,870

 

891
00:27:14,870 --> 00:27:17,350

 

892
00:27:17,350 --> 00:27:20,549

 

893
00:27:20,549 --> 00:27:23,350

 

894
00:27:23,350 --> 00:27:25,830

 

895
00:27:25,830 --> 00:27:27,590

 

896
00:27:27,590 --> 00:27:30,710

 

897
00:27:30,710 --> 00:27:32,389

 

898
00:27:32,389 --> 00:27:35,029

 

899
00:27:35,029 --> 00:27:36,070

 

900
00:27:36,070 --> 00:27:38,470

 

901
00:27:38,470 --> 00:27:38,480

 

902
00:27:38,480 --> 00:27:40,549


903
00:27:40,549 --> 00:27:41,750

 

904
00:27:41,750 --> 00:27:41,760

 

905
00:27:41,760 --> 00:27:43,510


906
00:27:43,510 --> 00:27:45,669

 

907
00:27:45,669 --> 00:27:47,350

 

908
00:27:47,350 --> 00:27:50,149

 

909
00:27:50,149 --> 00:27:52,870

 

910
00:27:52,870 --> 00:27:59,990

 

911
00:27:59,990 --> 00:28:03,669

 

912
00:28:03,669 --> 00:28:08,230

 

913
00:28:08,230 --> 00:28:08,240

 

914
00:28:08,240 --> 00:28:10,149


915
00:28:10,149 --> 00:28:11,990

 

916
00:28:11,990 --> 00:28:13,430

 

917
00:28:13,430 --> 00:28:15,029

 

918
00:28:15,029 --> 00:28:19,190

 

919
00:28:19,190 --> 00:28:21,990

 

920
00:28:21,990 --> 00:28:23,110

 

921
00:28:23,110 --> 00:28:25,590

 

922
00:28:25,590 --> 00:28:27,669

 

923
00:28:27,669 --> 00:28:30,070

 

924
00:28:30,070 --> 00:28:31,909

 

925
00:28:31,909 --> 00:28:33,830

 

926
00:28:33,830 --> 00:28:33,840

 

927
00:28:33,840 --> 00:28:34,710


928
00:28:34,710 --> 00:28:37,269

 

929
00:28:37,269 --> 00:28:39,430

 

930
00:28:39,430 --> 00:28:39,440

 

931
00:28:39,440 --> 00:28:41,830


932
00:28:41,830 --> 00:28:43,830

 

933
00:28:43,830 --> 00:28:46,389

 

934
00:28:46,389 --> 00:28:50,870

 

935
00:28:50,870 --> 00:28:50,880

 

936
00:28:50,880 --> 00:28:52,149


937
00:28:52,149 --> 00:28:54,070

 

938
00:28:54,070 --> 00:28:54,080

 

939
00:28:54,080 --> 00:28:55,510


940
00:28:55,510 --> 00:28:58,549

 

941
00:28:58,549 --> 00:29:00,149

 

942
00:29:00,149 --> 00:29:02,230

 

943
00:29:02,230 --> 00:29:05,909

 

944
00:29:05,909 --> 00:29:08,389

 

945
00:29:08,389 --> 00:29:11,190

 

946
00:29:11,190 --> 00:29:12,630

 

947
00:29:12,630 --> 00:29:15,110

 

948
00:29:15,110 --> 00:29:18,630

 

949
00:29:18,630 --> 00:29:20,470

 

950
00:29:20,470 --> 00:29:22,950

 

951
00:29:22,950 --> 00:29:25,110

 

952
00:29:25,110 --> 00:29:26,789

 

953
00:29:26,789 --> 00:29:26,799

 

954
00:29:26,799 --> 00:29:27,830


955
00:29:27,830 --> 00:29:29,990

 

956
00:29:29,990 --> 00:29:32,230

 

957
00:29:32,230 --> 00:29:34,950

 

958
00:29:34,950 --> 00:29:36,789

 

959
00:29:36,789 --> 00:29:39,110

 

960
00:29:39,110 --> 00:29:41,269

 

961
00:29:41,269 --> 00:29:43,350

 

962
00:29:43,350 --> 00:29:43,360

 

963
00:29:43,360 --> 00:29:44,230


964
00:29:44,230 --> 00:29:46,070

 

965
00:29:46,070 --> 00:29:48,149

 

966
00:29:48,149 --> 00:29:49,510

 

967
00:29:49,510 --> 00:29:51,669

 

968
00:29:51,669 --> 00:29:53,110

 

969
00:29:53,110 --> 00:29:55,190

 

970
00:29:55,190 --> 00:29:55,200

 

971
00:29:55,200 --> 00:29:56,549


972
00:29:56,549 --> 00:29:58,389

 

973
00:29:58,389 --> 00:29:58,399

 

974
00:29:58,399 --> 00:30:01,510


975
00:30:01,510 --> 00:30:03,350

 

976
00:30:03,350 --> 00:30:05,430

 

977
00:30:05,430 --> 00:30:08,310

 

978
00:30:08,310 --> 00:30:11,750

 

979
00:30:11,750 --> 00:30:14,230

 

980
00:30:14,230 --> 00:30:16,389

 

981
00:30:16,389 --> 00:30:18,549

 

982
00:30:18,549 --> 00:30:20,389

 

983
00:30:20,389 --> 00:30:22,950

 

984
00:30:22,950 --> 00:30:24,549

 

985
00:30:24,549 --> 00:30:26,230

 

986
00:30:26,230 --> 00:30:26,240

 

987
00:30:26,240 --> 00:30:27,190


988
00:30:27,190 --> 00:30:29,590

 

989
00:30:29,590 --> 00:30:31,830

 

990
00:30:31,830 --> 00:30:33,669

 

991
00:30:33,669 --> 00:30:35,669

 

992
00:30:35,669 --> 00:30:37,830

 

993
00:30:37,830 --> 00:30:39,590

 

994
00:30:39,590 --> 00:30:40,710

 

995
00:30:40,710 --> 00:30:42,389

 

996
00:30:42,389 --> 00:30:42,399

 

997
00:30:42,399 --> 00:30:43,669


998
00:30:43,669 --> 00:30:45,909

 

999
00:30:45,909 --> 00:30:49,669

 

1000
00:30:49,669 --> 00:30:51,430

 

1001
00:30:51,430 --> 00:30:55,430

 

1002
00:30:55,430 --> 00:30:56,950

 

1003
00:30:56,950 --> 00:30:56,960

 

1004
00:30:56,960 --> 00:30:59,750


1005
00:30:59,750 --> 00:31:03,350

 

1006
00:31:03,350 --> 00:31:06,389

 

1007
00:31:06,389 --> 00:31:09,590

 

1008
00:31:09,590 --> 00:31:11,990

 

1009
00:31:11,990 --> 00:31:14,389

 

1010
00:31:14,389 --> 00:31:14,399

 

1011
00:31:14,399 --> 00:31:15,190


1012
00:31:15,190 --> 00:31:15,200

 

1013
00:31:15,200 --> 00:31:15,440


1014
00:31:15,440 --> 00:31:15,450

 

1015
00:31:15,450 --> 00:31:16,950


1016
00:31:16,950 --> 00:31:21,430

 

1017
00:31:21,430 --> 00:31:23,190

 

1018
00:31:23,190 --> 00:31:24,870

 

1019
00:31:24,870 --> 00:31:27,509

 

1020
00:31:27,509 --> 00:31:29,830

 

1021
00:31:29,830 --> 00:31:31,750

 

1022
00:31:31,750 --> 00:31:34,470

 

1023
00:31:34,470 --> 00:31:36,230

 

1024
00:31:36,230 --> 00:31:37,990

 

1025
00:31:37,990 --> 00:31:39,990

 

1026
00:31:39,990 --> 00:31:42,070

 

1027
00:31:42,070 --> 00:31:44,630

 

1028
00:31:44,630 --> 00:31:44,640

 

1029
00:31:44,640 --> 00:31:45,669


1030
00:31:45,669 --> 00:31:46,950

 

1031
00:31:46,950 --> 00:31:48,950

 

1032
00:31:48,950 --> 00:31:51,669

 

1033
00:31:51,669 --> 00:31:53,990

 

1034
00:31:53,990 --> 00:31:55,110

 

1035
00:31:55,110 --> 00:31:56,630

 

1036
00:31:56,630 --> 00:31:59,509

 

1037
00:31:59,509 --> 00:31:59,519

 

1038
00:31:59,519 --> 00:32:00,789


1039
00:32:00,789 --> 00:32:02,870

 

1040
00:32:02,870 --> 00:32:03,990

 

1041
00:32:03,990 --> 00:32:06,950

 

1042
00:32:06,950 --> 00:32:08,389

 

1043
00:32:08,389 --> 00:32:09,750

 

1044
00:32:09,750 --> 00:32:14,470

 

1045
00:32:14,470 --> 00:32:14,480

 

1046
00:32:14,480 --> 00:32:16,710


1047
00:32:16,710 --> 00:32:19,750

 

1048
00:32:19,750 --> 00:32:21,350

 

1049
00:32:21,350 --> 00:32:23,110

 

1050
00:32:23,110 --> 00:32:24,470

 

1051
00:32:24,470 --> 00:32:26,789

 

1052
00:32:26,789 --> 00:32:28,389

 

1053
00:32:28,389 --> 00:32:32,310

 

1054
00:32:32,310 --> 00:32:32,320

 

1055
00:32:32,320 --> 00:32:35,029


1056
00:32:35,029 --> 00:32:38,070

 

1057
00:32:38,070 --> 00:32:41,110

 

1058
00:32:41,110 --> 00:32:43,269

 

1059
00:32:43,269 --> 00:32:45,430

 

1060
00:32:45,430 --> 00:32:47,990

 

1061
00:32:47,990 --> 00:32:49,590

 

1062
00:32:49,590 --> 00:32:51,269

 

1063
00:32:51,269 --> 00:32:51,279

 

1064
00:32:51,279 --> 00:32:53,029


1065
00:32:53,029 --> 00:32:54,389

 

1066
00:32:54,389 --> 00:32:56,149

 

1067
00:32:56,149 --> 00:32:56,159

 

1068
00:32:56,159 --> 00:32:58,789


1069
00:32:58,789 --> 00:33:01,750

 

1070
00:33:01,750 --> 00:33:03,509

 

1071
00:33:03,509 --> 00:33:05,909

 

1072
00:33:05,909 --> 00:33:09,029

 

1073
00:33:09,029 --> 00:33:12,549

 

1074
00:33:12,549 --> 00:33:14,549

 

1075
00:33:14,549 --> 00:33:18,389

 

1076
00:33:18,389 --> 00:33:19,990

 

1077
00:33:19,990 --> 00:33:23,029

 

1078
00:33:23,029 --> 00:33:24,070

 

1079
00:33:24,070 --> 00:33:27,269

 

1080
00:33:27,269 --> 00:33:28,950

 

1081
00:33:28,950 --> 00:33:31,190

 

1082
00:33:31,190 --> 00:33:31,200

 

1083
00:33:31,200 --> 00:33:32,070


1084
00:33:32,070 --> 00:33:33,350

 

1085
00:33:33,350 --> 00:33:33,360

 

1086
00:33:33,360 --> 00:33:35,430


1087
00:33:35,430 --> 00:33:37,430

 

1088
00:33:37,430 --> 00:33:39,590

 

1089
00:33:39,590 --> 00:33:42,230

 

1090
00:33:42,230 --> 00:33:45,110

 

1091
00:33:45,110 --> 00:33:47,029

 

1092
00:33:47,029 --> 00:33:47,039

 

1093
00:33:47,039 --> 00:33:48,070


1094
00:33:48,070 --> 00:33:50,310

 

1095
00:33:50,310 --> 00:33:53,590

 

1096
00:33:53,590 --> 00:33:56,070

 

1097
00:33:56,070 --> 00:33:57,830

 

1098
00:33:57,830 --> 00:34:00,470

 

1099
00:34:00,470 --> 00:34:04,310

 

1100
00:34:04,310 --> 00:34:05,590

 

1101
00:34:05,590 --> 00:34:07,430

 

