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uh so  we are  welcome back everybody  [Music]  and  we are  going to continue our discussion of um  computability theory  turing machines  and uh how to prove things undecidable  um  which is what we've been doing um so we  talked about this more advanced method  of proving things undecidable last  lecture called the computation history  method  which comes up in all sorts of  proofs of undecidability and usually  more uh  complex ones  such as for example the proof of  hilbert's tenth problem um  that i mentioned whether you want to if  you want to test  whether a polynomial has a solution  in integers  um  it's a reduction from atm just like  we've been doing all along those are the  all of those proofs are pretty much  reduction from something undecidable  this reduction from  you know from atm is a good as good a  starting point as any  and in um  and it uses the computation history  method so what the end of doing is given  a turing machine and an input you  construct a polynomial  that has several variables  and where  the  in order to get a an integer root an  integer solution of that polynomial um  one of the variables is going to have to  be assigned to some kind of encoding of  a computation history  of the turing machine of that of m on w  it's going to be some one of those  variables is going to be a computation  history  um an integer  which represents the computation history  for m on w um  and  uh the other variables are there to help  you kind of decode that  so that the polynomial can actually  check  and make a solution it becomes a  solution if that actually is a  legitimate computation history of m w so  it really uses the very same method that  we've been using all along but it's uh  it's just a lot it's  it's pretty hairy to construct that  polynomial and do the check  in the way that you need to do  so for the post correspondence problem  which we introduced last time it's  doing the check is relatively simple um  you know that the uh  the match is the is the computation  history and you know following the rules  of the match um is fairly simple to  construct that  uh post-correspondence problem instance  we talked about linearly down to  automata  you know we of course we define  configurations and and computation  histories along the way and proved some  certain problem other problems are  undecidable as well using the same  method  okay um  so today uh we're gonna shift gears  we're gonna in our last lecture on the  computability section of the course  we're going to talk about  something called the recursion theorem  which basically  gives turing machines the ability to  refer to themselves turing machines in  any program to do self-reference um so  that you can actually get at the code  of the turing machine or the code of the  program that you're writing um  even if that's not a built-in primitive  of the  programming language or the operating  system that you're working with it still  gives you that access  um and also we're gonna if we have time  at the end i'm gonna talk a little bit  about mathematical logic which is sort  of a nice application  of the recursion theorem and it's you  know it's a beautiful subject on its own  and it's something that i can give a  brief introduction to okay so today's  topic  is  um  about  self-reference  self-reproducing machines  and the broader topic called the  recursion theorem  um so let me introduce it with the what  i i would call the self-reproduction  paradox  and that is  suppose you have a factory  like a tesla effect or a car  manufacturing uh factory  um  [Music]  here's a picture  uh  of the factory and it's producing cars  all right so we have a factory that  makes cars  and um  what can we say about  the relative complexity  of the cars  compared with the factory in some  informal sense  um  so i would argue that you  would be reasonable to say  that  the  complexity of the factory is going to  have to be greater  than the complexity of the cars  that it makes  because not only does the factory have  to know how to make the cars so it has  to have all the instructions and  whatever things that go into a car it  has to be included in the you know it's  at least in some kind of  has been  in some sense represented in the factory  but the factory also has to have  other stuff  you know the robots and the machine and  the other manufacturing items uh tools  and so on for making the cars  so the factory has to have all the  complexity of a car incorporated plus  other things as well and on that for  that reason one could imagine that the  factory complexity is more than the  car's complexity  but now suppose you want to have a  factory  that makes factories  sort of imagine here's the picture  um  so or in general a machine that makes  copies of itself  well that seems at first glance  to be  uh impossible  um  because  not only does the factory obviously have  to have all of the instructions for what  a factory is like but it needs to have  all of the compo all of the the the  extra things that it would need to do  the manufacturing  and so for that reason it seems like a  um it's not possible  to have a machine make copies of itself  i mean you would run into the very  same problem  if i asked you to produce  a program and what in your favorite  language um  that prints out itself an exact copy of  the same code  i mean you can always write a program  which is going to print out some string  you know like hello world  that's easy because you just put hello  world into  you know some kind of a variable or some  sort of a um you know a table into the  program and say print that table but if  you want to make a print  if you want the program to print out a  copy of itself  you can't take the whole program and  stick that into a table because  um  you know the the program's gonna have to  be bigger than the table  you know and so you're gonna you're  gonna end up with some something  impossible happening because the program  an entire copy of the program can't fit  inside the program  um you just get uh you know cut the  program inside itself inside itself  inside itself forever um and so you get  we end up with like an infinite program  that way  uh so  if you know you just kind of  naively approach the problem for how to  make a program which is going to print  out a copy of itself  it's not so easy to do um  but hopefully  after today's lecture you will see that  it is possible and in fact how to do it  um  and  not only that is an idle  bit curious bit of curiosity but there  are actually applications for why you  might want to do that  mainly within mathematics um and in uh  computer science theory but there's even  kind of a real world application if you  will in a way too  so we'll get to that at the end  um so it seems as i'm saying impossible  to have a self-reproducing machine but  we know that  in the world there are things that make  copies of themselves  living things  um  so  uh it seems like a paradox no cells can  make copies exactly of themselves all  living things can make copies of  themselves um so how do they manage to  get around this paradox  well in fact  there's no paradox  because it is possible to make a machine  that self-reproduces that makes copies  of itself this has been known for  you know  for many years  probably it goes back to von neumann um  who wrote a famous paper on  self-reproducing machines  okay so self-reproducing machines are in  fact possible  [Music]  so let me give you an example of how you  would make a self-reproducing turing  machine  what do i mean by that  i'm going to i mean a machine i'm going  to call it self  which  ignores its input so on any input  you turn it on  machine grinds around for a while and  halts  with a description of itself on the tape  with the description of self its own  code  sitting on the table so very much like  producing a program  which would  uh print out its own code  that's really what we're doing  um  so for that we're going to first need a  little lemma which is a very simple  lemma but it looks more it looks worse  than it is  um  so uh let me just read it out to you and  then i'll explain what it's saying  because what it's  saying is extremely simple  so there's a computable function i'll  call it q that maps strings to strings  which will take any string w  and produce from w a turing machine  which will print w  okay that's all it does so  as you know if i give you a string w you  could produce a turing machine which  would have w  represented  in the um  states and transitions of the machine  so that if you turn the machine on the  machine will be will output w  i mean if i want you to put if you i  want you to give me a turing machine  that predict prints the string  1101 on the tape you could do that i  hope um  and i'm and if no matter what the string  was instead of one one zero zero or one  you know whatever it's you know  20 zeros and then five ones  you could do that too  and in fact um there's a simple  simple procedure that takes a string  and maps that onto a turing machine  which prints out that string  so that's that's a  computable function  um which basically takes a string and  converts it to something that evaluates  to that string  okay and um  i'm calling it q i don't know if this is  helpful to you or not it's kind of like  it converts the string w to w in quotes  so q stands for quotes  in a way  okay uh so if that's helpful maybe  um  then good um  but anyway p sub w is a turing machine  when you turn it on  it just prints out w and holds  okay and i can find piece of w from w  um  straightforward proof  okay so now i'm going to tell you  no  assuming that we have that  that  computable function q  um  i'm going to tell you how to make this  machine self  um and it's not complicated  um  the turing machine  um  a self is going to have two parts i'll  call a and b  here's a here's a schematic for the  machine  so here's self it's in two parts a and b  and what i mean by a and b it's like two  subroutines  uh self is going to start out running a  and when a is finished it's going to  pass control to b  and then b is going to finish the job  and when it's done you're going to have  the description self sitting here on the  tape  all right  so my what's left is to give you the  code for a and for b  okay um  [Music]  so a  is going to be something  super simple  um  a is going to be that machine  which prints out  b  prints out a description of b  um the one that i described up here  so remember p  sub w is the machine which prints out w  and p sub the description of b is simply  going to machine that has this string p  sub b  stored in its states and transitions you  turn on that machine and it prints out b  and then it's done  so this is a very simple image a is very  simple so here just to  here p sub b  um  is a part of a  and when it's done  b appears on the tape so that's at the  point when a has finished its work  now it's going to pass control to b  so we're not obviously done yet because  what we want is a and b both to be on  the tape not just b because self is a  combination of a and b together  um  so  i have to tell you how b  works to finish the job  um  so you might think  um as a first  you know given what we did for for  getting b on the tape that we'll uh get  a on the tape in the same way by putting  a copy of a inside b  so a copy of b is inside a and a copy of  a is inside b  um  and you know at some conceptual level  that seems like that might do the job  but it's really not that is not a  solution  because the fact that i can put b inside  a  um kind of forbids me from also putting  a inside b because that's going to be  the same kind of circular reasoning of  just putting a machine inside itself you  just can't do that because you're going  to end up with an infinitely big machine  that way  you know in fact if i'm putting b inside  a  a copy of b  in terms of its description inside a a  is really going to be much bigger than b  because it has all of b with inside it  plus other stuff  you know the states and transitions for  printing that  uh b out  so i can't have a b much bigger than b  and then b also much bigger than a so  something this is this is no good  we're going to have to do something else  so how is b now going to get a hold of a  um  and the uh  uh  the trick for doing that  without having a copy of a inside b  which we're not gonna which doesn't work  that's not going to that's not going to  be a good solution instead of the way  that b is going to get a it's going to  figure out what it's going to figure out  what a is  and how is it going to figure out what a  is  because if you remember  um  b can look and now look it can look at  the tape it sees some string there which  happens to be a description of itself  but it doesn't care it sees some string  on the tape  a is the machine that prints out that  string  a  is q  of this string  so b is simply going to compute q of  whatever it's whatever it sees on the  tape  that is a  okay so it's i don't know if you can  read this kind of small here  it's going to compute a from b sitting  on the tape so here is the instructions  for b it's going to compute q of the  tape contents which happens to be  the description of b but that's  irrelevant to b  b just sees some string on the tape it  computes q of that and that is the that  is a  because a is the machine which prints  out that string  then it's going to combine  a with b  doing whatever you know slight  interfacing that needs to happen i'm not  going to get into those details  to convert those two pieces into one  machine  which is the machine self  and then it's going to print out with  self on the tape as i'm going to  indicate over here  okay  so that's how  a turing machine  can print out a copy of its own  description  and leave it on the tape  so  and what's nice about this is nothing  specific about turing machines this is a  general  procedure  that allows any programming language  to print out a copy of its own code you  can even carry this out in english  as i'm going to do in the next slide  um  so here's a good question here there are  many possible turing machines that can  print out b  um that's right how do i know how do i  get the the particular one  what i have in mind that's that's a  little bit of a subtle question but but  that but it's a good question  um  i have in mind the the particular turing  machine that prints out b which is the  one that q produces  remember so we you have to refer back to  this lemma  this lemma produces a particular machine  that prints out b  from  from b  and that's the one that i'm going to use  for a and that's the one that b is going  to be able to obtain by running the q  algorithm  to figure out what a is so you have to  make sure you're being consistent it's a  little bit of a detail but but it's a  good question um  and why doesn't this create a circular  argument too well there is no certain so  that was another question i'm seeing  here on the on the tape well there's  there's no longer anything see b  does not have to have a stored within it  it figures out a  so in a sense you're going to write the  program the code for b first  b is just a simple program here it is  there's nothing circular about it says b  is a simple the code for b is take a  look at the tape compute q of  that um combine the result with whatever  was sitting on the tape from before and  print it out  i mean that's a piece of code which you  can just write i mean this will become  more clear hopefully  in our next slide where we talk about  the english implementation  but just just i don't want to rush to  that  so you know you could figure out b  without even knowing what a is b stands  alone  but then  you know because b is just a piece of  code that runs q  based on what it's what it sees on the  tape a  now you need to know what b is in order  to obtain a  because a has the code for b embedded  within a  as a string  so first you produce b  then you can figure out then you can  obtain a there's nothing circular in  this argument i don't know if that's  helpful to you but  you may need to mull it over or maybe  it'll be helpful from the next slide so  let's go there now  okay so as i'm saying you can implement  this in any language in particular you  can implement it in english  okay so let's just shift gears let's  talk about writing down english  instructions  and then i'll show um you know what  happens if you carry out those  instructions so  let's  start simple how about the sentence  hello write hello world  so you know um an obedient person  reading those instructions um would then  write hello world  um on their paper or or wherever  um  okay  fine you get the i hopefully you get you  get the idea um  uh so now what i'd like here's an  here is a  um  another  sentence  another instruction write the sentence  um  and  the uh  the reader uh the obedient reader would  then  okay write the sentence  this is the kind of thing i'm looking  for  here is a sentence which instructs  the reader to make an exact copy  but i don't like this one  even though it does in a sense what i'm  looking for  because  um it it kind of cheats  when it the when it has this  this refers to the entire sentence  itself  it's using  kind of implicitly here  a construct which a turing machine does  not have  the turing machine does not have  a way of accessing its own code  um  and in fact  really what the point of this uh whole  uh theorem that we're going to present  is that you don't need to have this  self-reference as a primitive  you can get that effectively  uh using um  the  procedure that i'm describing which will  give you the same effect so you don't  need it as a primitive you can you can  design some  software basically which will give you  the same effect  um  so let me show you how to do that in  english  so let's do uh uh  let's look at a slightly more oops so  cheating so turing machines don't have  the self-reference primitive so let's  look at another sentence here  write the following twice  the second time in quotes  and then hello world  so what do we get if we follow those  that instruction  um  well you get  hello world and then hello world again  now in quotes  okay  hopefully not too bad  but now i'm one step away for doing the  implementation of the self-reproducing  algorithm  in english  write the following twice the second  time in  quotes and then in quotes the same text  now following those instructions you get  well  write the following twice the second  time in quotes so that comes out here  and the second time you put it in quotes  just like you did with the hello world  but that's exactly  what  um  here the  the output is exactly what the input was  and  there's you know even though there is a  part of the  sentence here which refers to a  different part  so here the the the first part is  referring to the second part  never do i have to have the sense  referring to its entirety  you know uh  there's no part of the sentence here  that's pointing at the entire sentence  um  and so uh this here  manages to get the effect that i'm  looking for where the output is equal to  the the code um but doesn't have the  avoids having the self-reference  is in so it's one thing is referring to  something else but not referring back to  itself  okay um  so uh let me just see here  so here is sort of i'm sort of i'm gonna  have a  check in on this uh in a minute so i'm  gonna try to contrast what's happening  here with what's happening in that  turing machine itself that i had from  the previous slide  okay so a bunch of mulders over and i'm  going to give you a check in to see  this is a little bit of a challenging  check-in um but let's see if you can uh  figure weight your way through it  okay and basically it says so that  really what we're doing here is called  the recursion theorem as you'll see  we'll actually present the recursion  theorem formally on the next slide  um  but  here  in both of these cases we we kind of  have a template part and an action part  both in both cases we have there are two  parts to the instructions the template  and the action  okay so  i'm going to leave it to you to try to  imagine  which of those which is which  in each of these two examples  um  and then i'm going to ask you to pick um  in the turing machine which is the  action which is a template  and in the sentence which is the action  and which is the template the action is  the part that uh where there's some  interesting  sort of  stuff instructional stuff happening that  you have to carry out the template is  really basically just text  or just code just just uh just a string  okay so  let me uh pull up that poll  [Music]  see what you think  because i'm asking you now to to  indicate where is the action part in  both of those cases  what is the upper phrase and lower  phrase i mean of the sentence here so  write the following twice this is the  upper phrase and the part in quotations  is the lower phrase sorry  okay almost done here  five seconds  a few of you have not answered yet  answer it  one second to go  okay here we go  ending polling  so the uh the majority here is correct  um  uh  you know in i would say in the um  english sentence here  uh the action part is the first part  that's where you actually have  being directed to do something  the second half of the sentence the  lower part of the sentence is just a  template  written you know this is just some  string here there's no action really  being directed it happens to be the same  as the top but this could have been just  hello world this could have been  anything  and then the upper part acts on that so  the upper part is the action part so  it's the upper phrase that's the  relevant part now in the turing machine  it's kind of the uh in a sense it's the  other way around  the first part is really just the  template this the second part b  is is where you're doing some actually  work on the template um you're taking  that that uh  basically text which could be anything a  could be anything and your um  uh uh figure take looking at that  template and reconstructing what a was  from the from that uh string that  appears on the tape so b is actually one  the one that's doing the work so it's b  and the upper phrase that's part c is  correct  okay  um so let us uh  continue then um  [Music]  uh  oh  i want to mention here problem six on  the pset um  so you know your job really is to  implement this in  you know if you have a programming  language and you lot that you like you  can you you can it could be  python or whatever your favorite java  whatever you like you can implement this  if you don't know any programming  languages then just make up some sort of  pseudo programming language and  implement it there  let me point out that  um getting the quoting right  um  uh  it is a bit of a pain  um  because you have to kind of escape the  quotes and so on  i'm not going to be fussy about that so  you can still get full credit even if  you don't get the quoting part quite  correct do your best  um i think it's an interesting problem  uh to try to solve and if you you know  if you struggle with it for a while um  you know  you know it's slippery it's the kind of  thing you have to you know  uh you can easily spend a couple of  hours on this problem uh  because it's uh it's it's it's a bit  tricky to manage to make a program which  prints itself out  which is what the task of problem six is  on this on the pset but don't fuss about  too much on the quoting if that's the  only thing that's hanging you up try to  get the main structure of it which is  fairly simple actually um  and if you can't get the quoting part to  work you know i'll ask the graders not  to uh  penalize you for that part  um  uh  okay  uh  let's let let's then look at the sort of  the more formal version of what we've  just done and really kind of also taking  it to the next level  because being able to print out a copy  of yourself is um  is kind of a curiosity in a way  but you know the recursion theorem says  not only can you make a machine which  prints out a copy of itself but you can  actually make a machine which can obtain  a copy of itself and then do some  computation on that copy of itself that  on the on its own description which  actually turns out to be a useful thing  to do  uh once you have access to your own  description as we'll see from some  examples then you can do some perhaps  some interesting things with that  okay um  so basically what what what the  recursion theorem is a kind of a  compiler  it allows you to  have a new primitive when you're writing  turing machines which is compute your  own description  and the recursion theorem will implement  that for you  okay  so the technical form of the um  recursion theorem is going to look a  little bit um  counterintuitive perhaps uh let me put  it out there  if you if you struggle with a little bit  with the slide  don't sweat it the main thing to  remember and we'll see from examples is  that you can compute your own  description uh in a turing machine and  that's going to be allowed as code um  okay so what the way we're going to do  this is  uh  the what the recursion theorem does for  you is it says  um you can write  a piece of code  here let's call it piece of turing  machine code uh algorithm algorithmic  code called t  and t is going to get transformed into a  new machine r  and r is going to get provided  a copy of the program itself which is  just a description of r for free  but otherwise it's going to act exactly  like t so r is going to act exactly like  t except r is going to have provided  a copy of r  and that's what the theorem uh  shows you how to implement  so let me let me just see so for any  machine t there is a machine r  which for any w which is going to be the  input to  the input for r  uh r on w operates the same as t on the  input with w where it's given where it's  given r  so r is going to be  getting access to r without um it's  going to be configuring  obtaining r by by calculating it  maybe it'll be clear from the from the  proof uh i always struggle with how to  explain this clearly  so now the proof of this is going to be  very much like the proof uh from two  slides back uh except there's going to  be three parts  there's the the part t that you're going  to provide  and the t is going to be the turing  machine code that says get your own  description  um and you don't have to worry about how  that happens  uh  the compiler is going to add on the a  and b parts which is going to get the  whole description of the whole thing  which is r  and feed it into t  as if t  you know uh t had it as an input  so it's t is going to be allowed to get  its own description now and operate on  now opera it does its thing you know on  the input w  um  so the way it's going to work so t is  given  uh  a is going to be as before  the description of b now with t  so when a i when a is done  it's going to  produce bt sitting on the tape next to  the w  um  b is going to now figure out what a was  from the bt sitting on the tape  okay  and then it's going to combine that to  get abt which is r  and after that it passes control to t  um with  with w and now r sitting on the tape and  now t the  t is going to have its own description  but don't forget now t has been modified  to be r so it's not that t is going to  get t on the tape it's going to for this  to make sense this is this is going to  be now the new machine r and r is  getting is appears on the tape and now  the code that you provided t is going to  get is going to be now going to get to  operate on that  okay  if you didn't get that i'm not so  i'm not  uh worried um the main thing is that you  can use compute your own description  when you're describing turing machines  that's what this uh thing is uh telling  you  um  i think it'll be  oh there's a check in here  um  yeah so i don't know  let's let's do this one kind of quickly  here  can we use the recursion theorem to  design a  machine t  which instead of producing its own  description  um  um  recognizer recognizes accepts only only  its own description as an input  so the the language of this machine is  going to be simply the one string the  description of t  so can we make it a machine t which  does this no i and now i'm looking at  this checking  this t here is confusing with that t  it's not the same t  that's bad  um i should call it m  design a machine m where l of m is the  description of m  um  and can we use the recursion theorem to  do that  you know what i would ask you to do is  think about it in this context  you can  use  compute your own description when you're  writing the code for this machine  if you could do that could you make a  machine which just accepts strings which  happen to be their own description this  is supposed to be easy but  i think it ended up being a little bit  more complicated than i wanted  um  [Music]  okay launch polling  make your best guess  i think you will kind of see i was kind  of leading you leading you  leading you uh along the path here  yes so uh  i think you're pretty much all are  getting it maybe a few of you are uh  um  [Music]  unsure  um but anyway  let's just wrap this one up quickly uh  so that we can move on i think you're  pretty much so five seconds  i'm going to end it  so the correct answer is yes as i was  hinting at so let's uh i think maybe  this example would have been better  after the next example and then we're  going to have a break um  so here is a new proof that atm is  undecidable  but now using the recursion theorem  and this is going to give you a nice  example of how we use the recursion  theorem  in action  okay so remember  we spent  you know half a lecture  or more  with a