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okay everybody  let's begin  um welcome back  good to see you all here on zoom  um  so we're going to pick up with what we  had been discussing  um  last week  which was  um  a  an introduction to  np completeness so we're following on  our  description of  time complexity  we we started talking about the time  complexity classes  the class p  the non-deterministic classes the class  np p versus np problem and then leading  to this discussion of np completeness  and today we're going to prove the big  theorem in the field uh which really  kind of got things going uh back in the  early 1970s by cook and levin that uh  there was actually an mp complete  problem called that sat in particular is  an np complete problem and then we'll  also talk about threesat which is a  useful tool  um  so just to remember just to uh remember  we had this notion of np completeness  for  languages that be complete if it's in np  and everything else in np is polynomial  time reducible to it  and if an np complete problem  turns out to have a polynomial time  solution then every np problem has a  polynomial time solution  and that's part of the importance of np  completeness uh because since we  uh consider it unlikely that p equals np  uh  and that there probably are some  problems  that uh are in np but are not solvable  in polynomial time  that's  would me imply that an np-complete  problem would have that property and so  proving  a problem and being mp complete is  evidence very strong evidence that it  doesn't have a polynomial time uh  solution and so therefore it's we call  intractable uh it's a very difficult  problem  um  so um  the way we are going to typically show  problems np-complete is by reducing a  known a previously known np-complete  problem to that problem  often it's three-sat  as we've seen several examples already  or it could be some other uh example so  let's just survey briefly uh the things  that we've already  languages that we've already seen  which are mp complete  so we have the languages sat which is a  direct reduction from every np problem  and  so we're going to show that today  that every np language is polynomial  time reducible to sat which is in turn  reducible to three sat  and  we showed previously  that threesat is reducible to clique and  hampf  and  in recitation if you went to that they  showed that the subset sub problem and  the undirected hand path problems are  also  reducible from previously uh shown well  from uh from either from 3sat  or in the case of their undirected hand  path problem it's reducible from the  hand path problem  and the conclusion once we have  these two blue  reducibilities shown  then we know that all of these problems  are np-complete  um  so the class np basically breaks down  into the np complete problems  p problems problems that are in p and  then there might be problems in between  as well so there are some problems that  are not known to be in either category  and in fact there's a old theorem which  shows that if p is different from np  then there are actually problems that  are in the intermediate state  and um  of course  it's possible that p equals np  and then everything collapses down uh to  be the same with the tiny exception of  the um sigma star and empty set  languages which can never be uh complete  um  okay um so that's our quick review  here's a check in that i'm gonna use to  get us ready for the  big proof that we're gonna be spending  most of the lecture on about um  uh showing sat as np-complete but just  to define a little notation which you  might have maybe you've seen already um  but i'm going to do it in the form of a  check-in  so  i'm sure you've all seen the big sum  notation uh using a sigma  to represent a sum over um  over some set of possibilities  just as there is the big sigma notation  we have you can have other operations um  that apply to a a set of uh elements so  in this case we're going to be seeing  the big and and the big or  operation uh it's going to a notation  which is going to allow us to talk about  taking the end of many things or the ore  of many things um because we're we're  going to be building these boolean  formulas and so ands and orders are  going to be the uh operations that we're  going to be focusing on  and so just as an example just to  make sure you're we're all understanding  this notation if you have  uh two strings x and y  uh written out in terms of their  individual symbols so they're both of  length n so x is x1 to xn  y is y 1 to y n  and  now i write the following expression the  big and  for i ranging between 1 and n  of x i equal to y i if that big and is  true what does it tell us about x and y  and i'm just going to offer you two  possibilities that either x and y agree  in some symbol  or they are equal namely that they agree  on every symbol  and so  let's just pull that up as a  quick poll to get us going here  um  [Music]  i just want to make sure you understand  the notation because we're going to be  using that a lot  in describing the uh  polynomial time reduction  from  uh  languages in np to the sat language  okay i think most of you have got the  idea but  so let's finish this up quick  um  see another 15 seconds please  just to give you a chance to participate  um  okay so i'm gonna close this uh  close this poll  last chance let's call  all right  so yes  the big and  as most of you  can see the big and says that you know  x1 equals y1 and x2 equals y2 and x3  equals y3 so they're all  um  every symbol in x is equal to the  corresponding symbol in y  if we had a big or instead of a big and  the uh then a would be the correct  answer because then just there would be  some place where they agree instead of  every place where they they agree  um  okay  so let's um  then  start  to launch in to this big theorem  the proof itself  is a massive details  um with one underlying idea  and in fact it's an idea we've seen  before um but let's before we get ahead  of ourselves let's just understand what  we're trying to do  uh  so we want to show that this language  sat is np-complete remember what's that  is it's boolean formulas that are  satisfiable  uh  so we already showed that the set pro so  being mp complete means it has these two  features it's in np  and everything in np is reducible to it  so first of all sat is np as we've  already seen  the witness that shows a formula is  satisfiable is simply the satisfying  assignment that evaluates the true  so now we're going to pick  some language in np a  and show that a is polynomial time  reducible to  okay  so this is going to apply to any  language a and np so let a be some  language in np  it's decided by some non-deterministic  turing machine m in time into the k  that's what it means to be an np um i'm  going to make ignoring the constant  factors when we could carry that  throughout the proof it just would make  this the the description a little more  cumbersome and wouldn't change any of  the ideas but so let's just say it's  m runs in time into the k  and  recognizes or decides this language a in  uh  uh so it's an it's  in and it's a non-deterministic machine  so we've got to give a polynomial time  reduction from a to sad that's that's  what i have to  demonstrate to you  so what does that reduction look like  it's going to map strings  which may or may not be an a  to formulas which may or may not be  satisfied  that's what the reduction's got to do  so  saying  saying that out  more formally is  it's going to take some string w and map  it to a formula we'll call f sub m w  where m is the  machine that  um  decides a and w is is is the input  so f is going to map w to this formula  where the string is in the language  exactly when the formula is satisfiable  and the formula is going to depend upon  m and w that's why it just has it's  written in this way so we've seen that  kind of thing before  so basically my  my job  you know you know we have this language  a that's an np and now we have some  string w which might be an a or maybe  not is an a  and i have to  quickly in polynomial time now produce a  formula  which is going to be satisfiable exactly  when  the string w is in a  okay so how is that formula going to  work  how can i produce such a formula which i  mean i of course i don't know whether w  is in a or not um because i i'm just a  polynomial time reduction  and um a is an np language so  polynomial time probably is not enough  to solve whether w is in a so i've got  to make matt do that mapping without  knowing  the answer  and the idea is that um and this is  where you know we've seen things like  this before  the formula we're going to construct  simulates the machine on w  so in some way it's going to do that  simulation the trick is figuring out how  to what that means  um but  the the interpretation of that formula  is that in a sense describes and it says  though in my informal language that m  accepts w  very much like  you know  there's a lot of parallel here between  um  this construction and for example the  construction um the pcp construction  where we  made a an instant you know  given a machine and an  input we made a set of those dominoes  where finding a match forced you to  simulate the machine  here finding a satisfying assignment  is going to force you to simulate the  machine  [Music]  so this satisfying assignment is going  to be a computation history for mmw  and i'm just going to write that  computation history in a particular way  so it's going to have a somewhat  different encoding which is going to  help us to visualize what's going on  better  okay  um  so that's that's the  that's the  that's the approach that's the basic  idea of what we're trying to what we're  gonna do so i'm happy to take a minute  if that's  if you have any questions about this  part but um  otherwise i'll just move on to start  doing the actual construction of what  the formula looks like how is it going  to do that how is that formula going to  work  okay so no questions hopefully  we are uh all together  okay  so  first of all  let me describe what my computation  histories are going to look like  and and the situation is a little bit  different than what we had before  because  um when we were talking about the post  correspondence problem we had a  deterministic machine and now our  machine is  non-deterministic  okay so  um  we're going to call the object instead  of an accepting computation or  computation history we're just going to  we're going to call it a tableau or  sometimes an accepting tableau if you  want to  emphasize the accepting nature of it but  generally we're just going to call it a  tableau  so a tableau  is really an accepting computation  history  for  um for  for the machine  on an accepting branch  of its non-deterministic computation  so if m accepts w  it's got to have some accepting branch  and the  tableau  is going to be the sequence of  configurations that the machine goes  through on that accepting branch if  there are several accepting branches  there may be several tableaus there will  there will be several tableaus so  there's going to be one tableau for each  accepting branch of the  of the machine's computation on w  if the machine does not accept w there  won't be any tableaus  and so the whole point is that we're  going to make our formula  and  represent  the statement  that there is a tableau  and satisfying that formula is going to  correspond to filling out  the symbols in in the tableau to make it  a tableau  okay so here is a tableau um so  tableau is just again an accepting  computation history on some branch some  accepting branch of the machine's  computation  so and and it the rows of the ins  instead of writing the  computation history out linearly we're  going to represent it  in a table form where each configuration  is going to be on a separate row  all right  now the dimensions of that table are  going to be n to the k by n to the k  because the  machine  runs for n to the k steps  so  there's gonna if there's an accepting  branch it's going to accept within that  number of steps and we'll have enough  rows here to write down all of the  configurations that the machine goes  through one after the next row by row  each one having a configuration  uh in it  and uh  and then at the bottom there'll be an  accept  you know minor detail if the machine  accepts earlier we'll just say the  machine stays in the set once it's  it enters an accept  the machine does not change  from that point on so the rule of the  machine is  nothing changes and it just remains in  the same configuration from that point  on  okay um  so  um  important to understand what we're you  know if you're if you're not following  what i mean by a tableau  you're doomed  for this lecture  so  it it makes sense if you know for you to  ask a question  to understand what we mean by tableau  okay so just a few more elements here so  this is going to be the start  configuration for mw this would be an  accepting configuration here down at the  very last row  um  and you might imagine you know i i  think i filled out some  hypothetical  first step of the machine after the  start  where  maybe the machine was  a when it's in remember how we can  encode our configurations so this is the  machine is in state q0 looking at w uh  the first symbol of the input w1 and  maybe when it's  in q0 or looking in that first symbol  it moves to state q 7  and goes right  and then changes that w 1 to an a  and so now here is the head shown moved  1 position to the right in the new state  q7  i mean of course that depends on what  the machine is designed to do what the  what the transition function is but this  um  you know could possibly is  what happens so does the  okay so good does a tableau trace all  steps of all branches  no  the tableau corresponds to one accepting  branch  each different accepting branch is going  to have its own tableau so there might  be several different ways of filling out  that second row even of course the first  row is going to be in any tableau it's  got to be the same  um  you know  once i know w  here i've written down maybe i should  unpack this for you you know it's in the  start state here then here are the first  you know here are the n symbols of w w  is of length n so it's w 1 w 2 up to w n  and then i'm padding out the rest with  blanks  so i should have said that too but okay  um  so i want to make my table into the k by  n to the k because that's going to be  enough  to represent the entire  uh  um all of these  configurations  of m  uh running for most into the k steps  right because the machine  um even if it sends its head  moving to the right as fast as it  possibly can  it's never going to have um it's it's  never going to  go outside this box  if it only runs for n to the k steps so  this is going to be big enough to  represent the entire computation of  m assuming it's running for n to the k  time and  the input w is of like that  all right um  what is k so k is the running time of  the machine so we assumed from the  previous slide that m runs in time n to  the k  okay um  so a good question here how can the  tableau be a square table if we have a  low number of computation histories but  a lot of tape um  well i mean the low number of  computation history that that's a  problem you're not saying that well you  know there  each computation history  is in a different table  so this i'm representing a single  computation history here there may be  there are going to be many  configurations  um so i can't have a small number of  configurations using a lot of tape  because  you know i can only use one more cell in  one more one additional tape cell each  time i meant you know if each additional  step of the machine  is yes so there's really no difference  between this if you want to think of  this as a computation history that's  fine this is really just the standard  terminology that's used when you're  proving um  this this particular theorem typically  people talk about it as a tableau but  it's really just the computation history  um  the cue state i mean  the question about the what are these  states queue this is the way we  represent configurations  so this means the machine is in  uh state  they're not all going down the diagonal  the the states are going to zigzag here  through he through this picture here  depending upon how the head of the  machine moves  so you have to go back and review how we  represented configurations of the  machine remember the configuration is a  snapshot  of the machine at a given point  um  how do we know that m runs in polynomial  time we're assuming m runs in polynomial  we started off with a language that's in  np  so it's it's a non-deterministic  polynomial time machine and we're  picking one branch  that accepts  and writing down all the sequence  configurations the machine goes through  let's move on and maybe as  you know ask more questions as they come  to you and i'll pick out some to answer  if that's going to be helpful to others  um  okay  um  so uh  we're going to now construct  this  um  uh  formula  to say that m accepts w again that was  what we uh that's our goal  and it says that a tableau for m on w  exists  um  and basically what that means is we want  to say that it starts right  it ends right and everything in between  is right  and then we're going to need some other  stuff to talk about how we're going to  be encoding those symbols using boolean  variables  so those are going to be the four parts  here's the start starts right here it  ends right here it moves right and here  it talks about the encoding of the  symbols into boolean variables  so those are the four parts of this  formula that i'm going to describe  um  i hope you got your question  answered as to why the total number of  columns is into the k because it's just  big enough to fit the entire all  configurations if the is running for at  most end of the case steps  which is what we're assuming  okay  okay so now  just  getting back to that i'm going to  describe these different components now  on separate slots  um  let me start off with with this  component fee cell which  is sort of the most fundamental one  because it talks about how we're going  to be encoding  those symbols of the tableau  into the boolean variables  so again here's kind of our the picture  to have in mind uh of this tableau this  is this into the k by enter the k table  representing  um you know the some  accepting branch  of the machine's computation  if there is one  and so now  let me draw a  some one of the cells here i'm going to  magnify it  so this is the i  j cell  here  and le i'm going to there there are  going to be  a collection of boolean variables  associated with each one of the cells so  each one of the cells is going to have a  bunch of variables all to itself  and those are going to be basically  indicator variables they're going to  indicate  which symbol that cell gets to uh have  in it  okay so again picture here  um  the in this tableau  you know we don't know how it's going to  get filled out but however it gets  filled out each one of these cells  gets some symbol  um  and that symbol could either be a tape  alphabet symbol or it's symbol  representing a state  right that's the way we do our  configurations  so it could be a tape alphabet symbol  here  showing the magnification  maybe it's the tape alphabet symbol  which represents the blank symbol  or maybe it's a  represents some state  okay  now how am i going to encode that with  variables  so  this let me this is the collection of  variables that's going to apply to the  entire formula fees of mw  um  each cell is so each cell ij is going to  have um  a set of variables  one for each possible  symbol sigma  that's  um in the configuration alphabet namely  a tape symbol  or a state symbol  okay  um  [Music]  so i'm gonna have  well maybe this will become clear as i'm  writing it so  if i turn the the  uh the variable x i j sigma  equal to true  that's just a way of saying that cell i  j contains a sigma so if i have x  you know i j  a  that means the symbol contains an a so  i'm going to illustrate that now for you  so imagine you have lights representing  all the different  x i js  x i js for the different sigmas  i didn't say that well so  um  we're in cell i j so these are all x i j  variables they're x all x i j sigma  variables for the different sigmas  that can go in that can go in this uh  cell  so all different possible sigmas so this  is  gamma union q  okay so  now  um  if i have an a here  in that cell  so then the variable extra  a  is true  and i'm just helping you visualize that  i'm going to that's going to correspond  to turning the light on there's going to  be a light associated with each one of  these variables and it's be turned on  when that variable is true  similarly  if i have  the blank symbol is the thing that goes  in in that cell um then that variable  gets turned on i hope you can see it  maybe it's a little bit small on your  screen  um  the x i j blank  a variable is is true and similarly if  it's q seven here the x i j q seven  variable is true  okay so that that's the way we're going  to be encoding  the  contents of these cells  using these indicator variables  and now  we have to start  making some boolean logic to make sure  that those variables  reasonably represent  the cell con you know contents of these  cells so for example what would be the  first thing that comes to your mind  well we better not have  two lights  going on in any one of the cells because  then we have two symbols on that cells  and that's not allowed  each cell is going we want each cell to  have exactly one symbol and that  corresponds to each each each cell  having exactly one of its lights turned  on or equivalently each cell should have  exactly one of the variables be true  that's the very first part of the  formula is just going to say that so  let's let me show you how that what that  looks like  okay  so here we're talking about this free  cell  it says that there's exactly one light  on purcell  or in other words exactly one of the x  ij sigmas is true for each ij  um  so this is how i'm going to actually  express that using my boolean  formula  uh  i'm sort of color coding the different  parts of the formula which i'm writing  out to you here in english  so first i want to say there's a so in  every cell there's at least one light  that's on and there's almost one light  that's on  um so here's the green part this is  going to say at least one light is on so  i'm going to say that by taking  all of the very all of the symbols  that can appear in that cell  and and take taking in an or over all of  those different associated variables  so it's either got the first symbol on  or the second symbol is there or the or  dot or the last symbol is there one of  those has got to be there  i'm going to write this using my big or  notation  so for sigma  appearing in this  set of possibilities  one of those variables has got to be on  at least  that's what this or big word tells you  now i want to make sure  that there's at most one  that's on so that there are not there's  one at least one on but there are not  two that are on  so i'm gonna  have an additional part of the formula  here which says and i hope you can read  this  a little small  um  if i have two  different  symbols sigma and tau  that are  configuration  uh possible configuration symbols where  sigma and tau are not equal so that's  i'm reading it out to you if you can't  if it's too small for your screen  then i'm going to say it's not possible  so i have the negation  of x i j sigma and x i j tau so  saying in another way  it's not the case  that  that cell contains both sigma and tau  for any two symbols sigma and tau as  long as they're different  i'm going to take these two formulas and  add them together  and this tells me in the cell ij  is exactly one of the variables is true  exactly one of the lights is on and  that's going to represent which symbol  goes into that cell  and then i want to take the and over all  possible cells  to make sure that i'm going to now apply  that everywhere  and so i now i do an and  for i and j ranging between 1 and n to  the k to apply this logic throughout the  picture  yes sigma union asking sigma union q  contains the input output yes  this is not a sigma this is a gamma  gamma is the tape alphabet this is any  symbol including an input symbol um  from sigma  is going to be in in gamma so this is  any any symbol that can appear on the  tape  um okay and that's  this  this expression here that is the  expression feasible cell  okay  so here's a little check-in for you but  maybe before we jump into the check and  let's just make sure  maybe better to take some questions  and then we can ask a check-in  for you  you understand i mean if you're not  getting this  you should you should try to uh  try to figure out how to get it because  this is really just  the foundation it it only gets more it's  not it's not a very complicated proof  once you sort of get the idea what's  going on but if you're not getting this  part you won't be able to get the rest  okay i have no idea what  what that means but i'll read it out to  you guys this looks like a one-hot  tensor encoding same from common same  form commonly used in ml  okay  um  it's just an indicator i would just call  them indicator variables  um  why is it sigma union q  for the big and here  sigma union q  you mean gamma union q  this is wrong  okay  now i understand why everybody's upset  this should be a gamma not a sigma  uh there's a booboo  sorry about that  i don't know if i can fix that without  wrecking the whole slide but so i'm not  going to even try this symbol here  should be gamma not sigma  it's a typo  thanks for catching that  um  yes  so the question is  um  okay so  uh  feasible  is just trying to make sure that the  encoding  represents  setting a bunch of symbols into the  tableau  not so each cell is going to have one  symbol  exactly not two not zero  um so that's what pieces so if you've  satisfied if you set the variables to  satisfy feasible  then there's going to be one symbol on  in each  one in one symbol in each of those cells  um  now another question this is not a cnf  no this is not a cnf that's the second  half of the  that's going to be if we  hope we don't run out of time but i have  a way of converting general sat formulas  to cnfs um  and preserving satisfiability so we're  gonna do that reduction afterward  okay  let's let's  is it a little check to see if you  understand uh at some some level what's  going on  how many variables  does this formula actually have  is it order n order n square  enter the k  remember this  k is the running time of the machine  uh or n to the 2k  what do you think  so i i i mean for how many variables i  mean that in all of  uh  m sub m  yeah how many variables do we have all  together  in this formula if that's what the  question is  and here though here are the variables  so describing them here x i j sigma  okay i'm going to close this  so pick something  all right ending polling  one two three  okay  yeah so  um  okay so that's a good question um  so first of all the the correct answer  is in fact d it's order n to the 2k  um  now uh  i'm getting some questions about what  about the size of gamma and q  well those are going to be fixed  they depend only on the machine  but they don't depend on  n  so thinking about it in term  functionally in terms of n  that's going to be  a constant  multiplier um and so it's going to be  absorbed within the um the big o  uh so that's why we have uh you know  these are constant relative to n these  are fixed  um  they're not into the k possible symbols  there's a fixed number of symbols  depends only on the machine  so looking at a particular machine and  what happens when you look at large  inputs  so why is it d and not c  well don't forget how big is this table  um  this is n to the k by n to the k  so there are n's n to the 2k cell into  the k quantity squared or enter the 2k  cells here  um  and so there's a collection of variables  for each cell some fixed number  variables for each cell so that's why  it's order n to the 2k  good okay so let's uh  let's move on  i think we're actually oh no  um okay so we have one more  slide and i think then we have a break  after  uh so now let's next  uh talk about constructing  um  two more pieces of the  fisa bmw formula so we already got fisa  face of cell done  let's look at  fee sub start and fee sub except  and fistart  is going to tell us  that the start configuration has exactly  these  symbols  and  the accept  tells us  that the  bottom  configuration contains an accepting  state somewhere  so how we're going to write that down  well first of all i'm going to write  these down just cell by cell  so first of all fee start is going to  say  the cell 1 1  contains the q 0. i mean i know what the  start configuration  is it should be  because you know i  you know thinking of me or think you  know us as the reduction  the reduction is given m it's given w so  it knows what the start state is it  knows what the symbols of w  are  so it can it knows what that start  configuration is it's just q zero  followed by the n symbols of w followed  by blanks  so  it wants it wants the very first  um  cell in the in the left-hand corner here  to be a q0 the start state of the  machine which it knows so it's going to  say  x sub 1 1 q 0 that has to be turned on  so it's going to be and a bunch of a  bunch of variables here  and in order to satisfy fee sub start  all of those variables have to be set to  true  so that means  we have to have a q0 in that cell so now  we're going to do the next the next cell  here  the one two the next cell of the uh uh  start configuration so  fee start is going to have x12  so that's the next place contains w1  and so on  x13 contains w2  all the way  up to wn and then there's going to be a  bunch of additional  um  uh  parts which say that the  that we have blanks in the rest just  spelling out exactly all of the symbols  in that top row  because that's what the fiestart  um  uh  formula or sub formula of the of the  overall formula we're making looks like  okay  now let's take a look at fee accept  if you accept  because i'm just looking for  cue accept to appear somewhere  in that bottom row i'm going to do that  in terms of an or  so here is the variables now notice have  n to the k  because it's the last row in the table  so rho n to the k  here  and then i'm going to vary j from 1 to n  to the k so j the column number is going  to range from 1 to n to the k  and i'm looking for that q except so x  enter the kj which is where j is varying  and q accept  one of those has to be true  one of those has to be turned on and so  that's why it's a it's a big war  and that's my um  uh fee accept  peace okay  and now we'll take a little break and  feel free to ask me some more questions  um let me just start the our clock  and go grab yourself  some coffee or  ask me some questions happy to answer  them  why don't we check that q accept only  appears once  is it impossible for q except to appear  twice that's a great question  um  so that would definitely be a broken  configuration  if that happened because a configuration  can ha can have must have exactly one  state symbol appearing  the way we're going to enforce that is  is with the fee move  uh  part of the formula which we haven't  seen yet so p move is going to guarantee  that the machine is acting correctly so  that all of the rows of the tableau are  all legal configurations and they all  legally follow from the previous  so that really in a sense the hard heavy  lifting is coming in fee move  but it's really not that bad  um  somebody says out of curiosity how close  is this intuition proof to the actual  proof  this is the actual proof there's no i'm  not i'm not hiding anything  i mean you know we're being a little uh  loose here but you can turn this  this is this  not cutting any corners here this is  exactly how the proof goes um  and so uh  not not uh you're getting you're getting  the real deal here  somebody wanted to see the previous  slide so here we are whoops  is there something you want to ask  um  okay so fee cell says there's exactly  one symbol per cell  and the variables are set in a way in  terms of thinking of them as indicator  variables there's exactly one variable  set to true in each cell  so it's exactly one symbol per cell  that's what p cell the f cell tells you  what you know if you don't have that  then you have a mess  so you've got to start with that  and then with the other other things are  going to be additional conditions which  when satisfied going to enforce the rest  of the properties that we want  why would the proof fail if we  n to the k with 2 to the n to the k  so okay  i i presume you know  where does would this use the polynomial  running time of the machine uh of the of  m the nondeterministic machine  i mean if you had a an enormous tab we  have to show ultimately that this  reduction is a polynomial time reduction  and that's going to depend on how big  the tableau is because that's going to  tell us how big the formula we're  producing is fees of mw  if e sub mw is exponentially big we  don't have a prayer of being able to  output that formula in polynomial time  if there were  less than into the k steps do we repeat  the last configuration yeah that's what  i said if we have if the machine ends  early the last configuration just stays  there so we're going to  modify the definition of the machine  slightly so that you know the um so it  just stays  yeah  um  okay  um  yeah let me not take the other there's a  bunch of other questions some of them a  little on the technical side  um  let me uh maybe i'll try to address them  as they come along if it if it turns out  to work to do that  okay so uh the word the break is over  um  why don't we uh con is it possibly that  the encoding configuration will not fit  in  into the k  no no  so the question is is there a  possibility that the encoding of the  configuration won't fit into the k if  the machine runs for enter the k steps  the the configuration has to fit with an  end to the k because it can't use  anything more  within into the k steps to think about  it but no the ques the answer is there's  the configurations  if the machine runs time into the k  that whole um tableau is big enough to  write down the entire computation  history  all right so let's continue  fees of move  um  this is in a sense the part which is  going to tell us that  we started right  we ended right  um  fee cell says every every cell contains  one symbol and  now we have to say that the whole  interior is correct how are we going to  do that so these these are the parts  we've already done  and the way i'm going to describe that  is in terms of these kind of little  windows i'm calling neighborhoods  so  imagine here we have a two by three  rectangle  which i'm going to call a two by three  neighborhood  and  what i'm going to argue but i'm not  going to prove uh here i'm just going to  really state it but it's it's really  just a sort of  more or less obvious fact uh but the  proof the  book has the formal proof  that if any every one of these here is  legitimate is legal according to the  rules of the machine if every single  imagine  you know you have these um oops let me  put myself back on here so you can see  me  um if you have um  if you have here  every  um  two by three  window  you can take this as a window and you  slide that over the entire picture of  the tableau  and everything here looks okay as far as  the running of the machine so i'll say  what that means in a second  but if if everything looks locally fine  everywhere  then the whole tableau has to be  a valid tableau  in terms of the rules of m  um  maybe it's easier if i if i describe  what i mean by these being legal  um  so  these neighborhoods  um these two by three neighborhoods are  legal if they're consistent with m's  transition function so i'm going to  describe rather than i mean to do this  formally i would have to go through  kind of a process that we went like what  we did when we went through uh the  construction for the post correspondence  problem and see if the machine moves  left this thing's happens every move is  right that's i think that's  it's not really necessary you can kind  of get the idea very clearly by doing it  a little bit at a higher level um  so let's look at what i mean by a legal  neighborhood so this illegal  neighborhood is a setting of the values  the six values of this 2x3 neighborhood  in a way  which  doesn't violate m's rules  so for example  if m  when it's in state q7 reading a b  goes into state q3 and moves left  then this would be a legal neighborhood  because it shows  the head moving left  um the b becoming a c so reading a b i  should also say that it right converts  that b to a c  and and moves his head left into state  q3 so this would be a legal neighborhood  if that's the way  so being legal depends upon  um  it depends on the transition function of  the machine  so  given the transition function that's  going to tell you which of the legal  neighborhoods  so another legal neighborhood this would  always be a legal neighborhood is that  if nothing changes so that means the  head of the machine was somewhere else  and so  um  you know  whatever was on the tape in this step is  going to be the same stuff in the in  those places  one step later  um here's another possible legal  neighborhood um  is uh  if the head suddenly appears  you know on one of the on one of the  cells  either in the left or the right  that would correspond to the machine to  the machine moving its head from  somewhere off the nape off the  neighborhood into the neighborhood  you know in that step um so this could  be a legal neighborhood  provided the machine actually does move  its head left into a state q5 at some  point  under under some conditions  um  and here is another kind of a weird  illegal neighborhood if you have a b c  and then the a changes to a d  that could also be a legal neighborhood  if the machine  transition function allows  an a to get converted to a d when there  is some machine when there is some state  reading that a and that state also moves  its head left so it doesn't move into  this picture  so those are examples of legal  neighborhoods let me show you some  illegal neighborhoods  um just i'm doing this this is kind of a  proof by example now this is perhaps the  most intuitive part but i  claim that this is easy to turn this  into something airtight and formal  so this would be clearly illegal  if you're if you have  a piece of the  tape in the pre previous  step  where it's a b c and then suddenly the b  changes to a d the symbol on the tape  changes out of nowhere without having a  head nearby  uh to a different uh  to something else that could never  happen so that would be illegal  another thing would be illegal is if a  state appears from nowhere  that could never happen  um or if it just disappears that could  never happen  and here's another here's an interesting  one  if a state becomes two states  um don't forget the machine is  non-deterministic so the machine in  principle could move its head left  on one branch and move its head right on  a different branch but those would have  to be in different tableaus they can't  be in the same tableau  because that doesn't correspond to any  of the threads of the computation those  are multiple threads  um  and i say this because if you think  about  my claim which is going to put down over  here that if every  2x3 neighborhood is legal then the  tableau overall corresponds to a  computation history  this illustrates why it's not enough to  have a two by two two by two  neighborhood where you really need the  two by three  um  because if this was a two by two  neighborhood if you just look at these  four  this left most uh two by two  that could be  um  a legal neighborhood  if it was a two by two  if the rules of the machine allowed for  that and the right  four box  right four cells could also be a legal  neighborhood  so you could have something that looks  okay from the perspective of two by two  neighborhoods but  globally in terms of the overall tableau  is completely nonsensical because it has  multiple heads  um  but if you have a three two by three  neighborhood it's big enough to  prevent this situation from occurring  and  then you know you can check the details  uh and i think it's very plausible that  it guarantees  that the overall tableau is is is  legitimate if all of the two by three  neighborhoods are legal okay and so  that's what we're going to turn into  a um  a boolean expression  we're going to say for each cell  that the set  for each neighborhood so here's a  neighborhood at the iga location that's  i'm calling this sort of this position  here sort of the home location for that  um neighborhood  for each um  for each neighborhood  uh  it has to be set to one of the legal uh  possibilities  and there's again only a fixed number of  those because there's a fixed number of  possible symbols that can appear in  those cells  um so this is that fixed number to the  sixth power at most  and i'm going to say that  um  you know the cell in the upper left  which would be this one  is an r  and this one and this one here is an s  and this one is a t  and this and this one is a v if you just  trace down what the indices are telling  you  it says that  that piece of the tape that piece of the  tableau here is set  according to  one of the possible legal settings  and i'm just going to or over all of  those possible  fixed number of legal settings  and then i take a an and  over all possible tape cells over all  possible neighborhoods  and so that's going to be my fee move  and that's that's it  okay let's see can i explain again the  third example of illegal  so this one over here i presume is i'm  being asked about  well you know if the machine is in a  state q7  you know reading a c  you know the head has to move either  left or right so that at the next  you know the next um configuration  there's got to be a state symbol  appearing either in this cell  or in this cell  and here the tape you know the the the  the head has basically just vanished  with nowhere to you know  it's it's gone  that could not happen in the in the  you know  according to the rules of the machine  the way we talk about turing machines so  that's not possible so this would be an  illegal  um  a neighborhood  you want to prevent any of the ill bad  stuff from happening anywhere in here  it's only good stuff can be happening  locally and that guarantees the overall  picture is okay  um  do we have to check that the head  doesn't leave the tabo from the left  most or right yeah there are some little  details here like that so the question  is you know have to make sure that that  the  yeah you probably need to mark i think  the book probably does this correctly  you may have to mark the left and right  ends to make sure that i mean the right  end can is not a problem because the the  the  the machine can never go um  off the right end  and if you design the machine so that it  never moves its head off the left end  either which you can do then you  wouldn't have to worry about that  possibility but yeah otherwise you would  have to put some sort of delimiter here  to uh to enforce the mach the head not  moving off the left end so there are  some details like that too um  there will be two heads in the same row  no this can i don't know what you  somebody says there will be two heads in  the same row  please elaborate because  this is designed not to allow two heads  in the same row  could i go over the or for legal again  okay the or  so the the big or here  what i have in mind is i i take there's  gonna be first of all i look at the  machine and i look at the transition  function  and based on that  i write down  the list of all the legal two by three  neighborhoods so all the settings which  correspond to legal two way two by three  neighborhoods there's gonna be some  fixed number of those  you know  a hundred  there's a hundred possible legal  neighborhoods  of which you know i've written down here  four  but maybe there's some some number say a  hundred so now there's going to be an  ore  over those hundred different  possibilities it's either going to be  this legal neighborhood or some other  legal neighborhood or some other legal  neighborhood and for each one of those  legal neighborhoods  i'm going to say well  the variables are set according to that  legal neighborhood  or  the the variables are set according to  the next legal neighborhood or the  variables are set according to the next  legal neighborhood and do that 100 times  one of those has got to end up it's an  ore so one of them has to work  otherwise the formula fails and will be  false  because you're going to now end that  over all of the neighborhoods in the  picture  is it possible to have a head on the far  left of the configuration and one on the  far right  you mean a head over here  and a head over there i mean  how did the head get there can't happen  um  you know the head the head has to come  from a head above it um  if you're going to be worrying about the  you know details of the boundaries here  you know  all that's fixable so let's not  you know um  lose sight of the main idea i mean if  you understand the main idea you can fix  little details  so i want to make sure you understand  the main idea what's happening  okay so let's finish up this proof  um  so in summary  um we gave a reduction  from a to sat this is what we  needed  it was in those four pieces  and you really just need to argue that  that  formula we're building is not too big  it's and it's going to be basically the  size of the tableau if you look at what  we constructed  the number of variables is roughly the  size of the tableau  and the amount of logic that we're  putting into the formula is also going  to be a fixed amount of logic  independent of n  for each of the variables  in that tableau  and now somebody asked me about like  this how big the indices are  um  you know the indices  for the you know the x i value the inj  values um  technically they're going to be numbers  between 1 and n to the k  so you're going to have to write those  down and so that's going to be a slight  additional logarithmic cost um to write  those things down but it's not really  that interesting a point  um  and so  the overall f is going to be computable  in polynomial time because the output is  not very big and it's also not  complicated to write the output down  so that's the end of the that's the end  of the proof  um  i can take a couple of questions  we why can't we just check that the  whole this is a good question why can't  we just check that the whole row is  legal  um  you know you can check that you can  check that a row actually is a is a  configuration but you to check that the  row follows from the previous row  um  you know ultimately  the operation of a turing machine is is  a local  thing i mean hey  you know the way the the the way  it moves from one configuration to the  next depends locally on how the those  you know where the head is  and so really that's just another way of  the way i'm saying it is just really  checking the whole configuration but  just doing it locally  i don't know if that's satisfying to you  um  okay uh  why don't i move on because i just want  to make sure we have enough time to get  to the very last part which is a little  bit i'm afraid a little technical um so  we're going to kind of shift gears now  and talk about reducing sat to three set  and  um  let's let's see how it goes um i i don't  always have the most success with with  presenting this uh this little piece  because it's slightly a technical  argument but  um if you don't get it don't worry just  you have to accept that it's true um but  i'd like to show it to you just to makes  the whole presentation uh  you know complete in that sense okay um  so i'm going to give a reduction that  maps  general formulas to three cnf formulas  so that's how i map sat  to  if you remember threesat is  satisfiability but for  three cnfs so it's conjunctive normal  form you're in the form of those clauses  which are anded together and each clause  is an or of a bunch of literals which  are variables or negated variables  so i want to convert phi to phi prime  um which is a three cnf formula but  preserve the satisfiability  feed prime is not going to be logically  equivalent to fee  because i you know i could do that too i  can convert any formula to a logically  equivalent cnf formula maybe not even a  three second of it but yeah you won't be  able to get a 3c enough but you can get  a c and f  but um it might be exponentially larger  and that's not good enough i have to do  the reduction in polynomial time so i  can't generate a much larger formula  that's exponentially larger and so i'm  going to  do that by adding additional variables  so it won't be logically equivalent  because the new formula is going to have  additional variables in it  okay i'm going to kind of do it by  example  and let's see how that goes  so his fee  which is not  in  3c and f it's not even in cnf  because it's got you know  uh orrs of ands appearing which are not  allowed to happen in a cnf  okay so  how we're going to convert that  into a 3cnf formula preserving the  satisfiability and just working it  through with this example  i hope at least give you some idea of  how you of how to do the conversion in  general  so first of all i'm going to represent  this formula as a tree  using its natural tree structure  so you understand so a and b becomes a  and b written as a tree and then i take  or that with c  so i get the tree structure here in sort  of the natural way  and i'm going to label all of these  intermediate nodes which are associated  now with  operations  and  um i'm assuming also that the formula is  fully parenthesized  so that each operation i'm only thinking  about is applying just a two it's a  binary operation  um and let's let's ignore negations for  the minute because negations you can  always push those through down to the  leaves but  it's just going to make it too  complicated  so negations turn out not to be a  problem  so there's only going to be negations at  the level of the  inputs  not at the not negation operations in  the middle  okay so we have this tree structure here  and now i'm going to use these two  logical facts  you know  and i don't know if  you know you've probably all seen hands  and ors i hope otherwise it's going to  be really tough uh but there's also  other logical operators such as the  implication operator  where you have a implies b  thought of as a logical operator  operation  um and so that this is only this  requires that if a is true then b is  true  um  however if a is false  b can be anything  uh and similarly if b is true a can be  anything  um  the only thing that's prohibited is that  if a is true and b is false  that's the only thing that's uh  would be invalid  and so if you think about it that's  going to be equivalent to saying that  either a is false or b is true  one of those has to be  and that's going to be logically  equivalent to saying that a implies b  another logical equivalence maybe is  more familiar to you is just simply de  morgan's laws the mortgage law which  says that if you have the  a not of a and b that's uh equivalent to  saying  uh the not of a or the knot of b  okay i'm going to make use of both of  these  um  now  now here i want to i ran out of room on  the slide so i'm going to take myself  out of the picture here for for a minute  um  uh  i had no place else to put this  so here we have if you're going to think  of the and  in terms of its truth table  so here's a and b in terms of a and b  so if one and one is one  but all other settings of a and b  uh yield zero for the and  i'm going to represent those um  if you imagine a and b is going to be  called c  i'm going to represent this information  with uh four  um small formulas  which taken together you and them  together kind of force c to have the  correct behavior associated with a and b  okay um  so uh  so if a and b are both one and c is one  if a is zero  and b is one  um then  that so then it forces c to be zero  um  and  similarly you know every other setting  besides a and b being  being true  uh  for c to be false which is what you want  when you have and  so i'm gonna write this expression here  down with z1 being in the place of c  by just taking those four  uh expressions and ending them together  so this is exactly those same four  expressions written out linearly  now i want to do the same thing  for z2 but now that's written in terms  of an or  so it's a slightly different truth table  here up in this corner  uh  so now if either one is a one we get a  one result  and so  now  you know if a a and b are true you get  uh c is true  however if a is true and b is false that  still implies c is true  um so  i'm going to write down  uh those rules for specifying how  uh  z2  must be set  and each one of these things is going to  get converted into clauses  three clauses with three literals using  these uh using these um using these  rules over here  so  i'm going to do that for each zi  and lastly to make sure the whole thing  is satisfied which means there's an  output of one here i'm going to have one  clause associated  which says that z4 the output  is a one  now i can convert all of those  when i have a and b imply c that's  logically equivalent to not a or not b  or or c  and the way you can see that is really  by repeated app by application of these  uh rules here  running a little low on time  so  maybe you'll just have to check this  offline  but  quickly  a and b implies c using the first  equivalence is the not of this part or c  and then i can use the morgan to convert  that uh  knot of an and to an or of the knots  and then  and then  i can remove the parenthesis because  uh or is associative  and so i get a  uh  a clause which is what i need so each  one of these guys is really equivalent  to a clause and so i just get a bunch of  clauses and actually technically this  needs to be three um  copy of  three things here should it should be z4  or z4 or z4  which is a lot  um  so checking numbers so i realize my  checking is broken because i only  realized that last point just now as i  was talking so uh  the the actual  uh value that you get in terms of oh no  the number of clauses is correct  no i i i take it back this is fine  so if you understood what i was saying  hopefully you can see how big the  formula fif  fee prime  is in terms of the number of operations  and fee  so let's let's see how many people get  that  i i i acknowledge this maybe a little on  the technical side  okay i'm gonna close it  close it  down please enter your value  okay  yeah the correct answer is 4k plus one  um  because each one of these  operations  is going to end up being a row in this  picture  it's going each operation is going to  have a variable associated to it it's  going to become a row in this picture  and so then and each row is going to  have  four  um clauses which define  um  you know uh  what you need set what you need in order  to force the that that that variable to  have the right value corresponding to  that operation  okay and so then you need and you need  one extra clause here  uh for uh saying that the the this whole  thing evaluates to true  um  okay so that's all i wanted to uh  do today um we proved those two main  theorems  and now we know that there are  np-complete problems and all of the  other  problems that we can get from  by reductions from these problems um  are also going to be np-complete as long  as they're an np  okay so um that's that's it  feel free to  put some questions into the chat  or  move on to whatever else you're going to  be doing next okay so a good question  here is why is fee prime not logically  equivalent with this construction can't  be logically equivalent has  logically equivalent means that it gives  you exactly the same function as you if  you set the variables in the same way  you get the same result coming out  well  v prime has more variables than phi does  so it wouldn't even make sense to talk  about logical equivalence because there  are two functions on different numbers  of variables so in that sense it doesn't  really make sense  you know what you could say is that for  every setting of the  of the overlapping variables of the  variables that appear in both ph and v  prime so those are the original  variables of fee um there's going to  exist some setting of the new variables  which is going to make um  the uh  you know  yeah there's going to be there will  exist some setting of the new variables  which will make the the two formulas  agree but that's not the definition of  logic logical equivalence  okay so  why going back to the proof of status  the satisfiability proof and the legal  neighborhoods  could i go over why the number of legal  neighborhoods is polynomial  the number of legal neighborhoods is not  only polynomial it's constant  it depends only on the machine it does  not depend on n  so  because each cell  can have at most some fixed number of  you know can have the number of tape  symbols plus the number of state symbols  that's  you know that depends on the machine  only  so  uh now we have six six tape cells for  the six cells in a two by three  neighborhood so you're going to have  that number to the sixth power but it's  still it's a constant to the sixth power  still a constant doesn't depend on n  so it's not a question of even being  polynomial um it's it's a it's a  constant value it's a constant  multiplier uh if you want to think about  it in terms of the um  the size of the formula that's going to  result don't forget you know we're  trying to make a  formula  um which is  uh well the reduction has to be  polynomial  um it's a polynomial time reduction so  that means that as n increases the time  to calculate the reduction increases as  a polynomial  but  we're fixing m so m does not  change so therefore anything that  depends on m only is just going to be a  constant um impact on the formula it's  not going to be it doesn't depend on it  okay everybody  bye-bye see you  you