1102
00:34:07,430 --> 00:34:09,829

 

1103
00:34:09,829 --> 00:34:12,069

 

1104
00:34:12,069 --> 00:34:15,030

 

1105
00:34:15,030 --> 00:34:17,190

 

1106
00:34:17,190 --> 00:34:20,550

 

1107
00:34:20,550 --> 00:34:22,790

 

1108
00:34:22,790 --> 00:34:22,800

 

1109
00:34:22,800 --> 00:34:23,750


1110
00:34:23,750 --> 00:34:25,589

 

1111
00:34:25,589 --> 00:34:27,909

 

1112
00:34:27,909 --> 00:34:29,430

 

1113
00:34:29,430 --> 00:34:31,430

 

1114
00:34:31,430 --> 00:34:32,950

 

1115
00:34:32,950 --> 00:34:37,109

 

1116
00:34:37,109 --> 00:34:38,470

 

1117
00:34:38,470 --> 00:34:47,349

 

1118
00:34:47,349 --> 00:34:48,869

 

1119
00:34:48,869 --> 00:34:48,879

 

1120
00:34:48,879 --> 00:34:51,190


1121
00:34:51,190 --> 00:34:55,829

 

1122
00:34:55,829 --> 00:34:58,950

 

1123
00:34:58,950 --> 00:35:00,710

 

1124
00:35:00,710 --> 00:35:01,589

 

1125
00:35:01,589 --> 00:35:05,670

 

1126
00:35:05,670 --> 00:35:05,680

 

1127
00:35:05,680 --> 00:35:06,870


1128
00:35:06,870 --> 00:35:11,670

 

1129
00:35:11,670 --> 00:35:13,750

 

1130
00:35:13,750 --> 00:35:15,270

 

1131
00:35:15,270 --> 00:35:16,390

 

1132
00:35:16,390 --> 00:35:18,150

 

1133
00:35:18,150 --> 00:35:18,160

 

1134
00:35:18,160 --> 00:35:19,109


1135
00:35:19,109 --> 00:35:25,270

 

1136
00:35:25,270 --> 00:35:28,790

 

1137
00:35:28,790 --> 00:35:30,710

 

1138
00:35:30,710 --> 00:35:30,720

 

1139
00:35:30,720 --> 00:35:31,990


1140
00:35:31,990 --> 00:35:35,349

 

1141
00:35:35,349 --> 00:35:36,790

 

1142
00:35:36,790 --> 00:35:38,230

 

1143
00:35:38,230 --> 00:35:39,750

 

1144
00:35:39,750 --> 00:35:41,750

 

1145
00:35:41,750 --> 00:35:43,670

 

1146
00:35:43,670 --> 00:35:45,910

 

1147
00:35:45,910 --> 00:35:48,150

 

1148
00:35:48,150 --> 00:35:51,349

 

1149
00:35:51,349 --> 00:35:53,030

 

1150
00:35:53,030 --> 00:35:55,109

 

1151
00:35:55,109 --> 00:35:57,190

 

1152
00:35:57,190 --> 00:35:57,200

 

1153
00:35:57,200 --> 00:35:59,109


1154
00:35:59,109 --> 00:36:01,430

 

1155
00:36:01,430 --> 00:36:04,630

 

1156
00:36:04,630 --> 00:36:05,670

 

1157
00:36:05,670 --> 00:36:08,550

 

1158
00:36:08,550 --> 00:36:08,560

 

1159
00:36:08,560 --> 00:36:09,430


1160
00:36:09,430 --> 00:36:13,750

 

1161
00:36:13,750 --> 00:36:13,760

 

1162
00:36:13,760 --> 00:36:19,430


1163
00:36:19,430 --> 00:36:21,030

 

1164
00:36:21,030 --> 00:36:23,510

 

1165
00:36:23,510 --> 00:36:25,510

 

1166
00:36:25,510 --> 00:36:28,870

 

1167
00:36:28,870 --> 00:36:28,880

 

1168
00:36:28,880 --> 00:36:30,150


1169
00:36:30,150 --> 00:36:33,349

 

1170
00:36:33,349 --> 00:36:36,310

 

1171
00:36:36,310 --> 00:36:38,950

 

1172
00:36:38,950 --> 00:36:38,960

 

1173
00:36:38,960 --> 00:36:40,390


1174
00:36:40,390 --> 00:36:45,750

 

1175
00:36:45,750 --> 00:36:47,990

 

1176
00:36:47,990 --> 00:36:50,230

 

1177
00:36:50,230 --> 00:36:50,240

 

1178
00:36:50,240 --> 00:36:52,069


1179
00:36:52,069 --> 00:36:52,079

 

1180
00:36:52,079 --> 00:36:53,030


1181
00:36:53,030 --> 00:36:55,670

 

1182
00:36:55,670 --> 00:36:57,910

 

1183
00:36:57,910 --> 00:37:02,069

 

1184
00:37:02,069 --> 00:37:04,069

 

1185
00:37:04,069 --> 00:37:08,790

 

1186
00:37:08,790 --> 00:37:10,150

 

1187
00:37:10,150 --> 00:37:17,750

 

1188
00:37:17,750 --> 00:37:20,950

 

1189
00:37:20,950 --> 00:37:22,710

 

1190
00:37:22,710 --> 00:37:24,550

 

1191
00:37:24,550 --> 00:37:27,030

 

1192
00:37:27,030 --> 00:37:28,790

 

1193
00:37:28,790 --> 00:37:31,030

 

1194
00:37:31,030 --> 00:37:32,950

 

1195
00:37:32,950 --> 00:37:35,510

 

1196
00:37:35,510 --> 00:37:35,520

 

1197
00:37:35,520 --> 00:37:37,670


1198
00:37:37,670 --> 00:37:38,950

 

1199
00:37:38,950 --> 00:37:42,630

 

1200
00:37:42,630 --> 00:37:42,640

 

1201
00:37:42,640 --> 00:37:44,630


1202
00:37:44,630 --> 00:37:46,470

 

1203
00:37:46,470 --> 00:37:47,990

 

1204
00:37:47,990 --> 00:37:50,310

 

1205
00:37:50,310 --> 00:37:52,150

 

1206
00:37:52,150 --> 00:37:53,430

 

1207
00:37:53,430 --> 00:37:55,829

 

1208
00:37:55,829 --> 00:37:57,510

 

1209
00:37:57,510 --> 00:37:59,750

 

1210
00:37:59,750 --> 00:38:01,990

 

1211
00:38:01,990 --> 00:38:04,550

 

1212
00:38:04,550 --> 00:38:04,560

 

1213
00:38:04,560 --> 00:38:06,069


1214
00:38:06,069 --> 00:38:07,670

 

1215
00:38:07,670 --> 00:38:09,750

 

1216
00:38:09,750 --> 00:38:12,550

 

1217
00:38:12,550 --> 00:38:14,710

 

1218
00:38:14,710 --> 00:38:14,720

 

1219
00:38:14,720 --> 00:38:15,589


1220
00:38:15,589 --> 00:38:18,390

 

1221
00:38:18,390 --> 00:38:19,990

 

1222
00:38:19,990 --> 00:38:21,910

 

1223
00:38:21,910 --> 00:38:23,349

 

1224
00:38:23,349 --> 00:38:25,670

 

1225
00:38:25,670 --> 00:38:27,750

 

1226
00:38:27,750 --> 00:38:29,990

 

1227
00:38:29,990 --> 00:38:31,270

 

1228
00:38:31,270 --> 00:38:33,190

 

1229
00:38:33,190 --> 00:38:34,470

 

1230
00:38:34,470 --> 00:38:37,750

 

1231
00:38:37,750 --> 00:38:39,510

 

1232
00:38:39,510 --> 00:38:41,670

 

1233
00:38:41,670 --> 00:38:47,270

 

1234
00:38:47,270 --> 00:38:49,910

 

1235
00:38:49,910 --> 00:38:49,920

 

1236
00:38:49,920 --> 00:38:50,950


1237
00:38:50,950 --> 00:38:53,750

 

1238
00:38:53,750 --> 00:38:56,390

 

1239
00:38:56,390 --> 00:38:58,470

 

1240
00:38:58,470 --> 00:39:00,150

 

1241
00:39:00,150 --> 00:39:01,270

 

1242
00:39:01,270 --> 00:39:03,190

 

1243
00:39:03,190 --> 00:39:05,589

 

1244
00:39:05,589 --> 00:39:08,630

 

1245
00:39:08,630 --> 00:39:08,640

 

1246
00:39:08,640 --> 00:39:09,589


1247
00:39:09,589 --> 00:39:12,950

 

1248
00:39:12,950 --> 00:39:16,630

 

1249
00:39:16,630 --> 00:39:20,150

 

1250
00:39:20,150 --> 00:39:22,550

 

1251
00:39:22,550 --> 00:39:24,790

 

1252
00:39:24,790 --> 00:39:27,030

 

1253
00:39:27,030 --> 00:39:29,030

 

1254
00:39:29,030 --> 00:39:29,040

 

1255
00:39:29,040 --> 00:39:29,990


1256
00:39:29,990 --> 00:39:31,829

 

1257
00:39:31,829 --> 00:39:33,910

 

1258
00:39:33,910 --> 00:39:36,950

 

1259
00:39:36,950 --> 00:39:39,030

 

1260
00:39:39,030 --> 00:39:42,550

 

1261
00:39:42,550 --> 00:39:45,270

 

1262
00:39:45,270 --> 00:39:47,990

 

1263
00:39:47,990 --> 00:39:49,990

 

1264
00:39:49,990 --> 00:39:51,829

 

1265
00:39:51,829 --> 00:39:53,030

 

1266
00:39:53,030 --> 00:39:54,870

 

1267
00:39:54,870 --> 00:39:57,589

 

1268
00:39:57,589 --> 00:40:00,470

 

1269
00:40:00,470 --> 00:40:02,550

 

1270
00:40:02,550 --> 00:40:02,560

 

1271
00:40:02,560 --> 00:40:04,710


1272
00:40:04,710 --> 00:40:06,309

 

1273
00:40:06,309 --> 00:40:08,790

 

1274
00:40:08,790 --> 00:40:10,390

 

1275
00:40:10,390 --> 00:40:12,309

 

1276
00:40:12,309 --> 00:40:14,470

 

1277
00:40:14,470 --> 00:40:18,069

 

1278
00:40:18,069 --> 00:40:22,309

 

1279
00:40:22,309 --> 00:40:23,829

 

1280
00:40:23,829 --> 00:40:25,510

 

1281
00:40:25,510 --> 00:40:27,430

 

1282
00:40:27,430 --> 00:40:29,109

 

1283
00:40:29,109 --> 00:40:31,190

 

1284
00:40:31,190 --> 00:40:33,670

 

1285
00:40:33,670 --> 00:40:33,680

 

1286
00:40:33,680 --> 00:40:35,589


1287
00:40:35,589 --> 00:40:37,109

 

1288
00:40:37,109 --> 00:40:37,910

 

1289
00:40:37,910 --> 00:40:39,430

 

1290
00:40:39,430 --> 00:40:40,950

 

1291
00:40:40,950 --> 00:40:43,190

 

1292
00:40:43,190 --> 00:40:44,710

 

1293
00:40:44,710 --> 00:40:46,390

 

1294
00:40:46,390 --> 00:40:49,030

 

1295
00:40:49,030 --> 00:40:50,710

 

1296
00:40:50,710 --> 00:40:53,109

 

1297
00:40:53,109 --> 00:40:55,030

 

1298
00:40:55,030 --> 00:40:55,040

 

1299
00:40:55,040 --> 00:40:55,829


1300
00:40:55,829 --> 00:40:58,870

 

1301
00:40:58,870 --> 00:41:01,510

 

1302
00:41:01,510 --> 00:41:02,870

 

1303
00:41:02,870 --> 00:41:05,190

 

1304
00:41:05,190 --> 00:41:06,550

 

1305
00:41:06,550 --> 00:41:08,630

 

1306
00:41:08,630 --> 00:41:09,990

 

1307
00:41:09,990 --> 00:41:11,430

 

1308
00:41:11,430 --> 00:41:12,950

 