proof by diagonalization as our  first example of an undecidable problem  atm we you know subsequently showed  other things undecidable by re  uh by reduction but for the first very  first example we use that  diagonalization  um now i'm going to give you a new proof  okay so proof by contradiction assume we  have a turing machine that decides h  that decides atm it starts the same way  that the diagonalization proof went  but now i'm going to make a new a  machine  called r  so this is going to be diff this is  different from the earlier perf  r says on input w  i get my own description use the  recursion theorem that's the way these  things always start  now  i'm going to use h  now that i know my just my own  description  i'm going to feed  i can ask h  i can feed rw w was the input here i can  feed rw into h  to determine whether r accepts w that's  what h does  h will tell solving atm  h will tell  um  this tell this machine whether r accepts  w  r is the machine we're writing however  that is the machine that that is the  machine where we're currently uh that's  currently running  so h  uh r now uses h  and it knows what it's supposed to do  h is going to say well you're going to  accept w or you're not going to accept w  that's what h is assumed to be able to  do  but then what a what is r going to do  after that  r is going to do the opposite of what h  says it's going to do  so if r  h says r accepts w  then r is going to reject w  um  if  h says r  doesn't accept w it you know rejects it  by looping or whole thing doesn't matter  h just says it rejects  then uh r  it's  just  you know um it's just going to accept  so whatever h says r is going to show  that h is wrong so that's a  contradiction  it says that h cannot be  um deciding atm  so you know if you  step back and think about what this is  here that's that whole diagonalization  proof you know in in in one line  basically we've done that proof uh in a  different way though there is some  similarity here i don't want to say this  we've totally reinvented things and done  it totally differently but it's kind of  in some ways sort of gets at the essence  of the diagonalization in a certain  sense um but anyway it gives you kind of  a new very short proof that atm is  undecidable so i think that's kind of a  cool thing um so why don't we um  uh  take our little coffee break here and  you can feel free to ask questions  during the break  um i will start my timer going  and this will be back continuing lecture  in five minutes  okay so questions so we're getting some  questions on  when i said we don't have to worry about  the quotes um when we're making the  uh  solving uh problem six on the pset uh  somebody says can we just say print abc  instead of uh  print quote abc and it'll print abc yes  you can don't worry about the quotes if  you know  uh i think it's kind of  you'll see a challenge if you want to  try to make get the quotes to work but  it's also kind of a pain so um yeah you  can just kind of ignore the quotes and  i'll ask the graders to to mark that to  mark that for marks there so don't uh  that that's any reasonable  interpretation  will as long as you get the rest of the  concept right we'll be fine  um  let's see uh  somebody is asking in the recursion  theorem why doesn't t get the  description of t instead of the  description of r  because  we're producing the machine that's  running is r  back uh back in the previous uh slide  two slides ago  so if i if r got t it would not be  getting its own description be getting  the description of some other machine so  you need to think about what what we  actually need to have happen here in the  proof of the recursion theorem but it  needs to have r not t um  uh let's just see if i can  why does r do the opposite of what h  says  um  uh  why does r do the opposite of what says  well first of all i'm i'm the one who  gets to design r  so here's the co arc  you know we're assuming we have h i'm  going to design r  to do uh anything i want to satisfy the  proof  and r here is  designed to do the the um  [Music]  the opposite of h  so i'm asking r i'm programming r  to find out what it's what h predicts it  will do and then do the opposite you  know you know i maybe the situation is  sort of uh like this suppose um somebody  says uh  um you know i have a crystal ball  which is going to be like the role of  age he said oh really that's that's kind  of cool um i don't believe you but uh it  still sounds interesting um  and the person says yeah i can see the  future um  uh i you know i know what you're going  to do in five minutes um  uh and in fact i i i can see  that in five minutes you're gonna say  uh hello  and you can say well you know well i'm  you can think to yourself well this  person is nuts i'm not going to  do that i'm just not going to say hello  and then you know the  the genie there with the with the  crystal ball you know waits five minutes  and you don't say hello and then you've  proven that the crystal ball doesn't  work  it's very similar  um  okay uh  okay i'm not going to explain how we can  do the combining itself that's just it's  you know it i'm i just want to explain  at a high level uh  that's just going to be messy so  because i said we're somehow going to  combine and self  let me leave that as a conceptual level  um  okay how does this idea work for turing  machines  i don't really understand that question  how does this work idea work for turing  machines are decidable you'll have to  resend that because i don't understand  the question  um  can i explain use your own description  so when i when you write code that says  get your own description  um  after that code executes  the turing machine has  a appears on the tape like magic  um the description of  its own code  you say sitting next to the input to the  machine if because the machine may have  an is separate input from that  so the machine just magically gets its  own code and the the proof of the  recursion theorem implements that  so it's not magic after all  um  uh  i still don't understand the question  um  so for problem six is it enough to  attach code if you're yeah you can  attach code for problem set six problem  six um  and um  that's that's good enough and or if you  can explain it's also good well you like  usual do we worry about tabs and new  lines no  um  [Music]  uh  okay we're gonna have to defer the rest  of the questions don't forget we have a  zillion questions here which i didn't  get to uh  okay last last one here why does  programming artist do the opposite of h  is that a contradiction  well h is predicting that r accepts but  r doesn't accept so h is wrong  we're a little short on time let me skip  this one  uh you can look at this on your own  i mean this is proving sort of the cool  fact  that if you have a i'll just say a high  level if you have a  a program transformation  so if i have some method of transforming  one program to another program but it's  done by algorithm so an algorithmic way  that transforms one program to another  program there's always going to be some  program whose behavior is unchanged by  the transformation  that's called the fixed point theorem um  so there's some program whose behavior  doesn't change no matter how you try to  transform programs easy proof using the  recursion theorem um  you know you can  look at the slide  offline separately  you know if you'd like to see how that  goes it's pretty simple um  uh  here's another exercise uh from of the  recursion theorem um  um  so if i have a um  uh  uh  uh  let's say a minute  a turing machine is minimal  if um its description  is the shortest  among all turing machines  which behave the way it does which were  equivalent to it  um  you know  you know  when i was an undergraduate we had to uh  you know we had i took a programming  class  and some of us sort of enjoyed writing  short programs to uh  uh to carry out the exercises  um  you know probably these days that's  forbidden because it just encourages bad  programming style but anyway  um and  so  uh you know you kind of won if you found  the shortest solution for a given  programming exercise you know it was  heap sword i remember it was one of the  ones that we had to do  um  so  um  uh  you know this is sort of similar you  know you might you might imagine if  people try to find the shortest possible  universal turing machine  so shortest in our in our sense is in  terms of whatever encoding method we  have in mind  a machine is minimal if there is no  shorter program which does exactly has  which is equivalent using our descript  using our encoding uh system whatever it  is okay  so anything m is minimal if anything  that's  that has a shorter description has a  different language  um  okay so let's look at the uh collection  um of all  descriptions of minimal turing machines  and i'm going to prove that that  language is unrecognizable  i'm going to do it using the recursion  theorem it's kind of a cool exercise um  and you can actually use it to prove  something more powerful but but let's  focus on this  this theorem for for now  um  so  assume we have and it's also a little  nice exercise about enumerators um i  don't know how comfortable you ever got  with the numerators but uh let's i'm  trying to prove this language here is  not recognizable and so remember  enumerators  you can enumerate exactly all the  recognizable languages so i'm going to  assume i have an enumerator for this  language which just prints out all of  the  um  minimal touring machines so i have in  some enumerator it's a program that  prints out  the descriptions of all of the minimal  the shortest possible turing machines  and now i'm going to get a contradiction  so  we're going to take build this turing  machine r which gets its own description  and then it's going to start the  enumerator  um  until  so the it's not looking at the strings  that the numerator produces so this  enumerator is producing these minimal  touring machines one after the next  chunk chunk  chunk all these minimal touring machines  are coming out  and you keep looking at those until you  find one of them which is bigger than  yourself  and how do you know how big you are  yourself  from the description you're given by the  recursion there  so you can't keep up printing out these  uh turing machines until you find one  that's bigger than yourself  and then what do you do with that you  simulate that  so  um  now so  so what well the point is is that you're  going to be smaller  than that machine that you're simulating  because you waited to find a machine  that the enumerator produced which is  bigger than you  so you're going to assume you're going  to be simulating that machine that's  bigger than you  and so you're going to be doing exactly  what that machine does on every input  because you're always going to be  simulating that same machine on every  input  and so you're going to be equivalent to  that bigger machine  but that bigger machine  is supposed to be minimum because e is  producing it  but here you are exhibiting a machine  that's smaller than that um that's  allegedly minimal machine it couldn't be  minimal that's the contradiction  so  the language of r this machine i just  produced equals the language of b  because rs ends up simulating b  um  but r is smaller than b because  r waited until it found a bigger machine  that's bigger than it so b couldn't be  minimal  but b was one of the machines that the  enumerator produced that's a  contradiction  um  so let me uh  do a checking on this i i expect this is  going to cause some of you heartburn but  but let's let's do the best you can  um  uh  suppose i have this collection of  minimal touring machines and i take some  infinite subset of that  so and now i'm not demanding that i have  all of the minimal touring machines i  just have infinitely many minimal  touring machines  is it possible that subset whatever it  is  could be touring recognizable  now think about that  now you can you can have  languages which are not touring  recognizable that have  infinite touring recognizable subsets  could that be for this language  and maybe i'll give you um  it will be helpful to you to understand  and perhaps apply  the methodology that i gave you in this  proof here  and that might be helpful to you  but i can see  you know this is not uh this one is is a  bit of a struggle  okay let's end it  okay um  two seconds  just pick something  all right so the majority has the uh  correct answer here  um  um  so in fact  this proof  uh  would still work  if uh the enumerator was enumerating an  infinite subset  of minimal turing machines  because all you need is to wait until  one that's bigger than your appears  and that's you know  all that r needs to do is wait until one  that's bigger than r appears which will  certainly happen eventually if the  subset is infinite  and then r simulates that that simulates  that bigger machine and acts the same  way  thereby proving it could not be minimal  um  so it's exactly the same proof shows  that the answer here is no  um and it's a kind of a curiosity it's  not necessarily that easy to construct  languages  which not only are they not recognizable  but they have no recognizable subsets  infinite subsets obviously a finite  subset is going to be recognizable  because it would be decidable  as well  okay so let's anyway let's move on  some other applications  so first  uh a real world application  um  uh  somebody's asking for an example  of a language  uh of a recognizable uh subset  of a non-recognizable  language so starting out with something  which is not recognizable  and  um  um  uh  what can we come up with a quick  um  so  here's  um i don't know if this will help you so  i the question is can i give an example  of a non-recognizable language that has  a recognizable subset  uh so  i didn't prepare this but  let me see if this is uh helps helps you  know so let's take atm complement we  already showed that atm complement is  not recognizable  right so these are  the the sets of  pairs m and w where m does not accept w  okay  um  so  if i focus only on those m's  which are finite automata  which are a subclass of turing machines  then i can get the answer for those m's  so for the infinitely many cases where  the turing machine never writes on its  tape  i can it's even becomes decidable  okay so i don't know if that's helpful  um  but uh you can definitely find  cases where there are  undecidable languages that have  decidable subsets unrecognizable  languages that have recognizable  infinite subsets um but there's one  example where it's not true in the  previous slide  um  okay a lot of questions coming up here  yes  i'm seeing some other proposals here  if you take atm  complement and you union it with  just any all strings of ones just one  star assuming that one star  just strings the ones they're never  going to code for a turing machine um  that's still going to be unrecognizable  but then you can just throw away all of  those  those just strings of ones and you're  going to get an infinite subset oh wait  a minute  still unrecognizable that's not good  um  oh no yeah  i threw away the wrong stuff you throw  away the um  the descriptions of the turing machines  and you just have the one star strings  left  and so that becomes decidable even  regular  anyway i think i'm not sure i'm helping  you uh let's move on to other  applications so this is kind of a cure  curious application um  that actually is a real world where a  machine might want to get a copy of  itself and then do something perhaps  even  uh nefarious  with that uh with a copy of its own code  and that would be a computer virus  computer viruses up to you know  make copies of themselves  and then propagate them through the  internet or whatever uh media you have  to infect other computers and you know  i'm sure we all know how  you know about computer viruses um well  i mean they need to get copies of  themselves  in order to do that uh  to do the infection  um how do they do that  um  you know many of them operate in a way  where you know in in a  either in a language or or in a system  where they can make reference to their  own code either by looking at the um you  know the uh the the  um  the machine code or whatever uh direct  direct access to their own code there  are languages which allow and systems  which allow for that  but i'm not an expert on computer  viruses i would be shocked if they're  not some other  uh viruses that get access to their own  code by using something um you know in  the in the same s i using basically the  method of the recursion theorem this you  know i haven't done a systematic study  but i'm sure if you can't access to your  own code directly using some operating  system uh mechanism some primitive the  only other way is basically doing the  method that we just described  okay  so another application  um  is  in  a branch of mathematics called  mathematical logic  um  where i imagine that many of you have  heard of the work of girdle from the  earlier part of the 20th century where  they show that um  you can come up it's possible to  demonstrate  that there are true mathematical  statements but which cannot be proven to  be true  um  so proof  uh there might some be something like  maybe even you know questions of  interest to us you know like the p  versus np question which we will some at  some point uh look at in in a few weeks  um actually yeah maybe two or two two  weeks from now um  many unsolved mathematical problems and  you know people's wonder  you know maybe there's just no way to  prove them one way or the other  so um  in the 1930s when girl did his work he  shocked the mathematical community by  showing that there are truth that proof  it does not work for everything there  may be things that are true that you  cannot prove um  uh  you know hilbert in particular you know  from hilbert's 10th problem he was  dismayed by this result uh  because he believed that you know he had  earlier believed that anything that was  true you could prove um  uh so anyway let's let's just see how  let me i'm gonna sketch  how we actually go about doing that um  because we now have a kind of enough  technique to at least give you an idea  of how do you demonstrate that there are  true but unprovable statements and  actually even exhibit one  okay mathematical logic is the  mathematical study of mathematics itself  um  girdle's first incompleteness theorem as  we described is that in any reasonable  formal system of uh mathematical  probability there are going to be some  true statements that are not provable  um  and in order to sort of get this  sketch of the proof i shouldn't say  proof here we're gonna proof sketch  um  we're gonna basically use two  um properties of formal proof systems um  one is that  kind of the  kind of obvious property that you would  expect all provability systems to have  is that you can only prove true things  so if something is being proven  it's going to be true you can't prove  you can't prove anything false  if you if you can prove false things  your system of proof ability is bad  um  and the other thing is that um  proofs are checkable by machine  so if you write down your  your do your system in a formal way and  you have this formal notion of proofs  which underlies all mathematical  reasoning by the way this is completely  well accepted both of these properties  are accepted by mathematicians then in  principle you can convert any  mathematical proof  into a form that you can check it by  computer it might become much longer but  in principle you can put the proof into  a form where uh um a computer could  check the proof and the way we're going  to use  uh frame that you know in the way we've  been just talking about things is that  the language of all pairs of proof comma  statement being proved  so where the pi is a proof of the  statement fee  um that's a decidable thing so you can  check  by machine  whether pi is a valid proof of fee  your your proof checker  can say yes it's valid no not valid and  that's something you can do by algorithm  okay so those are the two assumptions  that we're going to make about our  system of proofs and that's all we're  going to need  um now the first conclusion which is i  think you know a good sort of little bit  exercise on the on  the kind of thinking we've been doing in  this course  number two checkability implies that the  set of provable statements  is recognizable  why  suppose i give you a um a statement that  has a proof  a provable statement  i'm not saying a true statement  necessarily that's going to be  you know  perhaps a larger class of statements but  the statements that do have proofs  that's a recognizable language  um  because you know if i give you a  statement  your recognizer is going to take that  statement and start looking through all  possible  proofs  it's going to look at string after  string as a candidate proof one after  the next some strings of course most  strings are just going to be junk but  every once in a while a proof is going  to come out that's going to be a valid  uh a string which is a valid proof of  something and then you're going to check  oh let me just see if that's a valid  proof and if it does prove the statement  i have in mind and if it is then i  accept that statement  and i go through statement by you know  is  the input is a  mathematical statement and that's going  to be accepted if the machine by looking  through all possible proofs finds one  and then it accepts that statement so  the collection of all statements that  have a proof that's recognizable  so similarly  if um  you can  if you take statements of the form  m and w  is not in  is in the complement of atm so m doesn't  accept w  if you take all statements of that form  where m doesn't accept so m and m  doesn't accept w or m w is in the  complement of atm  um  some of those statements may have proofs  so in your system some of them may not  have purse  if all of them had proofs  if you can prove every statement of this  form when it's true when it happens to  be true  obviously you can't prove the ones that  are not true but if you can prove all  the true verb statements of this kind  then atm complement would be turing  recognizable  because you can go through  you can just  uh you know for the same reason as above  but we know that that's false  so there must be some statement of this  kind which is  true  but does not have a proof because  otherwise atm complement would be  recognizable  um  okay so we've actually done the first  half  of um  of girdle's uh incompleteness theorem  the second half which we're going to  unfortunately  not have time to finish but let me just  give you the outline  is that we can use the recursion theorem  to give it specific see  what we showed here is that there is  some statement of this form which is  unprovable which is true but unprovable  doesn't exhibit a particular one  now the recursion theorem allows you to  give a particular one um  and  uh the the one it basically implements  girdles the cold so-called girdle  statement or girdle sentence that says  this statement is unprovable  um  and you can frame formalize that  um  uh precisely  and then that statement becomes true but  unprovable  let's let's just say why that is  because if the statement  were false  suppose this statement were false  then well then it would be provable  because the truth the truth says it's  not provable but if the statement were  provable then it had to be true  which would mean it would be unprovable  so the only viable outcome here is that  the statement is unprovable  um  and which is  it's therefore  that  that it's true that it's unprovable so  that this is a true statement but then  it has no proof  and  uh let me not go through it um because  again we're  unfortunately ran short on time but uh  you can implement this um using uh the  recursion theorem to make a to make a  particular machine r  where you cannot prove  um that r does not accept uh say some  string zero  um  so you can find a particular r using the  recursion theorem where it's impossible  to prove that r doesn't accept zero  um  even though by construction r does not  accept zero  uh so it's a little bit slippery there  because you have to understand what we  mean by proof within the formal system  versus versus our external form of  reasoning but  taking us a little bit far afield  uh so for those of you who care um  i hope this little digression was  interesting  as i mentioned at the beginning uh for  those of you don't care you don't have  to worry about it it's not going to be  on the midterm or on the final exam  you're not going to be responsible for  this last you know five minutes or so of  the lecture but i thought it's kind of  an interesting application of the  recursion theorem to a problem outside  of computer science in mathematical  logic  um  okay uh so uh  here's the entire the reasoning here  again i invite you to look at that um on  the slide uh  you know that i posted if you're curious  um  so anyway quick review of today is we  went through the self-reference and the  recursion theorem we gave a few  applications  and we did a sketch of girdle's first  incompleteness theorem in mathematical  logic  okay so that's all i'm going to have for  you today we're out of time  um and  i will um  take any questions  so getting back to the min turing  machine example somebody's asking how do  i know there's a turing machine that's  longer than the machine r that i'm  building  well there are infinitely many machines  in mintm or in the infinite subset of  minty m so eventually one of them is  going to have to be longer than the  machine that i'm machine that i'm  constructing because that's going to  that's a machine of some very specific  size  and so eventually there's going to have  to be one that's bigger that shows up  so this may be a similar question  give  now another similar question is about  how big is r uh and does r the size of r  in that previous thing here uh  i don't know if we want to go through  this but okay let's let's quickly look  at it  this machine r  has a fixed predetermined size  um its size does not depend on  b it depends on e  because it's going to be simulating e  but you know e might be a norm producing  very very long strings eventually it  will  so e is fixed  and then um  you know uh  and then r's size is fixed so eventually  r will find a machine that's bigger than  it is  um  let's let's look at  right  so this is a this is a good question i  was wondering if i would get questions  of this kind this is getting back to the  question in logic here at the end um  uh  yes because you know i said this  statement here is unprovable  but in a sense i proved it because how  do i know it's  it's true  and i'm gonna i gave you um you know an  argument for it and so you have to  differentiate between  uh the reasoning that we're providing  and the formal system that we're  reasoning about  and the formal system that we're  reasoning about is not capable of  proving this but we're outside that  formal system so that we can reason  about the formal system i know it's a  little bit perverse seeming um and  mathematical logic is a little tricky  because it has to deal with those kinds  of issues but um you know this is  arguing within any particular formal  system of provability  um  you know this is going to be true um and  but you know  uh  that's kind of an approximation to our  own thought process uh so it's it's it's  it's still it's slippery i i agree  um  so good question here would girdles  theorem still hold in formal systems  where we don't require that proofs of  statements are decidable so i'm not  saying that the proofs are decidable is  that you can prove  checkable so you can test whether a  proof is a proof  if you can't test whether a proof is a  proof i don't know of any people have  studied that that situation so that's a  little bit  of a trickier  trickier case i'm not sure what to say  about that um  can i give an example of two equivalent  turing machines where one has a shorter  description than the other  how do we define the length of this of  the description  well  we never really precisely defined our  encoding  system  but whatever encoding system you want to  use  is going to have it's going to  you know  represent turing machines as strings  and those strings are going to have a  length and so  it doesn't really matter which encoding  system you're going to use because this  statement is going to be true in any of  them  you know we could go through the  exercise of defining a particular  encoding system it's going to be pretty  tedious to do that but you can just  imagine  writing down the states the  transition function  you know etc etc etc as this big long  string  and that's going to be our encoding  system and then are going to be some  long machines some machines will have  long representations on other machines  that have short representations and you  know if there's going to be some machine  where you're going to introduce you can  introduce a bunch of useless states that  are never accessed so you can  you know expand  you you can kind of add  junk to the description of the machine  which is not going to do anything but  it's just going to make the  description of the machine unnecessarily  long  compared to what with with some other  description which is going to  compute the same thing but will be much  shorter  so you can certainly find  examples of pairs of the machines  pairs of machines that do the same thing  where one is much longer than the other  um  so i will then uh close the session here  and um  i will be uh very shortly on um on the  office hours link um  and uh see some of you there  you