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okay everybody  let's begin  um welcome back  good to see you all here on zoom  um  so we're going to pick up with what we  had been discussing  um  last week  which was  um  a  an introduction to  np completeness so we're following on  our  description of  time complexity  we we started talking about the time  complexity classes  the class p  the non-deterministic classes the class  np p versus np problem and then leading  to this discussion of np completeness  and today we're going to prove the big  theorem in the field uh which really  kind of got things going uh back in the  early 1970s by cook and levin that uh  there was actually an mp complete  problem called that sat in particular is  an np complete problem and then we'll  also talk about threesat which is a  useful tool  um  so just to remember just to uh remember  we had this notion of np completeness  for  languages that be complete if it's in np  and everything else in np is polynomial  time reducible to it  and if an np complete problem  turns out to have a polynomial time  solution then every np problem has a  polynomial time solution  and that's part of the importance of np  completeness uh because since we  uh consider it unlikely that p equals np  uh  and that there probably are some  problems  that uh are in np but are not solvable  in polynomial time  that's  would me imply that an np-complete  problem would have that property and so  proving  a problem and being mp complete is  evidence very strong evidence that it  doesn't have a polynomial time uh  solution and so therefore it's we call  intractable uh it's a very difficult  problem  um  so um  the way we are going to typically show  problems np-complete is by reducing a  known a previously known np-complete  problem to that problem  often it's three-sat  as we've seen several examples already  or it could be some other uh example so  let's just survey briefly uh the things  that we've already  languages that we've already seen  which are mp complete  so we have the languages sat which is a  direct reduction from every np problem  and  so we're going to show that today  that every np language is polynomial  time reducible to sat which is in turn  reducible to three sat  and  we showed previously  that threesat is reducible to clique and  hampf  and  in recitation if you went to that they  showed that the subset sub problem and  the undirected hand path problems are  also  reducible from previously uh shown well  from uh from either from 3sat  or in the case of their undirected hand  path problem it's reducible from the  hand path problem  and the conclusion once we have  these two blue  reducibilities shown  then we know that all of these problems  are np-complete  um  so the class np basically breaks down  into the np complete problems  p problems problems that are in p and  then there might be problems in between  as well so there are some problems that  are not known to be in either category  and in fact there's a old theorem which  shows that if p is different from np  then there are actually problems that  are in the intermediate state  and um  of course  it's possible that p equals np  and then everything collapses down uh to  be the same with the tiny exception of  the um sigma star and empty set  languages which can never be uh complete  um  okay um so that's our quick review  here's a check in that i'm gonna use to  get us ready for the  big proof that we're gonna be spending  most of the lecture on about um  uh showing sat as np-complete but just  to define a little notation which you  might have maybe you've seen already um  but i'm going to do it in the form of a  check-in  so  i'm sure you've all seen the big sum  notation uh using a sigma  to represent a sum over um  over some set of possibilities  just as there is the big sigma notation  we have you can have other operations um  that apply to a a set of uh elements so  in this case we're going to be seeing  the big and and the big or  operation uh it's going to a notation  which is going to allow us to talk about  taking the end of many things or the ore  of many things um because we're we're  going to be building these boolean  formulas and so ands and orders are  going to be the uh operations that we're  going to be focusing on  and so just as an example just to  make sure you're we're all understanding  this notation if you have  uh two strings x and y  uh written out in terms of their  individual symbols so they're both of  length n so x is x1 to xn  y is y 1 to y n  and  now i write the following expression the  big and  for i ranging between 1 and n  of x i equal to y i if that big and is  true what does it tell us about x and y  and i'm just going to offer you two  possibilities that either x and y agree  in some symbol  or they are equal namely that they agree  on every symbol  and so  let's just pull that up as a  quick poll to get us going here  um  [Music]  i just want to make sure you understand  the notation because we're going to be  using that a lot  in describing the uh  polynomial time reduction  from  uh  languages in np to the sat language  okay i think most of you have got the  idea but  so let's finish this up quick  um  see another 15 seconds please  just to give you a chance to participate  um  okay so i'm gonna close this uh  close this poll  last chance let's call  all right  so yes  the big and  as most of you  can see the big and says that you know  x1 equals y1 and x2 equals y2 and x3  equals y3 so they're all  um  every symbol in x is equal to the  corresponding symbol in y  if we had a big or instead of a big and  the uh then a would be the correct  answer because then just there would be  some place where they agree instead of  every place where they they agree  um  okay  so let's um  then  start  to launch in to this big theorem  the proof itself  is a massive details  um with one underlying idea  and in fact it's an idea we've seen  before um but let's before we get ahead  of ourselves let's just understand what  we're trying to do  uh  so we want to show that this language  sat is np-complete remember what's that  is it's boolean formulas that are  satisfiable  uh  so we already showed that the set pro so  being mp complete means it has these two  features it's in np  and everything in np is reducible to it  so first of all sat is np as we've  already seen  the witness that shows a formula is  satisfiable is simply the satisfying  assignment that evaluates the true  so now we're going to pick  some language in np a  and show that a is polynomial time  reducible to  okay  so this is going to apply to any  language a and np so let a be some  language in np  it's decided by some non-deterministic  turing machine m in time into the k  that's what it means to be an np um i'm  going to make ignoring the constant  factors when we could carry that  throughout the proof it just would make  this the the description a little more  cumbersome and wouldn't change any of  the ideas but so let's just say it's  m runs in time into the k  and  recognizes or decides this language a in  uh  uh so it's an it's  in and it's a non-deterministic machine  so we've got to give a polynomial time  reduction from a to sad that's that's  what i have to  demonstrate to you  so what does that reduction look like  it's going to map strings  which may or may not be an a  to formulas which may or may not be  satisfied  that's what the reduction's got to do  so  saying  saying that out  more formally is  it's going to take some string w and map  it to a formula we'll call f sub m w  where m is the  machine that  um  decides a and w is is is the input  so f is going to map w to this formula  where the string is in the language  exactly when the formula is satisfiable  and the formula is going to depend upon  m and w that's why it just has it's  written in this way so we've seen that  kind of thing before  so basically my  my job  you know you know we have this language  a that's an np and now we have some  string w which might be an a or maybe  not is an a  and i have to  quickly in polynomial time now produce a  formula  which is going to be satisfiable exactly  when  the string w is in a  okay so how is that formula going to  work  how can i produce such a formula which i  mean i of course i don't know whether w  is in a or not um because i i'm just a  polynomial time reduction  and um a is an np language so  polynomial time probably is not enough  to solve whether w is in a so i've got  to make matt do that mapping without  knowing  the answer  and the idea is that um and this is  where you know we've seen things like  this before  the formula we're going to construct  simulates the machine on w  so in some way it's going to do that  simulation the trick is figuring out how  to what that means  um but  the the interpretation of that formula  is that in a sense describes and it says  though in my informal language that m  accepts w  very much like  you know  there's a lot of parallel here between  um  this construction and for example the  construction um the pcp construction  where we  made a an instant you know  given a machine and an  input we made a set of those dominoes  where finding a match forced you to  simulate the machine  here finding a satisfying assignment  is going to force you to simulate the  machine  [Music]  so this satisfying assignment is going  to be a computation history for mmw  and i'm just going to write that  computation history in a particular way  so it's going to have a somewhat  different encoding which is going to  help us to visualize what's going on  better  okay  um  so that's that's the  that's the  that's the approach that's the basic  idea of what we're trying to what we're  gonna do so i'm happy to take a minute  if that's  if you have any questions about this  part but um  otherwise i'll just move on to start  doing the actual construction of what  the formula looks like how is it going  to do that how is that formula going to  work  okay so no questions hopefully  we are uh all together  okay  so  first of all  let me describe what my computation  histories are going to look like  and and the situation is a little bit  different than what we had before  because  um when we were talking about the post  correspondence problem we had a  deterministic machine and now our  machine is  non-deterministic  okay so  um  we're going to call the object instead  of an accepting computation or  computation history we're just going to  we're going to call it a tableau or  sometimes an accepting tableau if you  want to  emphasize the accepting nature of it but  generally we're just going to call it a  tableau  so a tableau  is really an accepting computation  history  for  um for  for the machine  on an accepting branch  of its non-deterministic computation  so if m accepts w  it's got to have some accepting branch  and the  tableau  is going to be the sequence of  configurations that the machine goes  through on that accepting branch if  there are several accepting branches  there may be several tableaus there will  there will be several tableaus so  there's going to be one tableau for each  accepting branch of the  of the machine's computation on w  if the machine does not accept w there  won't be any tableaus  and so the whole point is that we're  going to make our formula  and  represent  the statement  that there is a tableau  and satisfying that formula is going to  correspond to filling out  the symbols in in the tableau to make it  a tableau  okay so here is a tableau um so  tableau is just again an accepting  computation history on some branch some  accepting branch of the machine's  computation  so and and it the rows of the ins  instead of writing the  computation history out linearly we're  going to represent it  in a table form where each configuration  is going to be on a separate row  all right  now the dimensions of that table are  going to be n to the k by n to the k  because the  machine  runs for n to the k steps  so  there's gonna if there's an accepting  branch it's going to accept within that  number of steps and we'll have enough  rows here to write down all of the  configurations that the machine goes  through one after the next row by row  each one having a configuration  uh in it  and uh  and then at the bottom there'll be an  accept  you know minor detail if the machine  accepts earlier we'll just say the  machine stays in the set once it's  it enters an accept  the machine does not change  from that point on so the rule of the  machine is  nothing changes and it just remains in  the same configuration from that point  on  okay um  so  um  important to understand what we're you  know if you're if you're not following  what i mean by a tableau  you're doomed  for this lecture  so  it it makes sense if you know for you to  ask a question  to understand what we mean by tableau  okay so just a few more elements here so  this is going to be the start  configuration for mw this would be an  accepting configuration here down at the  very last row  um  and you might imagine you know i i  think i filled out some  hypothetical  first step of the machine after the  start  where  maybe the machine was  a when it's in remember how we can  encode our configurations so this is the  machine is in state q0 looking at w uh  the first symbol of the input w1 and  maybe when it's  in q0 or looking in that first symbol  it moves to state q 7  and goes right  and then changes that w 1 to an a  and so now here is the head shown moved  1 position to the right in the new state  q7  i mean of course that depends on what  the machine is designed to do what the  what the transition function is but this  um  you know could possibly is  what happens so does the  okay so good does a tableau trace all  steps of all branches  no  the tableau corresponds to one accepting  branch  each different accepting branch is going  to have its own tableau so there might  be several different ways of filling out  that second row even of course the first  row is going to be in any tableau it's  got to be the same  um  you know  once i know w  here i've written down maybe i should  unpack this for you you know it's in the  start state here then here are the first  you know here are the n symbols of w w  is of length n so it's w 1 w 2 up to w n  and then i'm padding out the rest with  blanks  so i should have said that too but okay  um  so i want to make my table into the k by  n to the k because that's going to be  enough  to represent the entire  uh  um all of these  configurations  of m  uh running for most into the k steps  right because the machine  um even if it sends its head  moving to the right as fast as it  possibly can  it's never going to have um it's it's  never going to  go outside this box  if it only runs for n to the k steps so  this is going to be big enough to  represent the entire computation of  m assuming it's running for n to the k  time and  the input w is of like that  all right um  what is k so k is the running time of  the machine so we assumed from the  previous slide that m runs in time n to  the k  okay um  so a good question here how can the  tableau be a square table if we have a  low number of computation histories but  a lot of tape um  well i mean the low number of  computation history that that's a  problem you're not saying that well you  know there  each computation history  is in a different table  so this i'm representing a single  computation history here there may be  there are going to be many  configurations  um so i can't have a small number of  configurations using a lot of tape  because  you know i can only use one more cell in  one more one additional tape cell each  time i meant you know if each additional  step of the machine  is yes so there's really no difference  between this if you want to think of  this as a computation history that's  fine this is really just the standard  terminology that's used when you're  proving um  this this particular theorem typically  people talk about it as a tableau but  it's really just the computation history  um  the cue state i mean  the question about the what are these  states queue this is the way we  represent configurations  so this means the machine is in  uh state  they're not all going down the diagonal  the the states are going to zigzag here  through he through this picture here  depending upon how the head of the  machine moves  so you have to go back and review how we  represented configurations of the  machine remember the configuration is a  snapshot  of the machine at a given point  um  how do we know that m runs in polynomial  time we're assuming m runs in polynomial  we started off with a language that's in  np  so it's it's a non-deterministic  polynomial time machine and we're  picking one branch  that accepts  and writing down all the sequence  configurations the machine goes through  let's move on and maybe as  you know ask more questions as they come  to you and i'll pick out some to answer  if that's going to be helpful to others  um  okay  um  so uh  we're going to now construct  this  um  uh  formula  to say that m accepts w again that was  what we uh that's our goal  and it says that a tableau for m on w  exists  um  and basically what that means is we want  to say that it starts right  it ends right and everything in between  is right  and then we're going to need some other  stuff to talk about how we're going to  be encoding those symbols using boolean  variables  so those are going to be the four parts  here's the start starts right here it  ends right here it moves right and here  it talks about the encoding of the  symbols into boolean variables  so those are the four parts of this  formula that i'm going to describe  um  i hope you got your question  answered as to why the total number of  columns is into the k because it's just  big enough to fit the entire all  configurations if the is running for at  most end of the case steps  which is what we're assuming  okay  okay so now  just  getting back to that i'm going to  describe these different components now  on separate slots  um  let me start off with with this  component fee cell which  is sort of the most fundamental one  because it talks about how we're going  to be encoding  those symbols of the tableau  into the boolean variables  so again here's kind of our the picture  to have in mind uh of this tableau this  is this into the k by enter the k table  representing  um you know the some  accepting branch  of the machine's computation  if there is one  and so now  let me draw a  some one of the cells here i'm going to  magnify it  so this is the i  j cell  here  and le i'm going to there there are  going to be  a collection of boolean variables  associated with each one of the cells so  each one of the cells is going to have a  bunch of variables all to itself  and those are going to be basically  indicator variables they're going to  indicate  which symbol that cell gets to uh have  in it  okay so again picture here  um  the in this tableau  you know we don't know how it's going to  get filled out but however it gets  filled out each one of these cells  gets some symbol  um  and that symbol could either be a tape  alphabet symbol or it's symbol  representing a state  right that's the way we do our  configurations  so it could be a tape alphabet symbol  here  showing the magnification  maybe it's the tape alphabet symbol  which represents the blank symbol  or maybe it's a  represents some state  okay  now how am i going to encode that with  variables  so  this let me this is the collection of  variables that's going to apply to the  entire formula fees of mw  um  each cell is so each cell ij is going to  have um  a set of variables  one for each possible  symbol sigma  that's  um in the configuration alphabet namely  a tape symbol  or a state symbol  okay  um  [Music]  so i'm gonna have  well maybe this will become clear as i'm  writing it so  if i turn the the  uh the variable x i j sigma  equal to true  that's just a way of saying that cell i  j contains a sigma so if i have x  you know i j  a  that means the symbol contains an a so  i'm going to illustrate that now for you  so imagine you have lights representing  all the different  x i js  x i js for the different sigmas  i didn't say that well so  um  we're in cell i j so these are all x i j  variables they're x all x i j sigma  variables for the different sigmas  that can go in that can go in this uh  cell  so all different possible sigmas so this  is  gamma union q  okay so  now  um  if i have an a here  in that cell  so then the variable extra  a  is true  and i'm just helping you visualize that  i'm going to that's going to correspond  to turning the light on there's going to  be a light associated with each one of  these variables and it's be turned on  when that variable is true  similarly  if i have  the blank symbol is the thing that goes  in in that cell um then that variable  gets turned on i hope you can see it  maybe it's a little bit small on your  screen  um  the x i j blank  a variable is is true and similarly if  it's q seven here the x i j q seven  variable is true  okay so that that's the way we're going  to be encoding  the  contents of these cells  using these indicator variables  and now  we have to start  making some boolean logic to make sure  that those variables  reasonably represent  the cell con you know contents of these  cells so for example what would be the  first thing that comes to your mind  well we better not have  two lights  going on in any one of the cells because  then we have two symbols on that cells  and that's not allowed  each cell is going we want each cell to  have exactly one symbol and that  corresponds to each each each cell  having exactly one of its lights turned  on or equivalently each cell should have  exactly one of the variables be true  that's the very first part of the  formula is just going to say that so  let's let me show you how that what that  looks like  okay  so here we're talking about this free  cell  it says that there's exactly one light  on purcell  or in other words exactly one of the x  ij sigmas is true for each ij  um  so this is how i'm going to actually  express that using my boolean  formula  uh  i'm sort of color coding the different  parts of the formula which i'm writing  out to you here in english  so first i want to say there's a so in  every cell there's at least one light  that's on and there's almost one light  that's on  um so here's the green part this is  going to say at least one light is on so  i'm going to say that by taking  all of the very all of the symbols  that can appear in that cell  and and take taking in an or over all of  those different associated variables  so it's either got the first symbol on  or the second symbol is there or the or  dot or the last symbol is there one of  those has got to be there  i'm going to write this using my big or  notation  so for sigma  appearing in this  set of possibilities  one of those variables has got to be on  at least  that's what this or big word tells you  now i want to make sure  that there's at most one  that's on so that there are not there's  one at least one on but there are not  two that are on  so i'm gonna  have an additional part of the formula  here which says and i hope you can read  this  a little small  um  if i have two  different  symbols sigma and tau  that are  configuration  uh possible configuration symbols where  sigma and tau are not equal so that's  i'm reading it out to you if you can't  if it's too small for your screen  then i'm going to say it's not possible  so i have the negation  of x i j sigma and x i j tau so  saying in another way  it's not the case  that  that cell contains both sigma and tau  for any two symbols sigma and tau as  long as they're different  i'm going to take these two formulas and  add them together  and this tells me in the cell ij  is exactly one of the variables is true  exactly one of the lights is on and  that's going to represent which symbol  goes into that cell  and then i want to take the and over all  possible cells  to make sure that i'm going to now apply  that everywhere  and so i now i do an and  for i and j ranging between 1 and n to  the k to apply this logic throughout the  picture  yes sigma union asking sigma union q  contains the input output yes  this is not a sigma this is a gamma  gamma is the tape alphabet this is any  symbol including an input symbol um  from sigma  is going to be in in gamma so this is  any any symbol that can appear on the  tape  um okay and that's  this  this expression here that is the  expression feasible cell  okay  so here's a little check-in for you but  maybe before we jump into the check and  let's just make sure  maybe better to take some questions  and then we can ask a check-in  for you  you understand i mean if you're not  getting this  you should you should try to uh  try to figure out how to get it because  this is really just  the foundation it it only gets more it's  not it's not a very complicated proof  once you sort of get the idea what's  going on but if you're not getting this  part you won't be able to get the rest  okay i have no idea what  what that means but i'll read it out to  you guys this looks like a one-hot  tensor encoding same from common same  form commonly used in ml  okay  um  it's just an indicator i would just call  them indicator variables  um  why is it sigma union q  for the big and here  sigma union q  you mean gamma union q  this is wrong  okay  now i understand why everybody's upset  this should be a gamma not a sigma  uh there's a booboo  sorry about that  i don't know if i can fix that without  wrecking the whole slide but so i'm not  going to even try this symbol here  should be gamma not sigma  it's a typo  thanks for catching that  um  yes  so the question is  um  okay so  uh  feasible  is just trying to make sure that the  encoding  represents  setting a bunch of symbols into the  tableau  not so each cell is going to have one  symbol  exactly not two not zero  um so that's what pieces so if you've  satisfied if you set the variables to  satisfy feasible  then there's going to be one symbol on  in each  one in one symbol in each of those cells  um  now another question this is not a cnf  no this is not a cnf that's the second  half of the  that's going to be if we  hope we don't run out of time but i have  a way of converting general sat formulas  to cnfs um  and preserving satisfiability so we're  gonna do that reduction afterward  okay  let's let's  is it a little check to see if you  understand uh at some some level what's  going on  how many variables  does this formula actually have  is it order n order n square  enter the k  remember this  k is the running time of the machine  uh or n to the 2k  what do you think  so i i i mean for how many variables i  mean that in all of  uh  m sub m  yeah how many variables do we have all  together  in this formula if that's what the  question is  and here though here are the variables  so describing them here x i j sigma  okay i'm going to close this  so pick something  all right ending polling  one two three  okay  yeah so  um  okay so that's a good question um  so first of all the the correct answer  is in fact d it's order n to the 2k  um  now uh  i'm getting some questions about what  about the size of gamma and q  well those are going to be fixed  they depend only on the machine  but they don't depend on  n  so thinking about it in term  functionally in terms of n  that's going to be  a constant  multiplier um and so it's going to be  absorbed within the um the big o  uh so that's why we have uh you know  these are constant relative to n these  are fixed  um  they're not into the k possible symbols  there's a fixed number of symbols  depends only on the machine  so looking at a particular machine and  what happens when you look at large  inputs  so why is it d and not c  well don't forget how big is this table  um  this is n to the k by n to the k  so there are n's n to the 2k cell into  the k quantity squared or enter the 2k  cells here  um  and so there's a collection of variables  for each cell some fixed number  variables for each cell so that's why  it's order n to the 2k  good okay so let's uh  let's move on  i think we're actually oh no  um okay so we have one more  slide and i think then we have a break  after  uh so now let's next  uh talk about constructing  um  two more pieces of the  fisa bmw formula so we already got fisa  face of cell done  let's look at  fee sub start and fee sub except  and fistart  is going to tell us  that the start configuration has exactly  these  symbols  and  the accept  tells us  that the  bottom  configuration contains an accepting  state somewhere  so how we're going to write that down  well first of all i'm going to write  these down just cell by cell  so first of all fee start is going to  say  the cell 1 1  contains the q 0. i mean i know what the  start configuration  is it should be  because you know i  you know thinking of me or think you  know us as the reduction  the reduction is given m it's given w so  it knows what the start state is it  knows what the symbols of w  are  so it can it knows what that start  