1309
00:41:12,950 --> 00:41:14,870

 

1310
00:41:14,870 --> 00:41:17,750

 

1311
00:41:17,750 --> 00:41:17,760

 

1312
00:41:17,760 --> 00:41:19,030


1313
00:41:19,030 --> 00:41:20,870

 

1314
00:41:20,870 --> 00:41:23,670

 

1315
00:41:23,670 --> 00:41:24,950

 

1316
00:41:24,950 --> 00:41:26,550

 

1317
00:41:26,550 --> 00:41:28,150

 

1318
00:41:28,150 --> 00:41:28,160

 

1319
00:41:28,160 --> 00:41:29,030


1320
00:41:29,030 --> 00:41:30,630

 

1321
00:41:30,630 --> 00:41:32,630

 

1322
00:41:32,630 --> 00:41:34,470

 

1323
00:41:34,470 --> 00:41:36,390

 

1324
00:41:36,390 --> 00:41:38,710

 

1325
00:41:38,710 --> 00:41:41,910

 

1326
00:41:41,910 --> 00:41:44,870

 

1327
00:41:44,870 --> 00:41:47,510

 

1328
00:41:47,510 --> 00:41:47,520

 

1329
00:41:47,520 --> 00:41:48,790


1330
00:41:48,790 --> 00:41:48,800

 

1331
00:41:48,800 --> 00:41:49,750


1332
00:41:49,750 --> 00:41:52,069

 

1333
00:41:52,069 --> 00:41:53,990

 

1334
00:41:53,990 --> 00:41:55,270

 

1335
00:41:55,270 --> 00:41:56,950

 

1336
00:41:56,950 --> 00:41:56,960

 

1337
00:41:56,960 --> 00:41:57,910


1338
00:41:57,910 --> 00:41:59,829

 

1339
00:41:59,829 --> 00:42:02,470

 

1340
00:42:02,470 --> 00:42:05,430

 

1341
00:42:05,430 --> 00:42:07,990

 

1342
00:42:07,990 --> 00:42:10,069

 

1343
00:42:10,069 --> 00:42:12,069

 

1344
00:42:12,069 --> 00:42:15,349

 

1345
00:42:15,349 --> 00:42:17,430

 

1346
00:42:17,430 --> 00:42:19,670

 

1347
00:42:19,670 --> 00:42:22,790

 

1348
00:42:22,790 --> 00:42:22,800

 

1349
00:42:22,800 --> 00:42:23,829


1350
00:42:23,829 --> 00:42:25,910

 

1351
00:42:25,910 --> 00:42:28,069

 

1352
00:42:28,069 --> 00:42:29,750

 

1353
00:42:29,750 --> 00:42:32,710

 

1354
00:42:32,710 --> 00:42:34,710

 

1355
00:42:34,710 --> 00:42:36,710

 

1356
00:42:36,710 --> 00:42:39,430

 

1357
00:42:39,430 --> 00:42:41,510

 

1358
00:42:41,510 --> 00:42:43,030

 

1359
00:42:43,030 --> 00:42:45,349

 

1360
00:42:45,349 --> 00:42:47,990

 

1361
00:42:47,990 --> 00:42:50,470

 

1362
00:42:50,470 --> 00:42:53,829

 

1363
00:42:53,829 --> 00:42:55,990

 

1364
00:42:55,990 --> 00:42:59,030

 

1365
00:42:59,030 --> 00:42:59,040

 

1366
00:42:59,040 --> 00:43:00,790


1367
00:43:00,790 --> 00:43:02,309

 

1368
00:43:02,309 --> 00:43:04,870

 

1369
00:43:04,870 --> 00:43:04,880

 

1370
00:43:04,880 --> 00:43:06,870


1371
00:43:06,870 --> 00:43:08,069

 

1372
00:43:08,069 --> 00:43:09,270

 

1373
00:43:09,270 --> 00:43:13,990

 

1374
00:43:13,990 --> 00:43:16,230

 

1375
00:43:16,230 --> 00:43:17,829

 

1376
00:43:17,829 --> 00:43:21,349

 

1377
00:43:21,349 --> 00:43:22,950

 

1378
00:43:22,950 --> 00:43:22,960

 

1379
00:43:22,960 --> 00:43:23,990


1380
00:43:23,990 --> 00:43:26,470

 

1381
00:43:26,470 --> 00:43:27,510

 

1382
00:43:27,510 --> 00:43:29,910

 

1383
00:43:29,910 --> 00:43:32,230

 

1384
00:43:32,230 --> 00:43:33,589

 

1385
00:43:33,589 --> 00:43:35,270

 

1386
00:43:35,270 --> 00:43:37,349

 

1387
00:43:37,349 --> 00:43:38,870

 

1388
00:43:38,870 --> 00:43:38,880

 

1389
00:43:38,880 --> 00:43:40,470


1390
00:43:40,470 --> 00:43:42,230

 

1391
00:43:42,230 --> 00:43:44,950

 

1392
00:43:44,950 --> 00:43:46,630

 

1393
00:43:46,630 --> 00:43:48,790

 

1394
00:43:48,790 --> 00:43:50,470

 

1395
00:43:50,470 --> 00:43:53,349

 

1396
00:43:53,349 --> 00:43:55,829

 

1397
00:43:55,829 --> 00:43:58,870

 

1398
00:43:58,870 --> 00:44:03,030

 

1399
00:44:03,030 --> 00:44:05,430

 

1400
00:44:05,430 --> 00:44:06,790

 

1401
00:44:06,790 --> 00:44:06,800

 

1402
00:44:06,800 --> 00:44:11,030


1403
00:44:11,030 --> 00:44:12,790

 

1404
00:44:12,790 --> 00:44:14,950

 

1405
00:44:14,950 --> 00:44:14,960

 

1406
00:44:14,960 --> 00:44:16,710


1407
00:44:16,710 --> 00:44:18,150

 

1408
00:44:18,150 --> 00:44:20,309

 

1409
00:44:20,309 --> 00:44:21,589

 

1410
00:44:21,589 --> 00:44:22,950

 

1411
00:44:22,950 --> 00:44:24,550

 

1412
00:44:24,550 --> 00:44:26,790

 

1413
00:44:26,790 --> 00:44:29,750

 

1414
00:44:29,750 --> 00:44:29,760

 

1415
00:44:29,760 --> 00:44:31,910


1416
00:44:31,910 --> 00:44:33,109

 

1417
00:44:33,109 --> 00:44:34,870

 

1418
00:44:34,870 --> 00:44:34,880

 

1419
00:44:34,880 --> 00:44:37,109


1420
00:44:37,109 --> 00:44:39,829

 

1421
00:44:39,829 --> 00:44:39,839

 

1422
00:44:39,839 --> 00:44:40,710


1423
00:44:40,710 --> 00:44:44,470

 

1424
00:44:44,470 --> 00:44:46,069

 

1425
00:44:46,069 --> 00:44:48,790

 

1426
00:44:48,790 --> 00:44:50,630

 

1427
00:44:50,630 --> 00:44:53,030

 

1428
00:44:53,030 --> 00:44:54,950

 

1429
00:44:54,950 --> 00:44:57,430

 

1430
00:44:57,430 --> 00:44:59,349

 

1431
00:44:59,349 --> 00:45:00,630

 

1432
00:45:00,630 --> 00:45:04,069

 

1433
00:45:04,069 --> 00:45:04,079

 

1434
00:45:04,079 --> 00:45:05,990


1435
00:45:05,990 --> 00:45:07,589

 

1436
00:45:07,589 --> 00:45:11,109

 

1437
00:45:11,109 --> 00:45:13,750

 

1438
00:45:13,750 --> 00:45:16,630

 

1439
00:45:16,630 --> 00:45:19,670

 

1440
00:45:19,670 --> 00:45:21,670

 

1441
00:45:21,670 --> 00:45:21,680

 

1442
00:45:21,680 --> 00:45:22,630


1443
00:45:22,630 --> 00:45:25,109

 

1444
00:45:25,109 --> 00:45:27,510

 

1445
00:45:27,510 --> 00:45:27,520

 

1446
00:45:27,520 --> 00:45:28,390


1447
00:45:28,390 --> 00:45:29,829

 

1448
00:45:29,829 --> 00:45:31,270

 

1449
00:45:31,270 --> 00:45:32,950

 

1450
00:45:32,950 --> 00:45:34,230

 

1451
00:45:34,230 --> 00:45:36,950

 

1452
00:45:36,950 --> 00:45:36,960

 

1453
00:45:36,960 --> 00:45:40,550


1454
00:45:40,550 --> 00:45:42,950

 

1455
00:45:42,950 --> 00:45:46,230

 

1456
00:45:46,230 --> 00:45:47,670

 

1457
00:45:47,670 --> 00:45:50,470

 

1458
00:45:50,470 --> 00:45:50,480

 

1459
00:45:50,480 --> 00:45:51,589


1460
00:45:51,589 --> 00:45:51,599

 

1461
00:45:51,599 --> 00:45:54,630


1462
00:45:54,630 --> 00:45:54,640

 

1463
00:45:54,640 --> 00:45:56,069


1464
00:45:56,069 --> 00:45:58,950

 

1465
00:45:58,950 --> 00:45:58,960

 

1466
00:45:58,960 --> 00:46:01,510


1467
00:46:01,510 --> 00:46:03,990

 

1468
00:46:03,990 --> 00:46:04,000

 

1469
00:46:04,000 --> 00:46:05,589


1470
00:46:05,589 --> 00:46:05,599

 

1471
00:46:05,599 --> 00:46:06,550


1472
00:46:06,550 --> 00:46:08,710

 

1473
00:46:08,710 --> 00:46:09,910

 

1474
00:46:09,910 --> 00:46:10,950

 

1475
00:46:10,950 --> 00:46:13,670

 

1476
00:46:13,670 --> 00:46:15,750

 

1477
00:46:15,750 --> 00:46:17,510

 

1478
00:46:17,510 --> 00:46:17,520

 

1479
00:46:17,520 --> 00:46:18,550


1480
00:46:18,550 --> 00:46:20,710

 

1481
00:46:20,710 --> 00:46:22,470

 

1482
00:46:22,470 --> 00:46:24,309

 

1483
00:46:24,309 --> 00:46:26,829

 

1484
00:46:26,829 --> 00:46:26,839

 

1485
00:46:26,839 --> 00:46:28,630


1486
00:46:28,630 --> 00:46:31,670

 

1487
00:46:31,670 --> 00:46:33,589

 

1488
00:46:33,589 --> 00:46:35,829

 

1489
00:46:35,829 --> 00:46:37,910

 

1490
00:46:37,910 --> 00:46:37,920

 

1491
00:46:37,920 --> 00:46:38,870


1492
00:46:38,870 --> 00:46:38,880

 

1493
00:46:38,880 --> 00:46:40,069


1494
00:46:40,069 --> 00:46:43,670

 

1495
00:46:43,670 --> 00:46:46,069

 

1496
00:46:46,069 --> 00:46:47,750

 

1497
00:46:47,750 --> 00:46:49,349

 

1498
00:46:49,349 --> 00:46:50,790

 

1499
00:46:50,790 --> 00:46:53,430

 

1500
00:46:53,430 --> 00:46:55,190

 

1501
00:46:55,190 --> 00:46:57,510

 

1502
00:46:57,510 --> 00:46:59,670

 

1503
00:46:59,670 --> 00:47:02,390

 

1504
00:47:02,390 --> 00:47:02,400

 

1505
00:47:02,400 --> 00:47:04,470


1506
00:47:04,470 --> 00:47:05,510

 

1507
00:47:05,510 --> 00:47:05,520

 

1508
00:47:05,520 --> 00:47:06,950


1509
00:47:06,950 --> 00:47:09,829

 

1510
00:47:09,829 --> 00:47:14,390

 

1511
00:47:14,390 --> 00:47:17,589

 

1512
00:47:17,589 --> 00:47:17,599

 

1513
00:47:17,599 --> 00:47:19,190


1514
00:47:19,190 --> 00:47:21,190

 

1515
00:47:21,190 --> 00:47:23,030

 