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uh so  we are  welcome back everybody  [Music]  and  we are  going to continue our discussion of um  computability theory  turing machines  and uh how to prove things undecidable  um  which is what we've been doing um so we  talked about this more advanced method  of proving things undecidable last  lecture called the computation history  method  which comes up in all sorts of  proofs of undecidability and usually  more uh  complex ones  such as for example the proof of  hilbert's tenth problem um  that i mentioned whether you want to if  you want to test  whether a polynomial has a solution  in integers  um  it's a reduction from atm just like  we've been doing all along those are the  all of those proofs are pretty much  reduction from something undecidable  this reduction from  you know from atm is a good as good a  starting point as any  and in um  and it uses the computation history  method so what the end of doing is given  a turing machine and an input you  construct a polynomial  that has several variables  and where  the  in order to get a an integer root an  integer solution of that polynomial um  one of the variables is going to have to  be assigned to some kind of encoding of  a computation history  of the turing machine of that of m on w  it's going to be some one of those  variables is going to be a computation  history  um an integer  which represents the computation history  for m on w um  and  uh the other variables are there to help  you kind of decode that  so that the polynomial can actually  check  and make a solution it becomes a  solution if that actually is a  legitimate computation history of m w so  it really uses the very same method that  we've been using all along but it's uh  it's just a lot it's  it's pretty hairy to construct that  polynomial and do the check  in the way that you need to do  so for the post correspondence problem  which we introduced last time it's  doing the check is relatively simple um  you know that the uh  the match is the is the computation  history and you know following the rules  of the match um is fairly simple to  construct that  uh post-correspondence problem instance  we talked about linearly down to  automata  you know we of course we define  configurations and and computation  histories along the way and proved some  certain problem other problems are  undecidable as well using the same  method  okay um  so today uh we're gonna shift gears  we're gonna in our last lecture on the  computability section of the course  we're going to talk about  something called the recursion theorem  which basically  gives turing machines the ability to  refer to themselves turing machines in  any program to do self-reference um so  that you can actually get at the code  of the turing machine or the code of the  program that you're writing um  even if that's not a built-in primitive  of the  programming language or the operating  system that you're working with it still  gives you that access  um and also we're gonna if we have time  at the end i'm gonna talk a little bit  about mathematical logic which is sort  of a nice application  of the recursion theorem and it's you  know it's a beautiful subject on its own  and it's something that i can give a  brief introduction to okay so today's  topic  is  um  about  self-reference  self-reproducing machines  and the broader topic called the  recursion theorem  um so let me introduce it with the what  i i would call the self-reproduction  paradox  and that is  suppose you have a factory  like a tesla effect or a car  manufacturing uh factory  um  [Music]  here's a picture  uh  of the factory and it's producing cars  all right so we have a factory that  makes cars  and um  what can we say about  the relative complexity  of the cars  compared with the factory in some  informal sense  um  so i would argue that you  would be reasonable to say  that  the  complexity of the factory is going to  have to be greater  than the complexity of the cars  that it makes  because not only does the factory have  to know how to make the cars so it has  to have all the instructions and  whatever things that go into a car it  has to be included in the you know it's  at least in some kind of  has been  in some sense represented in the factory  but the factory also has to have  other stuff  you know the robots and the machine and  the other manufacturing items uh tools  and so on for making the cars  so the factory has to have all the  complexity of a car incorporated plus  other things as well and on that for  that reason one could imagine that the  factory complexity is more than the  car's complexity  but now suppose you want to have a  factory  that makes factories  sort of imagine here's the picture  um  so or in general a machine that makes  copies of itself  well that seems at first glance  to be  uh impossible  um  because  not only does the factory obviously have  to have all of the instructions for what  a factory is like but it needs to have  all of the compo all of the the the  extra things that it would need to do  the manufacturing  and so for that reason it seems like a  um it's not possible  to have a machine make copies of itself  i mean you would run into the very  same problem  if i asked you to produce  a program and what in your favorite  language um  that prints out itself an exact copy of  the same code  i mean you can always write a program  which is going to print out some string  you know like hello world  that's easy because you just put hello  world into  you know some kind of a variable or some  sort of a um you know a table into the  program and say print that table but if  you want to make a print  if you want the program to print out a  copy of itself  you can't take the whole program and  stick that into a table because  um  you know the the program's gonna have to  be bigger than the table  you know and so you're gonna you're  gonna end up with some something  impossible happening because the program  an entire copy of the program can't fit  inside the program  um you just get uh you know cut the  program inside itself inside itself  inside itself forever um and so you get  we end up with like an infinite program  that way  uh so  if you know you just kind of  naively approach the problem for how to  make a program which is going to print  out a copy of itself  it's not so easy to do um  but hopefully  after today's lecture you will see that  it is possible and in fact how to do it  um  and  not only that is an idle  bit curious bit of curiosity but there  are actually applications for why you  might want to do that  mainly within mathematics um and in uh  computer science theory but there's even  kind of a real world application if you  will in a way too  so we'll get to that at the end  um so it seems as i'm saying impossible  to have a self-reproducing machine but  we know that  in the world there are things that make  copies of themselves  living things  um  so  uh it seems like a paradox no cells can  make copies exactly of themselves all  living things can make copies of  themselves um so how do they manage to  get around this paradox  well in fact  there's no paradox  because it is possible to make a machine  that self-reproduces that makes copies  of itself this has been known for  you know  for many years  probably it goes back to von neumann um  who wrote a famous paper on  self-reproducing machines  okay so self-reproducing machines are in  fact possible  [Music]  so let me give you an example of how you  would make a self-reproducing turing  machine  what do i mean by that  i'm going to i mean a machine i'm going  to call it self  which  ignores its input so on any input  you turn it on  machine grinds around for a while and  halts  with a description of itself on the tape  with the description of self its own  code  sitting on the table so very much like  producing a program  which would  uh print out its own code  that's really what we're doing  um  so for that we're going to first need a  little lemma which is a very simple  lemma but it looks more it looks worse  than it is  um  so uh let me just read it out to you and  then i'll explain what it's saying  because what it's  saying is extremely simple  so there's a computable function i'll  call it q that maps strings to strings  which will take any string w  and produce from w a turing machine  which will print w  okay that's all it does so  as you know if i give you a string w you  could produce a turing machine which  would have w  represented  in the um  states and transitions of the machine  so that if you turn the machine on the  machine will be will output w  i mean if i want you to put if you i  want you to give me a turing machine  that predict prints the string  1101 on the tape you could do that i  hope um  and i'm and if no matter what the string  was instead of one one zero zero or one  you know whatever it's you know  20 zeros and then five ones  you could do that too  and in fact um there's a simple  simple procedure that takes a string  and maps that onto a turing machine  which prints out that string  so that's that's a  computable function  um which basically takes a string and  converts it to something that evaluates  to that string  okay and um  i'm calling it q i don't know if this is  helpful to you or not it's kind of like  it converts the string w to w in quotes  so q stands for quotes  in a way  okay uh so if that's helpful maybe  um  then good um  but anyway p sub w is a turing machine  when you turn it on  it just prints out w and holds  okay and i can find piece of w from w  um  straightforward proof  okay so now i'm going to tell you  no  assuming that we have that  that  computable function q  um  i'm going to tell you how to make this  machine self  um and it's not complicated  um  the turing machine  um  a self is going to have two parts i'll  call a and b  here's a here's a schematic for the  machine  so here's self it's in two parts a and b  and what i mean by a and b it's like two  subroutines  uh self is going to start out running a  and when a is finished it's going to  pass control to b  and then b is going to finish the job  and when it's done you're going to have  the description self sitting here on the  tape  all right  so my what's left is to give you the  code for a and for b  okay um  [Music]  so a  is going to be something  super simple  um  a is going to be that machine  which prints out  b  prints out a description of b  um the one that i described up here  so remember p  sub w is the machine which prints out w  and p sub the description of b is simply  going to machine that has this string p  sub b  stored in its states and transitions you  turn on that machine and it prints out b  and then it's done  so this is a very simple image a is very  simple so here just to  here p sub b  um  is a part of a  and when it's done  b appears on the tape so that's at the  point when a has finished its work  now it's going to pass control to b  so we're not obviously done yet because  what we want is a and b both to be on  the tape not just b because self is a  combination of a and b together  um  so  i have to tell you how b  works to finish the job  um  so you might think  um as a first  you know given what we did for for  getting b on the tape that we'll uh get  a on the tape in the same way by putting  a copy of a inside b  so a copy of b is inside a and a copy of  a is inside b  um  and you know at some conceptual level  that seems like that might do the job  but it's really not that is not a  solution  because the fact that i can put b inside  a  um kind of forbids me from also putting  a inside b because that's going to be  the same kind of circular reasoning of  just putting a machine inside itself you  just can't do that because you're going  to end up with an infinitely big machine  that way  you know in fact if i'm putting b inside  a  a copy of b  in terms of its description inside a a  is really going to be much bigger than b  because it has all of b with inside it  plus other stuff  you know the states and transitions for  printing that  uh b out  so i can't have a b much bigger than b  and then b also much bigger than a so  something this is this is no good  we're going to have to do something else  so how is b now going to get a hold of a  um  and the uh  uh  the trick for doing that  without having a copy of a inside b  which we're not gonna which doesn't work  that's not going to that's not going to  be a good solution instead of the way  that b is going to get a it's going to  figure out what it's going to figure out  what a is  and how is it going to figure out what a  is  because if you remember  um  b can look and now look it can look at  the tape it sees some string there which  happens to be a description of itself  but it doesn't care it sees some string  on the tape  a is the machine that prints out that  string  a  is q  of this string  so b is simply going to compute q of  whatever it's whatever it sees on the  tape  that is a  okay so it's i don't know if you can  read this kind of small here  it's going to compute a from b sitting  on the tape so here is the instructions  for b it's going to compute q of the  tape contents which happens to be  the description of b but that's  irrelevant to b  b just sees some string on the tape it  computes q of that and that is the that  is a  because a is the machine which prints  out that string  then it's going to combine  a with b  doing whatever you know slight  interfacing that needs to happen i'm not  going to get into those details  to convert those two pieces into one  machine  which is the machine self  and then it's going to print out with  self on the tape as i'm going to  indicate over here  okay  so that's how  a turing machine  can print out a copy of its own  description  and leave it on the tape  so  and what's nice about this is nothing  specific about turing machines this is a  general  procedure  that allows any programming language  to print out a copy of its own code you  can even carry this out in english  as i'm going to do in the next slide  um  so here's a good question here there are  many possible turing machines that can  print out b  um that's right how do i know how do i  get the the particular one  what i have in mind that's that's a  little bit of a subtle question but but  that but it's a good question  um  i have in mind the the particular turing  machine that prints out b which is the  one that q produces  remember so we you have to refer back to  this lemma  this lemma produces a particular machine  that prints out b  from  from b  and that's the one that i'm going to use  for a and that's the one that b is going  to be able to obtain by running the q  algorithm  to figure out what a is so you have to  make sure you're being consistent it's a  little bit of a detail but but it's a  good question um  and why doesn't this create a circular  argument too well there is no certain so  that was another question i'm seeing  here on the on the tape well there's  there's no longer anything see b  does not have to have a stored within it  it figures out a  so in a sense you're going to write the  program the code for b first  b is just a simple program here it is  there's nothing circular about it says b  is a simple the code for b is take a  look at the tape compute q of  that um combine the result with whatever  was sitting on the tape from before and  print it out  i mean that's a piece of code which you  can just write i mean this will become  more clear hopefully  in our next slide where we talk about  the english implementation  but just just i don't want to rush to  that  so you know you could figure out b  without even knowing what a is b stands  alone  but then  you know because b is just a piece of  code that runs q  based on what it's what it sees on the  tape a  now you need to know what b is in order  to obtain a  because a has the code for b embedded  within a  as a string  so first you produce b  then you can figure out then you can  obtain a there's nothing circular in  this argument i don't know if that's  helpful to you but  you may need to mull it over or maybe  it'll be helpful from the next slide so  let's go there now  okay so as i'm saying you can implement  this in any language in particular you  can implement it in english  okay so let's just shift gears let's  talk about writing down english  instructions  and then i'll show um you know what  happens if you carry out those  instructions so  let's  start simple how about the sentence  hello write hello world  so you know um an obedient person  reading those instructions um would then  write hello world  um on their paper or or wherever  um  okay  fine you get the i hopefully you get you  get the idea um  uh so now what i'd like here's an  here is a  um  another  sentence  another instruction write the sentence  um  and  the uh  the reader uh the obedient reader would  then  okay write the sentence  this is the kind of thing i'm looking  for  here is a sentence which instructs  the reader to make an exact copy  but i don't like this one  even though it does in a sense what i'm  looking for  because  um it it kind of cheats  when it the when it has this  this refers to the entire sentence  itself  it's using  kind of implicitly here  a construct which a turing machine does  not have  the turing machine does not have  a way of accessing its own code  um  and in fact  really what the point of this uh whole  uh theorem that we're going to present  is that you don't need to have this  self-reference as a primitive  you can get that effectively  uh using um  the  procedure that i'm describing which will  give you the same effect so you don't  need it as a primitive you can you can  design some  software basically which will give you  the same effect  um  so let me show you how to do that in  english  so let's do uh uh  let's look at a slightly more oops so  cheating so turing machines don't have  the self-reference primitive so let's  look at another sentence here  write the following twice  the second time in quotes  and then hello world  so what do we get if we follow those  that instruction  um  well you get  hello world and then hello world again  now in quotes  okay  hopefully not too bad  but now i'm one step away for doing the  implementation of the self-reproducing  algorithm  in english  write the following twice the second  time in  quotes and then in quotes the same text  now following those instructions you get  well  write the following twice the second  time in quotes so that comes out here  and the second time you put it in quotes  just like you did with the hello world  but that's exactly  what  um  here the  the output is exactly what the input was  and  there's you know even though there is a  part of the  sentence here which refers to a  different part  so here the the the first part is  referring to the second part  never do i have to have the sense  referring to its entirety  you know uh  there's no part of the sentence here  that's pointing at the entire sentence  um  and so uh this here  manages to get the effect that i'm  looking for where the output is equal to  the the code um but doesn't have the  avoids having the self-reference  is in so it's one thing is referring to  something else but not referring back to  itself  okay um  so uh let me just see here  so here is sort of i'm sort of i'm gonna  have a  check in on this uh in a minute so i'm  gonna try to contrast what's happening  here with what's happening in that  turing machine itself that i had from  the previous slide  okay so a bunch of mulders over and i'm  going to give you a check in to see  this is a little bit of a challenging  check-in um but let's see if you can uh  figure weight your way through it  okay and basically it says so that  really what we're doing here is called  the recursion theorem as you'll see  we'll actually present the recursion  theorem formally on the next slide  um  but  here  in both of these cases we we kind of  have a template part and an action part  both in both cases we have there are two  parts to the instructions the template  and the action  okay so  i'm going to leave it to you to try to  imagine  which of those which is which  in each of these two examples  um  and then i'm going to ask you to pick um  in the turing machine which is the  action which is a template  and in the sentence which is the action  and which is the template the action is  the part that uh where there's some  interesting  sort of  stuff instructional stuff happening that  you have to carry out the template is  really basically just text  or just code just just uh just a string  okay so  let me uh pull up that poll  [Music]  see what you think  because i'm asking you now to to  indicate where is the action part in  both of those cases  what is the upper phrase and lower  phrase i mean of the sentence here so  write the following twice this is the  upper phrase and the part in quotations  is the lower phrase sorry  okay almost done here  five seconds  a few of you have not answered yet  answer it  one second to go  okay here we go  ending polling  so the uh the majority here is correct  um  uh  you know in i would say in the um  english sentence here  uh the action part is the first part  that's where you actually have  being directed to do something  the second half of the sentence the  lower part of the sentence is just a  template  written you know this is just some  string here there's no action really  being directed it happens to be the same  as the top but this could have been just  hello world this could have been  anything  and then the upper part acts on that so  the upper part is the action part so  it's the upper phrase that's the  relevant part now in the turing machine  it's kind of the uh in a sense it's the  other way around  the first part is really just the  template this the second part b  is is where you're doing some actually  work on the template um you're taking  that that uh  basically text which could be anything a  could be anything and your um  uh uh figure take looking at that  template and reconstructing what a was  from the from that uh string that  appears on the tape so b is actually one  the one that's doing the work so it's b  and the upper phrase that's part c is  correct  okay  um so let us uh  continue then um  [Music]  uh  oh  i want to mention here problem six on  the pset um  so you know your job really is to  implement this in  you know if you have a programming  language and you lot that you like you  can you you can it could be  python or whatever your favorite java  whatever you like you can implement this  if you don't know any programming  languages then just make up some sort of  pseudo programming language and  implement it there  let me point out that  um getting the quoting right  um  uh  it is a bit of a pain  um  because you have to kind of escape the  quotes and so on  i'm not going to be fussy about that so  you can still get full credit even if  you don't get the quoting part quite  correct do your best  um i think it's an interesting problem  uh to try to solve and if you you know  if you struggle with it for a while um  you know  you know it's slippery it's the kind of  thing you have to you know  uh you can easily spend a couple of  hours on this problem uh  because it's uh it's it's it's a bit  tricky to manage to make a program which  prints itself out  which is what the task of problem six is  on this on the pset but don't fuss about  too much on the quoting if that's the  only thing that's hanging you up try to  get the main structure of it which is  fairly simple actually um  and if you can't get the quoting part to  work you know i'll ask the graders not  to uh  penalize you for that part  um  uh  okay  uh  let's let let's then look at the sort of  the more formal version of what we've  just done and really kind of also taking  it to the next level  because being able to print out a copy  of yourself is um  is kind of a curiosity in a way  but you know the recursion theorem says  not only can you make a machine which  prints out a copy of itself but you can  actually make a machine which can obtain  a copy of itself and then do some  computation on that copy of itself that  on the on its own description which  actually turns out to be a useful thing  to do  uh once you have access to your own  description as we'll see from some  examples then you can do some perhaps  some interesting things with that  okay um  so basically what what what the  recursion theorem is a kind of a  compiler  it allows you to  have a new primitive when you're writing  turing machines which is compute your  own description  and the recursion theorem will implement  that for you  okay  so the technical form of the um  recursion theorem is going to look a  little bit um  counterintuitive perhaps uh let me put  it out there  if you if you struggle with a little bit  with the slide  don't sweat it the main thing to  remember and we'll see from examples is  that you can compute your own  description uh in a turing machine and  that's going to be allowed as code um  okay so what the way we're going to do  this is  uh  the what the recursion theorem does for  you is it says  um you can write  a piece of code  here let's call it piece of turing  machine code uh algorithm algorithmic  code called t  and t is going to get transformed into a  new machine r  and r is going to get provided  a copy of the program itself which is  just a description of r for free  but otherwise it's going to act exactly  like t so r is going to act exactly like  t except r is going to have provided  a copy of r  and that's what the theorem uh  shows you how to implement  so let me let me just see so for any  machine t there is a machine r  which for any w which is going to be the  input to  the input for r  uh r on w operates the same as t on the  input with w where it's given where it's  given r  so r is going to be  getting access to r without um it's  going to be configuring  obtaining r by by calculating it  maybe it'll be clear from the from the  proof uh i always struggle with how to  explain this clearly  so now the proof of this is going to be  very much like the proof uh from two  slides back uh except there's going to  be three parts  there's the the part t that you're going  to provide  and the t is going to be the turing  machine code that says get your own  description  um and you don't have to worry about how  that happens  uh  the compiler is going to add on the a  and b parts which is going to get the  whole description of the whole thing  which is r  and feed it into t  as if t  you know uh t had it as an input  so it's t is going to be allowed to get  its own description now and operate on  now opera it does its thing you know on  the input w  um  so the way it's going to work so t is  given  uh  a is going to be as before  the description of b now with t  so when a i when a is done  it's going to  produce bt sitting on the tape next to  the w  um  b is going to now figure out what a was  from the bt sitting on the tape  okay  and then it's going to combine that to  get abt which is r  and after that it passes control to t  um with  with w and now r sitting on the tape and  now t the  t is going to have its own description  but don't forget now t has been modified  to be r so it's not that t is going to  get t on the tape it's going to for this  to make sense this is this is going to  be now the new machine r and r is  getting is appears on the tape and now  the code that you provided t is going to  get is going to be now going to get to  operate on that  okay  if you didn't get that i'm not so  i'm not  uh worried um the main thing is that you  can use compute your own description  when you're describing turing machines  that's what this uh thing is uh telling  you  um  i think it'll be  oh there's a check in here  um  yeah so i don't know  let's let's do this one kind of quickly  here  can we use the recursion theorem to  design a  machine t  which instead of producing its own  description  um  um  recognizer recognizes accepts only only  its own description as an input  so the the language of this machine is  going to be simply the one string the  description of t  so can we make it a machine t which  does this no i and now i'm looking at  this checking  this t here is confusing with that t  it's not the same t  that's bad  um i should call it m  design a machine m where l of m is the  description of m  um  and can we use the recursion theorem to  do that  you know what i would ask you to do is  think about it in this context  you can  use  compute your own description when you're  writing the code for this machine  if you could do that could you make a  machine which just accepts strings which  happen to be their own description this  is supposed to be easy but  i think it ended up being a little bit  more complicated than i wanted  um  [Music]  okay launch polling  make your best guess  i think you will kind of see i was kind  of leading you leading you  leading you uh along the path here  yes so uh  i think you're pretty much all are  getting it maybe a few of you are uh  um  [Music]  unsure  um but anyway  let's just wrap this one up quickly uh  so that we can move on i think you're  pretty much so five seconds  i'm going to end it  so the correct answer is yes as i was  hinting at so let's uh i think maybe  this example would have been better  after the next example and then we're  going to have a break um  so here is a new proof that atm is  undecidable  but now using the recursion theorem  and this is going to give you a nice  example of how we use the recursion  theorem  in action  okay so remember  we spent  you know half a lecture  or more  with a