configuration is it's just q zero  followed by the n symbols of w followed  by blanks  so  it wants it wants the very first  um  cell in the in the left-hand corner here  to be a q0 the start state of the  machine which it knows so it's going to  say  x sub 1 1 q 0 that has to be turned on  so it's going to be and a bunch of a  bunch of variables here  and in order to satisfy fee sub start  all of those variables have to be set to  true  so that means  we have to have a q0 in that cell so now  we're going to do the next the next cell  here  the one two the next cell of the uh uh  start configuration so  fee start is going to have x12  so that's the next place contains w1  and so on  x13 contains w2  all the way  up to wn and then there's going to be a  bunch of additional  um  uh  parts which say that the  that we have blanks in the rest just  spelling out exactly all of the symbols  in that top row  because that's what the fiestart  um  uh  formula or sub formula of the of the  overall formula we're making looks like  okay  now let's take a look at fee accept  if you accept  because i'm just looking for  cue accept to appear somewhere  in that bottom row i'm going to do that  in terms of an or  so here is the variables now notice have  n to the k  because it's the last row in the table  so rho n to the k  here  and then i'm going to vary j from 1 to n  to the k so j the column number is going  to range from 1 to n to the k  and i'm looking for that q except so x  enter the kj which is where j is varying  and q accept  one of those has to be true  one of those has to be turned on and so  that's why it's a it's a big war  and that's my um  uh fee accept  peace okay  and now we'll take a little break and  feel free to ask me some more questions  um let me just start the our clock  and go grab yourself  some coffee or  ask me some questions happy to answer  them  why don't we check that q accept only  appears once  is it impossible for q except to appear  twice that's a great question  um  so that would definitely be a broken  configuration  if that happened because a configuration  can ha can have must have exactly one  state symbol appearing  the way we're going to enforce that is  is with the fee move  uh  part of the formula which we haven't  seen yet so p move is going to guarantee  that the machine is acting correctly so  that all of the rows of the tableau are  all legal configurations and they all  legally follow from the previous  so that really in a sense the hard heavy  lifting is coming in fee move  but it's really not that bad  um  somebody says out of curiosity how close  is this intuition proof to the actual  proof  this is the actual proof there's no i'm  not i'm not hiding anything  i mean you know we're being a little uh  loose here but you can turn this  this is this  not cutting any corners here this is  exactly how the proof goes um  and so uh  not not uh you're getting you're getting  the real deal here  somebody wanted to see the previous  slide so here we are whoops  is there something you want to ask  um  okay so fee cell says there's exactly  one symbol per cell  and the variables are set in a way in  terms of thinking of them as indicator  variables there's exactly one variable  set to true in each cell  so it's exactly one symbol per cell  that's what p cell the f cell tells you  what you know if you don't have that  then you have a mess  so you've got to start with that  and then with the other other things are  going to be additional conditions which  when satisfied going to enforce the rest  of the properties that we want  why would the proof fail if we  n to the k with 2 to the n to the k  so okay  i i presume you know  where does would this use the polynomial  running time of the machine uh of the of  m the nondeterministic machine  i mean if you had a an enormous tab we  have to show ultimately that this  reduction is a polynomial time reduction  and that's going to depend on how big  the tableau is because that's going to  tell us how big the formula we're  producing is fees of mw  if e sub mw is exponentially big we  don't have a prayer of being able to  output that formula in polynomial time  if there were  less than into the k steps do we repeat  the last configuration yeah that's what  i said if we have if the machine ends  early the last configuration just stays  there so we're going to  modify the definition of the machine  slightly so that you know the um so it  just stays  yeah  um  okay  um  yeah let me not take the other there's a  bunch of other questions some of them a  little on the technical side  um  let me uh maybe i'll try to address them  as they come along if it if it turns out  to work to do that  okay so uh the word the break is over  um  why don't we uh con is it possibly that  the encoding configuration will not fit  in  into the k  no no  so the question is is there a  possibility that the encoding of the  configuration won't fit into the k if  the machine runs for enter the k steps  the the configuration has to fit with an  end to the k because it can't use  anything more  within into the k steps to think about  it but no the ques the answer is there's  the configurations  if the machine runs time into the k  that whole um tableau is big enough to  write down the entire computation  history  all right so let's continue  fees of move  um  this is in a sense the part which is  going to tell us that  we started right  we ended right  um  fee cell says every every cell contains  one symbol and  now we have to say that the whole  interior is correct how are we going to  do that so these these are the parts  we've already done  and the way i'm going to describe that  is in terms of these kind of little  windows i'm calling neighborhoods  so  imagine here we have a two by three  rectangle  which i'm going to call a two by three  neighborhood  and  what i'm going to argue but i'm not  going to prove uh here i'm just going to  really state it but it's it's really  just a sort of  more or less obvious fact uh but the  proof the  book has the formal proof  that if any every one of these here is  legitimate is legal according to the  rules of the machine if every single  imagine  you know you have these um oops let me  put myself back on here so you can see  me  um if you have um  if you have here  every  um  two by three  window  you can take this as a window and you  slide that over the entire picture of  the tableau  and everything here looks okay as far as  the running of the machine so i'll say  what that means in a second  but if if everything looks locally fine  everywhere  then the whole tableau has to be  a valid tableau  in terms of the rules of m  um  maybe it's easier if i if i describe  what i mean by these being legal  um  so  these neighborhoods  um these two by three neighborhoods are  legal if they're consistent with m's  transition function so i'm going to  describe rather than i mean to do this  formally i would have to go through  kind of a process that we went like what  we did when we went through uh the  construction for the post correspondence  problem and see if the machine moves  left this thing's happens every move is  right that's i think that's  it's not really necessary you can kind  of get the idea very clearly by doing it  a little bit at a higher level um  so let's look at what i mean by a legal  neighborhood so this illegal  neighborhood is a setting of the values  the six values of this 2x3 neighborhood  in a way  which  doesn't violate m's rules  so for example  if m  when it's in state q7 reading a b  goes into state q3 and moves left  then this would be a legal neighborhood  because it shows  the head moving left  um the b becoming a c so reading a b i  should also say that it right converts  that b to a c  and and moves his head left into state  q3 so this would be a legal neighborhood  if that's the way  so being legal depends upon  um  it depends on the transition function of  the machine  so  given the transition function that's  going to tell you which of the legal  neighborhoods  so another legal neighborhood this would  always be a legal neighborhood is that  if nothing changes so that means the  head of the machine was somewhere else  and so  um  you know  whatever was on the tape in this step is  going to be the same stuff in the in  those places  one step later  um here's another possible legal  neighborhood um  is uh  if the head suddenly appears  you know on one of the on one of the  cells  either in the left or the right  that would correspond to the machine to  the machine moving its head from  somewhere off the nape off the  neighborhood into the neighborhood  you know in that step um so this could  be a legal neighborhood  provided the machine actually does move  its head left into a state q5 at some  point  under under some conditions  um  and here is another kind of a weird  illegal neighborhood if you have a b c  and then the a changes to a d  that could also be a legal neighborhood  if the machine  transition function allows  an a to get converted to a d when there  is some machine when there is some state  reading that a and that state also moves  its head left so it doesn't move into  this picture  so those are examples of legal  neighborhoods let me show you some  illegal neighborhoods  um just i'm doing this this is kind of a  proof by example now this is perhaps the  most intuitive part but i  claim that this is easy to turn this  into something airtight and formal  so this would be clearly illegal  if you're if you have  a piece of the  tape in the pre previous  step  where it's a b c and then suddenly the b  changes to a d the symbol on the tape  changes out of nowhere without having a  head nearby  uh to a different uh  to something else that could never  happen so that would be illegal  another thing would be illegal is if a  state appears from nowhere  that could never happen  um or if it just disappears that could  never happen  and here's another here's an interesting  one  if a state becomes two states  um don't forget the machine is  non-deterministic so the machine in  principle could move its head left  on one branch and move its head right on  a different branch but those would have  to be in different tableaus they can't  be in the same tableau  because that doesn't correspond to any  of the threads of the computation those  are multiple threads  um  and i say this because if you think  about  my claim which is going to put down over  here that if every  2x3 neighborhood is legal then the  tableau overall corresponds to a  computation history  this illustrates why it's not enough to  have a two by two two by two  neighborhood where you really need the  two by three  um  because if this was a two by two  neighborhood if you just look at these  four  this left most uh two by two  that could be  um  a legal neighborhood  if it was a two by two  if the rules of the machine allowed for  that and the right  four box  right four cells could also be a legal  neighborhood  so you could have something that looks  okay from the perspective of two by two  neighborhoods but  globally in terms of the overall tableau  is completely nonsensical because it has  multiple heads  um  but if you have a three two by three  neighborhood it's big enough to  prevent this situation from occurring  and  then you know you can check the details  uh and i think it's very plausible that  it guarantees  that the overall tableau is is is  legitimate if all of the two by three  neighborhoods are legal okay and so  that's what we're going to turn into  a um  a boolean expression  we're going to say for each cell  that the set  for each neighborhood so here's a  neighborhood at the iga location that's  i'm calling this sort of this position  here sort of the home location for that  um neighborhood  for each um  for each neighborhood  uh  it has to be set to one of the legal uh  possibilities  and there's again only a fixed number of  those because there's a fixed number of  possible symbols that can appear in  those cells  um so this is that fixed number to the  sixth power at most  and i'm going to say that  um  you know the cell in the upper left  which would be this one  is an r  and this one and this one here is an s  and this one is a t  and this and this one is a v if you just  trace down what the indices are telling  you  it says that  that piece of the tape that piece of the  tableau here is set  according to  one of the possible legal settings  and i'm just going to or over all of  those possible  fixed number of legal settings  and then i take a an and  over all possible tape cells over all  possible neighborhoods  and so that's going to be my fee move  and that's that's it  okay let's see can i explain again the  third example of illegal  so this one over here i presume is i'm  being asked about  well you know if the machine is in a  state q7  you know reading a c  you know the head has to move either  left or right so that at the next  you know the next um configuration  there's got to be a state symbol  appearing either in this cell  or in this cell  and here the tape you know the the the  the head has basically just vanished  with nowhere to you know  it's it's gone  that could not happen in the in the  you know  according to the rules of the machine  the way we talk about turing machines so  that's not possible so this would be an  illegal  um  a neighborhood  you want to prevent any of the ill bad  stuff from happening anywhere in here  it's only good stuff can be happening  locally and that guarantees the overall  picture is okay  um  do we have to check that the head  doesn't leave the tabo from the left  most or right yeah there are some little  details here like that so the question  is you know have to make sure that that  the  yeah you probably need to mark i think  the book probably does this correctly  you may have to mark the left and right  ends to make sure that i mean the right  end can is not a problem because the the  the  the machine can never go um  off the right end  and if you design the machine so that it  never moves its head off the left end  either which you can do then you  wouldn't have to worry about that  possibility but yeah otherwise you would  have to put some sort of delimiter here  to uh to enforce the mach the head not  moving off the left end so there are  some details like that too um  there will be two heads in the same row  no this can i don't know what you  somebody says there will be two heads in  the same row  please elaborate because  this is designed not to allow two heads  in the same row  could i go over the or for legal again  okay the or  so the the big or here  what i have in mind is i i take there's  gonna be first of all i look at the  machine and i look at the transition  function  and based on that  i write down  the list of all the legal two by three  neighborhoods so all the settings which  correspond to legal two way two by three  neighborhoods there's gonna be some  fixed number of those  you know  a hundred  there's a hundred possible legal  neighborhoods  of which you know i've written down here  four  but maybe there's some some number say a  hundred so now there's going to be an  ore  over those hundred different  possibilities it's either going to be  this legal neighborhood or some other  legal neighborhood or some other legal  neighborhood and for each one of those  legal neighborhoods  i'm going to say well  the variables are set according to that  legal neighborhood  or  the the variables are set according to  the next legal neighborhood or the  variables are set according to the next  legal neighborhood and do that 100 times  one of those has got to end up it's an  ore so one of them has to work  otherwise the formula fails and will be  false  because you're going to now end that  over all of the neighborhoods in the  picture  is it possible to have a head on the far  left of the configuration and one on the  far right  you mean a head over here  and a head over there i mean  how did the head get there can't happen  um  you know the head the head has to come  from a head above it um  if you're going to be worrying about the  you know details of the boundaries here  you know  all that's fixable so let's not  you know um  lose sight of the main idea i mean if  you understand the main idea you can fix  little details  so i want to make sure you understand  the main idea what's happening  okay so let's finish up this proof  um  so in summary  um we gave a reduction  from a to sat this is what we  needed  it was in those four pieces  and you really just need to argue that  that  formula we're building is not too big  it's and it's going to be basically the  size of the tableau if you look at what  we constructed  the number of variables is roughly the  size of the tableau  and the amount of logic that we're  putting into the formula is also going  to be a fixed amount of logic  independent of n  for each of the variables  in that tableau  and now somebody asked me about like  this how big the indices are  um  you know the indices  for the you know the x i value the inj  values um  technically they're going to be numbers  between 1 and n to the k  so you're going to have to write those  down and so that's going to be a slight  additional logarithmic cost um to write  those things down but it's not really  that interesting a point  um  and so  the overall f is going to be computable  in polynomial time because the output is  not very big and it's also not  complicated to write the output down  so that's the end of the that's the end  of the proof  um  i can take a couple of questions  we why can't we just check that the  whole this is a good question why can't  we just check that the whole row is  legal  um  you know you can check that you can  check that a row actually is a is a  configuration but you to check that the  row follows from the previous row  um  you know ultimately  the operation of a turing machine is is  a local  thing i mean hey  you know the way the the the way  it moves from one configuration to the  next depends locally on how the those  you know where the head is  and so really that's just another way of  the way i'm saying it is just really  checking the whole configuration but  just doing it locally  i don't know if that's satisfying to you  um  okay uh  why don't i move on because i just want  to make sure we have enough time to get  to the very last part which is a little  bit i'm afraid a little technical um so  we're going to kind of shift gears now  and talk about reducing sat to three set  and  um  let's let's see how it goes um i i don't  always have the most success with with  presenting this uh this little piece  because it's slightly a technical  argument but  um if you don't get it don't worry just  you have to accept that it's true um but  i'd like to show it to you just to makes  the whole presentation uh  you know complete in that sense okay um  so i'm going to give a reduction that  maps  general formulas to three cnf formulas  so that's how i map sat  to  if you remember threesat is  satisfiability but for  three cnfs so it's conjunctive normal  form you're in the form of those clauses  which are anded together and each clause  is an or of a bunch of literals which  are variables or negated variables  so i want to convert phi to phi prime  um which is a three cnf formula but  preserve the satisfiability  feed prime is not going to be logically  equivalent to fee  because i you know i could do that too i  can convert any formula to a logically  equivalent cnf formula maybe not even a  three second of it but yeah you won't be  able to get a 3c enough but you can get  a c and f  but um it might be exponentially larger  and that's not good enough i have to do  the reduction in polynomial time so i  can't generate a much larger formula  that's exponentially larger and so i'm  going to  do that by adding additional variables  so it won't be logically equivalent  because the new formula is going to have  additional variables in it  okay i'm going to kind of do it by  example  and let's see how that goes  so his fee  which is not  in  3c and f it's not even in cnf  because it's got you know  uh orrs of ands appearing which are not  allowed to happen in a cnf  okay so  how we're going to convert that  into a 3cnf formula preserving the  satisfiability and just working it  through with this example  i hope at least give you some idea of  how you of how to do the conversion in  general  so first of all i'm going to represent  this formula as a tree  using its natural tree structure  so you understand so a and b becomes a  and b written as a tree and then i take  or that with c  so i get the tree structure here in sort  of the natural way  and i'm going to label all of these  intermediate nodes which are associated  now with  operations  and  um i'm assuming also that the formula is  fully parenthesized  so that each operation i'm only thinking  about is applying just a two it's a  binary operation  um and let's let's ignore negations for  the minute because negations you can  always push those through down to the  leaves but  it's just going to make it too  complicated  so negations turn out not to be a  problem  so there's only going to be negations at  the level of the  inputs  not at the not negation operations in  the middle  okay so we have this tree structure here  and now i'm going to use these two  logical facts  you know  and i don't know if  you know you've probably all seen hands  and ors i hope otherwise it's going to  be really tough uh but there's also  other logical operators such as the  implication operator  where you have a implies b  thought of as a logical operator  operation  um and so that this is only this  requires that if a is true then b is  true  um  however if a is false  b can be anything  uh and similarly if b is true a can be  anything  um  the only thing that's prohibited is that  if a is true and b is false  that's the only thing that's uh  would be invalid  and so if you think about it that's  going to be equivalent to saying that  either a is false or b is true  one of those has to be  and that's going to be logically  equivalent to saying that a implies b  another logical equivalence maybe is  more familiar to you is just simply de  morgan's laws the mortgage law which  says that if you have the  a not of a and b that's uh equivalent to  saying  uh the not of a or the knot of b  okay i'm going to make use of both of  these  um  now  now here i want to i ran out of room on  the slide so i'm going to take myself  out of the picture here for for a minute  um  uh  i had no place else to put this  so here we have if you're going to think  of the and  in terms of its truth table  so here's a and b in terms of a and b  so if one and one is one  but all other settings of a and b  uh yield zero for the and  i'm going to represent those um  if you imagine a and b is going to be  called c  i'm going to represent this information  with uh four  um small formulas  which taken together you and them  together kind of force c to have the  correct behavior associated with a and b  okay um  so uh  so if a and b are both one and c is one  if a is zero  and b is one  um then  that so then it forces c to be zero  um  and  similarly you know every other setting  besides a and b being  being true  uh  for c to be false which is what you want  when you have and  so i'm gonna write this expression here  down with z1 being in the place of c  by just taking those four  uh expressions and ending them together  so this is exactly those same four  expressions written out linearly  now i want to do the same thing  for z2 but now that's written in terms  of an or  so it's a slightly different truth table  here up in this corner  uh  so now if either one is a one we get a  one result  and so  now  you know if a a and b are true you get  uh c is true  however if a is true and b is false that  still implies c is true  um so  i'm going to write down  uh those rules for specifying how  uh  z2  must be set  and each one of these things is going to  get converted into clauses  three clauses with three literals using  these uh using these um using these  rules over here  so  i'm going to do that for each zi  and lastly to make sure the whole thing  is satisfied which means there's an  output of one here i'm going to have one  clause associated  which says that z4 the output  is a one  now i can convert all of those  when i have a and b imply c that's  logically equivalent to not a or not b  or or c  and the way you can see that is really  by repeated app by application of these  uh rules here  running a little low on time  so  maybe you'll just have to check this  offline  but  quickly  a and b implies c using the first  equivalence is the not of this part or c  and then i can use the morgan to convert  that uh  knot of an and to an or of the knots  and then  and then  i can remove the parenthesis because  uh or is associative  and so i get a  uh  a clause which is what i need so each  one of these guys is really equivalent  to a clause and so i just get a bunch of  clauses and actually technically this  needs to be three um  copy of  three things here should it should be z4  or z4 or z4  which is a lot  um  so checking numbers so i realize my  checking is broken because i only  realized that last point just now as i  was talking so uh  the the actual  uh value that you get in terms of oh no  the number of clauses is correct  no i i i take it back this is fine  so if you understood what i was saying  hopefully you can see how big the  formula fif  fee prime  is in terms of the number of operations  and fee  so let's let's see how many people get  that  i i i acknowledge this maybe a little on  the technical side  okay i'm gonna close it  close it  down please enter your value  okay  yeah the correct answer is 4k plus one  um  because each one of these  operations  is going to end up being a row in this  picture  it's going each operation is going to  have a variable associated to it it's  going to become a row in this picture  and so then and each row is going to  have  four  um clauses which define  um  you know uh  what you need set what you need in order  to force the that that that variable to  have the right value corresponding to  that operation  okay and so then you need and you need  one extra clause here  uh for uh saying that the the this whole  thing evaluates to true  um  okay so that's all i wanted to uh  do today um we proved those two main  theorems  and now we know that there are  np-complete problems and all of the  other  problems that we can get from  by reductions from these problems um  are also going to be np-complete as long  as they're an np  okay so um that's that's it  feel free to  put some questions into the chat  or  move on to whatever else you're going to  be doing next okay so a good question  here is why is fee prime not logically  equivalent with this construction can't  be logically equivalent has  logically equivalent means that it gives  you exactly the same function as you if  you set the variables in the same way  you get the same result coming out  well  v prime has more variables than phi does  so it wouldn't even make sense to talk  about logical equivalence because there  are two functions on different numbers  of variables so in that sense it doesn't  really make sense  you know what you could say is that for  every setting of the  of the overlapping variables of the  variables that appear in both ph and v  prime so those are the original  variables of fee um there's going to  exist some setting of the new variables  which is going to make um  the uh  you know  yeah there's going to be there will  exist some setting of the new variables  which will make the the two formulas  agree but that's not the definition of  logic logical equivalence  okay so  why going back to the proof of status  the satisfiability proof and the legal  neighborhoods  could i go over why the number of legal  neighborhoods is polynomial  the number of legal neighborhoods is not  only polynomial it's constant  it depends only on the machine it does  not depend on n  so  because each cell  can have at most some fixed number of  you know can have the number of tape  symbols plus the number of state symbols  that's  you know that depends on the machine  only  so  uh now we have six six tape cells for  the six cells in a two by three  neighborhood so you're going to have  that number to the sixth power but it's  still it's a constant to the sixth power  still a constant doesn't depend on n  so it's not a question of even being  polynomial um it's it's a it's a  constant value it's a constant  multiplier uh if you want to think about  it in terms of the um  the size of the formula that's going to  result don't forget you know we're  trying to make a  formula  um which is  uh well the reduction has to be  polynomial  um it's a polynomial time reduction so  that means that as n increases the time  to calculate the reduction increases as  a polynomial  but  we're fixing m so m does not  change so therefore anything that  depends on m only is just going to be a  constant um impact on the formula it's  not going to be it doesn't depend on it  okay everybody  bye-bye see you  you
 