1516
00:47:23,030 --> 00:47:24,069

 

1517
00:47:24,069 --> 00:47:26,950

 

1518
00:47:26,950 --> 00:47:28,390

 

1519
00:47:28,390 --> 00:47:30,710

 

1520
00:47:30,710 --> 00:47:32,230

 

1521
00:47:32,230 --> 00:47:33,910

 

1522
00:47:33,910 --> 00:47:35,750

 

1523
00:47:35,750 --> 00:47:38,390

 

1524
00:47:38,390 --> 00:47:38,400

 

1525
00:47:38,400 --> 00:47:39,670


1526
00:47:39,670 --> 00:47:41,190

 

1527
00:47:41,190 --> 00:47:43,750

 

1528
00:47:43,750 --> 00:47:43,760

 

1529
00:47:43,760 --> 00:47:46,710


1530
00:47:46,710 --> 00:47:48,390

 

1531
00:47:48,390 --> 00:47:50,710

 

1532
00:47:50,710 --> 00:47:53,750

 

1533
00:47:53,750 --> 00:47:56,150

 

1534
00:47:56,150 --> 00:47:58,069

 

1535
00:47:58,069 --> 00:47:59,910

 

1536
00:47:59,910 --> 00:48:02,150

 

1537
00:48:02,150 --> 00:48:03,990

 

1538
00:48:03,990 --> 00:48:04,000

 

1539
00:48:04,000 --> 00:48:07,190


1540
00:48:07,190 --> 00:48:09,829

 

1541
00:48:09,829 --> 00:48:11,430

 

1542
00:48:11,430 --> 00:48:11,440

 

1543
00:48:11,440 --> 00:48:12,549


1544
00:48:12,549 --> 00:48:14,230

 

1545
00:48:14,230 --> 00:48:16,150

 

1546
00:48:16,150 --> 00:48:18,230

 

1547
00:48:18,230 --> 00:48:20,790

 

1548
00:48:20,790 --> 00:48:22,230

 

1549
00:48:22,230 --> 00:48:23,589

 

1550
00:48:23,589 --> 00:48:23,599

 

1551
00:48:23,599 --> 00:48:24,549


1552
00:48:24,549 --> 00:48:24,559

 

1553
00:48:24,559 --> 00:48:25,589


1554
00:48:25,589 --> 00:48:28,870

 

1555
00:48:28,870 --> 00:48:30,230

 

1556
00:48:30,230 --> 00:48:35,190

 

1557
00:48:35,190 --> 00:48:35,200

 

1558
00:48:35,200 --> 00:48:36,790


1559
00:48:36,790 --> 00:48:38,549

 

1560
00:48:38,549 --> 00:48:40,069

 

1561
00:48:40,069 --> 00:48:41,270

 

1562
00:48:41,270 --> 00:48:42,470

 

1563
00:48:42,470 --> 00:48:44,549

 

1564
00:48:44,549 --> 00:48:46,549

 

1565
00:48:46,549 --> 00:48:48,470

 

1566
00:48:48,470 --> 00:48:49,829

 

1567
00:48:49,829 --> 00:48:52,950

 

1568
00:48:52,950 --> 00:48:55,670

 

1569
00:48:55,670 --> 00:48:57,589

 

1570
00:48:57,589 --> 00:49:00,870

 

1571
00:49:00,870 --> 00:49:04,470

 

1572
00:49:04,470 --> 00:49:09,750

 

1573
00:49:09,750 --> 00:49:12,309

 

1574
00:49:12,309 --> 00:49:14,790

 

1575
00:49:14,790 --> 00:49:16,069

 

1576
00:49:16,069 --> 00:49:18,710

 

1577
00:49:18,710 --> 00:49:20,630

 

1578
00:49:20,630 --> 00:49:22,150

 

1579
00:49:22,150 --> 00:49:22,160

 

1580
00:49:22,160 --> 00:49:23,589


1581
00:49:23,589 --> 00:49:24,790

 

1582
00:49:24,790 --> 00:49:26,230

 

1583
00:49:26,230 --> 00:49:28,829

 

1584
00:49:28,829 --> 00:49:31,030

 

1585
00:49:31,030 --> 00:49:33,349

 

1586
00:49:33,349 --> 00:49:35,829

 

1587
00:49:35,829 --> 00:49:35,839

 

1588
00:49:35,839 --> 00:49:36,710


1589
00:49:36,710 --> 00:49:39,349

 

1590
00:49:39,349 --> 00:49:41,430

 

1591
00:49:41,430 --> 00:49:41,440

 

1592
00:49:41,440 --> 00:49:43,190


1593
00:49:43,190 --> 00:49:44,549

 

1594
00:49:44,549 --> 00:49:46,069

 

1595
00:49:46,069 --> 00:49:48,390

 

1596
00:49:48,390 --> 00:49:50,069

 

1597
00:49:50,069 --> 00:49:51,990

 

1598
00:49:51,990 --> 00:49:54,309

 

1599
00:49:54,309 --> 00:49:55,750

 

1600
00:49:55,750 --> 00:49:57,910

 

1601
00:49:57,910 --> 00:49:59,190

 

1602
00:49:59,190 --> 00:50:01,030

 

1603
00:50:01,030 --> 00:50:03,349

 

1604
00:50:03,349 --> 00:50:05,109

 

1605
00:50:05,109 --> 00:50:07,270

 

1606
00:50:07,270 --> 00:50:07,280

 

1607
00:50:07,280 --> 00:50:09,349


1608
00:50:09,349 --> 00:50:10,470

 

1609
00:50:10,470 --> 00:50:12,230

 

1610
00:50:12,230 --> 00:50:14,309

 

1611
00:50:14,309 --> 00:50:17,190

 

1612
00:50:17,190 --> 00:50:18,710

 

1613
00:50:18,710 --> 00:50:20,549

 

1614
00:50:20,549 --> 00:50:22,309

 

1615
00:50:22,309 --> 00:50:24,790

 

1616
00:50:24,790 --> 00:50:28,470

 

1617
00:50:28,470 --> 00:50:30,549

 

1618
00:50:30,549 --> 00:50:30,559

 

1619
00:50:30,559 --> 00:50:31,510


1620
00:50:31,510 --> 00:50:33,270

 

1621
00:50:33,270 --> 00:50:33,280

 

1622
00:50:33,280 --> 00:50:36,150


1623
00:50:36,150 --> 00:50:39,109

 

1624
00:50:39,109 --> 00:50:40,390

 

1625
00:50:40,390 --> 00:50:43,829

 

1626
00:50:43,829 --> 00:50:45,829

 

1627
00:50:45,829 --> 00:50:48,549

 

1628
00:50:48,549 --> 00:50:50,309

 

1629
00:50:50,309 --> 00:50:54,050

 

1630
00:50:54,050 --> 00:50:54,060

 

1631
00:50:54,060 --> 00:50:55,589


1632
00:50:55,589 --> 00:50:57,910

 

1633
00:50:57,910 --> 00:51:00,470

 

1634
00:51:00,470 --> 00:51:03,750

 

1635
00:51:03,750 --> 00:51:05,829

 

1636
00:51:05,829 --> 00:51:07,270

 

1637
00:51:07,270 --> 00:51:09,190

 

1638
00:51:09,190 --> 00:51:11,670

 

1639
00:51:11,670 --> 00:51:11,680

 

1640
00:51:11,680 --> 00:51:13,030


1641
00:51:13,030 --> 00:51:16,309

 

1642
00:51:16,309 --> 00:51:19,270

 

1643
00:51:19,270 --> 00:51:21,670

 

1644
00:51:21,670 --> 00:51:23,750

 

1645
00:51:23,750 --> 00:51:26,150

 

1646
00:51:26,150 --> 00:51:29,430

 

1647
00:51:29,430 --> 00:51:30,790

 

1648
00:51:30,790 --> 00:51:33,030

 

1649
00:51:33,030 --> 00:51:35,990

 

1650
00:51:35,990 --> 00:51:38,230

 

1651
00:51:38,230 --> 00:51:38,240

 

1652
00:51:38,240 --> 00:51:40,470


1653
00:51:40,470 --> 00:51:42,390

 

1654
00:51:42,390 --> 00:51:45,030

 

1655
00:51:45,030 --> 00:51:46,710

 

1656
00:51:46,710 --> 00:51:48,470

 

1657
00:51:48,470 --> 00:51:51,430

 

1658
00:51:51,430 --> 00:51:54,309

 

1659
00:51:54,309 --> 00:51:57,109

 

1660
00:51:57,109 --> 00:51:59,589

 

1661
00:51:59,589 --> 00:52:01,109

 

1662
00:52:01,109 --> 00:52:04,790

 

1663
00:52:04,790 --> 00:52:07,270

 

1664
00:52:07,270 --> 00:52:08,870

 

1665
00:52:08,870 --> 00:52:12,390

 

1666
00:52:12,390 --> 00:52:14,470

 

1667
00:52:14,470 --> 00:52:17,030

 

1668
00:52:17,030 --> 00:52:18,470

 

1669
00:52:18,470 --> 00:52:20,710

 

1670
00:52:20,710 --> 00:52:22,710

 

1671
00:52:22,710 --> 00:52:22,720

 

1672
00:52:22,720 --> 00:52:24,150


1673
00:52:24,150 --> 00:52:26,230

 

1674
00:52:26,230 --> 00:52:28,390

 

1675
00:52:28,390 --> 00:52:29,430

 

1676
00:52:29,430 --> 00:52:32,309

 

1677
00:52:32,309 --> 00:52:34,150

 

1678
00:52:34,150 --> 00:52:36,630

 

1679
00:52:36,630 --> 00:52:38,150

 

1680
00:52:38,150 --> 00:52:40,390

 

1681
00:52:40,390 --> 00:52:42,710

 

1682
00:52:42,710 --> 00:52:44,630

 

1683
00:52:44,630 --> 00:52:47,430

 

1684
00:52:47,430 --> 00:52:55,589

 

1685
00:52:55,589 --> 00:52:57,990

 

1686
00:52:57,990 --> 00:53:00,870

 

1687
00:53:00,870 --> 00:53:02,790

 

1688
00:53:02,790 --> 00:53:05,190

 

1689
00:53:05,190 --> 00:53:05,200

 

1690
00:53:05,200 --> 00:53:06,150


1691
00:53:06,150 --> 00:53:08,069

 

1692
00:53:08,069 --> 00:53:09,750

 

1693
00:53:09,750 --> 00:53:11,270

 

1694
00:53:11,270 --> 00:53:13,349

 

1695
00:53:13,349 --> 00:53:14,390

 

1696
00:53:14,390 --> 00:53:15,990

 

1697
00:53:15,990 --> 00:53:17,750

 

1698
00:53:17,750 --> 00:53:17,760

 

1699
00:53:17,760 --> 00:53:18,630


1700
00:53:18,630 --> 00:53:19,589

 

1701
00:53:19,589 --> 00:53:21,190

 

1702
00:53:21,190 --> 00:53:24,390

 

1703
00:53:24,390 --> 00:53:26,150

 

1704
00:53:26,150 --> 00:53:26,160

 

1705
00:53:26,160 --> 00:53:29,510


1706
00:53:29,510 --> 00:53:32,230

 

1707
00:53:32,230 --> 00:53:33,829

 

1708
00:53:33,829 --> 00:53:36,230

 

1709
00:53:36,230 --> 00:53:42,309

 

1710
00:53:42,309 --> 00:53:45,109

 

1711
00:53:45,109 --> 00:53:48,069

 

1712
00:53:48,069 --> 00:53:50,470

 

1713
00:53:50,470 --> 00:53:53,349

 

1714
00:53:53,349 --> 00:53:55,750

 

1715
00:53:55,750 --> 00:53:57,670

 

1716
00:53:57,670 --> 00:53:59,190

 

1717
00:53:59,190 --> 00:54:02,230

 

1718
00:54:02,230 --> 00:54:04,549

 

1719
00:54:04,549 --> 00:54:06,630

 

1720
00:54:06,630 --> 00:54:09,270

 

1721
00:54:09,270 --> 00:54:10,950

 