proof by diagonalization as our  first example of an undecidable problem  atm we you know subsequently showed  other things undecidable by re  uh by reduction but for the first very  first example we use that  diagonalization  um now i'm going to give you a new proof  okay so proof by contradiction assume we  have a turing machine that decides h  that decides atm it starts the same way  that the diagonalization proof went  but now i'm going to make a new a  machine  called r  so this is going to be diff this is  different from the earlier perf  r says on input w  i get my own description use the  recursion theorem that's the way these  things always start  now  i'm going to use h  now that i know my just my own  description  i'm going to feed  i can ask h  i can feed rw w was the input here i can  feed rw into h  to determine whether r accepts w that's  what h does  h will tell solving atm  h will tell  um  this tell this machine whether r accepts  w  r is the machine we're writing however  that is the machine that that is the  machine where we're currently uh that's  currently running  so h  uh r now uses h  and it knows what it's supposed to do  h is going to say well you're going to  accept w or you're not going to accept w  that's what h is assumed to be able to  do  but then what a what is r going to do  after that  r is going to do the opposite of what h  says it's going to do  so if r  h says r accepts w  then r is going to reject w  um  if  h says r  doesn't accept w it you know rejects it  by looping or whole thing doesn't matter  h just says it rejects  then uh r  it's  just  you know um it's just going to accept  so whatever h says r is going to show  that h is wrong so that's a  contradiction  it says that h cannot be  um deciding atm  so you know if you  step back and think about what this is  here that's that whole diagonalization  proof you know in in in one line  basically we've done that proof uh in a  different way though there is some  similarity here i don't want to say this  we've totally reinvented things and done  it totally differently but it's kind of  in some ways sort of gets at the essence  of the diagonalization in a certain  sense um but anyway it gives you kind of  a new very short proof that atm is  undecidable so i think that's kind of a  cool thing um so why don't we um  uh  take our little coffee break here and  you can feel free to ask questions  during the break  um i will start my timer going  and this will be back continuing lecture  in five minutes  okay so questions so we're getting some  questions on  when i said we don't have to worry about  the quotes um when we're making the  uh  solving uh problem six on the pset uh  somebody says can we just say print abc  instead of uh  print quote abc and it'll print abc yes  you can don't worry about the quotes if  you know  uh i think it's kind of  you'll see a challenge if you want to  try to make get the quotes to work but  it's also kind of a pain so um yeah you  can just kind of ignore the quotes and  i'll ask the graders to to mark that to  mark that for marks there so don't uh  that that's any reasonable  interpretation  will as long as you get the rest of the  concept right we'll be fine  um  let's see uh  somebody is asking in the recursion  theorem why doesn't t get the  description of t instead of the  description of r  because  we're producing the machine that's  running is r  back uh back in the previous uh slide  two slides ago  so if i if r got t it would not be  getting its own description be getting  the description of some other machine so  you need to think about what what we  actually need to have happen here in the  proof of the recursion theorem but it  needs to have r not t um  uh let's just see if i can  why does r do the opposite of what h  says  um  uh  why does r do the opposite of what says  well first of all i'm i'm the one who  gets to design r  so here's the co arc  you know we're assuming we have h i'm  going to design r  to do uh anything i want to satisfy the  proof  and r here is  designed to do the the um  [Music]  the opposite of h  so i'm asking r i'm programming r  to find out what it's what h predicts it  will do and then do the opposite you  know you know i maybe the situation is  sort of uh like this suppose um somebody  says uh  um you know i have a crystal ball  which is going to be like the role of  age he said oh really that's that's kind  of cool um i don't believe you but uh it  still sounds interesting um  and the person says yeah i can see the  future um  uh i you know i know what you're going  to do in five minutes um  uh and in fact i i i can see  that in five minutes you're gonna say  uh hello  and you can say well you know well i'm  you can think to yourself well this  person is nuts i'm not going to  do that i'm just not going to say hello  and then you know the  the genie there with the with the  crystal ball you know waits five minutes  and you don't say hello and then you've  proven that the crystal ball doesn't  work  it's very similar  um  okay uh  okay i'm not going to explain how we can  do the combining itself that's just it's  you know it i'm i just want to explain  at a high level uh  that's just going to be messy so  because i said we're somehow going to  combine and self  let me leave that as a conceptual level  um  okay how does this idea work for turing  machines  i don't really understand that question  how does this work idea work for turing  machines are decidable you'll have to  resend that because i don't understand  the question  um  can i explain use your own description  so when i when you write code that says  get your own description  um  after that code executes  the turing machine has  a appears on the tape like magic  um the description of  its own code  you say sitting next to the input to the  machine if because the machine may have  an is separate input from that  so the machine just magically gets its  own code and the the proof of the  recursion theorem implements that  so it's not magic after all  um  uh  i still don't understand the question  um  so for problem six is it enough to  attach code if you're yeah you can  attach code for problem set six problem  six um  and um  that's that's good enough and or if you  can explain it's also good well you like  usual do we worry about tabs and new  lines no  um  [Music]  uh  okay we're gonna have to defer the rest  of the questions don't forget we have a  zillion questions here which i didn't  get to uh  okay last last one here why does  programming artist do the opposite of h  is that a contradiction  well h is predicting that r accepts but  r doesn't accept so h is wrong  we're a little short on time let me skip  this one  uh you can look at this on your own  i mean this is proving sort of the cool  fact  that if you have a i'll just say a high  level if you have a  a program transformation  so if i have some method of transforming  one program to another program but it's  done by algorithm so an algorithmic way  that transforms one program to another  program there's always going to be some  program whose behavior is unchanged by  the transformation  that's called the fixed point theorem um  so there's some program whose behavior  doesn't change no matter how you try to  transform programs easy proof using the  recursion theorem um  you know you can  look at the slide  offline separately  you know if you'd like to see how that  goes it's pretty simple um  uh  here's another exercise uh from of the  recursion theorem um  um  so if i have a um  uh  uh  uh  let's say a minute  a turing machine is minimal  if um its description  is the shortest  among all turing machines  which behave the way it does which were  equivalent to it  um  you know  you know  when i was an undergraduate we had to uh  you know we had i took a programming  class  and some of us sort of enjoyed writing  short programs to uh  uh to carry out the exercises  um  you know probably these days that's  forbidden because it just encourages bad  programming style but anyway  um and  so  uh you know you kind of won if you found  the shortest solution for a given  programming exercise you know it was  heap sword i remember it was one of the  ones that we had to do  um  so  um  uh  you know this is sort of similar you  know you might you might imagine if  people try to find the shortest possible  universal turing machine  so shortest in our in our sense is in  terms of whatever encoding method we  have in mind  a machine is minimal if there is no  shorter program which does exactly has  which is equivalent using our descript  using our encoding uh system whatever it  is okay  so anything m is minimal if anything  that's  that has a shorter description has a  different language  um  okay so let's look at the uh collection  um of all  descriptions of minimal turing machines  and i'm going to prove that that  language is unrecognizable  i'm going to do it using the recursion  theorem it's kind of a cool exercise um  and you can actually use it to prove  something more powerful but but let's  focus on this  this theorem for for now  um  so  assume we have and it's also a little  nice exercise about enumerators um i  don't know how comfortable you ever got  with the numerators but uh let's i'm  trying to prove this language here is  not recognizable and so remember  enumerators  you can enumerate exactly all the  recognizable languages so i'm going to  assume i have an enumerator for this  language which just prints out all of  the  um  minimal touring machines so i have in  some enumerator it's a program that  prints out  the descriptions of all of the minimal  the shortest possible turing machines  and now i'm going to get a contradiction  so  we're going to take build this turing  machine r which gets its own description  and then it's going to start the  enumerator  um  until  so the it's not looking at the strings  that the numerator produces so this  enumerator is producing these minimal  touring machines one after the next  chunk chunk  chunk all these minimal touring machines  are coming out  and you keep looking at those until you  find one of them which is bigger than  yourself  and how do you know how big you are  yourself  from the description you're given by the  recursion there  so you can't keep up printing out these  uh turing machines until you find one  that's bigger than yourself  and then what do you do with that you  simulate that  so  um  now so  so what well the point is is that you're  going to be smaller  than that machine that you're simulating  because you waited to find a machine  that the enumerator produced which is  bigger than you  so you're going to assume you're going  to be simulating that machine that's  bigger than you  and so you're going to be doing exactly  what that machine does on every input  because you're always going to be  simulating that same machine on every  input  and so you're going to be equivalent to  that bigger machine  but that bigger machine  is supposed to be minimum because e is  producing it  but here you are exhibiting a machine  that's smaller than that um that's  allegedly minimal machine it couldn't be  minimal that's the contradiction  so  the language of r this machine i just  produced equals the language of b  because rs ends up simulating b  um  but r is smaller than b because  r waited until it found a bigger machine  that's bigger than it so b couldn't be  minimal  but b was one of the machines that the  enumerator produced that's a  contradiction  um  so let me uh  do a checking on this i i expect this is  going to cause some of you heartburn but  but let's let's do the best you can  um  uh  suppose i have this collection of  minimal touring machines and i take some  infinite subset of that  so and now i'm not demanding that i have  all of the minimal touring machines i  just have infinitely many minimal  touring machines  is it possible that subset whatever it  is  could be touring recognizable  now think about that  now you can you can have  languages which are not touring  recognizable that have  infinite touring recognizable subsets  could that be for this language  and maybe i'll give you um  it will be helpful to you to understand  and perhaps apply  the methodology that i gave you in this  proof here  and that might be helpful to you  but i can see  you know this is not uh this one is is a  bit of a struggle  okay let's end it  okay um  two seconds  just pick something  all right so the majority has the uh  correct answer here  um  um  so in fact  this proof  uh  would still work  if uh the enumerator was enumerating an  infinite subset  of minimal turing machines  because all you need is to wait until  one that's bigger than your appears  and that's you know  all that r needs to do is wait until one  that's bigger than r appears which will  certainly happen eventually if the  subset is infinite  and then r simulates that that simulates  that bigger machine and acts the same  way  thereby proving it could not be minimal  um  so it's exactly the same proof shows  that the answer here is no  um and it's a kind of a curiosity it's  not necessarily that easy to construct  languages  which not only are they not recognizable  but they have no recognizable subsets  infinite subsets obviously a finite  subset is going to be recognizable  because it would be decidable  as well  okay so let's anyway let's move on  some other applications  so first  uh a real world application  um  uh  somebody's asking for an example  of a language  uh of a recognizable uh subset  of a non-recognizable  language so starting out with something  which is not recognizable  and  um  um  uh  what can we come up with a quick  um  so  here's  um i don't know if this will help you so  i the question is can i give an example  of a non-recognizable language that has  a recognizable subset  uh so  i didn't prepare this but  let me see if this is uh helps helps you  know so let's take atm complement we  already showed that atm complement is  not recognizable  right so these are  the the sets of  pairs m and w where m does not accept w  okay  um  so  if i focus only on those m's  which are finite automata  which are a subclass of turing machines  then i can get the answer for those m's  so for the infinitely many cases where  the turing machine never writes on its  tape  i can it's even becomes decidable  okay so i don't know if that's helpful  um  but uh you can definitely find  cases where there are  undecidable languages that have  decidable subsets unrecognizable  languages that have recognizable  infinite subsets um but there's one  example where it's not true in the  previous slide  um  okay a lot of questions coming up here  yes  i'm seeing some other proposals here  if you take atm  complement and you union it with  just any all strings of ones just one  star assuming that one star  just strings the ones they're never  going to code for a turing machine um  that's still going to be unrecognizable  but then you can just throw away all of  those  those just strings of ones and you're  going to get an infinite subset oh wait  a minute  still unrecognizable that's not good  um  oh no yeah  i threw away the wrong stuff you throw  away the um  the descriptions of the turing machines  and you just have the one star strings  left  and so that becomes decidable even  regular  anyway i think i'm not sure i'm helping  you uh let's move on to other  applications so this is kind of a cure  curious application um  that actually is a real world where a  machine might want to get a copy of  itself and then do something perhaps  even  uh nefarious  with that uh with a copy of its own code  and that would be a computer virus  computer viruses up to you know  make copies of themselves  and then propagate them through the  internet or whatever uh media you have  to infect other computers and you know  i'm sure we all know how  you know about computer viruses um well  i mean they need to get copies of  themselves  in order to do that uh  to do the infection  um how do they do that  um  you know many of them operate in a way  where you know in in a  either in a language or or in a system  where they can make reference to their  own code either by looking at the um you  know the uh the the  um  the machine code or whatever uh direct  direct access to their own code there  are languages which allow and systems  which allow for that  but i'm not an expert on computer  viruses i would be shocked if they're  not some other  uh viruses that get access to their own  code by using something um you know in  the in the same s i using basically the  method of the recursion theorem this you  know i haven't done a systematic study  but i'm sure if you can't access to your  own code directly using some operating  system uh mechanism some primitive the  only other way is basically doing the  method that we just described  okay  so another application  um  is  in  a branch of mathematics called  mathematical logic  um  where i imagine that many of you have  heard of the work of girdle from the  earlier part of the 20th century where  they show that um  you can come up it's possible to  demonstrate  that there are true mathematical  statements but which cannot be proven to  be true  um  so proof  uh there might some be something like  maybe even you know questions of  interest to us you know like the p  versus np question which we will some at  some point uh look at in in a few weeks  um actually yeah maybe two or two two  weeks from now um  many unsolved mathematical problems and  you know people's wonder  you know maybe there's just no way to  prove them one way or the other  so um  in the 1930s when girl did his work he  shocked the mathematical community by  showing that there are truth that proof  it does not work for everything there  may be things that are true that you  cannot prove um  uh  you know hilbert in particular you know  from hilbert's 10th problem he was  dismayed by this result uh  because he believed that you know he had  earlier believed that anything that was  true you could prove um  uh so anyway let's let's just see how  let me i'm gonna sketch  how we actually go about doing that um  because we now have a kind of enough  technique to at least give you an idea  of how do you demonstrate that there are  true but unprovable statements and  actually even exhibit one  okay mathematical logic is the  mathematical study of mathematics itself  um  girdle's first incompleteness theorem as  we described is that in any reasonable  formal system of uh mathematical  probability there are going to be some  true statements that are not provable  um  and in order to sort of get this  sketch of the proof i shouldn't say  proof here we're gonna proof sketch  um  we're gonna basically use two  um properties of formal proof systems um  one is that  kind of the  kind of obvious property that you would  expect all provability systems to have  is that you can only prove true things  so if something is being proven  it's going to be true you can't prove  you can't prove anything false  if you if you can prove false things  your system of proof ability is bad  um  and the other thing is that um  proofs are checkable by machine  so if you write down your  your do your system in a formal way and  you have this formal notion of proofs  which underlies all mathematical  reasoning by the way this is completely  well accepted both of these properties  are accepted by mathematicians then in  principle you can convert any  mathematical proof  into a form that you can check it by  computer it might become much longer but  in principle you can put the proof into  a form where uh um a computer could  check the proof and the way we're going  to use  uh frame that you know in the way we've  been just talking about things is that  the language of all pairs of proof comma  statement being proved  so where the pi is a proof of the  statement fee  um that's a decidable thing so you can  check  by machine  whether pi is a valid proof of fee  your your proof checker  can say yes it's valid no not valid and  that's something you can do by algorithm  okay so those are the two assumptions  that we're going to make about our  system of proofs and that's all we're  going to need  um now the first conclusion which is i  think you know a good sort of little bit  exercise on the on  the kind of thinking we've been doing in  this course  number two checkability implies that the  set of provable statements  is recognizable  why  suppose i give you a um a statement that  has a proof  a provable statement  i'm not saying a true statement  necessarily that's going to be  you know  perhaps a larger class of statements but  the statements that do have proofs  that's a recognizable language  um  because you know if i give you a  statement  your recognizer is going to take that  statement and start looking through all  possible  proofs  it's going to look at string after  string as a candidate proof one after  the next some strings of course most  strings are just going to be junk but  every once in a while a proof is going  to come out that's going to be a valid  uh a string which is a valid proof of  something and then you're going to check  oh let me just see if that's a valid  proof and if it does prove the statement  i have in mind and if it is then i  accept that statement  and i go through statement by you know  is  the input is a  mathematical statement and that's going  to be accepted if the machine by looking  through all possible proofs finds one  and then it accepts that statement so  the collection of all statements that  have a proof that's recognizable  so similarly  if um  you can  if you take statements of the form  m and w  is not in  is in the complement of atm so m doesn't  accept w  if you take all statements of that form  where m doesn't accept so m and m  doesn't accept w or m w is in the  complement of atm  um  some of those statements may have proofs  so in your system some of them may not  have purse  if all of them had proofs  if you can prove every statement of this  form when it's true when it happens to  be true  obviously you can't prove the ones that  are not true but if you can prove all  the true verb statements of this kind  then atm complement would be turing  recognizable  because you can go through  you can just  uh you know for the same reason as above  but we know that that's false  so there must be some statement of this  kind which is  true  but does not have a proof because  otherwise atm complement would be  recognizable  um  okay so we've actually done the first  half  of um  of girdle's uh incompleteness theorem  the second half which we're going to  unfortunately  not have time to finish but let me just  give you the outline  is that we can use the recursion theorem  to give it specific see  what we showed here is that there is  some statement of this form which is  unprovable which is true but unprovable  doesn't exhibit a particular one  now the recursion theorem allows you to  give a particular one um  and  uh the the one it basically implements  girdles the cold so-called girdle  statement or girdle sentence that says  this statement is unprovable  um  and you can frame formalize that  um  uh precisely  and then that statement becomes true but  unprovable  let's let's just say why that is  because if the statement  were false  suppose this statement were false  then well then it would be provable  because the truth the truth says it's  not provable but if the statement were  provable then it had to be true  which would mean it would be unprovable  so the only viable outcome here is that  the statement is unprovable  um  and which is  it's therefore  that  that it's true that it's unprovable so  that this is a true statement but then  it has no proof  and  uh let me not go through it um because  again we're  unfortunately ran short on time but uh  you can implement this um using uh the  recursion theorem to make a to make a  particular machine r  where you cannot prove  um that r does not accept uh say some  string zero  um  so you can find a particular r using the  recursion theorem where it's impossible  to prove that r doesn't accept zero  um  even though by construction r does not  accept zero  uh so it's a little bit slippery there  because you have to understand what we  mean by proof within the formal system  versus versus our external form of  reasoning but  taking us a little bit far afield  uh so for those of you who care um  i hope this little digression was  interesting  as i mentioned at the beginning uh for  those of you don't care you don't have  to worry about it it's not going to be  on the midterm or on the final exam  you're not going to be responsible for  this last you know five minutes or so of  the lecture but i thought it's kind of  an interesting application of the  recursion theorem to a problem outside  of computer science in mathematical  logic  um  okay uh so uh  here's the entire the reasoning here  again i invite you to look at that um on  the slide uh  you know that i posted if you're curious  um  so anyway quick review of today is we  went through the self-reference and the  recursion theorem we gave a few  applications  and we did a sketch of girdle's first  incompleteness theorem in mathematical  logic  okay so that's all i'm going to have for  you today we're out of time  um and  i will um  take any questions  so getting back to the min turing  machine example somebody's asking how do  i know there's a turing machine that's  longer than the machine r that i'm  building  well there are infinitely many machines  in mintm or in the infinite subset of  minty m so eventually one of them is  going to have to be longer than the  machine that i'm machine that i'm  constructing because that's going to  that's a machine of some very specific  size  and so eventually there's going to have  to be one that's bigger that shows up  so this may be a similar question  give  now another similar question is about  how big is r uh and does r the size of r  in that previous thing here uh  i don't know if we want to go through  this but okay let's let's quickly look  at it  this machine r  has a fixed predetermined size  um its size does not depend on  b it depends on e  because it's going to be simulating e  but you know e might be a norm producing  very very long strings eventually it  will  so e is fixed  and then um  you know uh  and then r's size is fixed so eventually  r will find a machine that's bigger than  it is  um  let's let's look at  right  so this is a this is a good question i  was wondering if i would get questions  of this kind this is getting back to the  question in logic here at the end um  uh  yes because you know i said this  statement here is unprovable  but in a sense i proved it because how  do i know it's  it's true  and i'm gonna i gave you um you know an  argument for it and so you have to  differentiate between  uh the reasoning that we're providing  and the formal system that we're  reasoning about  and the formal system that we're  reasoning about is not capable of  proving this but we're outside that  formal system so that we can reason  about the formal system i know it's a  little bit perverse seeming um and  mathematical logic is a little tricky  because it has to deal with those kinds  of issues but um you know this is  arguing within any particular formal  system of provability  um  you know this is going to be true um and  but you know  uh  that's kind of an approximation to our  own thought process uh so it's it's it's  it's still it's slippery i i agree  um  so good question here would girdles  theorem still hold in formal systems  where we don't require that proofs of  statements are decidable so i'm not  saying that the proofs are decidable is  that you can prove  checkable so you can test whether a  proof is a proof  if you can't test whether a proof is a  proof i don't know of any people have  studied that that situation so that's a  little bit  of a trickier  trickier case i'm not sure what to say  about that um  can i give an example of two equivalent  turing machines where one has a shorter  description than the other  how do we define the length of this of  the description  well  we never really precisely defined our  encoding  system  but whatever encoding system you want to  use  is going to have it's going to  you know  represent turing machines as strings  and those strings are going to have a  length and so  it doesn't really matter which encoding  system you're going to use because this  statement is going to be true in any of  them  you know we could go through the  exercise of defining a particular  encoding system it's going to be pretty  tedious to do that but you can just  imagine  writing down the states the  transition function  you know etc etc etc as this big long  string  and that's going to be our encoding  system and then are going to be some  long machines some machines will have  long representations on other machines  that have short representations and you  know if there's going to be some machine  where you're going to introduce you can  introduce a bunch of useless states that  are never accessed so you can  you know expand  you you can kind of add  junk to the description of the machine  which is not going to do anything but  it's just going to make the  description of the machine unnecessarily  long  compared to what with with some other  description which is going to  compute the same thing but will be much  shorter  so you can certainly find  examples of pairs of the machines  pairs of machines that do the same thing  where one is much longer than the other  um  so i will then uh close the session here  and um  i will be uh very shortly on um on the  office hours link um  and uh see some of you there  you
 