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308
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310
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311
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319
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320
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321
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326
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327
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328
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329
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330
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331
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332
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337
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338
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339
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340
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341
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342
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343
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344
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345
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346
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347
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348
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349
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350
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351
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352
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353
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354
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355
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356
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357
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358
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359
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360
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361
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362
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363
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364
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365
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366
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367
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368
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369
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370
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371
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372
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373
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374
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375
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376
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377
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378
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379
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380
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381
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382
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383
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384
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385
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386
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387
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388
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389
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390
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391
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392
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393
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394
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395
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396
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397
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398
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399
00:13:22,959 --> 00:13:23,910


400
00:13:23,910 --> 00:13:26,310

 

401
00:13:26,310 --> 00:13:27,509

 

402
00:13:27,509 --> 00:13:28,949

 

403
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404
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405
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406
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407
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408
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409
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410
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411
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412
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413
00:13:45,519 --> 00:13:47,189


414
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415
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416
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417
00:13:53,600 --> 00:13:54,470


418
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419
00:13:54,480 --> 00:13:57,110


420
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421
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422
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423
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424
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425
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426
00:14:06,480 --> 00:14:07,750


427
00:14:07,750 --> 00:14:09,509

 

428
00:14:09,509 --> 00:14:11,750

 

429
00:14:11,750 --> 00:14:13,670

 

430
00:14:13,670 --> 00:14:14,829

 

431
00:14:14,829 --> 00:14:14,839

 

432
00:14:14,839 --> 00:14:16,790


433
00:14:16,790 --> 00:14:17,829

 

434
00:14:17,829 --> 00:14:17,839

 

435
00:14:17,839 --> 00:14:19,509


436
00:14:19,509 --> 00:14:21,910

 

437
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438
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439
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440
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441
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442
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443
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444
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445
00:14:34,399 --> 00:14:35,829


446
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447
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448
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449
00:14:40,000 --> 00:14:41,030


450
00:14:41,030 --> 00:14:41,040

 

451
00:14:41,040 --> 00:14:41,750


452
00:14:41,750 --> 00:14:43,269

 

453
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454
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455
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456
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457
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458
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459
00:15:01,350 --> 00:15:01,360

 

460
00:15:01,360 --> 00:15:02,470


461
00:15:02,470 --> 00:15:03,670

 

462
00:15:03,670 --> 00:15:05,430

 

463
00:15:05,430 --> 00:15:08,069

 

464
00:15:08,069 --> 00:15:09,750

 

465
00:15:09,750 --> 00:15:11,910

 

466
00:15:11,910 --> 00:15:13,509

 

467
00:15:13,509 --> 00:15:16,150

 

468
00:15:16,150 --> 00:15:18,389

 

469
00:15:18,389 --> 00:15:21,670

 

470
00:15:21,670 --> 00:15:23,750

 

471
00:15:23,750 --> 00:15:26,389

 

472
00:15:26,389 --> 00:15:29,350

 

473
00:15:29,350 --> 00:15:32,230

 

474
00:15:32,230 --> 00:15:32,240

 

475
00:15:32,240 --> 00:15:33,269


476
00:15:33,269 --> 00:15:33,279

 

477
00:15:33,279 --> 00:15:34,470


478
00:15:34,470 --> 00:15:35,670

 

479
00:15:35,670 --> 00:15:39,189

 

480
00:15:39,189 --> 00:15:41,189

 

481
00:15:41,189 --> 00:15:43,670

 

482
00:15:43,670 --> 00:15:46,629

 

483
00:15:46,629 --> 00:15:48,470

 

484
00:15:48,470 --> 00:15:52,230

 

485
00:15:52,230 --> 00:15:54,710

 

486
00:15:54,710 --> 00:15:57,350

 

487
00:15:57,350 --> 00:15:59,269

 

488
00:15:59,269 --> 00:15:59,279

 

489
00:15:59,279 --> 00:16:00,470


490
00:16:00,470 --> 00:16:03,590

 

491
00:16:03,590 --> 00:16:05,910

 

492
00:16:05,910 --> 00:16:07,990

 

493
00:16:07,990 --> 00:16:09,829

 

494
00:16:09,829 --> 00:16:12,629

 

495
00:16:12,629 --> 00:16:15,829

 

496
00:16:15,829 --> 00:16:16,790

 

497
00:16:16,790 --> 00:16:19,189

 

498
00:16:19,189 --> 00:16:22,949

 

499
00:16:22,949 --> 00:16:24,710

 

500
00:16:24,710 --> 00:16:24,720

 

501
00:16:24,720 --> 00:16:26,710


502
00:16:26,710 --> 00:16:30,310

 

503
00:16:30,310 --> 00:16:30,320

 

504
00:16:30,320 --> 00:16:31,269


505
00:16:31,269 --> 00:16:32,949

 

506
00:16:32,949 --> 00:16:35,509

 

507
00:16:35,509 --> 00:16:37,430

 

508
00:16:37,430 --> 00:16:39,670

 

509
00:16:39,670 --> 00:16:41,430

 

510
00:16:41,430 --> 00:16:44,550

 

511
00:16:44,550 --> 00:16:46,710

 

512
00:16:46,710 --> 00:16:47,829

 

513
00:16:47,829 --> 00:16:49,269

 

514
00:16:49,269 --> 00:16:50,949

 

515
00:16:50,949 --> 00:16:50,959

 

516
00:16:50,959 --> 00:16:52,389


517
00:16:52,389 --> 00:16:54,629

 

518
00:16:54,629 --> 00:16:56,470

 

519
00:16:56,470 --> 00:16:58,710

 

520
00:16:58,710 --> 00:17:00,629

 

521
00:17:00,629 --> 00:17:02,310

 

522
00:17:02,310 --> 00:17:03,910

 

523
00:17:03,910 --> 00:17:05,029

 

524
00:17:05,029 --> 00:17:06,949

 

525
00:17:06,949 --> 00:17:08,710

 

526
00:17:08,710 --> 00:17:08,720

 

527
00:17:08,720 --> 00:17:11,270


528
00:17:11,270 --> 00:17:13,510

 

529
00:17:13,510 --> 00:17:13,520

 

530
00:17:13,520 --> 00:17:14,390


531
00:17:14,390 --> 00:17:14,400

 

532
00:17:14,400 --> 00:17:15,909


533
00:17:15,909 --> 00:17:17,829

 

534
00:17:17,829 --> 00:17:18,870

 

535
00:17:18,870 --> 00:17:20,789

 

536
00:17:20,789 --> 00:17:22,150

 

537
00:17:22,150 --> 00:17:23,750

 

538
00:17:23,750 --> 00:17:23,760

 

539
00:17:23,760 --> 00:17:24,789


540
00:17:24,789 --> 00:17:27,909

 

541
00:17:27,909 --> 00:17:29,270

 

542
00:17:29,270 --> 00:17:33,350

 

543
00:17:33,350 --> 00:17:36,070

 

544
00:17:36,070 --> 00:17:37,029

 

545
00:17:37,029 --> 00:17:39,350

 

546
00:17:39,350 --> 00:17:41,270

 

547
00:17:41,270 --> 00:17:43,510

 

548
00:17:43,510 --> 00:17:43,520

 

549
00:17:43,520 --> 00:17:44,390


550
00:17:44,390 --> 00:17:47,510

 

551
00:17:47,510 --> 00:17:50,230

 

552
00:17:50,230 --> 00:17:50,240

 

553
00:17:50,240 --> 00:17:52,310


554
00:17:52,310 --> 00:17:54,390

 

555
00:17:54,390 --> 00:17:54,400

 

556
00:17:54,400 --> 00:17:55,430


557
00:17:55,430 --> 00:17:55,440

 

558
00:17:55,440 --> 00:17:56,310


559
00:17:56,310 --> 00:17:57,990

 

560
00:17:57,990 --> 00:18:01,110

 

561
00:18:01,110 --> 00:18:03,990

 

562
00:18:03,990 --> 00:18:07,590

 

563
00:18:07,590 --> 00:18:09,990

 

564
00:18:09,990 --> 00:18:11,750

 

565
00:18:11,750 --> 00:18:14,470

 

566
00:18:14,470 --> 00:18:16,789

 

567
00:18:16,789 --> 00:18:18,390

 

568
00:18:18,390 --> 00:18:22,630

 

569
00:18:22,630 --> 00:18:25,350

 

570
00:18:25,350 --> 00:18:27,430

 

571
00:18:27,430 --> 00:18:27,440

 

572
00:18:27,440 --> 00:18:28,470


573
00:18:28,470 --> 00:18:30,390

 

574
00:18:30,390 --> 00:18:32,390

 

575
00:18:32,390 --> 00:18:34,870

 

576
00:18:34,870 --> 00:18:34,880

 

577
00:18:34,880 --> 00:18:35,990


578
00:18:35,990 --> 00:18:38,310

 

579
00:18:38,310 --> 00:18:40,549

 

580
00:18:40,549 --> 00:18:43,029

 

581
00:18:43,029 --> 00:18:44,870

 

582
00:18:44,870 --> 00:18:44,880

 

583
00:18:44,880 --> 00:18:45,909


584
00:18:45,909 --> 00:18:48,950

 

585
00:18:48,950 --> 00:18:48,960

 

586
00:18:48,960 --> 00:18:51,510


587
00:18:51,510 --> 00:18:53,510

 

588
00:18:53,510 --> 00:18:55,350

 

589
00:18:55,350 --> 00:18:57,190

 

590
00:18:57,190 --> 00:18:59,110

 

591
00:18:59,110 --> 00:19:01,430

 

592
00:19:01,430 --> 00:19:02,789

 

593
00:19:02,789 --> 00:19:02,799

 

594
00:19:02,799 --> 00:19:03,909


595
00:19:03,909 --> 00:19:05,029

 

596
00:19:05,029 --> 00:19:06,630

 

597
00:19:06,630 --> 00:19:08,470

 

598
00:19:08,470 --> 00:19:10,390

 

599
00:19:10,390 --> 00:19:13,190

 

600
00:19:13,190 --> 00:19:15,510

 

601
00:19:15,510 --> 00:19:19,190

 

602
00:19:19,190 --> 00:19:20,950

 

603
00:19:20,950 --> 00:19:20,960

 

604
00:19:20,960 --> 00:19:23,029


605
00:19:23,029 --> 00:19:25,350

 

606
00:19:25,350 --> 00:19:25,360

 

607
00:19:25,360 --> 00:19:26,310


608
00:19:26,310 --> 00:19:28,390

 

609
00:19:28,390 --> 00:19:30,549

 

610
00:19:30,549 --> 00:19:30,559

 

611
00:19:30,559 --> 00:19:31,590


612
00:19:31,590 --> 00:19:34,230

 

613
00:19:34,230 --> 00:19:34,240

 

614
00:19:34,240 --> 00:19:35,029


615
00:19:35,029 --> 00:19:36,950

 

616
00:19:36,950 --> 00:19:36,960

 

617
00:19:36,960 --> 00:19:38,390


618
00:19:38,390 --> 00:19:39,590

 

619
00:19:39,590 --> 00:19:42,630

 

620
00:19:42,630 --> 00:19:45,029

 

621
00:19:45,029 --> 00:19:47,669

 

622
00:19:47,669 --> 00:19:49,909

 

623
00:19:49,909 --> 00:19:51,270

 

624
00:19:51,270 --> 00:19:53,830

 

625
00:19:53,830 --> 00:19:56,470

 

626
00:19:56,470 --> 00:19:58,950

 

627
00:19:58,950 --> 00:20:01,110

 

628
00:20:01,110 --> 00:20:02,470

 

629
00:20:02,470 --> 00:20:05,110

 

630
00:20:05,110 --> 00:20:07,669

 

631
00:20:07,669 --> 00:20:09,430

 

632
00:20:09,430 --> 00:20:12,710

 

633
00:20:12,710 --> 00:20:14,390

 

634
00:20:14,390 --> 00:20:16,789

 

635
00:20:16,789 --> 00:20:18,630

 

636
00:20:18,630 --> 00:20:21,669

 

637
00:20:21,669 --> 00:20:25,430

 

638
00:20:25,430 --> 00:20:31,590

 

639
00:20:31,590 --> 00:20:33,190

 

640
00:20:33,190 --> 00:20:35,590

 

641
00:20:35,590 --> 00:20:37,669

 

642
00:20:37,669 --> 00:20:40,310

 

643
00:20:40,310 --> 00:20:42,149

 

644
00:20:42,149 --> 00:20:43,590

 

645
00:20:43,590 --> 00:20:45,669

 

646
00:20:45,669 --> 00:20:47,029

 

647
00:20:47,029 --> 00:20:49,350

 

648
00:20:49,350 --> 00:20:51,430

 

649
00:20:51,430 --> 00:20:52,950

 

650
00:20:52,950 --> 00:20:54,870

 

651
00:20:54,870 --> 00:20:55,590

 

652
00:20:55,590 --> 00:20:55,600

 

653
00:20:55,600 --> 00:20:58,390


654
00:20:58,390 --> 00:21:00,470

 

655
00:21:00,470 --> 00:21:03,190

 

656
00:21:03,190 --> 00:21:03,200

 

657
00:21:03,200 --> 00:21:04,470


658
00:21:04,470 --> 00:21:06,630

 

659
00:21:06,630 --> 00:21:08,549

 

660
00:21:08,549 --> 00:21:10,870

 

661
00:21:10,870 --> 00:21:14,149

 

662
00:21:14,149 --> 00:21:16,310

 

663
00:21:16,310 --> 00:21:17,510

 

664
00:21:17,510 --> 00:21:19,430

 

665
00:21:19,430 --> 00:21:21,350

 

666
00:21:21,350 --> 00:21:22,870

 

667
00:21:22,870 --> 00:21:24,950

 

668
00:21:24,950 --> 00:21:27,270

 

669
00:21:27,270 --> 00:21:28,710

 

670
00:21:28,710 --> 00:21:31,510

 

671
00:21:31,510 --> 00:21:31,520

 

672
00:21:31,520 --> 00:21:35,270


673
00:21:35,270 --> 00:21:36,950

 

674
00:21:36,950 --> 00:21:38,710

 

675
00:21:38,710 --> 00:21:40,470

 

676
00:21:40,470 --> 00:21:42,549

 

677
00:21:42,549 --> 00:21:44,710

 