1722
00:54:10,950 --> 00:54:12,549

 

1723
00:54:12,549 --> 00:54:15,430

 

1724
00:54:15,430 --> 00:54:18,309

 

1725
00:54:18,309 --> 00:54:21,190

 

1726
00:54:21,190 --> 00:54:22,870

 

1727
00:54:22,870 --> 00:54:25,109

 

1728
00:54:25,109 --> 00:54:27,109

 

1729
00:54:27,109 --> 00:54:29,430

 

1730
00:54:29,430 --> 00:54:31,589

 

1731
00:54:31,589 --> 00:54:34,549

 

1732
00:54:34,549 --> 00:54:36,630

 

1733
00:54:36,630 --> 00:54:39,589

 

1734
00:54:39,589 --> 00:54:39,599

 

1735
00:54:39,599 --> 00:54:40,630


1736
00:54:40,630 --> 00:54:40,640

 

1737
00:54:40,640 --> 00:54:41,910


1738
00:54:41,910 --> 00:54:44,470

 

1739
00:54:44,470 --> 00:54:47,910

 

1740
00:54:47,910 --> 00:54:50,789

 

1741
00:54:50,789 --> 00:54:53,349

 

1742
00:54:53,349 --> 00:54:55,510

 

1743
00:54:55,510 --> 00:54:57,109

 

1744
00:54:57,109 --> 00:54:58,630

 

1745
00:54:58,630 --> 00:55:00,150

 

1746
00:55:00,150 --> 00:55:02,230

 

1747
00:55:02,230 --> 00:55:02,240

 

1748
00:55:02,240 --> 00:55:03,349


1749
00:55:03,349 --> 00:55:04,789

 

1750
00:55:04,789 --> 00:55:07,670

 

1751
00:55:07,670 --> 00:55:12,870

 

1752
00:55:12,870 --> 00:55:12,880

 

1753
00:55:12,880 --> 00:55:13,750


1754
00:55:13,750 --> 00:55:15,670

 

1755
00:55:15,670 --> 00:55:16,950

 

1756
00:55:16,950 --> 00:55:18,789

 

1757
00:55:18,789 --> 00:55:20,710

 

1758
00:55:20,710 --> 00:55:22,150

 

1759
00:55:22,150 --> 00:55:22,160

 

1760
00:55:22,160 --> 00:55:23,750


1761
00:55:23,750 --> 00:55:25,270

 

1762
00:55:25,270 --> 00:55:25,280

 

1763
00:55:25,280 --> 00:55:26,870


1764
00:55:26,870 --> 00:55:28,230

 

1765
00:55:28,230 --> 00:55:29,349

 

1766
00:55:29,349 --> 00:55:31,349

 

1767
00:55:31,349 --> 00:55:33,589

 

1768
00:55:33,589 --> 00:55:33,599

 

1769
00:55:33,599 --> 00:55:34,630


1770
00:55:34,630 --> 00:55:34,640

 

1771
00:55:34,640 --> 00:55:35,829


1772
00:55:35,829 --> 00:55:38,309

 

1773
00:55:38,309 --> 00:55:40,710

 

1774
00:55:40,710 --> 00:55:42,870

 

1775
00:55:42,870 --> 00:55:44,630

 

1776
00:55:44,630 --> 00:55:45,829

 

1777
00:55:45,829 --> 00:55:48,950

 

1778
00:55:48,950 --> 00:55:52,710

 

1779
00:55:52,710 --> 00:55:55,510

 

1780
00:55:55,510 --> 00:55:58,470

 

1781
00:55:58,470 --> 00:56:00,390

 

1782
00:56:00,390 --> 00:56:02,710

 

1783
00:56:02,710 --> 00:56:04,710

 

1784
00:56:04,710 --> 00:56:06,470

 

1785
00:56:06,470 --> 00:56:06,480

 

1786
00:56:06,480 --> 00:56:07,829


1787
00:56:07,829 --> 00:56:07,839

 

1788
00:56:07,839 --> 00:56:09,030


1789
00:56:09,030 --> 00:56:10,950

 

1790
00:56:10,950 --> 00:56:13,990

 

1791
00:56:13,990 --> 00:56:15,670

 

1792
00:56:15,670 --> 00:56:17,589

 

1793
00:56:17,589 --> 00:56:21,750

 

1794
00:56:21,750 --> 00:56:23,670

 

1795
00:56:23,670 --> 00:56:26,710

 

1796
00:56:26,710 --> 00:56:26,720

 

1797
00:56:26,720 --> 00:56:27,589


1798
00:56:27,589 --> 00:56:30,390

 

1799
00:56:30,390 --> 00:56:33,510

 

1800
00:56:33,510 --> 00:56:35,109

 

1801
00:56:35,109 --> 00:56:38,150

 

1802
00:56:38,150 --> 00:56:41,510

 

1803
00:56:41,510 --> 00:56:43,109

 

1804
00:56:43,109 --> 00:56:43,119

 

1805
00:56:43,119 --> 00:56:44,549


1806
00:56:44,549 --> 00:56:45,990

 

1807
00:56:45,990 --> 00:56:49,270

 

1808
00:56:49,270 --> 00:56:51,990

 

1809
00:56:51,990 --> 00:56:54,710

 

1810
00:56:54,710 --> 00:56:55,910

 

1811
00:56:55,910 --> 00:56:57,990

 

1812
00:56:57,990 --> 00:56:58,000

 

1813
00:56:58,000 --> 00:57:00,390


1814
00:57:00,390 --> 00:57:02,309

 

1815
00:57:02,309 --> 00:57:02,319

 

1816
00:57:02,319 --> 00:57:04,230


1817
00:57:04,230 --> 00:57:06,390

 

1818
00:57:06,390 --> 00:57:06,400

 

1819
00:57:06,400 --> 00:57:07,430


1820
00:57:07,430 --> 00:57:09,589

 

1821
00:57:09,589 --> 00:57:12,549

 

1822
00:57:12,549 --> 00:57:14,789

 

1823
00:57:14,789 --> 00:57:17,510

 

1824
00:57:17,510 --> 00:57:19,589

 

1825
00:57:19,589 --> 00:57:21,349

 

1826
00:57:21,349 --> 00:57:24,549

 

1827
00:57:24,549 --> 00:57:27,670

 

1828
00:57:27,670 --> 00:57:29,670

 

1829
00:57:29,670 --> 00:57:33,990

 

1830
00:57:33,990 --> 00:57:36,950

 

1831
00:57:36,950 --> 00:57:36,960

 

1832
00:57:36,960 --> 00:57:38,470


1833
00:57:38,470 --> 00:57:41,829

 

1834
00:57:41,829 --> 00:57:45,030

 

1835
00:57:45,030 --> 00:57:45,040

 

1836
00:57:45,040 --> 00:57:46,549


1837
00:57:46,549 --> 00:57:49,030

 

1838
00:57:49,030 --> 00:57:51,109

 

1839
00:57:51,109 --> 00:57:52,829

 

1840
00:57:52,829 --> 00:57:55,510

 

1841
00:57:55,510 --> 00:57:55,520

 

1842
00:57:55,520 --> 00:57:56,870


1843
00:57:56,870 --> 00:57:59,510

 

1844
00:57:59,510 --> 00:58:01,270

 

1845
00:58:01,270 --> 00:58:03,030

 

1846
00:58:03,030 --> 00:58:05,510

 

1847
00:58:05,510 --> 00:58:06,549

 

1848
00:58:06,549 --> 00:58:08,549

 

1849
00:58:08,549 --> 00:58:10,069

 

1850
00:58:10,069 --> 00:58:11,589

 

1851
00:58:11,589 --> 00:58:13,750

 

1852
00:58:13,750 --> 00:58:15,910

 

1853
00:58:15,910 --> 00:58:17,990

 

1854
00:58:17,990 --> 00:58:18,000

 

1855
00:58:18,000 --> 00:58:20,950


1856
00:58:20,950 --> 00:58:23,589

 

1857
00:58:23,589 --> 00:58:25,990

 

1858
00:58:25,990 --> 00:58:28,309

 

1859
00:58:28,309 --> 00:58:30,549

 

1860
00:58:30,549 --> 00:58:31,589

 

1861
00:58:31,589 --> 00:58:35,349

 

1862
00:58:35,349 --> 00:58:38,710

 

1863
00:58:38,710 --> 00:58:41,750

 

1864
00:58:41,750 --> 00:58:44,549

 

1865
00:58:44,549 --> 00:58:45,870

 

1866
00:58:45,870 --> 00:58:48,950

 

1867
00:58:48,950 --> 00:58:51,270

 

1868
00:58:51,270 --> 00:58:53,910

 

1869
00:58:53,910 --> 00:58:56,069

 

1870
00:58:56,069 --> 00:58:58,309

 

1871
00:58:58,309 --> 00:58:59,829

 

1872
00:58:59,829 --> 00:59:01,750

 

1873
00:59:01,750 --> 00:59:03,349

 

1874
00:59:03,349 --> 00:59:05,670

 

1875
00:59:05,670 --> 00:59:07,670

 

1876
00:59:07,670 --> 00:59:12,150

 

1877
00:59:12,150 --> 00:59:13,750

 

1878
00:59:13,750 --> 00:59:15,190

 

1879
00:59:15,190 --> 00:59:18,390

 

1880
00:59:18,390 --> 00:59:18,400

 

1881
00:59:18,400 --> 00:59:20,630


1882
00:59:20,630 --> 00:59:21,990

 

1883
00:59:21,990 --> 00:59:23,829

 

1884
00:59:23,829 --> 00:59:26,710

 

1885
00:59:26,710 --> 00:59:28,789

 

1886
00:59:28,789 --> 00:59:32,150

 

1887
00:59:32,150 --> 00:59:36,309

 

1888
00:59:36,309 --> 00:59:38,870

 

1889
00:59:38,870 --> 00:59:42,069

 

1890
00:59:42,069 --> 00:59:43,910

 

1891
00:59:43,910 --> 00:59:46,230

 

1892
00:59:46,230 --> 00:59:46,240

 

1893
00:59:46,240 --> 00:59:48,150


1894
00:59:48,150 --> 00:59:48,160

 

1895
00:59:48,160 --> 00:59:49,430


1896
00:59:49,430 --> 00:59:49,440

 

1897
00:59:49,440 --> 00:59:50,870


1898
00:59:50,870 --> 00:59:53,190

 

1899
00:59:53,190 --> 00:59:54,870

 

1900
00:59:54,870 --> 00:59:56,710

 

1901
00:59:56,710 --> 00:59:59,270

 

1902
00:59:59,270 --> 01:00:00,390

 

1903
01:00:00,390 --> 01:00:00,400

 

1904
01:00:00,400 --> 01:00:01,190


1905
01:00:01,190 --> 01:00:02,950

 

1906
01:00:02,950 --> 01:00:06,390

 

1907
01:00:06,390 --> 01:00:09,109

 

1908
01:00:09,109 --> 01:00:10,470

 

1909
01:00:10,470 --> 01:00:12,069

 

1910
01:00:12,069 --> 01:00:14,069

 

1911
01:00:14,069 --> 01:00:15,829

 

1912
01:00:15,829 --> 01:00:19,270

 

1913
01:00:19,270 --> 01:00:19,280

 

1914
01:00:19,280 --> 01:00:22,309


1915
01:00:22,309 --> 01:00:24,390

 

1916
01:00:24,390 --> 01:00:26,390

 

1917
01:00:26,390 --> 01:00:26,400

 

1918
01:00:26,400 --> 01:00:27,589


1919
01:00:27,589 --> 01:00:29,990

 

1920
01:00:29,990 --> 01:00:31,910

 

1921
01:00:31,910 --> 01:00:33,670

 

1922
01:00:33,670 --> 01:00:36,470

 

1923
01:00:36,470 --> 01:00:37,990

 

1924
01:00:37,990 --> 01:00:39,910

 

1925
01:00:39,910 --> 01:00:42,230

 

1926
01:00:42,230 --> 01:00:42,240

 

1927
01:00:42,240 --> 01:00:43,510


1928
01:00:43,510 --> 01:00:45,750

 