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441
00:14:03,189 --> 00:14:06,629

 

442
00:14:06,629 --> 00:14:07,910

 

443
00:14:07,910 --> 00:14:10,550

 

444
00:14:10,550 --> 00:14:12,949

 

445
00:14:12,949 --> 00:14:14,870

 

446
00:14:14,870 --> 00:14:16,949

 

447
00:14:16,949 --> 00:14:19,269

 

448
00:14:19,269 --> 00:14:21,110

 

449
00:14:21,110 --> 00:14:21,120

 

450
00:14:21,120 --> 00:14:23,030


451
00:14:23,030 --> 00:14:25,110

 

452
00:14:25,110 --> 00:14:28,870

 

453
00:14:28,870 --> 00:14:30,949

 

454
00:14:30,949 --> 00:14:34,230

 

455
00:14:34,230 --> 00:14:37,110

 

456
00:14:37,110 --> 00:14:39,430

 

457
00:14:39,430 --> 00:14:41,750

 

458
00:14:41,750 --> 00:14:43,829

 

459
00:14:43,829 --> 00:14:47,430

 

460
00:14:47,430 --> 00:14:47,440

 

461
00:14:47,440 --> 00:14:48,790


462
00:14:48,790 --> 00:14:48,800

 

463
00:14:48,800 --> 00:14:49,990


464
00:14:49,990 --> 00:14:52,230

 

465
00:14:52,230 --> 00:14:55,990

 

466
00:14:55,990 --> 00:14:56,000

 

467
00:14:56,000 --> 00:14:56,790


468
00:14:56,790 --> 00:14:58,790

 

469
00:14:58,790 --> 00:15:01,189

 

470
00:15:01,189 --> 00:15:03,430

 

471
00:15:03,430 --> 00:15:05,750

 

472
00:15:05,750 --> 00:15:08,069

 

473
00:15:08,069 --> 00:15:14,069

 

474
00:15:14,069 --> 00:15:17,030

 