678
00:21:44,710 --> 00:21:46,789

 

679
00:21:46,789 --> 00:21:48,230

 

680
00:21:48,230 --> 00:21:50,789

 

681
00:21:50,789 --> 00:21:52,789

 

682
00:21:52,789 --> 00:21:54,310

 

683
00:21:54,310 --> 00:21:55,990

 

684
00:21:55,990 --> 00:21:58,230

 

685
00:21:58,230 --> 00:21:59,830

 

686
00:21:59,830 --> 00:22:01,669

 

687
00:22:01,669 --> 00:22:01,679

 

688
00:22:01,679 --> 00:22:02,870


689
00:22:02,870 --> 00:22:05,750

 

690
00:22:05,750 --> 00:22:05,760

 

691
00:22:05,760 --> 00:22:07,510


692
00:22:07,510 --> 00:22:09,270

 

693
00:22:09,270 --> 00:22:11,510

 

694
00:22:11,510 --> 00:22:14,710

 

695
00:22:14,710 --> 00:22:14,720

 

696
00:22:14,720 --> 00:22:15,669


697
00:22:15,669 --> 00:22:17,590

 

698
00:22:17,590 --> 00:22:19,990

 

699
00:22:19,990 --> 00:22:22,070

 

700
00:22:22,070 --> 00:22:23,350

 

701
00:22:23,350 --> 00:22:24,950

 

702
00:22:24,950 --> 00:22:26,630

 

703
00:22:26,630 --> 00:22:29,270

 

704
00:22:29,270 --> 00:22:31,270

 

705
00:22:31,270 --> 00:22:34,310

 

706
00:22:34,310 --> 00:22:36,390

 

707
00:22:36,390 --> 00:22:36,400

 

708
00:22:36,400 --> 00:22:37,669


709
00:22:37,669 --> 00:22:37,679

 

710
00:22:37,679 --> 00:22:38,630


711
00:22:38,630 --> 00:22:38,640

 

712
00:22:38,640 --> 00:22:41,510


713
00:22:41,510 --> 00:22:43,029

 

714
00:22:43,029 --> 00:22:45,029

 

715
00:22:45,029 --> 00:22:45,039

 

716
00:22:45,039 --> 00:22:46,789


717
00:22:46,789 --> 00:22:46,799

 

718
00:22:46,799 --> 00:22:48,950


719
00:22:48,950 --> 00:22:48,960

 

720
00:22:48,960 --> 00:22:49,750


721
00:22:49,750 --> 00:22:49,760

 

722
00:22:49,760 --> 00:22:50,950


723
00:22:50,950 --> 00:22:53,270

 

724
00:22:53,270 --> 00:22:55,990

 

725
00:22:55,990 --> 00:22:58,390

 

726
00:22:58,390 --> 00:22:58,400

 

727
00:22:58,400 --> 00:23:01,029


728
00:23:01,029 --> 00:23:01,039

 

729
00:23:01,039 --> 00:23:02,630


730
00:23:02,630 --> 00:23:04,710

 

731
00:23:04,710 --> 00:23:06,710

 

732
00:23:06,710 --> 00:23:09,270

 

733
00:23:09,270 --> 00:23:10,950

 

734
00:23:10,950 --> 00:23:12,230

 

735
00:23:12,230 --> 00:23:14,230

 

736
00:23:14,230 --> 00:23:16,789

 

737
00:23:16,789 --> 00:23:16,799

 

738
00:23:16,799 --> 00:23:17,990


739
00:23:17,990 --> 00:23:19,990

 

740
00:23:19,990 --> 00:23:22,310

 

741
00:23:22,310 --> 00:23:24,870

 

742
00:23:24,870 --> 00:23:26,789

 

743
00:23:26,789 --> 00:23:30,470

 

744
00:23:30,470 --> 00:23:33,510

 

745
00:23:33,510 --> 00:23:37,190

 

746
00:23:37,190 --> 00:23:37,200

 

747
00:23:37,200 --> 00:23:38,470


748
00:23:38,470 --> 00:23:40,390

 

749
00:23:40,390 --> 00:23:42,230

 

750
00:23:42,230 --> 00:23:44,310

 

751
00:23:44,310 --> 00:23:47,310

 

752
00:23:47,310 --> 00:23:50,149

 

753
00:23:50,149 --> 00:23:52,310

 

754
00:23:52,310 --> 00:23:54,630

 

755
00:23:54,630 --> 00:23:54,640

 

756
00:23:54,640 --> 00:23:56,950


757
00:23:56,950 --> 00:23:58,230

 

758
00:23:58,230 --> 00:23:58,240

 

759
00:23:58,240 --> 00:23:58,950


760
00:23:58,950 --> 00:24:00,870

 

761
00:24:00,870 --> 00:24:03,269

 

762
00:24:03,269 --> 00:24:06,789

 

763
00:24:06,789 --> 00:24:06,799

 

764
00:24:06,799 --> 00:24:07,830


765
00:24:07,830 --> 00:24:09,510

 

766
00:24:09,510 --> 00:24:11,430

 

767
00:24:11,430 --> 00:24:13,110

 

768
00:24:13,110 --> 00:24:14,470

 

769
00:24:14,470 --> 00:24:16,830

 

770
00:24:16,830 --> 00:24:19,830

 

771
00:24:19,830 --> 00:24:24,789

 

772
00:24:24,789 --> 00:24:26,310

 

773
00:24:26,310 --> 00:24:28,630

 

774
00:24:28,630 --> 00:24:31,990

 

775
00:24:31,990 --> 00:24:32,000

 

776
00:24:32,000 --> 00:24:33,669


777
00:24:33,669 --> 00:24:35,990

 

778
00:24:35,990 --> 00:24:37,909

 

779
00:24:37,909 --> 00:24:40,789

 

780
00:24:40,789 --> 00:24:42,630

 

781
00:24:42,630 --> 00:24:43,830

 

782
00:24:43,830 --> 00:24:46,870

 

783
00:24:46,870 --> 00:24:49,190

 

784
00:24:49,190 --> 00:24:51,190

 

785
00:24:51,190 --> 00:24:52,950

 

786
00:24:52,950 --> 00:24:54,789

 

787
00:24:54,789 --> 00:24:54,799

 

788
00:24:54,799 --> 00:24:55,830


789
00:24:55,830 --> 00:24:58,230

 

790
00:24:58,230 --> 00:25:01,510

 

791
00:25:01,510 --> 00:25:04,310

 

792
00:25:04,310 --> 00:25:07,430

 

793
00:25:07,430 --> 00:25:08,789

 

794
00:25:08,789 --> 00:25:12,870

 

795
00:25:12,870 --> 00:25:14,630

 

796
00:25:14,630 --> 00:25:16,310

 

797
00:25:16,310 --> 00:25:16,320

 

798
00:25:16,320 --> 00:25:17,350


799
00:25:17,350 --> 00:25:21,909

 

800
00:25:21,909 --> 00:25:23,669

 

801
00:25:23,669 --> 00:25:26,470

 

802
00:25:26,470 --> 00:25:26,480

 

803
00:25:26,480 --> 00:25:27,510


804
00:25:27,510 --> 00:25:29,269

 

805
00:25:29,269 --> 00:25:30,310

 

806
00:25:30,310 --> 00:25:31,830

 

807
00:25:31,830 --> 00:25:34,789

 

808
00:25:34,789 --> 00:25:37,110

 

809
00:25:37,110 --> 00:25:37,120

 

810
00:25:37,120 --> 00:25:38,470


811
00:25:38,470 --> 00:25:39,990

 

812
00:25:39,990 --> 00:25:41,990

 

813
00:25:41,990 --> 00:25:44,070

 

814
00:25:44,070 --> 00:25:45,269

 

815
00:25:45,269 --> 00:25:45,279

 

816
00:25:45,279 --> 00:25:46,549


817
00:25:46,549 --> 00:25:48,230

 

818
00:25:48,230 --> 00:25:48,240

 

819
00:25:48,240 --> 00:25:49,269


820
00:25:49,269 --> 00:25:53,110

 

821
00:25:53,110 --> 00:25:54,470

 

822
00:25:54,470 --> 00:25:57,269

 

823
00:25:57,269 --> 00:25:59,190

 

824
00:25:59,190 --> 00:26:02,230

 

825
00:26:02,230 --> 00:26:02,240

 

826
00:26:02,240 --> 00:26:03,350


827
00:26:03,350 --> 00:26:05,669

 

828
00:26:05,669 --> 00:26:05,679

 

829
00:26:05,679 --> 00:26:07,029


830
00:26:07,029 --> 00:26:07,039

 

831
00:26:07,039 --> 00:26:07,830


832
00:26:07,830 --> 00:26:11,590

 

833
00:26:11,590 --> 00:26:13,510

 

834
00:26:13,510 --> 00:26:17,029

 

835
00:26:17,029 --> 00:26:17,039

 

836
00:26:17,039 --> 00:26:20,149


837
00:26:20,149 --> 00:26:24,149

 

838
00:26:24,149 --> 00:26:26,149

 

839
00:26:26,149 --> 00:26:27,830

 

840
00:26:27,830 --> 00:26:30,789

 

841
00:26:30,789 --> 00:26:33,269

 

842
00:26:33,269 --> 00:26:33,279

 

843
00:26:33,279 --> 00:26:34,149


844
00:26:34,149 --> 00:26:38,390

 

845
00:26:38,390 --> 00:26:39,990

 

846
00:26:39,990 --> 00:26:43,909

 

847
00:26:43,909 --> 00:26:43,919

 

848
00:26:43,919 --> 00:26:44,870


849
00:26:44,870 --> 00:26:44,880

 

850
00:26:44,880 --> 00:26:45,520


851
00:26:45,520 --> 00:26:45,530

 

852
00:26:45,530 --> 00:26:47,350


853
00:26:47,350 --> 00:26:49,830

 

854
00:26:49,830 --> 00:26:52,230

 

855
00:26:52,230 --> 00:26:53,830

 

856
00:26:53,830 --> 00:26:55,990

 

857
00:26:55,990 --> 00:26:59,510

 

858
00:26:59,510 --> 00:27:01,669

 

859
00:27:01,669 --> 00:27:04,070

 

860
00:27:04,070 --> 00:27:07,750

 

861
00:27:07,750 --> 00:27:09,830

 

862
00:27:09,830 --> 00:27:09,840

 

863
00:27:09,840 --> 00:27:10,630


864
00:27:10,630 --> 00:27:12,710

 

865
00:27:12,710 --> 00:27:14,950

 

866
00:27:14,950 --> 00:27:18,789

 

867
00:27:18,789 --> 00:27:20,389

 

868
00:27:20,389 --> 00:27:22,149

 

869
00:27:22,149 --> 00:27:25,669

 

870
00:27:25,669 --> 00:27:27,350

 

871
00:27:27,350 --> 00:27:27,360

 

872
00:27:27,360 --> 00:27:28,789


873
00:27:28,789 --> 00:27:31,750

 

874
00:27:31,750 --> 00:27:33,830

 

875
00:27:33,830 --> 00:27:36,070

 

876
00:27:36,070 --> 00:27:39,830

 

877
00:27:39,830 --> 00:27:39,840

 

878
00:27:39,840 --> 00:27:40,630


879
00:27:40,630 --> 00:27:43,029

 

880
00:27:43,029 --> 00:27:43,039

 

881
00:27:43,039 --> 00:27:43,909


882
00:27:43,909 --> 00:27:47,350

 

883
00:27:47,350 --> 00:27:48,389

 

884
00:27:48,389 --> 00:27:48,399

 

885
00:27:48,399 --> 00:27:49,190


886
00:27:49,190 --> 00:27:49,200

 

887
00:27:49,200 --> 00:27:50,230


888
00:27:50,230 --> 00:27:55,510

 

889
00:27:55,510 --> 00:27:56,950

 

890
00:27:56,950 --> 00:28:01,269

 

891
00:28:01,269 --> 00:28:01,279

 

892
00:28:01,279 --> 00:28:02,070


893
00:28:02,070 --> 00:28:03,430

 

894
00:28:03,430 --> 00:28:05,990

 

895
00:28:05,990 --> 00:28:07,430

 

896
00:28:07,430 --> 00:28:09,110

 

897
00:28:09,110 --> 00:28:10,630

 

898
00:28:10,630 --> 00:28:12,950

 

899
00:28:12,950 --> 00:28:15,430

 

900
00:28:15,430 --> 00:28:15,440

 

901
00:28:15,440 --> 00:28:16,710


902
00:28:16,710 --> 00:28:18,230

 

903
00:28:18,230 --> 00:28:19,990

 

904
00:28:19,990 --> 00:28:23,669

 

905
00:28:23,669 --> 00:28:24,870

 

906
00:28:24,870 --> 00:28:26,310

 

907
00:28:26,310 --> 00:28:26,320

 

908
00:28:26,320 --> 00:28:27,430


909
00:28:27,430 --> 00:28:27,440

 

910
00:28:27,440 --> 00:28:28,549


911
00:28:28,549 --> 00:28:30,789

 

912
00:28:30,789 --> 00:28:33,750

 

913
00:28:33,750 --> 00:28:36,470

 

914
00:28:36,470 --> 00:28:39,110

 

915
00:28:39,110 --> 00:28:41,029

 

916
00:28:41,029 --> 00:28:42,470

 

917
00:28:42,470 --> 00:28:42,480

 

918
00:28:42,480 --> 00:28:43,669


919
00:28:43,669 --> 00:28:45,669

 

920
00:28:45,669 --> 00:28:48,470

 

921
00:28:48,470 --> 00:28:49,590

 

922
00:28:49,590 --> 00:28:51,350

 

923
00:28:51,350 --> 00:28:54,470

 

924
00:28:54,470 --> 00:28:56,950

 

925
00:28:56,950 --> 00:28:59,909

 

926
00:28:59,909 --> 00:29:01,990

 

927
00:29:01,990 --> 00:29:03,990

 

928
00:29:03,990 --> 00:29:07,110

 

929
00:29:07,110 --> 00:29:09,269

 

930
00:29:09,269 --> 00:29:10,549

 

931
00:29:10,549 --> 00:29:12,710

 

932
00:29:12,710 --> 00:29:14,310

 

933
00:29:14,310 --> 00:29:15,669

 

934
00:29:15,669 --> 00:29:17,190

 

935
00:29:17,190 --> 00:29:19,110

 

936
00:29:19,110 --> 00:29:21,350

 

937
00:29:21,350 --> 00:29:23,269

 

938
00:29:23,269 --> 00:29:25,909

 

939
00:29:25,909 --> 00:29:29,510

 

940
00:29:29,510 --> 00:29:30,950

 

941
00:29:30,950 --> 00:29:32,950

 

942
00:29:32,950 --> 00:29:34,789

 

943
00:29:34,789 --> 00:29:36,310

 

944
00:29:36,310 --> 00:29:36,320

 

945
00:29:36,320 --> 00:29:38,549


946
00:29:38,549 --> 00:29:40,070

 

947
00:29:40,070 --> 00:29:40,080

 

948
00:29:40,080 --> 00:29:41,110


949
00:29:41,110 --> 00:29:43,029

 

950
00:29:43,029 --> 00:29:44,549

 

951
00:29:44,549 --> 00:29:46,950

 

952
00:29:46,950 --> 00:29:54,950

 

953
00:29:54,950 --> 00:29:54,960

 

954
00:29:54,960 --> 00:29:56,070


955
00:29:56,070 --> 00:29:57,669

 

956
00:29:57,669 --> 00:30:00,630

 

957
00:30:00,630 --> 00:30:00,640

 

958
00:30:00,640 --> 00:30:02,230


959
00:30:02,230 --> 00:30:02,240

 

960
00:30:02,240 --> 00:30:04,389


961
00:30:04,389 --> 00:30:06,070

 

962
00:30:06,070 --> 00:30:07,590

 

963
00:30:07,590 --> 00:30:09,430

 

964
00:30:09,430 --> 00:30:11,750

 

965
00:30:11,750 --> 00:30:13,830

 

966
00:30:13,830 --> 00:30:15,909

 

967
00:30:15,909 --> 00:30:18,549

 

968
00:30:18,549 --> 00:30:21,110

 

969
00:30:21,110 --> 00:30:23,350

 

970
00:30:23,350 --> 00:30:26,230

 

971
00:30:26,230 --> 00:30:29,430

 

972
00:30:29,430 --> 00:30:31,990

 

973
00:30:31,990 --> 00:30:35,669

 

974
00:30:35,669 --> 00:30:38,070

 

975
00:30:38,070 --> 00:30:40,070

 

976
00:30:40,070 --> 00:30:43,590

 

977
00:30:43,590 --> 00:30:45,590

 

978
00:30:45,590 --> 00:30:47,430

 

979
00:30:47,430 --> 00:30:49,830

 

980
00:30:49,830 --> 00:30:49,840

 

981
00:30:49,840 --> 00:30:52,549


982
00:30:52,549 --> 00:30:54,070

 

983
00:30:54,070 --> 00:30:55,669

 

984
00:30:55,669 --> 00:30:58,470

 

985
00:30:58,470 --> 00:31:01,110

 

986
00:31:01,110 --> 00:31:02,389

 

987
00:31:02,389 --> 00:31:05,269

 

988
00:31:05,269 --> 00:31:06,950

 

989
00:31:06,950 --> 00:31:09,110

 

990
00:31:09,110 --> 00:31:11,509

 

991
00:31:11,509 --> 00:31:13,990

 

992
00:31:13,990 --> 00:31:15,830

 

993
00:31:15,830 --> 00:31:17,430

 

994
00:31:17,430 --> 00:31:19,509

 

995
00:31:19,509 --> 00:31:22,070

 

996
00:31:22,070 --> 00:31:22,080

 

997
00:31:22,080 --> 00:31:23,350


998
00:31:23,350 --> 00:31:25,029

 

999
00:31:25,029 --> 00:31:25,039

 

1000
00:31:25,039 --> 00:31:26,710


1001
00:31:26,710 --> 00:31:28,389

 

1002
00:31:28,389 --> 00:31:28,399

 

1003
00:31:28,399 --> 00:31:29,830


1004
00:31:29,830 --> 00:31:33,590

 

1005
00:31:33,590 --> 00:31:34,710

 

1006
00:31:34,710 --> 00:31:34,720

 

1007
00:31:34,720 --> 00:31:36,149


1008
00:31:36,149 --> 00:31:38,630

 

1009
00:31:38,630 --> 00:31:40,549

 

1010
00:31:40,549 --> 00:31:42,070

 

1011
00:31:42,070 --> 00:31:44,870

 

1012
00:31:44,870 --> 00:31:47,269

 

1013
00:31:47,269 --> 00:31:49,110

 

1014
00:31:49,110 --> 00:31:53,190

 

1015
00:31:53,190 --> 00:31:54,950

 

1016
00:31:54,950 --> 00:31:56,710

 

1017
00:31:56,710 --> 00:31:56,720

 

1018
00:31:56,720 --> 00:31:57,509


1019
00:31:57,509 --> 00:32:00,630

 

1020
00:32:00,630 --> 00:32:02,870

 

1021
00:32:02,870 --> 00:32:06,549

 

1022
00:32:06,549 --> 00:32:08,230

 

1023
00:32:08,230 --> 00:32:10,070

 

1024
00:32:10,070 --> 00:32:14,950

 

1025
00:32:14,950 --> 00:32:18,310

 

1026
00:32:18,310 --> 00:32:20,310

 

1027
00:32:20,310 --> 00:32:22,470

 

1028
00:32:22,470 --> 00:32:24,389

 

1029
00:32:24,389 --> 00:32:27,190

 

1030
00:32:27,190 --> 00:32:28,470

 

1031
00:32:28,470 --> 00:32:30,310

 

1032
00:32:30,310 --> 00:32:33,269

 

1033
00:32:33,269 --> 00:32:35,509

 

1034
00:32:35,509 --> 00:32:37,909

 

1035
00:32:37,909 --> 00:32:41,990

 

1036
00:32:41,990 --> 00:32:42,000

 

1037
00:32:42,000 --> 00:32:45,590


1038
00:32:45,590 --> 00:32:48,870

 

1039
00:32:48,870 --> 00:32:51,509

 

1040
00:32:51,509 --> 00:32:54,470

 

1041
00:32:54,470 --> 00:32:56,470

 

1042
00:32:56,470 --> 00:32:59,669

 

1043
00:32:59,669 --> 00:33:00,870

 

1044
00:33:00,870 --> 00:33:03,509

 

1045
00:33:03,509 --> 00:33:05,509

 

1046
00:33:05,509 --> 00:33:05,519

 

1047
00:33:05,519 --> 00:33:08,830


1048
00:33:08,830 --> 00:33:11,029

 

1049
00:33:11,029 --> 00:33:11,039

 

1050
00:33:11,039 --> 00:33:11,830


1051
00:33:11,830 --> 00:33:13,669

 

1052
00:33:13,669 --> 00:33:16,389

 

1053
00:33:16,389 --> 00:33:16,399

 

1054
00:33:16,399 --> 00:33:17,269


1055
00:33:17,269 --> 00:33:19,909

 

1056
00:33:19,909 --> 00:33:21,669

 

1057
00:33:21,669 --> 00:33:23,830

 

1058
00:33:23,830 --> 00:33:25,909

 

1059
00:33:25,909 --> 00:33:29,190

 

1060
00:33:29,190 --> 00:33:32,070

 

1061
00:33:32,070 --> 00:33:33,750

 

1062
00:33:33,750 --> 00:33:37,190

 

1063
00:33:37,190 --> 00:33:40,630

 

1064
00:33:40,630 --> 00:33:42,950

 

1065
00:33:42,950 --> 00:33:44,789

 

1066
00:33:44,789 --> 00:33:47,509

 

1067
00:33:47,509 --> 00:33:48,950

 

1068
00:33:48,950 --> 00:33:50,389

 

1069
00:33:50,389 --> 00:33:52,310

 

1070
00:33:52,310 --> 00:34:01,190

 

1071
00:34:01,190 --> 00:34:03,029

 

1072
00:34:03,029 --> 00:34:05,029

 

1073
00:34:05,029 --> 00:34:07,509

 

1074
00:34:07,509 --> 00:34:10,149

 

1075
00:34:10,149 --> 00:34:12,470

 

1076
00:34:12,470 --> 00:34:12,480

 

1077
00:34:12,480 --> 00:34:13,589


1078
00:34:13,589 --> 00:34:13,599

 

1079
00:34:13,599 --> 00:34:14,869


1080
00:34:14,869 --> 00:34:16,310

 

1081
00:34:16,310 --> 00:34:19,190

 

1082
00:34:19,190 --> 00:34:19,200

 

1083
00:34:19,200 --> 00:34:23,349


1084
00:34:23,349 --> 00:34:25,829

 

1085
00:34:25,829 --> 00:34:30,149

 

1086
00:34:30,149 --> 00:34:31,829

 

1087
00:34:31,829 --> 00:34:34,710

 

1088
00:34:34,710 --> 00:34:36,710

 

1089
00:34:36,710 --> 00:34:36,720

 

1090
00:34:36,720 --> 00:34:37,750


1091
00:34:37,750 --> 00:34:39,990

 

1092
00:34:39,990 --> 00:34:43,190

 

1093
00:34:43,190 --> 00:34:45,510

 

1094
00:34:45,510 --> 00:34:47,109

 

1095
00:34:47,109 --> 00:34:48,389

 

1096
00:34:48,389 --> 00:34:50,310

 

1097
00:34:50,310 --> 00:34:52,310

 

1098
00:34:52,310 --> 00:34:54,629

 

1099
00:34:54,629 --> 00:34:56,470

 

1100
00:34:56,470 --> 00:34:58,390

 

1101
00:34:58,390 --> 00:34:58,400

 

1102
00:34:58,400 --> 00:35:04,630


1103
00:35:04,630 --> 00:35:04,640

 

1104
00:35:04,640 --> 00:35:05,430


1105
00:35:05,430 --> 00:35:07,030

 

1106
00:35:07,030 --> 00:35:07,040

 

1107
00:35:07,040 --> 00:35:08,550


1108
00:35:08,550 --> 00:35:10,790

 

1109
00:35:10,790 --> 00:35:10,800

 

1110
00:35:10,800 --> 00:35:13,589


1111
00:35:13,589 --> 00:35:13,599

 

1112
00:35:13,599 --> 00:35:15,750


1113
00:35:15,750 --> 00:35:17,430

 

1114
00:35:17,430 --> 00:35:17,440

 

1115
00:35:17,440 --> 00:35:18,870


1116
00:35:18,870 --> 00:35:18,880

 

1117
00:35:18,880 --> 00:35:20,470


1118
00:35:20,470 --> 00:35:22,150

 

1119
00:35:22,150 --> 00:35:22,160

 

1120
00:35:22,160 --> 00:35:23,270


1121
00:35:23,270 --> 00:35:25,109

 

1122
00:35:25,109 --> 00:35:25,119

 

1123
00:35:25,119 --> 00:35:26,150


1124
00:35:26,150 --> 00:35:29,349

 

1125
00:35:29,349 --> 00:35:31,349

 

1126
00:35:31,349 --> 00:35:34,150

 

1127
00:35:34,150 --> 00:35:36,470

 

1128
00:35:36,470 --> 00:35:38,550

 

1129
00:35:38,550 --> 00:35:39,510

 

1130
00:35:39,510 --> 00:35:42,630

 

1131
00:35:42,630 --> 00:35:42,640

 

1132
00:35:42,640 --> 00:35:43,349


1133
00:35:43,349 --> 00:35:45,829

 