1929
01:00:45,750 --> 01:00:48,549

 

1930
01:00:48,549 --> 01:00:51,270

 

1931
01:00:51,270 --> 01:00:51,280

 

1932
01:00:51,280 --> 01:00:52,549


1933
01:00:52,549 --> 01:00:55,750

 

1934
01:00:55,750 --> 01:00:58,549

 

1935
01:00:58,549 --> 01:01:01,030

 

1936
01:01:01,030 --> 01:01:03,190

 

1937
01:01:03,190 --> 01:01:03,200

 

1938
01:01:03,200 --> 01:01:04,870


1939
01:01:04,870 --> 01:01:06,230

 

1940
01:01:06,230 --> 01:01:06,240

 

1941
01:01:06,240 --> 01:01:07,829


1942
01:01:07,829 --> 01:01:08,950

 

1943
01:01:08,950 --> 01:01:10,870

 

1944
01:01:10,870 --> 01:01:13,109

 

1945
01:01:13,109 --> 01:01:15,190

 

1946
01:01:15,190 --> 01:01:16,230

 

1947
01:01:16,230 --> 01:01:18,549

 

1948
01:01:18,549 --> 01:01:20,470

 

1949
01:01:20,470 --> 01:01:21,829

 

1950
01:01:21,829 --> 01:01:23,910

 

1951
01:01:23,910 --> 01:01:25,829

 

1952
01:01:25,829 --> 01:01:25,839

 

1953
01:01:25,839 --> 01:01:26,950


1954
01:01:26,950 --> 01:01:29,349

 

1955
01:01:29,349 --> 01:01:31,910

 

1956
01:01:31,910 --> 01:01:33,510

 

1957
01:01:33,510 --> 01:01:35,910

 

1958
01:01:35,910 --> 01:01:38,230

 

1959
01:01:38,230 --> 01:01:41,430

 

1960
01:01:41,430 --> 01:01:43,190

 

1961
01:01:43,190 --> 01:01:45,109

 

1962
01:01:45,109 --> 01:01:48,390

 

1963
01:01:48,390 --> 01:01:49,829

 

1964
01:01:49,829 --> 01:01:52,150

 

1965
01:01:52,150 --> 01:01:53,990

 

1966
01:01:53,990 --> 01:01:55,510

 

1967
01:01:55,510 --> 01:01:57,990

 

1968
01:01:57,990 --> 01:02:00,630

 

1969
01:02:00,630 --> 01:02:01,910

 

1970
01:02:01,910 --> 01:02:04,150

 

1971
01:02:04,150 --> 01:02:05,510

 

1972
01:02:05,510 --> 01:02:08,309

 

1973
01:02:08,309 --> 01:02:11,510

 

1974
01:02:11,510 --> 01:02:13,670

 

1975
01:02:13,670 --> 01:02:16,309

 

1976
01:02:16,309 --> 01:02:19,109

 

1977
01:02:19,109 --> 01:02:20,580

 

1978
01:02:20,580 --> 01:02:20,590

 

1979
01:02:20,590 --> 01:02:22,150


1980
01:02:22,150 --> 01:02:22,160

 

1981
01:02:22,160 --> 01:02:22,950


1982
01:02:22,950 --> 01:02:24,309

 

1983
01:02:24,309 --> 01:02:26,870

 

1984
01:02:26,870 --> 01:02:28,309

 

1985
01:02:28,309 --> 01:02:30,230

 

1986
01:02:30,230 --> 01:02:31,910

 

1987
01:02:31,910 --> 01:02:33,829

 

1988
01:02:33,829 --> 01:02:33,839

 

1989
01:02:33,839 --> 01:02:34,950


1990
01:02:34,950 --> 01:02:34,960

 

1991
01:02:34,960 --> 01:02:36,309


1992
01:02:36,309 --> 01:02:38,870

 

1993
01:02:38,870 --> 01:02:41,750

 

1994
01:02:41,750 --> 01:02:44,950

 

1995
01:02:44,950 --> 01:02:46,630

 

1996
01:02:46,630 --> 01:02:50,470

 

1997
01:02:50,470 --> 01:02:53,190

 

1998
01:02:53,190 --> 01:02:54,230

 

1999
01:02:54,230 --> 01:02:57,670

 

2000
01:02:57,670 --> 01:03:00,069

 

2001
01:03:00,069 --> 01:03:00,079

 

2002
01:03:00,079 --> 01:03:04,069


2003
01:03:04,069 --> 01:03:05,589

 

2004
01:03:05,589 --> 01:03:07,029

 

2005
01:03:07,029 --> 01:03:09,029

 

2006
01:03:09,029 --> 01:03:11,190

 

2007
01:03:11,190 --> 01:03:11,200

 

2008
01:03:11,200 --> 01:03:17,750


2009
01:03:17,750 --> 01:03:17,760

 

2010
01:03:17,760 --> 01:03:18,549


2011
01:03:18,549 --> 01:03:20,630

 

2012
01:03:20,630 --> 01:03:23,670

 

2013
01:03:23,670 --> 01:03:25,270

 

2014
01:03:25,270 --> 01:03:26,950

 

2015
01:03:26,950 --> 01:03:28,870

 

2016
01:03:28,870 --> 01:03:28,880

 

2017
01:03:28,880 --> 01:03:30,470


2018
01:03:30,470 --> 01:03:32,150

 

2019
01:03:32,150 --> 01:03:33,670

 

2020
01:03:33,670 --> 01:03:35,510

 

2021
01:03:35,510 --> 01:03:39,270

 

2022
01:03:39,270 --> 01:03:42,390

 

2023
01:03:42,390 --> 01:03:44,390

 

2024
01:03:44,390 --> 01:03:47,190

 

2025
01:03:47,190 --> 01:03:48,390

 

2026
01:03:48,390 --> 01:03:51,109

 

2027
01:03:51,109 --> 01:03:52,309

 

2028
01:03:52,309 --> 01:03:53,910

 

2029
01:03:53,910 --> 01:03:56,069

 

2030
01:03:56,069 --> 01:03:58,630

 

2031
01:03:58,630 --> 01:04:01,270

 

2032
01:04:01,270 --> 01:04:03,430

 

2033
01:04:03,430 --> 01:04:05,910

 

2034
01:04:05,910 --> 01:04:07,430

 

2035
01:04:07,430 --> 01:04:10,710

 

2036
01:04:10,710 --> 01:04:13,109

 

2037
01:04:13,109 --> 01:04:17,270

 

2038
01:04:17,270 --> 01:04:18,470

 

2039
01:04:18,470 --> 01:04:19,750

 

2040
01:04:19,750 --> 01:04:21,670

 

2041
01:04:21,670 --> 01:04:25,109

 

2042
01:04:25,109 --> 01:04:27,190

 

2043
01:04:27,190 --> 01:04:30,470

 

2044
01:04:30,470 --> 01:04:32,470

 

2045
01:04:32,470 --> 01:04:33,670

 

2046
01:04:33,670 --> 01:04:36,710

 

2047
01:04:36,710 --> 01:04:39,829

 

2048
01:04:39,829 --> 01:04:42,710

 

2049
01:04:42,710 --> 01:04:45,109

 

2050
01:04:45,109 --> 01:04:48,069

 

2051
01:04:48,069 --> 01:04:49,109

 

2052
01:04:49,109 --> 01:04:51,670

 

2053
01:04:51,670 --> 01:04:53,349

 

2054
01:04:53,349 --> 01:04:55,750

 

2055
01:04:55,750 --> 01:04:57,670

 

2056
01:04:57,670 --> 01:04:58,950

 

2057
01:04:58,950 --> 01:05:01,670

 

2058
01:05:01,670 --> 01:05:04,150

 

2059
01:05:04,150 --> 01:05:07,349

 

2060
01:05:07,349 --> 01:05:10,789

 

2061
01:05:10,789 --> 01:05:12,230

 

2062
01:05:12,230 --> 01:05:13,910

 

2063
01:05:13,910 --> 01:05:16,069

 

2064
01:05:16,069 --> 01:05:18,309

 

2065
01:05:18,309 --> 01:05:20,230

 

2066
01:05:20,230 --> 01:05:22,470

 

2067
01:05:22,470 --> 01:05:24,549

 

2068
01:05:24,549 --> 01:05:26,390

 

2069
01:05:26,390 --> 01:05:28,230

 

2070
01:05:28,230 --> 01:05:30,150

 

2071
01:05:30,150 --> 01:05:31,990

 

2072
01:05:31,990 --> 01:05:34,470

 

2073
01:05:34,470 --> 01:05:36,470

 

2074
01:05:36,470 --> 01:05:38,630

 

2075
01:05:38,630 --> 01:05:38,640

 

2076
01:05:38,640 --> 01:05:39,910


2077
01:05:39,910 --> 01:05:41,910

 

2078
01:05:41,910 --> 01:05:44,710

 

2079
01:05:44,710 --> 01:05:47,670

 

2080
01:05:47,670 --> 01:05:49,430

 

2081
01:05:49,430 --> 01:05:49,440

 

2082
01:05:49,440 --> 01:05:51,190


2083
01:05:51,190 --> 01:05:53,029

 

2084
01:05:53,029 --> 01:05:55,190

 

2085
01:05:55,190 --> 01:05:57,829

 

2086
01:05:57,829 --> 01:06:01,270

 

2087
01:06:01,270 --> 01:06:01,280

 

2088
01:06:01,280 --> 01:06:02,150


2089
01:06:02,150 --> 01:06:04,549

 

2090
01:06:04,549 --> 01:06:06,470

 

2091
01:06:06,470 --> 01:06:06,480

 

2092
01:06:06,480 --> 01:06:09,910


2093
01:06:09,910 --> 01:06:11,990

 

2094
01:06:11,990 --> 01:06:13,430

 

2095
01:06:13,430 --> 01:06:15,029

 

2096
01:06:15,029 --> 01:06:15,039

 

2097
01:06:15,039 --> 01:06:17,990


2098
01:06:17,990 --> 01:06:19,910

 

2099
01:06:19,910 --> 01:06:21,750

 

2100
01:06:21,750 --> 01:06:23,910

 

2101
01:06:23,910 --> 01:06:25,589

 

2102
01:06:25,589 --> 01:06:27,190

 

2103
01:06:27,190 --> 01:06:29,270

 

2104
01:06:29,270 --> 01:06:31,270

 

2105
01:06:31,270 --> 01:06:32,950

 

2106
01:06:32,950 --> 01:06:32,960

 

2107
01:06:32,960 --> 01:06:42,630


2108
01:06:42,630 --> 01:06:44,789

 

2109
01:06:44,789 --> 01:06:47,270

 

2110
01:06:47,270 --> 01:06:47,280

 

2111
01:06:47,280 --> 01:06:48,309


2112
01:06:48,309 --> 01:06:51,430

 

2113
01:06:51,430 --> 01:06:52,789

 

2114
01:06:52,789 --> 01:06:52,799

 

2115
01:06:52,799 --> 01:06:56,069


2116
01:06:56,069 --> 01:06:56,079

 

2117
01:06:56,079 --> 01:06:57,349


2118
01:06:57,349 --> 01:07:05,430

 

2119
01:07:05,430 --> 01:07:07,349

 

2120
01:07:07,349 --> 01:07:09,910

 

2121
01:07:09,910 --> 01:07:12,150

 

2122
01:07:12,150 --> 01:07:15,190

 

2123
01:07:15,190 --> 01:07:18,069

 

2124
01:07:18,069 --> 01:07:18,079

 

2125
01:07:18,079 --> 01:07:18,440


2126
01:07:18,440 --> 01:07:18,450

 

2127
01:07:18,450 --> 01:07:20,630


2128
01:07:20,630 --> 01:07:23,029

 

2129
01:07:23,029 --> 01:07:24,390

 

2130
01:07:24,390 --> 01:07:27,190

 

2131
01:07:27,190 --> 01:07:29,349

 

2132
01:07:29,349 --> 01:07:29,359

 

2133
01:07:29,359 --> 01:07:31,829


2134
01:07:31,829 --> 01:07:34,950

 