475
00:15:17,030 --> 00:15:18,710

 

476
00:15:18,710 --> 00:15:18,720

 

477
00:15:18,720 --> 00:15:20,389


478
00:15:20,389 --> 00:15:23,189

 

479
00:15:23,189 --> 00:15:24,790

 

480
00:15:24,790 --> 00:15:26,710

 

481
00:15:26,710 --> 00:15:26,720

 

482
00:15:26,720 --> 00:15:28,310


483
00:15:28,310 --> 00:15:31,110

 

484
00:15:31,110 --> 00:15:31,120

 

485
00:15:31,120 --> 00:15:31,910


486
00:15:31,910 --> 00:15:33,990

 

487
00:15:33,990 --> 00:15:35,590

 

488
00:15:35,590 --> 00:15:37,910

 

489
00:15:37,910 --> 00:15:40,230

 

490
00:15:40,230 --> 00:15:41,430

 

491
00:15:41,430 --> 00:15:42,949

 

492
00:15:42,949 --> 00:15:45,670

 

493
00:15:45,670 --> 00:15:48,550

 

494
00:15:48,550 --> 00:15:48,560

 

495
00:15:48,560 --> 00:15:49,990


496
00:15:49,990 --> 00:15:51,590

 

497
00:15:51,590 --> 00:15:53,670

 

498
00:15:53,670 --> 00:15:55,269

 

499
00:15:55,269 --> 00:15:57,670

 

500
00:15:57,670 --> 00:15:58,949

 

501
00:15:58,949 --> 00:16:00,710

 

502
00:16:00,710 --> 00:16:01,829

 

503
00:16:01,829 --> 00:16:03,350

 

504
00:16:03,350 --> 00:16:05,509

 

505
00:16:05,509 --> 00:16:07,670

 

506
00:16:07,670 --> 00:16:11,749

 

507
00:16:11,749 --> 00:16:16,870

 

508
00:16:16,870 --> 00:16:23,350

 

509
00:16:23,350 --> 00:16:23,360

 

510
00:16:23,360 --> 00:16:24,710


511
00:16:24,710 --> 00:16:27,269

 

512
00:16:27,269 --> 00:16:27,279

 

513
00:16:27,279 --> 00:16:28,790


514
00:16:28,790 --> 00:16:30,949

 

515
00:16:30,949 --> 00:16:33,269

 

516
00:16:33,269 --> 00:16:35,110

 

517
00:16:35,110 --> 00:16:36,310

 

518
00:16:36,310 --> 00:16:38,230

 

519
00:16:38,230 --> 00:16:39,990

 

520
00:16:39,990 --> 00:16:42,230

 

521
00:16:42,230 --> 00:16:44,790

 

522
00:16:44,790 --> 00:16:46,310

 

523
00:16:46,310 --> 00:16:46,320

 

524
00:16:46,320 --> 00:16:47,269


525
00:16:47,269 --> 00:16:49,189

 

526
00:16:49,189 --> 00:16:49,199

 

527
00:16:49,199 --> 00:16:51,189


528
00:16:51,189 --> 00:16:52,870

 

529
00:16:52,870 --> 00:16:55,430

 

530
00:16:55,430 --> 00:16:56,949

 

531
00:16:56,949 --> 00:16:58,790

 

532
00:16:58,790 --> 00:17:00,150

 

533
00:17:00,150 --> 00:17:02,389

 

534
00:17:02,389 --> 00:17:02,399

 

535
00:17:02,399 --> 00:17:07,669


536
00:17:07,669 --> 00:17:07,679

 

537
00:17:07,679 --> 00:17:08,390


538
00:17:08,390 --> 00:17:09,669

 

539
00:17:09,669 --> 00:17:15,029

 

540
00:17:15,029 --> 00:17:17,829

 

541
00:17:17,829 --> 00:17:19,429

 

542
00:17:19,429 --> 00:17:19,439

 

543
00:17:19,439 --> 00:17:21,029


544
00:17:21,029 --> 00:17:27,590

 

545
00:17:27,590 --> 00:17:29,430

 

546
00:17:29,430 --> 00:17:31,190

 

547
00:17:31,190 --> 00:17:34,150

 

548
00:17:34,150 --> 00:17:35,909

 

549
00:17:35,909 --> 00:17:38,950

 

550
00:17:38,950 --> 00:17:41,909

 

551
00:17:41,909 --> 00:17:43,190

 

552
00:17:43,190 --> 00:17:44,549

 

553
00:17:44,549 --> 00:17:46,789

 

554
00:17:46,789 --> 00:17:49,669

 

555
00:17:49,669 --> 00:17:51,110

 

556
00:17:51,110 --> 00:17:53,110

 

557
00:17:53,110 --> 00:17:55,750

 

558
00:17:55,750 --> 00:17:57,750

 

559
00:17:57,750 --> 00:17:59,590

 

560
00:17:59,590 --> 00:18:01,750

 

561
00:18:01,750 --> 00:18:03,909

 

562
00:18:03,909 --> 00:18:06,549

 

563
00:18:06,549 --> 00:18:09,190

 

564
00:18:09,190 --> 00:18:09,200

 

565
00:18:09,200 --> 00:18:10,470


566
00:18:10,470 --> 00:18:13,990

 

567
00:18:13,990 --> 00:18:15,350

 

568
00:18:15,350 --> 00:18:16,789

 

569
00:18:16,789 --> 00:18:21,909

 

570
00:18:21,909 --> 00:18:21,919

 

571
00:18:21,919 --> 00:18:22,950


572
00:18:22,950 --> 00:18:24,710

 

573
00:18:24,710 --> 00:18:27,270

 

574
00:18:27,270 --> 00:18:29,190

 

575
00:18:29,190 --> 00:18:29,200

 

576
00:18:29,200 --> 00:18:30,470


577
00:18:30,470 --> 00:18:33,350

 

578
00:18:33,350 --> 00:18:33,360

 

579
00:18:33,360 --> 00:18:34,870


580
00:18:34,870 --> 00:18:36,310

 

581
00:18:36,310 --> 00:18:38,230

 

582
00:18:38,230 --> 00:18:38,240

 

583
00:18:38,240 --> 00:18:39,190


584
00:18:39,190 --> 00:18:39,200

 

585
00:18:39,200 --> 00:18:41,590


586
00:18:41,590 --> 00:18:44,630

 

587
00:18:44,630 --> 00:18:46,549

 

588
00:18:46,549 --> 00:18:49,270

 

589
00:18:49,270 --> 00:18:52,710

 

590
00:18:52,710 --> 00:18:52,720

 

591
00:18:52,720 --> 00:18:56,230


592
00:18:56,230 --> 00:18:58,070

 

593
00:18:58,070 --> 00:18:59,830

 

594
00:18:59,830 --> 00:19:01,270

 

595
00:19:01,270 --> 00:19:02,870

 

596
00:19:02,870 --> 00:19:04,710

 

597
00:19:04,710 --> 00:19:06,150

 

598
00:19:06,150 --> 00:19:07,750

 

599
00:19:07,750 --> 00:19:09,510

 

600
00:19:09,510 --> 00:19:09,520

 

601
00:19:09,520 --> 00:19:12,070


602
00:19:12,070 --> 00:19:15,430

 

603
00:19:15,430 --> 00:19:17,669

 

604
00:19:17,669 --> 00:19:21,909

 

605
00:19:21,909 --> 00:19:23,830

 

606
00:19:23,830 --> 00:19:25,029

 

607
00:19:25,029 --> 00:19:27,750

 

608
00:19:27,750 --> 00:19:29,510

 

609
00:19:29,510 --> 00:19:29,520

 

610
00:19:29,520 --> 00:19:30,630


611
00:19:30,630 --> 00:19:32,150

 

612
00:19:32,150 --> 00:19:34,150

 

613
00:19:34,150 --> 00:19:36,390

 

614
00:19:36,390 --> 00:19:38,870

 

615
00:19:38,870 --> 00:19:38,880

 

616
00:19:38,880 --> 00:19:40,310


617
00:19:40,310 --> 00:19:42,230

 

618
00:19:42,230 --> 00:19:44,310

 

619
00:19:44,310 --> 00:19:45,990

 

620
00:19:45,990 --> 00:19:49,350

 

621
00:19:49,350 --> 00:19:50,950

 

622
00:19:50,950 --> 00:19:52,870

 

623
00:19:52,870 --> 00:19:53,909

 

624
00:19:53,909 --> 00:19:55,909

 

625
00:19:55,909 --> 00:19:58,789

 

626
00:19:58,789 --> 00:20:01,350

 

627
00:20:01,350 --> 00:20:03,190

 

628
00:20:03,190 --> 00:20:05,110

 

629
00:20:05,110 --> 00:20:09,430

 

630
00:20:09,430 --> 00:20:11,350

 

631
00:20:11,350 --> 00:20:13,430

 

632
00:20:13,430 --> 00:20:16,070

 

633
00:20:16,070 --> 00:20:18,830

 

634
00:20:18,830 --> 00:20:22,149

 

635
00:20:22,149 --> 00:20:24,230

 

636
00:20:24,230 --> 00:20:26,710

 

637
00:20:26,710 --> 00:20:28,470

 

638
00:20:28,470 --> 00:20:30,630

 

639
00:20:30,630 --> 00:20:32,310

 

640
00:20:32,310 --> 00:20:33,830

 

641
00:20:33,830 --> 00:20:35,669

 

642
00:20:35,669 --> 00:20:37,909

 

643
00:20:37,909 --> 00:20:37,919

 

644
00:20:37,919 --> 00:20:39,669


645
00:20:39,669 --> 00:20:41,190

 

646
00:20:41,190 --> 00:20:44,549

 

647
00:20:44,549 --> 00:20:44,559

 

648
00:20:44,559 --> 00:20:45,590


649
00:20:45,590 --> 00:20:46,630

 

650
00:20:46,630 --> 00:20:48,230

 

651
00:20:48,230 --> 00:20:50,230

 

652
00:20:50,230 --> 00:20:51,669

 

653
00:20:51,669 --> 00:20:53,669

 

654
00:20:53,669 --> 00:20:56,230

 

655
00:20:56,230 --> 00:20:58,149

 

656
00:20:58,149 --> 00:21:01,990

 

657
00:21:01,990 --> 00:21:03,350

 

658
00:21:03,350 --> 00:21:05,190

 

659
00:21:05,190 --> 00:21:07,270

 

660
00:21:07,270 --> 00:21:08,870

 

661
00:21:08,870 --> 00:21:10,710

 

662
00:21:10,710 --> 00:21:11,909

 

663
00:21:11,909 --> 00:21:13,830

 

664
00:21:13,830 --> 00:21:15,350

 

665
00:21:15,350 --> 00:21:17,190

 

666
00:21:17,190 --> 00:21:19,510

 

667
00:21:19,510 --> 00:21:21,110

 

668
00:21:21,110 --> 00:21:22,950

 

669
00:21:22,950 --> 00:21:25,190

 

670
00:21:25,190 --> 00:21:27,430

 

671
00:21:27,430 --> 00:21:29,750

 

672
00:21:29,750 --> 00:21:29,760

 

673
00:21:29,760 --> 00:21:31,669


674
00:21:31,669 --> 00:21:34,630

 

675
00:21:34,630 --> 00:21:35,990

 

676
00:21:35,990 --> 00:21:38,070

 

677
00:21:38,070 --> 00:21:38,080

 

678
00:21:38,080 --> 00:21:39,029


679
00:21:39,029 --> 00:21:41,430

 

680
00:21:41,430 --> 00:21:45,110

 

681
00:21:45,110 --> 00:21:47,990

 

682
00:21:47,990 --> 00:21:52,630

 

683
00:21:52,630 --> 00:21:54,390

 

684
00:21:54,390 --> 00:21:58,630

 

685
00:21:58,630 --> 00:21:58,640

 

686
00:21:58,640 --> 00:22:00,870


687
00:22:00,870 --> 00:22:00,880

 

688
00:22:00,880 --> 00:22:02,070


689
00:22:02,070 --> 00:22:03,990

 

690
00:22:03,990 --> 00:22:06,310

 

691
00:22:06,310 --> 00:22:09,990

 

692
00:22:09,990 --> 00:22:11,270

 

693
00:22:11,270 --> 00:22:11,280

 

694
00:22:11,280 --> 00:22:12,950


695
00:22:12,950 --> 00:22:12,960

 

696
00:22:12,960 --> 00:22:13,990


697
00:22:13,990 --> 00:22:14,000

 

698
00:22:14,000 --> 00:22:16,070


699
00:22:16,070 --> 00:22:19,830

 

700
00:22:19,830 --> 00:22:19,840

 

701
00:22:19,840 --> 00:22:20,870


702
00:22:20,870 --> 00:22:20,880

 

703
00:22:20,880 --> 00:22:22,070


704
00:22:22,070 --> 00:22:24,470

 

705
00:22:24,470 --> 00:22:27,270

 

706
00:22:27,270 --> 00:22:27,280

 

707
00:22:27,280 --> 00:22:28,710


708
00:22:28,710 --> 00:22:30,710

 

709
00:22:30,710 --> 00:22:32,630

 

710
00:22:32,630 --> 00:22:32,640

 

711
00:22:32,640 --> 00:22:33,909


712
00:22:33,909 --> 00:22:36,950

 

713
00:22:36,950 --> 00:22:40,390

 

714
00:22:40,390 --> 00:22:41,990

 

715
00:22:41,990 --> 00:22:43,510

 

716
00:22:43,510 --> 00:22:44,710

 

717
00:22:44,710 --> 00:22:44,720

 

718
00:22:44,720 --> 00:22:46,230


719
00:22:46,230 --> 00:22:49,190

 

720
00:22:49,190 --> 00:22:52,310

 

721
00:22:52,310 --> 00:22:54,630

 

722
00:22:54,630 --> 00:22:54,640

 

723
00:22:54,640 --> 00:22:55,750


724
00:22:55,750 --> 00:22:57,029

 

725
00:22:57,029 --> 00:22:59,430

 

726
00:22:59,430 --> 00:23:01,830

 

727
00:23:01,830 --> 00:23:03,350

 

728
00:23:03,350 --> 00:23:05,430

 

729
00:23:05,430 --> 00:23:10,950

 

730
00:23:10,950 --> 00:23:10,960

 

731
00:23:10,960 --> 00:23:11,909


732
00:23:11,909 --> 00:23:12,870

 

733
00:23:12,870 --> 00:23:15,110

 

734
00:23:15,110 --> 00:23:16,870

 

735
00:23:16,870 --> 00:23:19,350

 

736
00:23:19,350 --> 00:23:22,549

 

737
00:23:22,549 --> 00:23:24,870

 

738
00:23:24,870 --> 00:23:27,190

 

739
00:23:27,190 --> 00:23:27,200

 

740
00:23:27,200 --> 00:23:28,230


741
00:23:28,230 --> 00:23:30,230

 

742
00:23:30,230 --> 00:23:31,909

 

743
00:23:31,909 --> 00:23:34,070

 

744
00:23:34,070 --> 00:23:35,510

 

745
00:23:35,510 --> 00:23:37,190

 

746
00:23:37,190 --> 00:23:38,950

 

747
00:23:38,950 --> 00:23:38,960

 

748
00:23:38,960 --> 00:23:40,870


749
00:23:40,870 --> 00:23:42,149

 

750
00:23:42,149 --> 00:23:42,159

 

751
00:23:42,159 --> 00:23:43,110


752
00:23:43,110 --> 00:23:45,750

 

753
00:23:45,750 --> 00:23:48,070

 

754
00:23:48,070 --> 00:23:49,990

 

755
00:23:49,990 --> 00:23:51,909

 

756
00:23:51,909 --> 00:23:53,990

 

757
00:23:53,990 --> 00:23:55,750

 

758
00:23:55,750 --> 00:23:58,549

 

759
00:23:58,549 --> 00:24:00,630

 

760
00:24:00,630 --> 00:24:02,710

 

761
00:24:02,710 --> 00:24:04,390

 

762
00:24:04,390 --> 00:24:04,400

 

763
00:24:04,400 --> 00:24:06,070


764
00:24:06,070 --> 00:24:07,269

 

765
00:24:07,269 --> 00:24:10,390

 

766
00:24:10,390 --> 00:24:13,750

 

767
00:24:13,750 --> 00:24:13,760

 

768
00:24:13,760 --> 00:24:15,350


769
00:24:15,350 --> 00:24:17,430

 

770
00:24:17,430 --> 00:24:19,830

 

771
00:24:19,830 --> 00:24:22,870

 

772
00:24:22,870 --> 00:24:22,880

 

773
00:24:22,880 --> 00:24:24,549


774
00:24:24,549 --> 00:24:27,909

 

775
00:24:27,909 --> 00:24:29,990

 

776
00:24:29,990 --> 00:24:30,830

 

777
00:24:30,830 --> 00:24:37,350

 

778
00:24:37,350 --> 00:24:41,510

 

779
00:24:41,510 --> 00:24:41,520

 

780
00:24:41,520 --> 00:24:42,630


781
00:24:42,630 --> 00:24:43,909

 

782
00:24:43,909 --> 00:24:45,909

 

783
00:24:45,909 --> 00:24:48,470

 

784
00:24:48,470 --> 00:24:51,430

 

785
00:24:51,430 --> 00:24:53,430

 

786
00:24:53,430 --> 00:24:53,440

 

787
00:24:53,440 --> 00:24:54,230


788
00:24:54,230 --> 00:24:54,240

 

789
00:24:54,240 --> 00:24:55,350


790
00:24:55,350 --> 00:24:56,789

 

791
00:24:56,789 --> 00:25:00,549

 

792
00:25:00,549 --> 00:25:00,559

 

793
00:25:00,559 --> 00:25:01,430


794
00:25:01,430 --> 00:25:03,669

 

795
00:25:03,669 --> 00:25:04,950

 

796
00:25:04,950 --> 00:25:06,390

 

797
00:25:06,390 --> 00:25:07,669

 

798
00:25:07,669 --> 00:25:09,590

 

799
00:25:09,590 --> 00:25:11,750

 

800
00:25:11,750 --> 00:25:14,149

 