1134
00:35:45,829 --> 00:35:48,310

 

1135
00:35:48,310 --> 00:35:49,430

 

1136
00:35:49,430 --> 00:35:50,790

 

1137
00:35:50,790 --> 00:35:53,109

 

1138
00:35:53,109 --> 00:35:55,829

 

1139
00:35:55,829 --> 00:35:58,550

 

1140
00:35:58,550 --> 00:36:00,230

 

1141
00:36:00,230 --> 00:36:04,310

 

1142
00:36:04,310 --> 00:36:04,320

 

1143
00:36:04,320 --> 00:36:05,270


1144
00:36:05,270 --> 00:36:07,910

 

1145
00:36:07,910 --> 00:36:09,750

 

1146
00:36:09,750 --> 00:36:12,790

 

1147
00:36:12,790 --> 00:36:17,510

 

1148
00:36:17,510 --> 00:36:19,990

 

1149
00:36:19,990 --> 00:36:24,790

 

1150
00:36:24,790 --> 00:36:27,430

 

1151
00:36:27,430 --> 00:36:28,710

 

1152
00:36:28,710 --> 00:36:29,829

 

1153
00:36:29,829 --> 00:36:31,910

 

1154
00:36:31,910 --> 00:36:34,630

 

1155
00:36:34,630 --> 00:36:36,790

 

1156
00:36:36,790 --> 00:36:40,630

 

1157
00:36:40,630 --> 00:36:42,630

 

1158
00:36:42,630 --> 00:36:42,640

 

1159
00:36:42,640 --> 00:36:43,670


1160
00:36:43,670 --> 00:36:45,349

 

1161
00:36:45,349 --> 00:36:47,589

 

1162
00:36:47,589 --> 00:36:47,599

 

1163
00:36:47,599 --> 00:36:48,870


1164
00:36:48,870 --> 00:36:50,710

 

1165
00:36:50,710 --> 00:36:56,470

 

1166
00:36:56,470 --> 00:36:58,390

 

1167
00:36:58,390 --> 00:37:02,310

 

1168
00:37:02,310 --> 00:37:05,430

 

1169
00:37:05,430 --> 00:37:08,069

 

1170
00:37:08,069 --> 00:37:10,390

 

1171
00:37:10,390 --> 00:37:12,950

 

1172
00:37:12,950 --> 00:37:12,960

 

1173
00:37:12,960 --> 00:37:14,790


1174
00:37:14,790 --> 00:37:15,910

 

1175
00:37:15,910 --> 00:37:15,920

 

1176
00:37:15,920 --> 00:37:21,430


1177
00:37:21,430 --> 00:37:23,670

 

1178
00:37:23,670 --> 00:37:25,589

 

1179
00:37:25,589 --> 00:37:28,230

 

1180
00:37:28,230 --> 00:37:28,240

 

1181
00:37:28,240 --> 00:37:29,349


1182
00:37:29,349 --> 00:37:32,310

 

1183
00:37:32,310 --> 00:37:34,069

 

1184
00:37:34,069 --> 00:37:37,270

 

1185
00:37:37,270 --> 00:37:40,470

 

1186
00:37:40,470 --> 00:37:42,630

 

1187
00:37:42,630 --> 00:37:45,030

 

1188
00:37:45,030 --> 00:37:45,040

 

1189
00:37:45,040 --> 00:37:45,750


1190
00:37:45,750 --> 00:37:47,430

 

1191
00:37:47,430 --> 00:37:49,510

 

1192
00:37:49,510 --> 00:37:50,870

 

1193
00:37:50,870 --> 00:37:52,310

 

1194
00:37:52,310 --> 00:37:55,430

 

1195
00:37:55,430 --> 00:37:59,109

 

1196
00:37:59,109 --> 00:38:03,910

 

1197
00:38:03,910 --> 00:38:06,069

 

1198
00:38:06,069 --> 00:38:07,829

 

1199
00:38:07,829 --> 00:38:07,839

 

1200
00:38:07,839 --> 00:38:11,109


1201
00:38:11,109 --> 00:38:13,589

 

1202
00:38:13,589 --> 00:38:15,430

 

1203
00:38:15,430 --> 00:38:20,069

 

1204
00:38:20,069 --> 00:38:22,550

 

1205
00:38:22,550 --> 00:38:23,990

 

1206
00:38:23,990 --> 00:38:24,000

 

1207
00:38:24,000 --> 00:38:31,190


1208
00:38:31,190 --> 00:38:34,069

 

1209
00:38:34,069 --> 00:38:37,349

 

1210
00:38:37,349 --> 00:38:37,359

 

1211
00:38:37,359 --> 00:38:38,390


1212
00:38:38,390 --> 00:38:41,190

 

1213
00:38:41,190 --> 00:38:43,910

 

1214
00:38:43,910 --> 00:38:47,030

 

1215
00:38:47,030 --> 00:38:49,030

 

1216
00:38:49,030 --> 00:38:49,040

 

1217
00:38:49,040 --> 00:38:49,910


1218
00:38:49,910 --> 00:38:51,589

 

1219
00:38:51,589 --> 00:38:53,270

 

1220
00:38:53,270 --> 00:38:55,109

 

1221
00:38:55,109 --> 00:39:00,150

 

1222
00:39:00,150 --> 00:39:02,470

 

1223
00:39:02,470 --> 00:39:03,670

 

1224
00:39:03,670 --> 00:39:05,990

 

1225
00:39:05,990 --> 00:39:09,109

 

1226
00:39:09,109 --> 00:39:10,790

 

1227
00:39:10,790 --> 00:39:10,800

 

1228
00:39:10,800 --> 00:39:11,829


1229
00:39:11,829 --> 00:39:14,870

 

1230
00:39:14,870 --> 00:39:17,589

 

1231
00:39:17,589 --> 00:39:17,599

 

1232
00:39:17,599 --> 00:39:19,349


1233
00:39:19,349 --> 00:39:21,910

 

1234
00:39:21,910 --> 00:39:25,430

 

1235
00:39:25,430 --> 00:39:28,390

 

1236
00:39:28,390 --> 00:39:30,069

 

1237
00:39:30,069 --> 00:39:33,270

 

1238
00:39:33,270 --> 00:39:35,270

 

1239
00:39:35,270 --> 00:39:36,950

 

1240
00:39:36,950 --> 00:39:39,829

 

1241
00:39:39,829 --> 00:39:39,839

 

1242
00:39:39,839 --> 00:39:40,829


1243
00:39:40,829 --> 00:39:40,839

 

1244
00:39:40,839 --> 00:39:42,470


1245
00:39:42,470 --> 00:39:42,480

 

1246
00:39:42,480 --> 00:39:43,990


1247
00:39:43,990 --> 00:39:45,430

 

1248
00:39:45,430 --> 00:39:46,470

 

1249
00:39:46,470 --> 00:39:47,750

 

1250
00:39:47,750 --> 00:39:47,760

 

1251
00:39:47,760 --> 00:39:49,589


1252
00:39:49,589 --> 00:39:51,750

 

1253
00:39:51,750 --> 00:39:53,990

 

1254
00:39:53,990 --> 00:39:56,310

 

1255
00:39:56,310 --> 00:39:57,829

 

1256
00:39:57,829 --> 00:40:01,030

 

1257
00:40:01,030 --> 00:40:02,950

 

1258
00:40:02,950 --> 00:40:02,960

 

1259
00:40:02,960 --> 00:40:03,750


1260
00:40:03,750 --> 00:40:06,230

 

1261
00:40:06,230 --> 00:40:09,270

 

1262
00:40:09,270 --> 00:40:10,870

 

1263
00:40:10,870 --> 00:40:13,109

 

1264
00:40:13,109 --> 00:40:14,390

 

1265
00:40:14,390 --> 00:40:16,390

 

1266
00:40:16,390 --> 00:40:19,109

 

1267
00:40:19,109 --> 00:40:21,910

 

1268
00:40:21,910 --> 00:40:23,349

 

1269
00:40:23,349 --> 00:40:27,430

 

1270
00:40:27,430 --> 00:40:27,440

 

1271
00:40:27,440 --> 00:40:28,230


1272
00:40:28,230 --> 00:40:30,470

 

1273
00:40:30,470 --> 00:40:33,670

 

1274
00:40:33,670 --> 00:40:36,230

 

1275
00:40:36,230 --> 00:40:37,670

 

1276
00:40:37,670 --> 00:40:37,680

 

1277
00:40:37,680 --> 00:40:38,390


1278
00:40:38,390 --> 00:40:41,030

 

1279
00:40:41,030 --> 00:40:41,040

 

1280
00:40:41,040 --> 00:40:42,150


1281
00:40:42,150 --> 00:40:45,030

 

1282
00:40:45,030 --> 00:40:47,349

 

1283
00:40:47,349 --> 00:40:49,349

 

1284
00:40:49,349 --> 00:40:49,359

 

1285
00:40:49,359 --> 00:40:50,710


1286
00:40:50,710 --> 00:40:54,950

 

1287
00:40:54,950 --> 00:40:56,790

 

1288
00:40:56,790 --> 00:40:58,950

 

1289
00:40:58,950 --> 00:41:01,829

 

1290
00:41:01,829 --> 00:41:03,829

 

1291
00:41:03,829 --> 00:41:03,839

 

1292
00:41:03,839 --> 00:41:05,430


1293
00:41:05,430 --> 00:41:06,870

 

1294
00:41:06,870 --> 00:41:09,990

 

1295
00:41:09,990 --> 00:41:11,829

 

1296
00:41:11,829 --> 00:41:11,839

 

1297
00:41:11,839 --> 00:41:12,950


1298
00:41:12,950 --> 00:41:16,550

 

1299
00:41:16,550 --> 00:41:18,630

 

1300
00:41:18,630 --> 00:41:21,750

 

1301
00:41:21,750 --> 00:41:25,750

 

1302
00:41:25,750 --> 00:41:27,030

 

1303
00:41:27,030 --> 00:41:29,910

 

1304
00:41:29,910 --> 00:41:31,430

 

1305
00:41:31,430 --> 00:41:33,670

 

1306
00:41:33,670 --> 00:41:35,430

 

1307
00:41:35,430 --> 00:41:35,440

 

1308
00:41:35,440 --> 00:41:36,550


1309
00:41:36,550 --> 00:41:36,560

 

1310
00:41:36,560 --> 00:41:39,190


1311
00:41:39,190 --> 00:41:41,349

 

1312
00:41:41,349 --> 00:41:43,670

 

1313
00:41:43,670 --> 00:41:45,829

 

1314
00:41:45,829 --> 00:41:48,470

 

1315
00:41:48,470 --> 00:41:52,390

 

1316
00:41:52,390 --> 00:41:52,400

 

1317
00:41:52,400 --> 00:41:54,630


1318
00:41:54,630 --> 00:41:54,640

 

1319
00:41:54,640 --> 00:41:56,309


1320
00:41:56,309 --> 00:41:58,790

 

1321
00:41:58,790 --> 00:42:02,870

 

1322
00:42:02,870 --> 00:42:02,880

 

1323
00:42:02,880 --> 00:42:03,750


1324
00:42:03,750 --> 00:42:06,470

 

1325
00:42:06,470 --> 00:42:08,150

 

1326
00:42:08,150 --> 00:42:10,870

 

1327
00:42:10,870 --> 00:42:14,550

 

1328
00:42:14,550 --> 00:42:16,870

 

1329
00:42:16,870 --> 00:42:19,030

 

1330
00:42:19,030 --> 00:42:22,230

 

1331
00:42:22,230 --> 00:42:23,670

 

1332
00:42:23,670 --> 00:42:27,030

 

1333
00:42:27,030 --> 00:42:29,349

 

1334
00:42:29,349 --> 00:42:29,359

 

1335
00:42:29,359 --> 00:42:30,390


1336
00:42:30,390 --> 00:42:33,030

 

1337
00:42:33,030 --> 00:42:35,510

 

1338
00:42:35,510 --> 00:42:37,430

 

1339
00:42:37,430 --> 00:42:40,309

 

1340
00:42:40,309 --> 00:42:43,030

 

1341
00:42:43,030 --> 00:42:44,470

 

1342
00:42:44,470 --> 00:42:47,030

 

1343
00:42:47,030 --> 00:42:49,270

 

1344
00:42:49,270 --> 00:42:52,390

 

1345
00:42:52,390 --> 00:42:55,910

 

1346
00:42:55,910 --> 00:42:58,550

 

1347
00:42:58,550 --> 00:43:00,150

 

1348
00:43:00,150 --> 00:43:01,349

 

1349
00:43:01,349 --> 00:43:03,829

 

1350
00:43:03,829 --> 00:43:14,710

 

1351
00:43:14,710 --> 00:43:17,190

 

1352
00:43:17,190 --> 00:43:19,190

 

1353
00:43:19,190 --> 00:43:20,790

 

1354
00:43:20,790 --> 00:43:20,800

 

1355
00:43:20,800 --> 00:43:22,710


1356
00:43:22,710 --> 00:43:26,470

 

1357
00:43:26,470 --> 00:43:27,910

 

1358
00:43:27,910 --> 00:43:29,910

 

1359
00:43:29,910 --> 00:43:32,550

 

1360
00:43:32,550 --> 00:43:32,560

 

1361
00:43:32,560 --> 00:43:34,309


1362
00:43:34,309 --> 00:43:36,069

 

1363
00:43:36,069 --> 00:43:36,079

 

1364
00:43:36,079 --> 00:43:37,829


1365
00:43:37,829 --> 00:43:39,589

 

1366
00:43:39,589 --> 00:43:42,870

 

1367
00:43:42,870 --> 00:43:44,790

 

1368
00:43:44,790 --> 00:43:46,550

 

1369
00:43:46,550 --> 00:43:48,870

 

1370
00:43:48,870 --> 00:43:48,880

 

1371
00:43:48,880 --> 00:43:50,390


1372
00:43:50,390 --> 00:43:51,829

 

1373
00:43:51,829 --> 00:43:54,790

 

1374
00:43:54,790 --> 00:43:57,589

 

1375
00:43:57,589 --> 00:44:01,589

 

1376
00:44:01,589 --> 00:44:03,589

 

1377
00:44:03,589 --> 00:44:06,230

 

1378
00:44:06,230 --> 00:44:08,470

 

1379
00:44:08,470 --> 00:44:10,630

 

1380
00:44:10,630 --> 00:44:12,630

 

1381
00:44:12,630 --> 00:44:12,640

 

1382
00:44:12,640 --> 00:44:18,470


1383
00:44:18,470 --> 00:44:20,230

 

1384
00:44:20,230 --> 00:44:21,910

 

1385
00:44:21,910 --> 00:44:21,920

 

1386
00:44:21,920 --> 00:44:23,109


1387
00:44:23,109 --> 00:44:25,510

 

1388
00:44:25,510 --> 00:44:27,990

 

1389
00:44:27,990 --> 00:44:30,550

 

1390
00:44:30,550 --> 00:44:32,550

 

1391
00:44:32,550 --> 00:44:33,990

 

1392
00:44:33,990 --> 00:44:35,510

 

1393
00:44:35,510 --> 00:44:38,390

 

1394
00:44:38,390 --> 00:44:40,550

 

1395
00:44:40,550 --> 00:44:42,230

 

1396
00:44:42,230 --> 00:44:43,910

 

1397
00:44:43,910 --> 00:44:45,190

 

1398
00:44:45,190 --> 00:44:49,109

 

1399
00:44:49,109 --> 00:44:52,470

 

1400
00:44:52,470 --> 00:44:52,480

 

1401
00:44:52,480 --> 00:44:54,470


1402
00:44:54,470 --> 00:44:58,150

 

1403
00:44:58,150 --> 00:45:01,990

 

1404
00:45:01,990 --> 00:45:03,910

 

1405
00:45:03,910 --> 00:45:05,430

 

1406
00:45:05,430 --> 00:45:08,790

 

1407
00:45:08,790 --> 00:45:10,870

 

1408
00:45:10,870 --> 00:45:13,190

 

1409
00:45:13,190 --> 00:45:15,910

 

1410
00:45:15,910 --> 00:45:17,030

 

1411
00:45:17,030 --> 00:45:18,470

 

1412
00:45:18,470 --> 00:45:20,870

 

1413
00:45:20,870 --> 00:45:22,150

 

1414
00:45:22,150 --> 00:45:23,910

 

1415
00:45:23,910 --> 00:45:26,950

 

1416
00:45:26,950 --> 00:45:29,109

 

1417
00:45:29,109 --> 00:45:31,190

 

1418
00:45:31,190 --> 00:45:36,790

 

1419
00:45:36,790 --> 00:45:38,870

 

1420
00:45:38,870 --> 00:45:40,550

 

1421
00:45:40,550 --> 00:45:42,790

 

1422
00:45:42,790 --> 00:45:45,990

 

1423
00:45:45,990 --> 00:45:48,150

 

1424
00:45:48,150 --> 00:45:50,470

 

1425
00:45:50,470 --> 00:45:52,470

 

1426
00:45:52,470 --> 00:45:55,750

 

1427
00:45:55,750 --> 00:45:57,109

 

1428
00:45:57,109 --> 00:45:58,950

 

1429
00:45:58,950 --> 00:46:00,150

 

1430
00:46:00,150 --> 00:46:02,230

 

1431
00:46:02,230 --> 00:46:04,390

 

1432
00:46:04,390 --> 00:46:06,630

 

1433
00:46:06,630 --> 00:46:11,670

 

1434
00:46:11,670 --> 00:46:12,790

 

1435
00:46:12,790 --> 00:46:15,270

 

1436
00:46:15,270 --> 00:46:16,790

 

1437
00:46:16,790 --> 00:46:19,109

 

1438
00:46:19,109 --> 00:46:22,150

 

1439
00:46:22,150 --> 00:46:23,910

 

1440
00:46:23,910 --> 00:46:25,510

 

1441
00:46:25,510 --> 00:46:28,309

 

1442
00:46:28,309 --> 00:46:29,670

 

1443
00:46:29,670 --> 00:46:29,680

 

1444
00:46:29,680 --> 00:46:31,750


1445
00:46:31,750 --> 00:46:31,760

 

1446
00:46:31,760 --> 00:46:35,190


1447
00:46:35,190 --> 00:46:35,200

 

1448
00:46:35,200 --> 00:46:37,190


1449
00:46:37,190 --> 00:46:37,200

 

1450
00:46:37,200 --> 00:46:46,790


1451
00:46:46,790 --> 00:46:49,190

 

1452
00:46:49,190 --> 00:46:50,470

 

1453
00:46:50,470 --> 00:46:52,309

 

1454
00:46:52,309 --> 00:46:52,319

 

1455
00:46:52,319 --> 00:46:53,990


1456
00:46:53,990 --> 00:46:57,030

 

1457
00:46:57,030 --> 00:46:59,349

 

1458
00:46:59,349 --> 00:47:00,870

 

1459
00:47:00,870 --> 00:47:04,069

 

1460
00:47:04,069 --> 00:47:04,079

 

1461
00:47:04,079 --> 00:47:06,150


1462
00:47:06,150 --> 00:47:08,870

 

1463
00:47:08,870 --> 00:47:10,950

 

1464
00:47:10,950 --> 00:47:10,960

 

1465
00:47:10,960 --> 00:47:11,910


1466
00:47:11,910 --> 00:47:14,870

 

1467
00:47:14,870 --> 00:47:15,910

 

1468
00:47:15,910 --> 00:47:16,950

 

1469
00:47:16,950 --> 00:47:18,390

 

1470
00:47:18,390 --> 00:47:20,710

 

1471
00:47:20,710 --> 00:47:23,750

 

1472
00:47:23,750 --> 00:47:25,990

 

1473
00:47:25,990 --> 00:47:27,510

 

1474
00:47:27,510 --> 00:47:29,430

 

1475
00:47:29,430 --> 00:47:31,510

 

1476
00:47:31,510 --> 00:47:34,230

 

1477
00:47:34,230 --> 00:47:36,390

 

1478
00:47:36,390 --> 00:47:39,270

 

1479
00:47:39,270 --> 00:47:42,710

 

1480
00:47:42,710 --> 00:47:44,390

 

1481
00:47:44,390 --> 00:47:44,400

 

1482
00:47:44,400 --> 00:47:46,230


1483
00:47:46,230 --> 00:47:48,470

 

1484
00:47:48,470 --> 00:47:50,150

 

1485
00:47:50,150 --> 00:47:50,160

 

1486
00:47:50,160 --> 00:47:51,510


1487
00:47:51,510 --> 00:47:54,870

 

1488
00:47:54,870 --> 00:47:57,589

 

1489
00:47:57,589 --> 00:47:59,589

 

1490
00:47:59,589 --> 00:48:01,109

 

1491
00:48:01,109 --> 00:48:01,119

 

1492
00:48:01,119 --> 00:48:02,069


1493
00:48:02,069 --> 00:48:04,549

 

1494
00:48:04,549 --> 00:48:07,109

 

1495
00:48:07,109 --> 00:48:08,870

 

1496
00:48:08,870 --> 00:48:11,030

 

1497
00:48:11,030 --> 00:48:12,950

 

1498
00:48:12,950 --> 00:48:15,430

 

1499
00:48:15,430 --> 00:48:18,870

 

1500
00:48:18,870 --> 00:48:20,549

 

1501
00:48:20,549 --> 00:48:22,790

 

1502
00:48:22,790 --> 00:48:22,800

 

1503
00:48:22,800 --> 00:48:23,589


1504
00:48:23,589 --> 00:48:26,470

 

1505
00:48:26,470 --> 00:48:26,480

 

1506
00:48:26,480 --> 00:48:27,990


1507
00:48:27,990 --> 00:48:29,510

 

1508
00:48:29,510 --> 00:48:29,520

 

1509
00:48:29,520 --> 00:48:31,510


1510
00:48:31,510 --> 00:48:31,520

 

1511
00:48:31,520 --> 00:48:32,710


1512
00:48:32,710 --> 00:48:33,990

 

1513
00:48:33,990 --> 00:48:35,910

 

1514
00:48:35,910 --> 00:48:37,430

 

1515
00:48:37,430 --> 00:48:39,109

 

1516
00:48:39,109 --> 00:48:41,910

 

1517
00:48:41,910 --> 00:48:43,109

 

1518
00:48:43,109 --> 00:48:45,349

 

1519
00:48:45,349 --> 00:48:49,190

 

1520
00:48:49,190 --> 00:48:51,270

 

1521
00:48:51,270 --> 00:48:53,750

 

1522
00:48:53,750 --> 00:48:53,760

 

1523
00:48:53,760 --> 00:48:54,950


1524
00:48:54,950 --> 00:48:57,430

 

1525
00:48:57,430 --> 00:48:59,190

 

1526
00:48:59,190 --> 00:48:59,200

 

1527
00:48:59,200 --> 00:49:00,829


1528
00:49:00,829 --> 00:49:04,309

 

1529
00:49:04,309 --> 00:49:06,069

 

1530
00:49:06,069 --> 00:49:06,079

 

1531
00:49:06,079 --> 00:49:07,750


1532
00:49:07,750 --> 00:49:07,760

 

1533
00:49:07,760 --> 00:49:08,870


1534
00:49:08,870 --> 00:49:10,230

 

1535
00:49:10,230 --> 00:49:10,240

 

1536
00:49:10,240 --> 00:49:11,829


1537
00:49:11,829 --> 00:49:14,950

 

1538
00:49:14,950 --> 00:49:17,750

 

1539
00:49:17,750 --> 00:49:19,430

 

1540
00:49:19,430 --> 00:49:22,470

 

1541
00:49:22,470 --> 00:49:24,549

 

1542
00:49:24,549 --> 00:49:26,230

 

1543
00:49:26,230 --> 00:49:29,510

 

1544
00:49:29,510 --> 00:49:29,520

 

1545
00:49:29,520 --> 00:49:30,630


1546
00:49:30,630 --> 00:49:32,950

 

1547
00:49:32,950 --> 00:49:34,549

 

1548
00:49:34,549 --> 00:49:38,069

 

1549
00:49:38,069 --> 00:49:38,079

 

1550
00:49:38,079 --> 00:49:39,589


1551
00:49:39,589 --> 00:49:41,910

 

1552
00:49:41,910 --> 00:49:44,470

 

1553
00:49:44,470 --> 00:49:44,480

 

1554
00:49:44,480 --> 00:49:46,950


1555
00:49:46,950 --> 00:49:46,960

 

1556
00:49:46,960 --> 00:49:47,829


1557
00:49:47,829 --> 00:49:49,990

 

1558
00:49:49,990 --> 00:49:52,710

 

1559
00:49:52,710 --> 00:49:55,030

 

1560
00:49:55,030 --> 00:49:56,790

 

1561
00:49:56,790 --> 00:49:59,109

 

1562
00:49:59,109 --> 00:50:01,829

 

1563
00:50:01,829 --> 00:50:03,670

 

1564
00:50:03,670 --> 00:50:07,030

 

1565
00:50:07,030 --> 00:50:08,710

 

1566
00:50:08,710 --> 00:50:10,390

 

1567
00:50:10,390 --> 00:50:12,069

 

1568
00:50:12,069 --> 00:50:14,150

 

1569
00:50:14,150 --> 00:50:15,829

 

1570
00:50:15,829 --> 00:50:18,790

 

1571
00:50:18,790 --> 00:50:23,030

 

1572
00:50:23,030 --> 00:50:25,750

 

1573
00:50:25,750 --> 00:50:27,109

 

1574
00:50:27,109 --> 00:50:29,750

 