2135
01:07:34,950 --> 01:07:38,950

 

2136
01:07:38,950 --> 01:07:45,829

 

2137
01:07:45,829 --> 01:07:47,430

 

2138
01:07:47,430 --> 01:07:50,549

 

2139
01:07:50,549 --> 01:07:50,559

 

2140
01:07:50,559 --> 01:07:51,670


2141
01:07:51,670 --> 01:07:54,069

 

2142
01:07:54,069 --> 01:07:56,870

 

2143
01:07:56,870 --> 01:07:59,670

 

2144
01:07:59,670 --> 01:07:59,680

 

2145
01:07:59,680 --> 01:08:00,549


2146
01:08:00,549 --> 01:08:01,910

 

2147
01:08:01,910 --> 01:08:04,870

 

2148
01:08:04,870 --> 01:08:06,470

 

2149
01:08:06,470 --> 01:08:08,630

 

2150
01:08:08,630 --> 01:08:09,750

 

2151
01:08:09,750 --> 01:08:12,950

 

2152
01:08:12,950 --> 01:08:15,589

 

2153
01:08:15,589 --> 01:08:17,669

 

2154
01:08:17,669 --> 01:08:18,789

 

2155
01:08:18,789 --> 01:08:21,510

 

2156
01:08:21,510 --> 01:08:23,349

 

2157
01:08:23,349 --> 01:08:24,630

 

2158
01:08:24,630 --> 01:08:26,789

 

2159
01:08:26,789 --> 01:08:28,950

 

2160
01:08:28,950 --> 01:08:31,269

 

2161
01:08:31,269 --> 01:08:35,590

 

2162
01:08:35,590 --> 01:08:38,630

 

2163
01:08:38,630 --> 01:08:39,829

 

2164
01:08:39,829 --> 01:08:39,839

 

2165
01:08:39,839 --> 01:08:41,110


2166
01:08:41,110 --> 01:08:43,749

 

2167
01:08:43,749 --> 01:08:46,149

 

2168
01:08:46,149 --> 01:08:48,070

 

2169
01:08:48,070 --> 01:08:50,390

 

2170
01:08:50,390 --> 01:08:51,990

 

2171
01:08:51,990 --> 01:08:54,070

 

2172
01:08:54,070 --> 01:08:56,309

 

2173
01:08:56,309 --> 01:08:58,229

 

2174
01:08:58,229 --> 01:09:01,030

 

2175
01:09:01,030 --> 01:09:03,669

 

2176
01:09:03,669 --> 01:09:06,789

 

2177
01:09:06,789 --> 01:09:09,349

 

2178
01:09:09,349 --> 01:09:10,630

 

2179
01:09:10,630 --> 01:09:12,630

 

2180
01:09:12,630 --> 01:09:14,070

 

2181
01:09:14,070 --> 01:09:16,390

 

2182
01:09:16,390 --> 01:09:19,829

 

2183
01:09:19,829 --> 01:09:22,070

 

2184
01:09:22,070 --> 01:09:24,789

 

2185
01:09:24,789 --> 01:09:26,470

 

2186
01:09:26,470 --> 01:09:28,470

 

2187
01:09:28,470 --> 01:09:30,149

 

2188
01:09:30,149 --> 01:09:33,269

 

2189
01:09:33,269 --> 01:09:36,229

 

2190
01:09:36,229 --> 01:09:38,229

 

2191
01:09:38,229 --> 01:09:41,910

 

2192
01:09:41,910 --> 01:09:45,590

 

2193
01:09:45,590 --> 01:09:47,189

 

2194
01:09:47,189 --> 01:09:49,669

 

2195
01:09:49,669 --> 01:09:52,229

 

2196
01:09:52,229 --> 01:09:52,239

 

2197
01:09:52,239 --> 01:09:53,590


2198
01:09:53,590 --> 01:09:53,600

 

2199
01:09:53,600 --> 01:09:56,149


2200
01:09:56,149 --> 01:09:57,830

 

2201
01:09:57,830 --> 01:09:59,990

 

2202
01:09:59,990 --> 01:10:02,950

 

2203
01:10:02,950 --> 01:10:02,960

 

2204
01:10:02,960 --> 01:10:04,390


2205
01:10:04,390 --> 01:10:07,189

 

2206
01:10:07,189 --> 01:10:10,470

 

2207
01:10:10,470 --> 01:10:12,790

 

2208
01:10:12,790 --> 01:10:15,030

 

2209
01:10:15,030 --> 01:10:17,910

 

2210
01:10:17,910 --> 01:10:20,070

 

2211
01:10:20,070 --> 01:10:21,590

 

2212
01:10:21,590 --> 01:10:24,470

 

2213
01:10:24,470 --> 01:10:24,480

 

2214
01:10:24,480 --> 01:10:25,430


2215
01:10:25,430 --> 01:10:29,189

 

2216
01:10:29,189 --> 01:10:30,550

 

2217
01:10:30,550 --> 01:10:30,560

 

2218
01:10:30,560 --> 01:10:32,550


2219
01:10:32,550 --> 01:10:34,870

 

2220
01:10:34,870 --> 01:10:36,470

 

2221
01:10:36,470 --> 01:10:39,430

 

2222
01:10:39,430 --> 01:10:42,070

 

2223
01:10:42,070 --> 01:10:45,430

 

2224
01:10:45,430 --> 01:10:48,310

 

2225
01:10:48,310 --> 01:10:50,630

 

2226
01:10:50,630 --> 01:10:52,709

 

2227
01:10:52,709 --> 01:10:54,550

 

2228
01:10:54,550 --> 01:10:56,709

 

2229
01:10:56,709 --> 01:10:58,630

 

2230
01:10:58,630 --> 01:11:00,950

 

2231
01:11:00,950 --> 01:11:03,189

 

2232
01:11:03,189 --> 01:11:04,870

 

2233
01:11:04,870 --> 01:11:06,950

 

2234
01:11:06,950 --> 01:11:09,669

 

2235
01:11:09,669 --> 01:11:10,870

 

2236
01:11:10,870 --> 01:11:14,470

 

2237
01:11:14,470 --> 01:11:14,480

 

2238
01:11:14,480 --> 01:11:15,590


2239
01:11:15,590 --> 01:11:16,790

 

2240
01:11:16,790 --> 01:11:19,270

 

2241
01:11:19,270 --> 01:11:21,030

 

2242
01:11:21,030 --> 01:11:23,590

 

2243
01:11:23,590 --> 01:11:25,110

 

2244
01:11:25,110 --> 01:11:28,149

 

2245
01:11:28,149 --> 01:11:31,189

 

2246
01:11:31,189 --> 01:11:33,030

 

2247
01:11:33,030 --> 01:11:35,270

 

2248
01:11:35,270 --> 01:11:36,790

 

2249
01:11:36,790 --> 01:11:36,800

 

2250
01:11:36,800 --> 01:11:37,990


2251
01:11:37,990 --> 01:11:39,750

 

2252
01:11:39,750 --> 01:11:41,030

 

2253
01:11:41,030 --> 01:11:43,510

 

2254
01:11:43,510 --> 01:11:45,669

 

2255
01:11:45,669 --> 01:11:47,990

 

2256
01:11:47,990 --> 01:11:51,189

 

2257
01:11:51,189 --> 01:11:52,790

 

2258
01:11:52,790 --> 01:11:55,270

 

2259
01:11:55,270 --> 01:11:58,310

 

2260
01:11:58,310 --> 01:11:59,750

 

2261
01:11:59,750 --> 01:12:02,709

 

2262
01:12:02,709 --> 01:12:06,070

 

2263
01:12:06,070 --> 01:12:09,510

 

2264
01:12:09,510 --> 01:12:09,520

 

2265
01:12:09,520 --> 01:12:11,910


2266
01:12:11,910 --> 01:12:13,830

 

2267
01:12:13,830 --> 01:12:13,840

 

2268
01:12:13,840 --> 01:12:15,510


2269
01:12:15,510 --> 01:12:17,030

 

2270
01:12:17,030 --> 01:12:19,110

 

2271
01:12:19,110 --> 01:12:26,630

 

2272
01:12:26,630 --> 01:12:28,070

 

2273
01:12:28,070 --> 01:12:29,430

 

2274
01:12:29,430 --> 01:12:30,870

 

2275
01:12:30,870 --> 01:12:32,470

 

2276
01:12:32,470 --> 01:12:32,480

 

2277
01:12:32,480 --> 01:12:34,950


2278
01:12:34,950 --> 01:12:36,470

 

2279
01:12:36,470 --> 01:12:39,110

 

2280
01:12:39,110 --> 01:12:40,470

 

2281
01:12:40,470 --> 01:12:42,149

 

2282
01:12:42,149 --> 01:12:43,189

 

2283
01:12:43,189 --> 01:12:45,270

 

2284
01:12:45,270 --> 01:12:47,030

 

2285
01:12:47,030 --> 01:12:50,630

 

2286
01:12:50,630 --> 01:12:54,310

 

2287
01:12:54,310 --> 01:12:56,630

 

2288
01:12:56,630 --> 01:13:00,310

 

2289
01:13:00,310 --> 01:13:03,030

 

2290
01:13:03,030 --> 01:13:05,430

 

2291
01:13:05,430 --> 01:13:07,990

 

2292
01:13:07,990 --> 01:13:10,149

 

2293
01:13:10,149 --> 01:13:12,070

 

2294
01:13:12,070 --> 01:13:13,669

 

2295
01:13:13,669 --> 01:13:16,390

 

2296
01:13:16,390 --> 01:13:18,070

 

2297
01:13:18,070 --> 01:13:20,470

 

2298
01:13:20,470 --> 01:13:22,390

 

2299
01:13:22,390 --> 01:13:22,400

 

2300
01:13:22,400 --> 01:13:23,910


2301
01:13:23,910 --> 01:13:26,070

 

2302
01:13:26,070 --> 01:13:28,870

 

2303
01:13:28,870 --> 01:13:30,310

 

2304
01:13:30,310 --> 01:13:30,320

 

2305
01:13:30,320 --> 01:13:31,270


2306
01:13:31,270 --> 01:13:33,350

 

2307
01:13:33,350 --> 01:13:35,510

 

2308
01:13:35,510 --> 01:13:37,750

 

2309
01:13:37,750 --> 01:13:39,030

 

2310
01:13:39,030 --> 01:13:41,590

 

2311
01:13:41,590 --> 01:13:43,189

 

2312
01:13:43,189 --> 01:13:46,070

 

2313
01:13:46,070 --> 01:13:47,510

 

2314
01:13:47,510 --> 01:13:50,390

 

2315
01:13:50,390 --> 01:13:53,830

 

2316
01:13:53,830 --> 01:13:56,229

 

2317
01:13:56,229 --> 01:13:58,870

 

2318
01:13:58,870 --> 01:14:01,590

 

2319
01:14:01,590 --> 01:14:01,600

 

2320
01:14:01,600 --> 01:14:02,550


2321
01:14:02,550 --> 01:14:02,560

 

2322
01:14:02,560 --> 01:14:03,430


2323
01:14:03,430 --> 01:14:05,750

 

2324
01:14:05,750 --> 01:14:06,790

 

2325
01:14:06,790 --> 01:14:10,709

 

2326
01:14:10,709 --> 01:14:12,390

 

2327
01:14:12,390 --> 01:14:14,630

 

2328
01:14:14,630 --> 01:14:16,470

 

2329
01:14:16,470 --> 01:14:19,030

 

2330
01:14:19,030 --> 01:14:22,470

 

2331
01:14:22,470 --> 01:14:24,390

 

2332
01:14:24,390 --> 01:14:25,750

 

2333
01:14:25,750 --> 01:14:25,760

 

2334
01:14:25,760 --> 01:14:27,270


2335
01:14:27,270 --> 01:14:29,110

 

2336
01:14:29,110 --> 01:14:30,550

 

2337
01:14:30,550 --> 01:14:30,560

 

2338
01:14:30,560 --> 01:14:31,430


2339
01:14:31,430 --> 01:14:32,950

 

2340
01:14:32,950 --> 01:14:34,310

 

2341
01:14:34,310 --> 01:14:36,149

 