801
00:25:14,149 --> 00:25:16,630

 

802
00:25:16,630 --> 00:25:17,909

 

803
00:25:17,909 --> 00:25:19,029

 

804
00:25:19,029 --> 00:25:21,430

 

805
00:25:21,430 --> 00:25:21,440

 

806
00:25:21,440 --> 00:25:22,470


807
00:25:22,470 --> 00:25:25,350

 

808
00:25:25,350 --> 00:25:27,110

 

809
00:25:27,110 --> 00:25:29,669

 

810
00:25:29,669 --> 00:25:32,630

 

811
00:25:32,630 --> 00:25:34,870

 

812
00:25:34,870 --> 00:25:36,630

 

813
00:25:36,630 --> 00:25:38,630

 

814
00:25:38,630 --> 00:25:38,640

 

815
00:25:38,640 --> 00:25:41,350


816
00:25:41,350 --> 00:25:43,510

 

817
00:25:43,510 --> 00:25:46,950

 

818
00:25:46,950 --> 00:25:50,149

 

819
00:25:50,149 --> 00:25:51,269

 

820
00:25:51,269 --> 00:25:53,269

 

821
00:25:53,269 --> 00:25:55,430

 

822
00:25:55,430 --> 00:25:57,350

 

823
00:25:57,350 --> 00:25:58,950

 

824
00:25:58,950 --> 00:26:01,190

 

825
00:26:01,190 --> 00:26:03,590

 

826
00:26:03,590 --> 00:26:06,789

 

827
00:26:06,789 --> 00:26:08,310

 

828
00:26:08,310 --> 00:26:12,230

 

829
00:26:12,230 --> 00:26:16,789

 

830
00:26:16,789 --> 00:26:18,630

 

831
00:26:18,630 --> 00:26:19,909

 

832
00:26:19,909 --> 00:26:21,350

 

833
00:26:21,350 --> 00:26:22,710

 

834
00:26:22,710 --> 00:26:24,549

 

835
00:26:24,549 --> 00:26:24,559

 

836
00:26:24,559 --> 00:26:25,430


837
00:26:25,430 --> 00:26:25,440

 

838
00:26:25,440 --> 00:26:26,470


839
00:26:26,470 --> 00:26:26,480

 

840
00:26:26,480 --> 00:26:27,350


841
00:26:27,350 --> 00:26:29,830

 

842
00:26:29,830 --> 00:26:33,110

 

843
00:26:33,110 --> 00:26:35,510

 

844
00:26:35,510 --> 00:26:38,470

 

845
00:26:38,470 --> 00:26:40,310

 

846
00:26:40,310 --> 00:26:41,669

 

847
00:26:41,669 --> 00:26:43,190

 

848
00:26:43,190 --> 00:26:43,200

 

849
00:26:43,200 --> 00:26:44,310


850
00:26:44,310 --> 00:26:46,310

 

851
00:26:46,310 --> 00:26:48,470

 

852
00:26:48,470 --> 00:26:48,480

 

853
00:26:48,480 --> 00:26:49,510


854
00:26:49,510 --> 00:26:53,669

 

855
00:26:53,669 --> 00:26:55,029

 

856
00:26:55,029 --> 00:26:56,950

 

857
00:26:56,950 --> 00:26:58,630

 

858
00:26:58,630 --> 00:27:00,230

 

859
00:27:00,230 --> 00:27:03,110

 

860
00:27:03,110 --> 00:27:03,120

 

861
00:27:03,120 --> 00:27:04,310


862
00:27:04,310 --> 00:27:05,510

 

863
00:27:05,510 --> 00:27:08,070

 

864
00:27:08,070 --> 00:27:10,230

 

865
00:27:10,230 --> 00:27:12,470

 

866
00:27:12,470 --> 00:27:16,470

 

867
00:27:16,470 --> 00:27:17,590

 

868
00:27:17,590 --> 00:27:20,830

 

869
00:27:20,830 --> 00:27:20,840

 

870
00:27:20,840 --> 00:27:23,590


871
00:27:23,590 --> 00:27:30,710

 

872
00:27:30,710 --> 00:27:32,870

 

873
00:27:32,870 --> 00:27:35,990

 

874
00:27:35,990 --> 00:27:44,710

 

875
00:27:44,710 --> 00:27:45,909

 

876
00:27:45,909 --> 00:27:47,430

 

877
00:27:47,430 --> 00:27:48,870

 

878
00:27:48,870 --> 00:27:51,590

 

879
00:27:51,590 --> 00:28:00,310

 

880
00:28:00,310 --> 00:28:06,950

 

881
00:28:06,950 --> 00:28:10,389

 

882
00:28:10,389 --> 00:28:12,230

 

883
00:28:12,230 --> 00:28:15,669

 

884
00:28:15,669 --> 00:28:17,190

 

885
00:28:17,190 --> 00:28:18,950

 

886
00:28:18,950 --> 00:28:22,549

 

887
00:28:22,549 --> 00:28:25,110

 

888
00:28:25,110 --> 00:28:25,120

 

889
00:28:25,120 --> 00:28:27,510


890
00:28:27,510 --> 00:28:27,520

 

891
00:28:27,520 --> 00:28:28,470


892
00:28:28,470 --> 00:28:31,269

 

893
00:28:31,269 --> 00:28:33,190

 

894
00:28:33,190 --> 00:28:35,909

 

895
00:28:35,909 --> 00:28:38,389

 

896
00:28:38,389 --> 00:28:40,470

 

897
00:28:40,470 --> 00:28:42,310

 

898
00:28:42,310 --> 00:28:45,190

 

899
00:28:45,190 --> 00:28:45,200

 

900
00:28:45,200 --> 00:28:47,350


901
00:28:47,350 --> 00:28:49,029

 

902
00:28:49,029 --> 00:28:50,789

 

903
00:28:50,789 --> 00:28:52,630

 

904
00:28:52,630 --> 00:28:54,310

 

905
00:28:54,310 --> 00:28:55,510

 

906
00:28:55,510 --> 00:28:55,520

 

907
00:28:55,520 --> 00:28:57,029


908
00:28:57,029 --> 00:28:59,190

 

909
00:28:59,190 --> 00:29:01,909

 

910
00:29:01,909 --> 00:29:03,350

 

911
00:29:03,350 --> 00:29:05,830

 

912
00:29:05,830 --> 00:29:07,830

 

913
00:29:07,830 --> 00:29:09,590

 

914
00:29:09,590 --> 00:29:11,269

 

915
00:29:11,269 --> 00:29:13,990

 

916
00:29:13,990 --> 00:29:16,149

 

917
00:29:16,149 --> 00:29:18,549

 

918
00:29:18,549 --> 00:29:21,110

 

919
00:29:21,110 --> 00:29:23,110

 

920
00:29:23,110 --> 00:29:27,110

 

921
00:29:27,110 --> 00:29:29,190

 

922
00:29:29,190 --> 00:29:31,990

 

923
00:29:31,990 --> 00:29:34,710

 

924
00:29:34,710 --> 00:29:36,710

 

925
00:29:36,710 --> 00:29:38,870

 

926
00:29:38,870 --> 00:29:40,549

 

927
00:29:40,549 --> 00:29:40,559

 

928
00:29:40,559 --> 00:29:41,909


929
00:29:41,909 --> 00:29:41,919

 

930
00:29:41,919 --> 00:29:42,870


931
00:29:42,870 --> 00:29:45,590

 

932
00:29:45,590 --> 00:29:47,350

 

933
00:29:47,350 --> 00:29:47,360

 

934
00:29:47,360 --> 00:29:50,310


935
00:29:50,310 --> 00:29:50,320

 

936
00:29:50,320 --> 00:29:51,990


937
00:29:51,990 --> 00:29:52,000

 

938
00:29:52,000 --> 00:29:52,950


939
00:29:52,950 --> 00:29:55,350

 

940
00:29:55,350 --> 00:29:57,990

 

941
00:29:57,990 --> 00:30:00,070

 

942
00:30:00,070 --> 00:30:01,909

 

943
00:30:01,909 --> 00:30:03,110

 

944
00:30:03,110 --> 00:30:04,630

 

945
00:30:04,630 --> 00:30:06,630

 

946
00:30:06,630 --> 00:30:09,510

 

947
00:30:09,510 --> 00:30:11,830

 

948
00:30:11,830 --> 00:30:13,110

 

949
00:30:13,110 --> 00:30:15,269

 

950
00:30:15,269 --> 00:30:16,549

 

951
00:30:16,549 --> 00:30:18,389

 

952
00:30:18,389 --> 00:30:20,830

 

953
00:30:20,830 --> 00:30:24,230

 

954
00:30:24,230 --> 00:30:24,240

 

955
00:30:24,240 --> 00:30:25,510


956
00:30:25,510 --> 00:30:25,520

 

957
00:30:25,520 --> 00:30:26,549


958
00:30:26,549 --> 00:30:28,389

 

959
00:30:28,389 --> 00:30:28,399

 

960
00:30:28,399 --> 00:30:29,510


961
00:30:29,510 --> 00:30:31,590

 

962
00:30:31,590 --> 00:30:33,430

 

963
00:30:33,430 --> 00:30:35,590

 

964
00:30:35,590 --> 00:30:37,190

 

965
00:30:37,190 --> 00:30:38,950

 

966
00:30:38,950 --> 00:30:41,110

 

967
00:30:41,110 --> 00:30:43,510

 

968
00:30:43,510 --> 00:30:45,830

 

969
00:30:45,830 --> 00:30:48,230

 

970
00:30:48,230 --> 00:30:49,190

 

971
00:30:49,190 --> 00:30:50,950

 

972
00:30:50,950 --> 00:30:53,110

 

973
00:30:53,110 --> 00:30:54,950

 

974
00:30:54,950 --> 00:30:56,950

 

975
00:30:56,950 --> 00:30:59,269

 

976
00:30:59,269 --> 00:31:01,509

 

977
00:31:01,509 --> 00:31:03,029

 

978
00:31:03,029 --> 00:31:04,950

 

979
00:31:04,950 --> 00:31:07,590

 

980
00:31:07,590 --> 00:31:08,950

 

981
00:31:08,950 --> 00:31:10,950

 

982
00:31:10,950 --> 00:31:12,310

 

983
00:31:12,310 --> 00:31:14,549

 

984
00:31:14,549 --> 00:31:16,310

 

985
00:31:16,310 --> 00:31:18,149

 

986
00:31:18,149 --> 00:31:19,590

 

987
00:31:19,590 --> 00:31:21,830

 

988
00:31:21,830 --> 00:31:21,840

 

989
00:31:21,840 --> 00:31:23,029


990
00:31:23,029 --> 00:31:23,039

 

991
00:31:23,039 --> 00:31:24,230


992
00:31:24,230 --> 00:31:24,240

 

993
00:31:24,240 --> 00:31:25,110


994
00:31:25,110 --> 00:31:25,120

 

995
00:31:25,120 --> 00:31:26,470


996
00:31:26,470 --> 00:31:29,029

 

997
00:31:29,029 --> 00:31:32,549

 

998
00:31:32,549 --> 00:31:34,789

 

999
00:31:34,789 --> 00:31:36,789

 

1000
00:31:36,789 --> 00:31:39,509

 

1001
00:31:39,509 --> 00:31:42,470

 

1002
00:31:42,470 --> 00:31:45,430

 

1003
00:31:45,430 --> 00:31:47,190

 

1004
00:31:47,190 --> 00:31:48,870

 

1005
00:31:48,870 --> 00:31:51,110

 

1006
00:31:51,110 --> 00:31:53,909

 

1007
00:31:53,909 --> 00:31:55,909

 

1008
00:31:55,909 --> 00:31:58,950

 

1009
00:31:58,950 --> 00:32:00,950

 

1010
00:32:00,950 --> 00:32:02,470

 

1011
00:32:02,470 --> 00:32:03,509

 

1012
00:32:03,509 --> 00:32:05,509

 

1013
00:32:05,509 --> 00:32:06,950

 

1014
00:32:06,950 --> 00:32:08,870

 

1015
00:32:08,870 --> 00:32:10,710

 

1016
00:32:10,710 --> 00:32:12,230

 

1017
00:32:12,230 --> 00:32:13,830

 

1018
00:32:13,830 --> 00:32:14,950

 

1019
00:32:14,950 --> 00:32:14,960

 

1020
00:32:14,960 --> 00:32:16,149


1021
00:32:16,149 --> 00:32:18,389

 

1022
00:32:18,389 --> 00:32:19,909

 

1023
00:32:19,909 --> 00:32:22,310

 

1024
00:32:22,310 --> 00:32:23,669

 

1025
00:32:23,669 --> 00:32:25,590

 

1026
00:32:25,590 --> 00:32:27,269

 

1027
00:32:27,269 --> 00:32:27,279

 

1028
00:32:27,279 --> 00:32:28,950


1029
00:32:28,950 --> 00:32:31,669

 

1030
00:32:31,669 --> 00:32:33,509

 

1031
00:32:33,509 --> 00:32:36,549

 

1032
00:32:36,549 --> 00:32:39,110

 

1033
00:32:39,110 --> 00:32:40,389

 

1034
00:32:40,389 --> 00:32:42,149

 

1035
00:32:42,149 --> 00:32:43,909

 

1036
00:32:43,909 --> 00:32:45,750

 

1037
00:32:45,750 --> 00:32:48,310

 

1038
00:32:48,310 --> 00:32:49,350

 

1039
00:32:49,350 --> 00:32:51,430

 

1040
00:32:51,430 --> 00:32:54,870

 

1041
00:32:54,870 --> 00:32:57,269

 

1042
00:32:57,269 --> 00:32:58,470

 

1043
00:32:58,470 --> 00:32:58,480

 

1044
00:32:58,480 --> 00:33:01,430


1045
00:33:01,430 --> 00:33:02,950

 

1046
00:33:02,950 --> 00:33:04,549

 

1047
00:33:04,549 --> 00:33:06,470

 

1048
00:33:06,470 --> 00:33:08,230

 

1049
00:33:08,230 --> 00:33:10,230

 

1050
00:33:10,230 --> 00:33:13,029

 

1051
00:33:13,029 --> 00:33:15,909

 

1052
00:33:15,909 --> 00:33:19,110

 

1053
00:33:19,110 --> 00:33:21,669

 

1054
00:33:21,669 --> 00:33:26,149

 

1055
00:33:26,149 --> 00:33:27,990

 

1056
00:33:27,990 --> 00:33:31,029

 

1057
00:33:31,029 --> 00:33:32,950

 

1058
00:33:32,950 --> 00:33:35,750

 

1059
00:33:35,750 --> 00:33:39,509

 

1060
00:33:39,509 --> 00:33:41,430

 

1061
00:33:41,430 --> 00:33:43,430

 

1062
00:33:43,430 --> 00:33:45,110

 

1063
00:33:45,110 --> 00:33:47,110

 

1064
00:33:47,110 --> 00:33:49,430

 

1065
00:33:49,430 --> 00:33:51,350

 

1066
00:33:51,350 --> 00:33:52,950

 

1067
00:33:52,950 --> 00:33:54,710

 

1068
00:33:54,710 --> 00:33:58,789

 

1069
00:33:58,789 --> 00:34:00,630

 

1070
00:34:00,630 --> 00:34:02,070

 

1071
00:34:02,070 --> 00:34:04,070

 

1072
00:34:04,070 --> 00:34:07,029

 

1073
00:34:07,029 --> 00:34:09,589

 

1074
00:34:09,589 --> 00:34:12,550

 

1075
00:34:12,550 --> 00:34:14,230

 

1076
00:34:14,230 --> 00:34:16,310

 

1077
00:34:16,310 --> 00:34:19,109

 

1078
00:34:19,109 --> 00:34:20,550

 

1079
00:34:20,550 --> 00:34:22,389

 

1080
00:34:22,389 --> 00:34:24,710

 

1081
00:34:24,710 --> 00:34:26,069

 

1082
00:34:26,069 --> 00:34:27,990

 

1083
00:34:27,990 --> 00:34:29,270

 

1084
00:34:29,270 --> 00:34:30,710

 

1085
00:34:30,710 --> 00:34:32,550

 

1086
00:34:32,550 --> 00:34:32,560

 

1087
00:34:32,560 --> 00:34:34,310


1088
00:34:34,310 --> 00:34:36,389

 

1089
00:34:36,389 --> 00:34:37,990

 

1090
00:34:37,990 --> 00:34:38,000

 

1091
00:34:38,000 --> 00:34:38,950


1092
00:34:38,950 --> 00:34:40,950

 

1093
00:34:40,950 --> 00:34:42,790

 

1094
00:34:42,790 --> 00:34:44,550

 

1095
00:34:44,550 --> 00:34:45,669

 

1096
00:34:45,669 --> 00:34:47,349

 

1097
00:34:47,349 --> 00:34:48,790

 

1098
00:34:48,790 --> 00:34:51,669

 

1099
00:34:51,669 --> 00:34:53,349

 

1100
00:34:53,349 --> 00:34:56,710

 

1101
00:34:56,710 --> 00:34:59,030

 

1102
00:34:59,030 --> 00:35:00,630

 

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

 

1104
00:35:00,640 --> 00:35:01,430


1105
00:35:01,430 --> 00:35:03,190

 

1106
00:35:03,190 --> 00:35:03,200

 

1107
00:35:03,200 --> 00:35:04,550


1108
00:35:04,550 --> 00:35:04,560

 

1109
00:35:04,560 --> 00:35:06,790


1110
00:35:06,790 --> 00:35:09,349

 

1111
00:35:09,349 --> 00:35:12,069

 

1112
00:35:12,069 --> 00:35:14,550

 

1113
00:35:14,550 --> 00:35:15,510

 

1114
00:35:15,510 --> 00:35:17,589

 

1115
00:35:17,589 --> 00:35:19,430

 

1116
00:35:19,430 --> 00:35:19,440

 

1117
00:35:19,440 --> 00:35:20,710


1118
00:35:20,710 --> 00:35:24,069

 

1119
00:35:24,069 --> 00:35:29,670

 

1120
00:35:29,670 --> 00:35:29,680

 