1575
00:50:29,750 --> 00:50:34,870

 

1576
00:50:34,870 --> 00:50:35,910

 

1577
00:50:35,910 --> 00:50:35,920

 

1578
00:50:35,920 --> 00:50:37,190


1579
00:50:37,190 --> 00:50:39,589

 

1580
00:50:39,589 --> 00:50:41,030

 

1581
00:50:41,030 --> 00:50:42,390

 

1582
00:50:42,390 --> 00:50:47,990

 

1583
00:50:47,990 --> 00:50:52,309

 

1584
00:50:52,309 --> 00:50:56,150

 

1585
00:50:56,150 --> 00:50:57,670

 

1586
00:50:57,670 --> 00:50:59,990

 

1587
00:50:59,990 --> 00:51:03,030

 

1588
00:51:03,030 --> 00:51:04,630

 

1589
00:51:04,630 --> 00:51:06,150

 

1590
00:51:06,150 --> 00:51:08,230

 

1591
00:51:08,230 --> 00:51:11,030

 

1592
00:51:11,030 --> 00:51:12,549

 

1593
00:51:12,549 --> 00:51:15,349

 

1594
00:51:15,349 --> 00:51:15,359

 

1595
00:51:15,359 --> 00:51:16,309


1596
00:51:16,309 --> 00:51:17,990

 

1597
00:51:17,990 --> 00:51:19,589

 

1598
00:51:19,589 --> 00:51:19,599

 

1599
00:51:19,599 --> 00:51:20,470


1600
00:51:20,470 --> 00:51:21,829

 

1601
00:51:21,829 --> 00:51:23,270

 

1602
00:51:23,270 --> 00:51:23,280

 

1603
00:51:23,280 --> 00:51:24,470


1604
00:51:24,470 --> 00:51:26,309

 

1605
00:51:26,309 --> 00:51:28,230

 

1606
00:51:28,230 --> 00:51:30,309

 

1607
00:51:30,309 --> 00:51:33,190

 

1608
00:51:33,190 --> 00:51:34,309

 

1609
00:51:34,309 --> 00:51:34,319

 

1610
00:51:34,319 --> 00:51:35,670


1611
00:51:35,670 --> 00:51:36,470

 

1612
00:51:36,470 --> 00:51:38,950

 

1613
00:51:38,950 --> 00:51:40,870

 

1614
00:51:40,870 --> 00:51:42,150

 

1615
00:51:42,150 --> 00:51:46,309

 

1616
00:51:46,309 --> 00:51:48,829

 

1617
00:51:48,829 --> 00:51:50,790

 

1618
00:51:50,790 --> 00:51:52,150

 

1619
00:51:52,150 --> 00:51:55,430

 

1620
00:51:55,430 --> 00:51:58,710

 

1621
00:51:58,710 --> 00:51:58,720

 

1622
00:51:58,720 --> 00:51:59,910


1623
00:51:59,910 --> 00:52:03,589

 

1624
00:52:03,589 --> 00:52:06,069

 

1625
00:52:06,069 --> 00:52:07,910

 

1626
00:52:07,910 --> 00:52:09,990

 

1627
00:52:09,990 --> 00:52:12,309

 

1628
00:52:12,309 --> 00:52:14,870

 

1629
00:52:14,870 --> 00:52:16,549

 

1630
00:52:16,549 --> 00:52:18,470

 

1631
00:52:18,470 --> 00:52:21,270

 

1632
00:52:21,270 --> 00:52:21,280

 

1633
00:52:21,280 --> 00:52:22,069


1634
00:52:22,069 --> 00:52:24,230

 

1635
00:52:24,230 --> 00:52:24,240

 

1636
00:52:24,240 --> 00:52:25,829


1637
00:52:25,829 --> 00:52:27,430

 

1638
00:52:27,430 --> 00:52:30,069

 

1639
00:52:30,069 --> 00:52:32,630

 

1640
00:52:32,630 --> 00:52:34,870

 

1641
00:52:34,870 --> 00:52:36,309

 

1642
00:52:36,309 --> 00:52:39,109

 

1643
00:52:39,109 --> 00:52:41,750

 

1644
00:52:41,750 --> 00:52:43,510

 

1645
00:52:43,510 --> 00:52:45,589

 

1646
00:52:45,589 --> 00:52:47,829

 

1647
00:52:47,829 --> 00:52:49,670

 

1648
00:52:49,670 --> 00:52:51,030

 

1649
00:52:51,030 --> 00:52:52,230

 

1650
00:52:52,230 --> 00:52:53,829

 

1651
00:52:53,829 --> 00:52:55,829

 

1652
00:52:55,829 --> 00:52:57,910

 

1653
00:52:57,910 --> 00:53:00,390

 

1654
00:53:00,390 --> 00:53:02,230

 

1655
00:53:02,230 --> 00:53:06,069

 

1656
00:53:06,069 --> 00:53:08,309

 

1657
00:53:08,309 --> 00:53:10,790

 

1658
00:53:10,790 --> 00:53:12,230

 

1659
00:53:12,230 --> 00:53:14,390

 

1660
00:53:14,390 --> 00:53:14,400

 

1661
00:53:14,400 --> 00:53:15,270


1662
00:53:15,270 --> 00:53:17,829

 

1663
00:53:17,829 --> 00:53:20,870

 

1664
00:53:20,870 --> 00:53:23,750

 

1665
00:53:23,750 --> 00:53:25,109

 

1666
00:53:25,109 --> 00:53:27,030

 

1667
00:53:27,030 --> 00:53:28,790

 

1668
00:53:28,790 --> 00:53:31,030

 

1669
00:53:31,030 --> 00:53:32,470

 

1670
00:53:32,470 --> 00:53:34,950

 

1671
00:53:34,950 --> 00:53:36,790

 

1672
00:53:36,790 --> 00:53:38,790

 

1673
00:53:38,790 --> 00:53:40,390

 

1674
00:53:40,390 --> 00:53:42,150

 

1675
00:53:42,150 --> 00:53:42,160

 

1676
00:53:42,160 --> 00:53:43,190


1677
00:53:43,190 --> 00:53:47,990

 

1678
00:53:47,990 --> 00:53:49,750

 

1679
00:53:49,750 --> 00:53:51,910

 

1680
00:53:51,910 --> 00:53:54,710

 

1681
00:53:54,710 --> 00:53:56,630

 

1682
00:53:56,630 --> 00:53:58,470

 

1683
00:53:58,470 --> 00:54:00,309

 

1684
00:54:00,309 --> 00:54:02,150

 

1685
00:54:02,150 --> 00:54:04,069

 

1686
00:54:04,069 --> 00:54:05,829

 

1687
00:54:05,829 --> 00:54:07,430

 

1688
00:54:07,430 --> 00:54:07,440

 

1689
00:54:07,440 --> 00:54:08,390


1690
00:54:08,390 --> 00:54:11,430

 

1691
00:54:11,430 --> 00:54:11,440

 

1692
00:54:11,440 --> 00:54:13,349


1693
00:54:13,349 --> 00:54:15,829

 

1694
00:54:15,829 --> 00:54:17,829

 

1695
00:54:17,829 --> 00:54:20,150

 

1696
00:54:20,150 --> 00:54:21,990

 

1697
00:54:21,990 --> 00:54:24,710

 

1698
00:54:24,710 --> 00:54:27,510

 

1699
00:54:27,510 --> 00:54:29,270

 

1700
00:54:29,270 --> 00:54:31,030

 

1701
00:54:31,030 --> 00:54:32,390

 

1702
00:54:32,390 --> 00:54:32,400

 

1703
00:54:32,400 --> 00:54:33,270


1704
00:54:33,270 --> 00:54:35,270

 

1705
00:54:35,270 --> 00:54:37,670

 

1706
00:54:37,670 --> 00:54:37,680

 

1707
00:54:37,680 --> 00:54:38,870


1708
00:54:38,870 --> 00:54:41,990

 

1709
00:54:41,990 --> 00:54:43,430

 

1710
00:54:43,430 --> 00:54:43,440

 

1711
00:54:43,440 --> 00:54:44,870


1712
00:54:44,870 --> 00:54:46,390

 

1713
00:54:46,390 --> 00:54:48,870

 

1714
00:54:48,870 --> 00:54:50,630

 

1715
00:54:50,630 --> 00:54:52,549

 

1716
00:54:52,549 --> 00:54:53,829

 

1717
00:54:53,829 --> 00:54:55,510

 

1718
00:54:55,510 --> 00:54:55,520

 

1719
00:54:55,520 --> 00:54:56,950


1720
00:54:56,950 --> 00:54:59,349

 

1721
00:54:59,349 --> 00:55:01,670

 

1722
00:55:01,670 --> 00:55:03,430

 

1723
00:55:03,430 --> 00:55:05,589

 

1724
00:55:05,589 --> 00:55:07,430

 

1725
00:55:07,430 --> 00:55:09,190

 

1726
00:55:09,190 --> 00:55:09,200

 

1727
00:55:09,200 --> 00:55:11,270


1728
00:55:11,270 --> 00:55:12,710

 

1729
00:55:12,710 --> 00:55:15,030

 

1730
00:55:15,030 --> 00:55:17,589

 

1731
00:55:17,589 --> 00:55:17,599

 

1732
00:55:17,599 --> 00:55:19,030


1733
00:55:19,030 --> 00:55:21,670

 

1734
00:55:21,670 --> 00:55:24,150

 

1735
00:55:24,150 --> 00:55:25,589

 

1736
00:55:25,589 --> 00:55:28,870

 

1737
00:55:28,870 --> 00:55:31,990

 

1738
00:55:31,990 --> 00:55:34,710

 

1739
00:55:34,710 --> 00:55:36,789

 

1740
00:55:36,789 --> 00:55:39,109

 

1741
00:55:39,109 --> 00:55:41,430

 

1742
00:55:41,430 --> 00:55:45,030

 

1743
00:55:45,030 --> 00:55:46,309

 

1744
00:55:46,309 --> 00:55:48,470

 

1745
00:55:48,470 --> 00:55:51,030

 

1746
00:55:51,030 --> 00:55:53,349

 

1747
00:55:53,349 --> 00:55:55,510

 

1748
00:55:55,510 --> 00:55:57,109

 

1749
00:55:57,109 --> 00:56:00,150

 

1750
00:56:00,150 --> 00:56:02,230

 

1751
00:56:02,230 --> 00:56:02,240

 

1752
00:56:02,240 --> 00:56:03,030


1753
00:56:03,030 --> 00:56:05,990

 

1754
00:56:05,990 --> 00:56:06,000

 

1755
00:56:06,000 --> 00:56:07,349


1756
00:56:07,349 --> 00:56:09,430

 

1757
00:56:09,430 --> 00:56:11,589

 

1758
00:56:11,589 --> 00:56:13,829

 

1759
00:56:13,829 --> 00:56:14,950

 

1760
00:56:14,950 --> 00:56:17,430

 

1761
00:56:17,430 --> 00:56:19,430

 

1762
00:56:19,430 --> 00:56:21,030

 

1763
00:56:21,030 --> 00:56:21,040

 

1764
00:56:21,040 --> 00:56:22,150


1765
00:56:22,150 --> 00:56:24,230

 

1766
00:56:24,230 --> 00:56:25,750

 

1767
00:56:25,750 --> 00:56:26,870

 

1768
00:56:26,870 --> 00:56:29,109

 

1769
00:56:29,109 --> 00:56:30,710

 

1770
00:56:30,710 --> 00:56:32,710

 

1771
00:56:32,710 --> 00:56:34,630

 

1772
00:56:34,630 --> 00:56:34,640

 

1773
00:56:34,640 --> 00:56:35,990


1774
00:56:35,990 --> 00:56:37,270

 

1775
00:56:37,270 --> 00:56:39,430

 

1776
00:56:39,430 --> 00:56:41,510

 

1777
00:56:41,510 --> 00:56:42,710

 

1778
00:56:42,710 --> 00:56:46,309

 

1779
00:56:46,309 --> 00:56:48,549

 

1780
00:56:48,549 --> 00:56:49,910

 

1781
00:56:49,910 --> 00:56:52,390

 

1782
00:56:52,390 --> 00:56:55,670

 

1783
00:56:55,670 --> 00:56:58,230

 

1784
00:56:58,230 --> 00:57:02,390

 

1785
00:57:02,390 --> 00:57:06,630

 

1786
00:57:06,630 --> 00:57:10,470

 

1787
00:57:10,470 --> 00:57:13,030

 

1788
00:57:13,030 --> 00:57:15,109

 

1789
00:57:15,109 --> 00:57:17,349

 

1790
00:57:17,349 --> 00:57:18,950

 

1791
00:57:18,950 --> 00:57:20,710

 

1792
00:57:20,710 --> 00:57:22,630

 

1793
00:57:22,630 --> 00:57:25,190

 

1794
00:57:25,190 --> 00:57:26,630

 

1795
00:57:26,630 --> 00:57:29,910

 

1796
00:57:29,910 --> 00:57:33,109

 

1797
00:57:33,109 --> 00:57:34,950

 

1798
00:57:34,950 --> 00:57:37,109

 

1799
00:57:37,109 --> 00:57:39,750

 

1800
00:57:39,750 --> 00:57:42,069

 

1801
00:57:42,069 --> 00:57:45,349

 

1802
00:57:45,349 --> 00:57:46,789

 

1803
00:57:46,789 --> 00:57:48,390

 

1804
00:57:48,390 --> 00:57:51,190

 

1805
00:57:51,190 --> 00:57:52,150

 

1806
00:57:52,150 --> 00:57:53,430

 

1807
00:57:53,430 --> 00:57:54,950

 

1808
00:57:54,950 --> 00:57:57,109

 

1809
00:57:57,109 --> 00:57:57,119

 

1810
00:57:57,119 --> 00:57:58,630


1811
00:57:58,630 --> 00:57:58,640

 

1812
00:57:58,640 --> 00:58:01,030


1813
00:58:01,030 --> 00:58:02,309

 

1814
00:58:02,309 --> 00:58:04,789

 

1815
00:58:04,789 --> 00:58:06,630

 

1816
00:58:06,630 --> 00:58:08,230

 

1817
00:58:08,230 --> 00:58:10,390

 

1818
00:58:10,390 --> 00:58:13,349

 

1819
00:58:13,349 --> 00:58:13,359

 

1820
00:58:13,359 --> 00:58:16,470


1821
00:58:16,470 --> 00:58:18,950

 

1822
00:58:18,950 --> 00:58:20,710

 

1823
00:58:20,710 --> 00:58:22,230

 

1824
00:58:22,230 --> 00:58:23,829

 

1825
00:58:23,829 --> 00:58:26,630

 

1826
00:58:26,630 --> 00:58:26,640

 

1827
00:58:26,640 --> 00:58:27,589


1828
00:58:27,589 --> 00:58:29,589

 

1829
00:58:29,589 --> 00:58:31,670

 

1830
00:58:31,670 --> 00:58:33,670

 

1831
00:58:33,670 --> 00:58:36,390

 

1832
00:58:36,390 --> 00:58:38,950

 

1833
00:58:38,950 --> 00:58:38,960

 

1834
00:58:38,960 --> 00:58:40,230


1835
00:58:40,230 --> 00:58:44,150

 

1836
00:58:44,150 --> 00:58:45,670

 

1837
00:58:45,670 --> 00:58:47,750

 

1838
00:58:47,750 --> 00:58:49,430

 

1839
00:58:49,430 --> 00:58:51,589

 

1840
00:58:51,589 --> 00:58:52,710

 

1841
00:58:52,710 --> 00:58:54,390

 

1842
00:58:54,390 --> 00:58:56,870

 

1843
00:58:56,870 --> 00:58:59,349

 

1844
00:58:59,349 --> 00:59:01,510

 

1845
00:59:01,510 --> 00:59:05,750

 

1846
00:59:05,750 --> 00:59:07,430

 

1847
00:59:07,430 --> 00:59:09,270

 

1848
00:59:09,270 --> 00:59:10,870

 

1849
00:59:10,870 --> 00:59:12,630

 

1850
00:59:12,630 --> 00:59:14,789

 

1851
00:59:14,789 --> 00:59:17,030

 

1852
00:59:17,030 --> 00:59:21,030

 

1853
00:59:21,030 --> 00:59:23,510

 

1854
00:59:23,510 --> 00:59:24,710

 

1855
00:59:24,710 --> 00:59:27,430

 

1856
00:59:27,430 --> 00:59:30,630

 

1857
00:59:30,630 --> 00:59:32,390

 

1858
00:59:32,390 --> 00:59:33,990

 

1859
00:59:33,990 --> 00:59:34,000

 

1860
00:59:34,000 --> 00:59:35,430


1861
00:59:35,430 --> 00:59:36,950

 

1862
00:59:36,950 --> 00:59:39,030

 

1863
00:59:39,030 --> 00:59:42,069

 

1864
00:59:42,069 --> 00:59:44,230

 

1865
00:59:44,230 --> 00:59:46,150

 

1866
00:59:46,150 --> 00:59:47,589

 

1867
00:59:47,589 --> 00:59:49,270

 

1868
00:59:49,270 --> 00:59:50,150

 

1869
00:59:50,150 --> 00:59:52,150

 

1870
00:59:52,150 --> 00:59:53,589

 

1871
00:59:53,589 --> 00:59:53,599

 

1872
00:59:53,599 --> 00:59:54,710


1873
00:59:54,710 --> 00:59:56,230

 

1874
00:59:56,230 --> 00:59:56,240

 

1875
00:59:56,240 --> 00:59:57,589


1876
00:59:57,589 --> 00:59:59,430

 

1877
00:59:59,430 --> 01:00:01,589

 

1878
01:00:01,589 --> 01:00:01,599

 

1879
01:00:01,599 --> 01:00:02,390


1880
01:00:02,390 --> 01:00:03,829

 

1881
01:00:03,829 --> 01:00:05,990

 

1882
01:00:05,990 --> 01:00:07,750

 

1883
01:00:07,750 --> 01:00:09,190

 

1884
01:00:09,190 --> 01:00:11,030

 

1885
01:00:11,030 --> 01:00:12,470

 

1886
01:00:12,470 --> 01:00:15,349

 

1887
01:00:15,349 --> 01:00:17,190

 

1888
01:00:17,190 --> 01:00:18,549

 

1889
01:00:18,549 --> 01:00:18,559

 

1890
01:00:18,559 --> 01:00:19,510


1891
01:00:19,510 --> 01:00:21,510

 

1892
01:00:21,510 --> 01:00:23,349

 

1893
01:00:23,349 --> 01:00:24,789

 

1894
01:00:24,789 --> 01:00:28,150

 

1895
01:00:28,150 --> 01:00:30,710

 

1896
01:00:30,710 --> 01:00:32,870

 

1897
01:00:32,870 --> 01:00:34,870

 

1898
01:00:34,870 --> 01:00:34,880

 

1899
01:00:34,880 --> 01:00:35,910


1900
01:00:35,910 --> 01:00:37,589

 

1901
01:00:37,589 --> 01:00:39,990

 

1902
01:00:39,990 --> 01:00:40,000

 

1903
01:00:40,000 --> 01:00:46,390


1904
01:00:46,390 --> 01:00:48,470

 

1905
01:00:48,470 --> 01:00:50,390

 

1906
01:00:50,390 --> 01:00:54,870

 

1907
01:00:54,870 --> 01:00:56,549

 

1908
01:00:56,549 --> 01:00:59,190

 

1909
01:00:59,190 --> 01:01:01,990

 

1910
01:01:01,990 --> 01:01:02,000

 

1911
01:01:02,000 --> 01:01:03,430


1912
01:01:03,430 --> 01:01:05,430

 

1913
01:01:05,430 --> 01:01:08,870

 

1914
01:01:08,870 --> 01:01:10,630

 

1915
01:01:10,630 --> 01:01:12,870

 

1916
01:01:12,870 --> 01:01:15,430

 

1917
01:01:15,430 --> 01:01:18,309

 

1918
01:01:18,309 --> 01:01:19,750

 

1919
01:01:19,750 --> 01:01:21,750

 

1920
01:01:21,750 --> 01:01:23,510

 

1921
01:01:23,510 --> 01:01:24,870

 

1922
01:01:24,870 --> 01:01:26,870

 

1923
01:01:26,870 --> 01:01:29,270

 

1924
01:01:29,270 --> 01:01:31,910

 

1925
01:01:31,910 --> 01:01:31,920

 

1926
01:01:31,920 --> 01:01:33,750


1927
01:01:33,750 --> 01:01:35,430

 

1928
01:01:35,430 --> 01:01:37,510

 

1929
01:01:37,510 --> 01:01:40,710

 

1930
01:01:40,710 --> 01:01:40,720

 

1931
01:01:40,720 --> 01:01:41,670


1932
01:01:41,670 --> 01:01:43,750

 

1933
01:01:43,750 --> 01:01:45,510

 

1934
01:01:45,510 --> 01:01:45,520

 

1935
01:01:45,520 --> 01:01:46,789


1936
01:01:46,789 --> 01:01:49,270

 

1937
01:01:49,270 --> 01:01:50,789

 

1938
01:01:50,789 --> 01:01:52,549

 

1939
01:01:52,549 --> 01:01:54,710

 

1940
01:01:54,710 --> 01:01:56,309

 

1941
01:01:56,309 --> 01:01:57,750

 

1942
01:01:57,750 --> 01:01:59,270

 

1943
01:01:59,270 --> 01:02:01,109

 

1944
01:02:01,109 --> 01:02:03,270

 

1945
01:02:03,270 --> 01:02:05,029

 

1946
01:02:05,029 --> 01:02:06,870

 

1947
01:02:06,870 --> 01:02:08,230

 

1948
01:02:08,230 --> 01:02:09,750

 

1949
01:02:09,750 --> 01:02:12,069

 

1950
01:02:12,069 --> 01:02:12,079

 

1951
01:02:12,079 --> 01:02:14,230


1952
01:02:14,230 --> 01:02:15,910

 

1953
01:02:15,910 --> 01:02:18,870

 

1954
01:02:18,870 --> 01:02:21,910

 

1955
01:02:21,910 --> 01:02:23,829

 

1956
01:02:23,829 --> 01:02:25,910

 

1957
01:02:25,910 --> 01:02:27,190

 

1958
01:02:27,190 --> 01:02:28,789

 

1959
01:02:28,789 --> 01:02:31,510

 

1960
01:02:31,510 --> 01:02:32,710

 

1961
01:02:32,710 --> 01:02:34,390

 

1962
01:02:34,390 --> 01:02:34,400

 

1963
01:02:34,400 --> 01:02:36,630


1964
01:02:36,630 --> 01:02:37,910

 

1965
01:02:37,910 --> 01:02:39,829

 

1966
01:02:39,829 --> 01:02:41,670

 

1967
01:02:41,670 --> 01:02:43,349

 

1968
01:02:43,349 --> 01:02:46,390

 

1969
01:02:46,390 --> 01:02:48,309

 

1970
01:02:48,309 --> 01:02:49,589

 

1971
01:02:49,589 --> 01:02:49,599

 

1972
01:02:49,599 --> 01:02:50,390


1973
01:02:50,390 --> 01:02:54,230

 

1974
01:02:54,230 --> 01:02:55,910

 

1975
01:02:55,910 --> 01:02:57,109

 

1976
01:02:57,109 --> 01:02:58,630

 

1977
01:02:58,630 --> 01:02:58,640

 

1978
01:02:58,640 --> 01:02:59,670


1979
01:02:59,670 --> 01:02:59,680

 

1980
01:02:59,680 --> 01:03:03,670


1981
01:03:03,670 --> 01:03:05,430

 

1982
01:03:05,430 --> 01:03:07,349

 

1983
01:03:07,349 --> 01:03:09,670

 

1984
01:03:09,670 --> 01:03:12,870

 

1985
01:03:12,870 --> 01:03:12,880

 

1986
01:03:12,880 --> 01:03:14,789


1987
01:03:14,789 --> 01:03:16,789

 

1988
01:03:16,789 --> 01:03:18,870

 

1989
01:03:18,870 --> 01:03:19,910

 

1990
01:03:19,910 --> 01:03:21,990

 

1991
01:03:21,990 --> 01:03:24,069

 

1992
01:03:24,069 --> 01:03:25,990

 

1993
01:03:25,990 --> 01:03:29,510

 

1994
01:03:29,510 --> 01:03:31,190

 

1995
01:03:31,190 --> 01:03:33,990

 

1996
01:03:33,990 --> 01:03:35,910

 

1997
01:03:35,910 --> 01:03:37,829

 

1998
01:03:37,829 --> 01:03:39,510

 

1999
01:03:39,510 --> 01:03:41,349

 

2000
01:03:41,349 --> 01:03:41,359

 

2001
01:03:41,359 --> 01:03:42,630


2002
01:03:42,630 --> 01:03:44,630

 

2003
01:03:44,630 --> 01:03:46,390

 

2004
01:03:46,390 --> 01:03:47,910

 

2005
01:03:47,910 --> 01:03:49,990

 

2006
01:03:49,990 --> 01:03:54,390

 

2007
01:03:54,390 --> 01:03:55,750

 

2008
01:03:55,750 --> 01:04:00,710

 

2009
01:04:00,710 --> 01:04:00,720

 

2010
01:04:00,720 --> 01:04:01,670


2011
01:04:01,670 --> 01:04:01,680

 

2012
01:04:01,680 --> 01:04:04,230


2013
01:04:04,230 --> 01:04:07,190

 

2014
01:04:07,190 --> 01:04:08,950

 