2342
01:14:36,149 --> 01:14:37,830

 

2343
01:14:37,830 --> 01:14:37,840

 

2344
01:14:37,840 --> 01:14:39,750


2345
01:14:39,750 --> 01:14:42,790

 

2346
01:14:42,790 --> 01:14:46,310

 

2347
01:14:46,310 --> 01:14:48,310

 

2348
01:14:48,310 --> 01:14:50,149

 

2349
01:14:50,149 --> 01:14:52,070

 

2350
01:14:52,070 --> 01:14:55,669

 

2351
01:14:55,669 --> 01:14:57,750

 

2352
01:14:57,750 --> 01:14:59,350

 

2353
01:14:59,350 --> 01:15:01,030

 

2354
01:15:01,030 --> 01:15:02,310

 

2355
01:15:02,310 --> 01:15:03,990

 

2356
01:15:03,990 --> 01:15:06,229

 

2357
01:15:06,229 --> 01:15:09,189

 

2358
01:15:09,189 --> 01:15:11,110

 

2359
01:15:11,110 --> 01:15:13,510

 

2360
01:15:13,510 --> 01:15:15,910

 

2361
01:15:15,910 --> 01:15:18,470

 

2362
01:15:18,470 --> 01:15:20,950

 

2363
01:15:20,950 --> 01:15:22,310

 

2364
01:15:22,310 --> 01:15:22,320

 

2365
01:15:22,320 --> 01:15:27,669


2366
01:15:27,669 --> 01:15:29,590

 

2367
01:15:29,590 --> 01:15:31,910

 

2368
01:15:31,910 --> 01:15:33,350

 

2369
01:15:33,350 --> 01:15:36,149

 

2370
01:15:36,149 --> 01:15:37,750

 

2371
01:15:37,750 --> 01:15:39,030

 

2372
01:15:39,030 --> 01:15:41,110

 

2373
01:15:41,110 --> 01:15:42,630

 

2374
01:15:42,630 --> 01:15:44,229

 

2375
01:15:44,229 --> 01:15:45,910

 

2376
01:15:45,910 --> 01:15:45,920

 

2377
01:15:45,920 --> 01:15:46,870


2378
01:15:46,870 --> 01:15:46,880

 

2379
01:15:46,880 --> 01:15:48,470


2380
01:15:48,470 --> 01:15:52,310

 

2381
01:15:52,310 --> 01:15:55,110

 

2382
01:15:55,110 --> 01:15:57,270

 

2383
01:15:57,270 --> 01:15:57,280

 

2384
01:15:57,280 --> 01:15:59,430


2385
01:15:59,430 --> 01:16:01,590

 

2386
01:16:01,590 --> 01:16:03,350

 

2387
01:16:03,350 --> 01:16:05,510

 

2388
01:16:05,510 --> 01:16:08,550

 

2389
01:16:08,550 --> 01:16:10,310

 

2390
01:16:10,310 --> 01:16:12,149

 

2391
01:16:12,149 --> 01:16:13,590

 

2392
01:16:13,590 --> 01:16:15,910

 

2393
01:16:15,910 --> 01:16:15,920

 

2394
01:16:15,920 --> 01:16:17,270


2395
01:16:17,270 --> 01:16:19,510

 

2396
01:16:19,510 --> 01:16:20,790

 

2397
01:16:20,790 --> 01:16:20,800

 

2398
01:16:20,800 --> 01:16:21,910


2399
01:16:21,910 --> 01:16:24,709

 

2400
01:16:24,709 --> 01:16:26,870

 

2401
01:16:26,870 --> 01:16:28,870

 

2402
01:16:28,870 --> 01:16:32,630

 

2403
01:16:32,630 --> 01:16:35,189

 

2404
01:16:35,189 --> 01:16:39,590

 

2405
01:16:39,590 --> 01:16:42,070

 

2406
01:16:42,070 --> 01:16:45,669

 

2407
01:16:45,669 --> 01:16:47,350

 

2408
01:16:47,350 --> 01:16:49,669

 

2409
01:16:49,669 --> 01:16:52,470

 

2410
01:16:52,470 --> 01:16:54,070

 

2411
01:16:54,070 --> 01:16:56,470

 

2412
01:16:56,470 --> 01:16:57,910

 

2413
01:16:57,910 --> 01:16:59,669

 

2414
01:16:59,669 --> 01:17:01,669

 

2415
01:17:01,669 --> 01:17:03,430

 

2416
01:17:03,430 --> 01:17:15,510

 

2417
01:17:15,510 --> 01:17:15,520

 

2418
01:17:15,520 --> 01:17:16,709


2419
01:17:16,709 --> 01:17:19,189

 

2420
01:17:19,189 --> 01:17:20,870

 

2421
01:17:20,870 --> 01:17:24,149

 

2422
01:17:24,149 --> 01:17:24,159

 

2423
01:17:24,159 --> 01:17:25,110


2424
01:17:25,110 --> 01:17:27,270

 

2425
01:17:27,270 --> 01:17:29,750

 

2426
01:17:29,750 --> 01:17:31,430

 

2427
01:17:31,430 --> 01:17:33,030

 

2428
01:17:33,030 --> 01:17:35,350

 

2429
01:17:35,350 --> 01:17:36,950

 

2430
01:17:36,950 --> 01:17:38,630

 

2431
01:17:38,630 --> 01:17:40,229

 

2432
01:17:40,229 --> 01:17:41,669

 

2433
01:17:41,669 --> 01:17:43,189

 

2434
01:17:43,189 --> 01:17:43,199

 

2435
01:17:43,199 --> 01:17:44,470


2436
01:17:44,470 --> 01:17:47,030

 

2437
01:17:47,030 --> 01:17:48,550

 

2438
01:17:48,550 --> 01:17:52,790

 

2439
01:17:52,790 --> 01:17:54,870

 

2440
01:17:54,870 --> 01:17:57,510

 

2441
01:17:57,510 --> 01:17:59,510

 

2442
01:17:59,510 --> 01:18:02,709

 

2443
01:18:02,709 --> 01:18:04,470

 

2444
01:18:04,470 --> 01:18:07,750

 

2445
01:18:07,750 --> 01:18:07,760

 

2446
01:18:07,760 --> 01:18:11,270


2447
01:18:11,270 --> 01:18:11,280

 

2448
01:18:11,280 --> 01:18:12,950


2449
01:18:12,950 --> 01:18:14,310

 

2450
01:18:14,310 --> 01:18:16,390

 

2451
01:18:16,390 --> 01:18:17,750

 

2452
01:18:17,750 --> 01:18:20,149

 

2453
01:18:20,149 --> 01:18:20,159

 

2454
01:18:20,159 --> 01:18:21,830


2455
01:18:21,830 --> 01:18:23,430

 

2456
01:18:23,430 --> 01:18:24,830

 

2457
01:18:24,830 --> 01:18:26,790

 

2458
01:18:26,790 --> 01:18:28,149

 

2459
01:18:28,149 --> 01:18:31,030

 

2460
01:18:31,030 --> 01:18:35,350

 

2461
01:18:35,350 --> 01:18:36,709

 

2462
01:18:36,709 --> 01:18:39,590

 

2463
01:18:39,590 --> 01:18:41,270

 

2464
01:18:41,270 --> 01:18:41,280

 

2465
01:18:41,280 --> 01:18:43,669


2466
01:18:43,669 --> 01:18:46,950

 

2467
01:18:46,950 --> 01:18:48,390

 

2468
01:18:48,390 --> 01:18:48,400

 

2469
01:18:48,400 --> 01:18:49,430


2470
01:18:49,430 --> 01:18:49,440

 

2471
01:18:49,440 --> 01:18:50,149


2472
01:18:50,149 --> 01:18:52,390

 

2473
01:18:52,390 --> 01:18:52,400

 

2474
01:18:52,400 --> 01:18:53,990


2475
01:18:53,990 --> 01:18:55,830

 

2476
01:18:55,830 --> 01:18:58,310

 

2477
01:18:58,310 --> 01:19:00,870

 

2478
01:19:00,870 --> 01:19:02,470

 

2479
01:19:02,470 --> 01:19:04,390

 

2480
01:19:04,390 --> 01:19:05,830

 

2481
01:19:05,830 --> 01:19:08,470

 

2482
01:19:08,470 --> 01:19:09,910

 

2483
01:19:09,910 --> 01:19:12,470

 

2484
01:19:12,470 --> 01:19:15,030

 

2485
01:19:15,030 --> 01:19:17,270

 

2486
01:19:17,270 --> 01:19:18,709

 

2487
01:19:18,709 --> 01:19:18,719

 

2488
01:19:18,719 --> 01:19:22,550


2489
01:19:22,550 --> 01:19:25,830

 

2490
01:19:25,830 --> 01:19:28,310

 

2491
01:19:28,310 --> 01:19:30,470

 

2492
01:19:30,470 --> 01:19:32,950

 

2493
01:19:32,950 --> 01:19:35,030

 

2494
01:19:35,030 --> 01:19:36,950

 

2495
01:19:36,950 --> 01:19:38,630

 

2496
01:19:38,630 --> 01:19:41,110

 

2497
01:19:41,110 --> 01:19:43,270

 

2498
01:19:43,270 --> 01:19:43,280

 

2499
01:19:43,280 --> 01:19:45,189


2500
01:19:45,189 --> 01:19:47,669

 

2501
01:19:47,669 --> 01:19:50,870

 

2502
01:19:50,870 --> 01:19:53,669

 

2503
01:19:53,669 --> 01:19:55,270

 

2504
01:19:55,270 --> 01:19:57,669

 

2505
01:19:57,669 --> 01:19:57,679

 

2506
01:19:57,679 --> 01:19:58,630


2507
01:19:58,630 --> 01:20:00,550

 

2508
01:20:00,550 --> 01:20:01,750

 

2509
01:20:01,750 --> 01:20:05,430

 

2510
01:20:05,430 --> 01:20:08,629

 

2511
01:20:08,629 --> 01:20:09,669

 

2512
01:20:09,669 --> 01:20:11,990

 

2513
01:20:11,990 --> 01:20:13,750

 

2514
01:20:13,750 --> 01:20:14,550

 

2515
01:20:14,550 --> 01:20:17,030

 

2516
01:20:17,030 --> 01:20:18,709

 

2517
01:20:18,709 --> 01:20:20,550

 

2518
01:20:20,550 --> 01:20:22,390

 

2519
01:20:22,390 --> 01:20:23,510

 

2520
01:20:23,510 --> 01:20:25,510

 

2521
01:20:25,510 --> 01:20:26,790

 

2522
01:20:26,790 --> 01:20:29,669

 

2523
01:20:29,669 --> 01:20:31,030

 

2524
01:20:31,030 --> 01:20:32,709

 

2525
01:20:32,709 --> 01:20:34,390

 

2526
01:20:34,390 --> 01:20:37,110

 

2527
01:20:37,110 --> 01:20:39,430

 

2528
01:20:39,430 --> 01:20:41,350

 

2529
01:20:41,350 --> 01:20:44,790

 

2530
01:20:44,790 --> 01:20:48,229

 

2531
01:20:48,229 --> 01:20:50,149

 

2532
01:20:50,149 --> 01:20:52,390

 

2533
01:20:52,390 --> 01:20:53,830

 

2534
01:20:53,830 --> 01:20:55,990

 

2535
01:20:55,990 --> 01:20:57,910

 

2536
01:20:57,910 --> 01:21:01,189

 

2537
01:21:01,189 --> 01:21:03,189

 

2538
01:21:03,189 --> 01:21:06,310

 

2539
01:21:06,310 --> 01:21:08,950

 

2540
01:21:08,950 --> 01:21:11,430

 

2541
01:21:11,430 --> 01:21:13,270

 

2542
01:21:13,270 --> 01:21:14,790

 

2543
01:21:14,790 --> 01:21:16,950

 

2544
01:21:16,950 --> 01:21:20,390

 

2545
01:21:20,390 --> 01:21:24,310

 

2546
01:21:24,310 --> 01:21:26,790

 

2547
01:21:26,790 --> 01:21:29,840