2015
01:04:08,950 --> 01:04:11,190

 

2016
01:04:11,190 --> 01:04:12,470

 

2017
01:04:12,470 --> 01:04:13,910

 

2018
01:04:13,910 --> 01:04:16,549

 

2019
01:04:16,549 --> 01:04:18,870

 

2020
01:04:18,870 --> 01:04:21,190

 

2021
01:04:21,190 --> 01:04:24,069

 

2022
01:04:24,069 --> 01:04:27,349

 

2023
01:04:27,349 --> 01:04:29,750

 

2024
01:04:29,750 --> 01:04:29,760

 

2025
01:04:29,760 --> 01:04:31,270


2026
01:04:31,270 --> 01:04:35,589

 

2027
01:04:35,589 --> 01:04:37,750

 

2028
01:04:37,750 --> 01:04:37,760

 

2029
01:04:37,760 --> 01:04:39,829


2030
01:04:39,829 --> 01:04:41,589

 

2031
01:04:41,589 --> 01:04:43,910

 

2032
01:04:43,910 --> 01:04:46,150

 

2033
01:04:46,150 --> 01:04:48,710

 

2034
01:04:48,710 --> 01:04:51,910

 

2035
01:04:51,910 --> 01:04:53,910

 

2036
01:04:53,910 --> 01:04:57,829

 

2037
01:04:57,829 --> 01:05:01,829

 

2038
01:05:01,829 --> 01:05:04,390

 

2039
01:05:04,390 --> 01:05:07,270

 

2040
01:05:07,270 --> 01:05:09,029

 

2041
01:05:09,029 --> 01:05:10,390

 

2042
01:05:10,390 --> 01:05:12,710

 

2043
01:05:12,710 --> 01:05:15,990

 

2044
01:05:15,990 --> 01:05:18,789

 

2045
01:05:18,789 --> 01:05:20,870

 

2046
01:05:20,870 --> 01:05:22,470

 

2047
01:05:22,470 --> 01:05:23,910

 

2048
01:05:23,910 --> 01:05:26,470

 

2049
01:05:26,470 --> 01:05:27,829

 

2050
01:05:27,829 --> 01:05:30,309

 

2051
01:05:30,309 --> 01:05:32,549

 

2052
01:05:32,549 --> 01:05:35,190

 

2053
01:05:35,190 --> 01:05:36,710

 

2054
01:05:36,710 --> 01:05:40,309

 

2055
01:05:40,309 --> 01:05:41,990

 

2056
01:05:41,990 --> 01:05:43,910

 

2057
01:05:43,910 --> 01:05:46,069

 

2058
01:05:46,069 --> 01:05:47,270

 

2059
01:05:47,270 --> 01:05:47,280

 

2060
01:05:47,280 --> 01:05:48,950


2061
01:05:48,950 --> 01:05:51,589

 

2062
01:05:51,589 --> 01:05:52,870

 

2063
01:05:52,870 --> 01:05:54,309

 

2064
01:05:54,309 --> 01:05:54,319

 

2065
01:05:54,319 --> 01:05:55,829


2066
01:05:55,829 --> 01:05:58,950

 

2067
01:05:58,950 --> 01:06:01,029

 

2068
01:06:01,029 --> 01:06:03,670

 

2069
01:06:03,670 --> 01:06:07,109

 

2070
01:06:07,109 --> 01:06:08,230

 

2071
01:06:08,230 --> 01:06:10,870

 

2072
01:06:10,870 --> 01:06:13,829

 

2073
01:06:13,829 --> 01:06:15,750

 

2074
01:06:15,750 --> 01:06:17,589

 

2075
01:06:17,589 --> 01:06:18,950

 

2076
01:06:18,950 --> 01:06:20,789

 

2077
01:06:20,789 --> 01:06:20,799

 

2078
01:06:20,799 --> 01:06:21,829


2079
01:06:21,829 --> 01:06:23,349

 

2080
01:06:23,349 --> 01:06:25,910

 

2081
01:06:25,910 --> 01:06:28,870

 

2082
01:06:28,870 --> 01:06:32,710

 

2083
01:06:32,710 --> 01:06:35,510

 

2084
01:06:35,510 --> 01:06:37,670

 

2085
01:06:37,670 --> 01:06:39,750

 

2086
01:06:39,750 --> 01:06:41,270

 

2087
01:06:41,270 --> 01:06:43,430

 

2088
01:06:43,430 --> 01:06:45,670

 

2089
01:06:45,670 --> 01:06:47,910

 

2090
01:06:47,910 --> 01:06:47,920

 

2091
01:06:47,920 --> 01:06:49,349


2092
01:06:49,349 --> 01:06:49,359

 

2093
01:06:49,359 --> 01:06:50,150


2094
01:06:50,150 --> 01:06:52,870

 

2095
01:06:52,870 --> 01:06:55,750

 

2096
01:06:55,750 --> 01:06:57,829

 

2097
01:06:57,829 --> 01:06:59,910

 

2098
01:06:59,910 --> 01:07:01,670

 

2099
01:07:01,670 --> 01:07:03,990

 

2100
01:07:03,990 --> 01:07:05,750

 

2101
01:07:05,750 --> 01:07:07,430

 

2102
01:07:07,430 --> 01:07:09,190

 

2103
01:07:09,190 --> 01:07:10,150

 

2104
01:07:10,150 --> 01:07:10,160

 

2105
01:07:10,160 --> 01:07:11,430


2106
01:07:11,430 --> 01:07:13,029

 

2107
01:07:13,029 --> 01:07:13,039

 

2108
01:07:13,039 --> 01:07:15,829


2109
01:07:15,829 --> 01:07:17,750

 

2110
01:07:17,750 --> 01:07:19,349

 

2111
01:07:19,349 --> 01:07:19,359

 

2112
01:07:19,359 --> 01:07:20,470


2113
01:07:20,470 --> 01:07:23,190

 

2114
01:07:23,190 --> 01:07:24,870

 

2115
01:07:24,870 --> 01:07:27,270

 

2116
01:07:27,270 --> 01:07:28,630

 

2117
01:07:28,630 --> 01:07:30,390

 

2118
01:07:30,390 --> 01:07:31,510

 

2119
01:07:31,510 --> 01:07:33,430

 

2120
01:07:33,430 --> 01:07:35,750

 

2121
01:07:35,750 --> 01:07:37,349

 

2122
01:07:37,349 --> 01:07:39,510

 

2123
01:07:39,510 --> 01:07:41,270

 

2124
01:07:41,270 --> 01:07:43,750

 

2125
01:07:43,750 --> 01:07:48,630

 

2126
01:07:48,630 --> 01:07:50,789

 

2127
01:07:50,789 --> 01:07:50,799

 

2128
01:07:50,799 --> 01:07:51,910


2129
01:07:51,910 --> 01:07:54,870

 

2130
01:07:54,870 --> 01:07:57,990

 

2131
01:07:57,990 --> 01:07:58,000

 

2132
01:07:58,000 --> 01:07:59,430


2133
01:07:59,430 --> 01:07:59,440

 

2134
01:07:59,440 --> 01:08:02,630


2135
01:08:02,630 --> 01:08:04,710

 

2136
01:08:04,710 --> 01:08:06,789

 

2137
01:08:06,789 --> 01:08:09,190

 

2138
01:08:09,190 --> 01:08:09,200

 

2139
01:08:09,200 --> 01:08:11,190


2140
01:08:11,190 --> 01:08:11,200

 

2141
01:08:11,200 --> 01:08:12,309


2142
01:08:12,309 --> 01:08:14,630

 

2143
01:08:14,630 --> 01:08:16,950

 

2144
01:08:16,950 --> 01:08:19,110

 

2145
01:08:19,110 --> 01:08:20,550

 

2146
01:08:20,550 --> 01:08:22,309

 

2147
01:08:22,309 --> 01:08:24,470

 

2148
01:08:24,470 --> 01:08:27,590

 

2149
01:08:27,590 --> 01:08:29,990

 

2150
01:08:29,990 --> 01:08:31,669

 

2151
01:08:31,669 --> 01:08:36,470

 

2152
01:08:36,470 --> 01:08:38,070

 

2153
01:08:38,070 --> 01:08:39,990

 

2154
01:08:39,990 --> 01:08:42,309

 

2155
01:08:42,309 --> 01:08:44,229

 

2156
01:08:44,229 --> 01:08:47,669

 

2157
01:08:47,669 --> 01:08:47,679

 

2158
01:08:47,679 --> 01:08:48,550


2159
01:08:48,550 --> 01:08:54,309

 

2160
01:08:54,309 --> 01:08:55,749

 

2161
01:08:55,749 --> 01:08:55,759

 

2162
01:08:55,759 --> 01:08:56,789


2163
01:08:56,789 --> 01:08:56,799

 

2164
01:08:56,799 --> 01:08:57,910


2165
01:08:57,910 --> 01:08:57,920

 

2166
01:08:57,920 --> 01:09:00,309


2167
01:09:00,309 --> 01:09:02,390

 

2168
01:09:02,390 --> 01:09:03,829

 

2169
01:09:03,829 --> 01:09:05,829

 

2170
01:09:05,829 --> 01:09:05,839

 

2171
01:09:05,839 --> 01:09:07,430


2172
01:09:07,430 --> 01:09:07,440

 

2173
01:09:07,440 --> 01:09:08,550


2174
01:09:08,550 --> 01:09:10,870

 

2175
01:09:10,870 --> 01:09:13,269

 

2176
01:09:13,269 --> 01:09:14,789

 

2177
01:09:14,789 --> 01:09:17,269

 

2178
01:09:17,269 --> 01:09:21,910

 

2179
01:09:21,910 --> 01:09:24,550

 

2180
01:09:24,550 --> 01:09:27,349

 

2181
01:09:27,349 --> 01:09:30,390

 

2182
01:09:30,390 --> 01:09:35,110

 

2183
01:09:35,110 --> 01:09:37,030

 

2184
01:09:37,030 --> 01:09:38,229

 

2185
01:09:38,229 --> 01:09:41,189

 

2186
01:09:41,189 --> 01:09:43,189

 

2187
01:09:43,189 --> 01:09:46,070

 

2188
01:09:46,070 --> 01:09:48,470

 

2189
01:09:48,470 --> 01:09:51,030

 

2190
01:09:51,030 --> 01:09:55,830

 

2191
01:09:55,830 --> 01:09:57,750

 

2192
01:09:57,750 --> 01:10:00,390

 

2193
01:10:00,390 --> 01:10:03,910

 

2194
01:10:03,910 --> 01:10:05,910

 

2195
01:10:05,910 --> 01:10:07,669

 

2196
01:10:07,669 --> 01:10:09,030

 

2197
01:10:09,030 --> 01:10:12,709

 

2198
01:10:12,709 --> 01:10:12,719

 

2199
01:10:12,719 --> 01:10:14,310


2200
01:10:14,310 --> 01:10:14,320

 

2201
01:10:14,320 --> 01:10:15,270


2202
01:10:15,270 --> 01:10:17,669

 

2203
01:10:17,669 --> 01:10:19,590

 

2204
01:10:19,590 --> 01:10:20,790

 

2205
01:10:20,790 --> 01:10:20,800

 

2206
01:10:20,800 --> 01:10:22,149


2207
01:10:22,149 --> 01:10:24,630

 

2208
01:10:24,630 --> 01:10:26,310

 

2209
01:10:26,310 --> 01:10:28,709

 

2210
01:10:28,709 --> 01:10:32,390

 

2211
01:10:32,390 --> 01:10:34,950

 

2212
01:10:34,950 --> 01:10:38,070

 

2213
01:10:38,070 --> 01:10:40,070

 

2214
01:10:40,070 --> 01:10:42,709

 

2215
01:10:42,709 --> 01:10:44,550

 

2216
01:10:44,550 --> 01:10:46,950

 

2217
01:10:46,950 --> 01:10:48,149

 

2218
01:10:48,149 --> 01:10:49,830

 

2219
01:10:49,830 --> 01:10:52,550

 

2220
01:10:52,550 --> 01:10:52,560

 

2221
01:10:52,560 --> 01:10:54,870


2222
01:10:54,870 --> 01:10:57,350

 

2223
01:10:57,350 --> 01:10:59,270

 

2224
01:10:59,270 --> 01:11:00,229

 

2225
01:11:00,229 --> 01:11:00,239

 

2226
01:11:00,239 --> 01:11:01,669


2227
01:11:01,669 --> 01:11:04,229

 

2228
01:11:04,229 --> 01:11:06,149

 

2229
01:11:06,149 --> 01:11:08,790

 

2230
01:11:08,790 --> 01:11:11,110

 

2231
01:11:11,110 --> 01:11:14,709

 

2232
01:11:14,709 --> 01:11:17,830

 

2233
01:11:17,830 --> 01:11:21,110

 

2234
01:11:21,110 --> 01:11:21,120

 

2235
01:11:21,120 --> 01:11:21,990


2236
01:11:21,990 --> 01:11:22,000

 

2237
01:11:22,000 --> 01:11:23,510


2238
01:11:23,510 --> 01:11:25,350

 

2239
01:11:25,350 --> 01:11:27,030

 

2240
01:11:27,030 --> 01:11:29,510

 

2241
01:11:29,510 --> 01:11:32,229

 

2242
01:11:32,229 --> 01:11:35,110

 

2243
01:11:35,110 --> 01:11:36,709

 

2244
01:11:36,709 --> 01:11:36,719

 

2245
01:11:36,719 --> 01:11:37,430


2246
01:11:37,430 --> 01:11:39,590

 

2247
01:11:39,590 --> 01:11:41,350

 

2248
01:11:41,350 --> 01:11:42,950

 

2249
01:11:42,950 --> 01:11:45,030

 

2250
01:11:45,030 --> 01:11:47,430

 

2251
01:11:47,430 --> 01:11:50,229

 

2252
01:11:50,229 --> 01:11:52,630

 

2253
01:11:52,630 --> 01:11:55,510

 

2254
01:11:55,510 --> 01:11:58,149

 

2255
01:11:58,149 --> 01:12:01,510

 

2256
01:12:01,510 --> 01:12:03,030

 

2257
01:12:03,030 --> 01:12:04,550

 

2258
01:12:04,550 --> 01:12:06,950

 

2259
01:12:06,950 --> 01:12:09,030

 

2260
01:12:09,030 --> 01:12:11,030

 

2261
01:12:11,030 --> 01:12:11,040

 

2262
01:12:11,040 --> 01:12:11,910


2263
01:12:11,910 --> 01:12:13,189

 

2264
01:12:13,189 --> 01:12:13,199

 

2265
01:12:13,199 --> 01:12:14,310


2266
01:12:14,310 --> 01:12:14,320

 

2267
01:12:14,320 --> 01:12:16,149


2268
01:12:16,149 --> 01:12:16,159

 

2269
01:12:16,159 --> 01:12:17,750


2270
01:12:17,750 --> 01:12:20,310

 

2271
01:12:20,310 --> 01:12:25,669

 

2272
01:12:25,669 --> 01:12:27,990

 

2273
01:12:27,990 --> 01:12:29,270

 

2274
01:12:29,270 --> 01:12:32,229

 

2275
01:12:32,229 --> 01:12:33,189

 

2276
01:12:33,189 --> 01:12:34,709

 

2277
01:12:34,709 --> 01:12:37,590

 

2278
01:12:37,590 --> 01:12:39,590

 

2279
01:12:39,590 --> 01:12:41,189

 

2280
01:12:41,189 --> 01:12:41,199

 

2281
01:12:41,199 --> 01:12:42,149


2282
01:12:42,149 --> 01:12:43,910

 

2283
01:12:43,910 --> 01:12:45,990

 

2284
01:12:45,990 --> 01:12:48,229

 

2285
01:12:48,229 --> 01:12:50,149

 

2286
01:12:50,149 --> 01:12:53,270

 

2287
01:12:53,270 --> 01:12:54,470

 

2288
01:12:54,470 --> 01:12:58,070

 

2289
01:12:58,070 --> 01:13:00,149

 

2290
01:13:00,149 --> 01:13:01,590

 

2291
01:13:01,590 --> 01:13:01,600

 

2292
01:13:01,600 --> 01:13:02,790


2293
01:13:02,790 --> 01:13:05,590

 

2294
01:13:05,590 --> 01:13:07,590

 

2295
01:13:07,590 --> 01:13:09,430

 

2296
01:13:09,430 --> 01:13:12,070

 

2297
01:13:12,070 --> 01:13:13,590

 

2298
01:13:13,590 --> 01:13:16,630

 

2299
01:13:16,630 --> 01:13:18,950

 

2300
01:13:18,950 --> 01:13:21,669

 

2301
01:13:21,669 --> 01:13:23,669

 

2302
01:13:23,669 --> 01:13:25,669

 

2303
01:13:25,669 --> 01:13:27,830

 

2304
01:13:27,830 --> 01:13:29,990

 

2305
01:13:29,990 --> 01:13:32,470

 

2306
01:13:32,470 --> 01:13:33,430

 

2307
01:13:33,430 --> 01:13:34,870

 

2308
01:13:34,870 --> 01:13:34,880

 

2309
01:13:34,880 --> 01:13:35,750


2310
01:13:35,750 --> 01:13:38,470

 

2311
01:13:38,470 --> 01:13:41,669

 

2312
01:13:41,669 --> 01:13:43,990

 

2313
01:13:43,990 --> 01:13:44,830

 

2314
01:13:44,830 --> 01:13:50,229

 

2315
01:13:50,229 --> 01:13:50,239

 

2316
01:13:50,239 --> 01:13:52,470


2317
01:13:52,470 --> 01:13:55,430

 

2318
01:13:55,430 --> 01:13:55,440

 

2319
01:13:55,440 --> 01:13:56,790


2320
01:13:56,790 --> 01:14:00,790

 

2321
01:14:00,790 --> 01:14:00,800

 

2322
01:14:00,800 --> 01:14:02,790


2323
01:14:02,790 --> 01:14:04,550

 

2324
01:14:04,550 --> 01:14:04,560

 

2325
01:14:04,560 --> 01:14:06,950


2326
01:14:06,950 --> 01:14:08,390

 

2327
01:14:08,390 --> 01:14:09,990

 

2328
01:14:09,990 --> 01:14:11,990

 

2329
01:14:11,990 --> 01:14:15,189

 

2330
01:14:15,189 --> 01:14:15,199

 

2331
01:14:15,199 --> 01:14:15,990


2332
01:14:15,990 --> 01:14:16,000

 

2333
01:14:16,000 --> 01:14:18,070


2334
01:14:18,070 --> 01:14:21,270

 

2335
01:14:21,270 --> 01:14:21,280

 

2336
01:14:21,280 --> 01:14:22,229


2337
01:14:22,229 --> 01:14:24,149

 

2338
01:14:24,149 --> 01:14:25,910

 

2339
01:14:25,910 --> 01:14:28,630

 

2340
01:14:28,630 --> 01:14:30,790

 

2341
01:14:30,790 --> 01:14:34,149

 

2342
01:14:34,149 --> 01:14:35,910

 

2343
01:14:35,910 --> 01:14:37,669

 

2344
01:14:37,669 --> 01:14:41,590

 

2345
01:14:41,590 --> 01:14:43,990

 

2346
01:14:43,990 --> 01:14:44,000

 

2347
01:14:44,000 --> 01:14:45,030


2348
01:14:45,030 --> 01:14:49,110

 

2349
01:14:49,110 --> 01:14:51,750

 

2350
01:14:51,750 --> 01:14:51,760

 

2351
01:14:51,760 --> 01:14:52,790


2352
01:14:52,790 --> 01:14:54,070

 

2353
01:14:54,070 --> 01:14:56,310

 

2354
01:14:56,310 --> 01:14:56,320

 

2355
01:14:56,320 --> 01:14:57,590


2356
01:14:57,590 --> 01:14:59,510

 

2357
01:14:59,510 --> 01:15:02,470

 

2358
01:15:02,470 --> 01:15:04,149

 

2359
01:15:04,149 --> 01:15:05,590

 

2360
01:15:05,590 --> 01:15:08,870

 

2361
01:15:08,870 --> 01:15:11,270

 

2362
01:15:11,270 --> 01:15:13,590

 

2363
01:15:13,590 --> 01:15:13,600

 

2364
01:15:13,600 --> 01:15:14,310


2365
01:15:14,310 --> 01:15:15,910

 

2366
01:15:15,910 --> 01:15:18,149

 

2367
01:15:18,149 --> 01:15:20,630

 

2368
01:15:20,630 --> 01:15:22,870

 

2369
01:15:22,870 --> 01:15:24,790

 

2370
01:15:24,790 --> 01:15:26,790

 

2371
01:15:26,790 --> 01:15:29,030

 

2372
01:15:29,030 --> 01:15:30,870

 

2373
01:15:30,870 --> 01:15:34,070

 

2374
01:15:34,070 --> 01:15:34,080

 

2375
01:15:34,080 --> 01:15:34,870


2376
01:15:34,870 --> 01:15:37,510

 

2377
01:15:37,510 --> 01:15:39,510

 

2378
01:15:39,510 --> 01:15:41,110

 

2379
01:15:41,110 --> 01:15:42,870

 

2380
01:15:42,870 --> 01:15:45,110

 

2381
01:15:45,110 --> 01:15:46,390

 

2382
01:15:46,390 --> 01:15:49,110

 

2383
01:15:49,110 --> 01:15:51,590

 

2384
01:15:51,590 --> 01:15:53,430

 

2385
01:15:53,430 --> 01:15:55,350

 

2386
01:15:55,350 --> 01:15:56,709

 

2387
01:15:56,709 --> 01:15:59,110

 

2388
01:15:59,110 --> 01:16:02,470

 

2389
01:16:02,470 --> 01:16:04,830

 

2390
01:16:04,830 --> 01:16:06,790

 

2391
01:16:06,790 --> 01:16:11,750

 

2392
01:16:11,750 --> 01:16:12,950

 

2393
01:16:12,950 --> 01:16:14,870

 

2394
01:16:14,870 --> 01:16:17,110

 

2395
01:16:17,110 --> 01:16:18,709

 

2396
01:16:18,709 --> 01:16:23,510

 

2397
01:16:23,510 --> 01:16:25,430

 

2398
01:16:25,430 --> 01:16:27,510

 

2399
01:16:27,510 --> 01:16:29,189

 

2400
01:16:29,189 --> 01:16:29,199

 

2401
01:16:29,199 --> 01:16:30,310


2402
01:16:30,310 --> 01:16:32,790

 

2403
01:16:32,790 --> 01:16:34,870

 

2404
01:16:34,870 --> 01:16:36,950

 

2405
01:16:36,950 --> 01:16:40,630

 

2406
01:16:40,630 --> 01:16:42,709

 

2407
01:16:42,709 --> 01:16:45,110

 

2408
01:16:45,110 --> 01:16:45,120

 

2409
01:16:45,120 --> 01:16:47,030


2410
01:16:47,030 --> 01:16:49,270

 

2411
01:16:49,270 --> 01:16:52,310

 

2412
01:16:52,310 --> 01:16:54,550

 

2413
01:16:54,550 --> 01:16:56,790

 

2414
01:16:56,790 --> 01:16:56,800

 

2415
01:16:56,800 --> 01:16:57,590


2416
01:16:57,590 --> 01:16:59,030

 

2417
01:16:59,030 --> 01:16:59,040

 

2418
01:16:59,040 --> 01:17:00,070


2419
01:17:00,070 --> 01:17:00,080

 

2420
01:17:00,080 --> 01:17:00,950


2421
01:17:00,950 --> 01:17:04,310

 

2422
01:17:04,310 --> 01:17:06,470

 

2423
01:17:06,470 --> 01:17:07,910

 

2424
01:17:07,910 --> 01:17:09,990

 

2425
01:17:09,990 --> 01:17:11,830

 

2426
01:17:11,830 --> 01:17:14,229

 

2427
01:17:14,229 --> 01:17:15,669

 

2428
01:17:15,669 --> 01:17:18,390

 

2429
01:17:18,390 --> 01:17:19,669

 

2430
01:17:19,669 --> 01:17:21,990

 

2431
01:17:21,990 --> 01:17:25,030

 

2432
01:17:25,030 --> 01:17:26,870

 

2433
01:17:26,870 --> 01:17:28,870

 

2434
01:17:28,870 --> 01:17:30,709

 

2435
01:17:30,709 --> 01:17:30,719

 

2436
01:17:30,719 --> 01:17:32,229


2437
01:17:32,229 --> 01:17:34,950

 

2438
01:17:34,950 --> 01:17:37,270

 

2439
01:17:37,270 --> 01:17:37,280

 

2440
01:17:37,280 --> 01:17:39,350


2441
01:17:39,350 --> 01:17:42,149

 

2442
01:17:42,149 --> 01:17:45,669

 

2443
01:17:45,669 --> 01:17:47,669

 

2444
01:17:47,669 --> 01:17:50,229

 

2445
01:17:50,229 --> 01:17:50,239

 

2446
01:17:50,239 --> 01:17:54,310


2447
01:17:54,310 --> 01:17:56,830

 

2448
01:17:56,830 --> 01:17:59,030

 

2449
01:17:59,030 --> 01:18:01,110

 

2450
01:18:01,110 --> 01:18:03,669

 

2451
01:18:03,669 --> 01:18:07,990

 

2452
01:18:07,990 --> 01:18:10,790

 

2453
01:18:10,790 --> 01:18:26,630

 

2454
01:18:26,630 --> 01:18:26,640

 

2455
01:18:26,640 --> 01:18:28,719