1121
00:35:29,680 --> 00:35:30,390


1122
00:35:30,390 --> 00:35:31,750

 

1123
00:35:31,750 --> 00:35:35,109

 

1124
00:35:35,109 --> 00:35:37,750

 

1125
00:35:37,750 --> 00:35:39,829

 

1126
00:35:39,829 --> 00:35:42,790

 

1127
00:35:42,790 --> 00:35:45,030

 

1128
00:35:45,030 --> 00:35:47,109

 

1129
00:35:47,109 --> 00:35:48,950

 

1130
00:35:48,950 --> 00:35:51,030

 

1131
00:35:51,030 --> 00:35:53,030

 

1132
00:35:53,030 --> 00:35:54,950

 

1133
00:35:54,950 --> 00:35:57,030

 

1134
00:35:57,030 --> 00:36:00,150

 

1135
00:36:00,150 --> 00:36:01,990

 

1136
00:36:01,990 --> 00:36:04,470

 

1137
00:36:04,470 --> 00:36:06,470

 

1138
00:36:06,470 --> 00:36:06,480

 

1139
00:36:06,480 --> 00:36:07,510


1140
00:36:07,510 --> 00:36:09,430

 

1141
00:36:09,430 --> 00:36:10,310

 

1142
00:36:10,310 --> 00:36:13,030

 

1143
00:36:13,030 --> 00:36:14,550

 

1144
00:36:14,550 --> 00:36:15,910

 

1145
00:36:15,910 --> 00:36:18,310

 

1146
00:36:18,310 --> 00:36:18,320

 

1147
00:36:18,320 --> 00:36:19,190


1148
00:36:19,190 --> 00:36:19,200

 

1149
00:36:19,200 --> 00:36:20,390


1150
00:36:20,390 --> 00:36:21,750

 

1151
00:36:21,750 --> 00:36:24,069

 

1152
00:36:24,069 --> 00:36:24,079

 

1153
00:36:24,079 --> 00:36:25,349


1154
00:36:25,349 --> 00:36:27,270

 

1155
00:36:27,270 --> 00:36:28,790

 

1156
00:36:28,790 --> 00:36:28,800

 

1157
00:36:28,800 --> 00:36:29,589


1158
00:36:29,589 --> 00:36:31,430

 

1159
00:36:31,430 --> 00:36:33,190

 

1160
00:36:33,190 --> 00:36:35,430

 

1161
00:36:35,430 --> 00:36:37,030

 

1162
00:36:37,030 --> 00:36:37,040

 

1163
00:36:37,040 --> 00:36:39,589


1164
00:36:39,589 --> 00:36:39,599

 

1165
00:36:39,599 --> 00:36:40,630


1166
00:36:40,630 --> 00:36:40,640

 

1167
00:36:40,640 --> 00:36:41,589


1168
00:36:41,589 --> 00:36:45,270

 

1169
00:36:45,270 --> 00:36:48,550

 

1170
00:36:48,550 --> 00:36:50,870

 

1171
00:36:50,870 --> 00:36:52,870

 

1172
00:36:52,870 --> 00:36:56,790

 

1173
00:36:56,790 --> 00:36:59,270

 

1174
00:36:59,270 --> 00:37:01,510

 

1175
00:37:01,510 --> 00:37:02,630

 

1176
00:37:02,630 --> 00:37:04,630

 

1177
00:37:04,630 --> 00:37:06,069

 

1178
00:37:06,069 --> 00:37:07,109

 

1179
00:37:07,109 --> 00:37:09,190

 

1180
00:37:09,190 --> 00:37:11,670

 

1181
00:37:11,670 --> 00:37:13,510

 

1182
00:37:13,510 --> 00:37:13,520

 

1183
00:37:13,520 --> 00:37:15,750


1184
00:37:15,750 --> 00:37:17,589

 

1185
00:37:17,589 --> 00:37:18,550

 

1186
00:37:18,550 --> 00:37:20,390

 

1187
00:37:20,390 --> 00:37:23,270

 

1188
00:37:23,270 --> 00:37:24,230

 

1189
00:37:24,230 --> 00:37:24,240

 

1190
00:37:24,240 --> 00:37:25,109


1191
00:37:25,109 --> 00:37:27,349

 

1192
00:37:27,349 --> 00:37:29,990

 

1193
00:37:29,990 --> 00:37:31,750

 

1194
00:37:31,750 --> 00:37:33,910

 

1195
00:37:33,910 --> 00:37:35,990

 

1196
00:37:35,990 --> 00:37:38,310

 

1197
00:37:38,310 --> 00:37:39,589

 

1198
00:37:39,589 --> 00:37:41,670

 

1199
00:37:41,670 --> 00:37:41,680

 

1200
00:37:41,680 --> 00:37:42,020


1201
00:37:42,020 --> 00:37:42,030

 

1202
00:37:42,030 --> 00:37:44,470


1203
00:37:44,470 --> 00:37:47,990

 

1204
00:37:47,990 --> 00:37:51,349

 

1205
00:37:51,349 --> 00:37:53,589

 

1206
00:37:53,589 --> 00:37:55,430

 

1207
00:37:55,430 --> 00:37:58,630

 

1208
00:37:58,630 --> 00:38:00,390

 

1209
00:38:00,390 --> 00:38:02,230

 

1210
00:38:02,230 --> 00:38:06,069

 

1211
00:38:06,069 --> 00:38:06,079

 

1212
00:38:06,079 --> 00:38:08,680


1213
00:38:08,680 --> 00:38:08,690

 

1214
00:38:08,690 --> 00:38:10,550


1215
00:38:10,550 --> 00:38:10,560

 

1216
00:38:10,560 --> 00:38:11,589


1217
00:38:11,589 --> 00:38:13,510

 

1218
00:38:13,510 --> 00:38:15,589

 

1219
00:38:15,589 --> 00:38:16,790

 

1220
00:38:16,790 --> 00:38:20,390

 

1221
00:38:20,390 --> 00:38:23,270

 

1222
00:38:23,270 --> 00:38:25,430

 

1223
00:38:25,430 --> 00:38:28,069

 

1224
00:38:28,069 --> 00:38:29,510

 

1225
00:38:29,510 --> 00:38:31,030

 

1226
00:38:31,030 --> 00:38:34,470

 

1227
00:38:34,470 --> 00:38:37,190

 

1228
00:38:37,190 --> 00:38:37,200

 

1229
00:38:37,200 --> 00:38:38,710


1230
00:38:38,710 --> 00:38:40,870

 

1231
00:38:40,870 --> 00:38:42,390

 

1232
00:38:42,390 --> 00:38:44,230

 

1233
00:38:44,230 --> 00:38:44,240

 

1234
00:38:44,240 --> 00:38:45,270


1235
00:38:45,270 --> 00:38:46,870

 

1236
00:38:46,870 --> 00:38:48,790

 

1237
00:38:48,790 --> 00:38:50,069

 

1238
00:38:50,069 --> 00:38:51,750

 

1239
00:38:51,750 --> 00:38:53,109

 

1240
00:38:53,109 --> 00:38:55,670

 

1241
00:38:55,670 --> 00:38:58,710

 

1242
00:38:58,710 --> 00:39:00,950

 

1243
00:39:00,950 --> 00:39:02,790

 

1244
00:39:02,790 --> 00:39:05,190

 

1245
00:39:05,190 --> 00:39:06,550

 

1246
00:39:06,550 --> 00:39:06,560

 

1247
00:39:06,560 --> 00:39:09,030


1248
00:39:09,030 --> 00:39:13,349

 

1249
00:39:13,349 --> 00:39:15,109

 

1250
00:39:15,109 --> 00:39:16,950

 

1251
00:39:16,950 --> 00:39:19,030

 

1252
00:39:19,030 --> 00:39:21,510

 

1253
00:39:21,510 --> 00:39:22,870

 

1254
00:39:22,870 --> 00:39:22,880

 

1255
00:39:22,880 --> 00:39:23,910


1256
00:39:23,910 --> 00:39:25,349

 

1257
00:39:25,349 --> 00:39:26,790

 

1258
00:39:26,790 --> 00:39:29,109

 

1259
00:39:29,109 --> 00:39:31,270

 

1260
00:39:31,270 --> 00:39:33,190

 

1261
00:39:33,190 --> 00:39:34,950

 

1262
00:39:34,950 --> 00:39:36,630

 

1263
00:39:36,630 --> 00:39:36,640

 

1264
00:39:36,640 --> 00:39:37,349


1265
00:39:37,349 --> 00:39:39,829

 

1266
00:39:39,829 --> 00:39:41,349

 

1267
00:39:41,349 --> 00:39:41,359

 

1268
00:39:41,359 --> 00:39:42,710


1269
00:39:42,710 --> 00:39:45,990

 

1270
00:39:45,990 --> 00:39:48,150

 

1271
00:39:48,150 --> 00:39:51,510

 

1272
00:39:51,510 --> 00:39:53,750

 

1273
00:39:53,750 --> 00:39:56,630

 

1274
00:39:56,630 --> 00:39:58,150

 

1275
00:39:58,150 --> 00:40:00,710

 

1276
00:40:00,710 --> 00:40:01,990

 

1277
00:40:01,990 --> 00:40:02,000

 

1278
00:40:02,000 --> 00:40:03,510


1279
00:40:03,510 --> 00:40:06,390

 

1280
00:40:06,390 --> 00:40:06,400

 

1281
00:40:06,400 --> 00:40:07,990


1282
00:40:07,990 --> 00:40:10,870

 

1283
00:40:10,870 --> 00:40:12,069

 

1284
00:40:12,069 --> 00:40:14,390

 

1285
00:40:14,390 --> 00:40:15,750

 

1286
00:40:15,750 --> 00:40:17,589

 

1287
00:40:17,589 --> 00:40:20,390

 

1288
00:40:20,390 --> 00:40:23,670

 

1289
00:40:23,670 --> 00:40:25,109

 

1290
00:40:25,109 --> 00:40:28,470

 

1291
00:40:28,470 --> 00:40:30,390

 

1292
00:40:30,390 --> 00:40:30,400

 

1293
00:40:30,400 --> 00:40:31,270


1294
00:40:31,270 --> 00:40:32,790

 

1295
00:40:32,790 --> 00:40:34,710

 

1296
00:40:34,710 --> 00:40:36,309

 

1297
00:40:36,309 --> 00:40:39,190

 

1298
00:40:39,190 --> 00:40:40,309

 

1299
00:40:40,309 --> 00:40:42,390

 

1300
00:40:42,390 --> 00:40:45,750

 

1301
00:40:45,750 --> 00:40:45,760

 

1302
00:40:45,760 --> 00:40:46,710


1303
00:40:46,710 --> 00:40:46,720

 

1304
00:40:46,720 --> 00:40:47,510


1305
00:40:47,510 --> 00:40:49,270

 

1306
00:40:49,270 --> 00:40:51,030

 

1307
00:40:51,030 --> 00:40:53,589

 

1308
00:40:53,589 --> 00:40:55,670

 

1309
00:40:55,670 --> 00:40:57,670

 

1310
00:40:57,670 --> 00:40:57,680

 

1311
00:40:57,680 --> 00:40:58,710


1312
00:40:58,710 --> 00:40:58,720

 

1313
00:40:58,720 --> 00:40:59,510


1314
00:40:59,510 --> 00:41:02,309

 

1315
00:41:02,309 --> 00:41:04,630

 

1316
00:41:04,630 --> 00:41:06,390

 

1317
00:41:06,390 --> 00:41:06,400

 

1318
00:41:06,400 --> 00:41:08,150


1319
00:41:08,150 --> 00:41:10,470

 

1320
00:41:10,470 --> 00:41:13,349

 

1321
00:41:13,349 --> 00:41:15,190

 

1322
00:41:15,190 --> 00:41:16,870

 

1323
00:41:16,870 --> 00:41:19,270

 

1324
00:41:19,270 --> 00:41:21,990

 

1325
00:41:21,990 --> 00:41:24,790

 

1326
00:41:24,790 --> 00:41:25,990

 

1327
00:41:25,990 --> 00:41:27,589

 

1328
00:41:27,589 --> 00:41:29,589

 

1329
00:41:29,589 --> 00:41:31,109

 

1330
00:41:31,109 --> 00:41:33,270

 

1331
00:41:33,270 --> 00:41:35,109

 

1332
00:41:35,109 --> 00:41:37,109

 

1333
00:41:37,109 --> 00:41:39,670

 

1334
00:41:39,670 --> 00:41:41,190

 

1335
00:41:41,190 --> 00:41:44,309

 

1336
00:41:44,309 --> 00:41:44,319

 

1337
00:41:44,319 --> 00:41:46,390


1338
00:41:46,390 --> 00:41:48,630

 

1339
00:41:48,630 --> 00:41:50,309

 

1340
00:41:50,309 --> 00:41:51,510

 

1341
00:41:51,510 --> 00:41:55,829

 

1342
00:41:55,829 --> 00:41:58,630

 

1343
00:41:58,630 --> 00:42:14,470

 

1344
00:42:14,470 --> 00:42:16,309

 

1345
00:42:16,309 --> 00:42:17,829

 

1346
00:42:17,829 --> 00:42:19,430

 

1347
00:42:19,430 --> 00:42:22,470

 

1348
00:42:22,470 --> 00:42:22,480

 

1349
00:42:22,480 --> 00:42:23,990


1350
00:42:23,990 --> 00:42:27,270

 

1351
00:42:27,270 --> 00:42:31,430

 

1352
00:42:31,430 --> 00:42:33,190

 

1353
00:42:33,190 --> 00:42:36,230

 

1354
00:42:36,230 --> 00:42:38,950

 

1355
00:42:38,950 --> 00:42:40,470

 

1356
00:42:40,470 --> 00:42:42,230

 

1357
00:42:42,230 --> 00:42:43,750

 

1358
00:42:43,750 --> 00:42:45,589

 

1359
00:42:45,589 --> 00:42:48,230

 

1360
00:42:48,230 --> 00:42:50,069

 

1361
00:42:50,069 --> 00:42:52,069

 

1362
00:42:52,069 --> 00:42:54,710

 

1363
00:42:54,710 --> 00:42:56,470

 

1364
00:42:56,470 --> 00:42:56,480

 

1365
00:42:56,480 --> 00:42:57,990


1366
00:42:57,990 --> 00:42:59,829

 

1367
00:42:59,829 --> 00:43:02,150

 

1368
00:43:02,150 --> 00:43:02,160

 

1369
00:43:02,160 --> 00:43:03,270


1370
00:43:03,270 --> 00:43:06,150

 

1371
00:43:06,150 --> 00:43:08,470

 

1372
00:43:08,470 --> 00:43:11,109

 

1373
00:43:11,109 --> 00:43:13,349

 

1374
00:43:13,349 --> 00:43:15,510

 

1375
00:43:15,510 --> 00:43:15,520

 

1376
00:43:15,520 --> 00:43:16,790


1377
00:43:16,790 --> 00:43:18,710

 

1378
00:43:18,710 --> 00:43:20,790

 

1379
00:43:20,790 --> 00:43:24,710

 

1380
00:43:24,710 --> 00:43:26,390

 

1381
00:43:26,390 --> 00:43:28,790

 

1382
00:43:28,790 --> 00:43:30,150

 

1383
00:43:30,150 --> 00:43:31,829

 

1384
00:43:31,829 --> 00:43:33,349

 

1385
00:43:33,349 --> 00:43:34,950

 

1386
00:43:34,950 --> 00:43:36,470

 

1387
00:43:36,470 --> 00:43:39,430

 

1388
00:43:39,430 --> 00:43:41,510

 

1389
00:43:41,510 --> 00:43:43,510

 

1390
00:43:43,510 --> 00:43:43,520

 

1391
00:43:43,520 --> 00:43:45,750


1392
00:43:45,750 --> 00:43:45,760

 

1393
00:43:45,760 --> 00:43:48,790


1394
00:43:48,790 --> 00:43:48,800

 

1395
00:43:48,800 --> 00:43:56,630


1396
00:43:56,630 --> 00:43:59,030

 

1397
00:43:59,030 --> 00:44:00,390

 

1398
00:44:00,390 --> 00:44:01,990

 

1399
00:44:01,990 --> 00:44:03,670

 

1400
00:44:03,670 --> 00:44:05,589

 

1401
00:44:05,589 --> 00:44:07,270

 

1402
00:44:07,270 --> 00:44:10,150

 

1403
00:44:10,150 --> 00:44:10,160

 

1404
00:44:10,160 --> 00:44:11,190


1405
00:44:11,190 --> 00:44:13,349

 

1406
00:44:13,349 --> 00:44:16,570

 

1407
00:44:16,570 --> 00:44:16,580

 

1408
00:44:16,580 --> 00:44:18,470


1409
00:44:18,470 --> 00:44:19,910

 

1410
00:44:19,910 --> 00:44:22,390

 

1411
00:44:22,390 --> 00:44:25,190

 

1412
00:44:25,190 --> 00:44:27,510

 

1413
00:44:27,510 --> 00:44:29,990

 

1414
00:44:29,990 --> 00:44:33,109

 

1415
00:44:33,109 --> 00:44:35,349

 

1416
00:44:35,349 --> 00:44:38,470

 

1417
00:44:38,470 --> 00:44:39,990

 

1418
00:44:39,990 --> 00:44:41,990

 

1419
00:44:41,990 --> 00:44:44,630

 

1420
00:44:44,630 --> 00:44:46,870

 

1421
00:44:46,870 --> 00:44:48,630

 

1422
00:44:48,630 --> 00:44:50,390

 

1423
00:44:50,390 --> 00:44:53,430

 

1424
00:44:53,430 --> 00:44:56,550

 

1425
00:44:56,550 --> 00:44:59,430

 

1426
00:44:59,430 --> 00:45:02,390

 

1427
00:45:02,390 --> 00:45:05,349

 

1428
00:45:05,349 --> 00:45:07,430

 

1429
00:45:07,430 --> 00:45:09,030

 

1430
00:45:09,030 --> 00:45:11,270

 

1431
00:45:11,270 --> 00:45:13,829

 

1432
00:45:13,829 --> 00:45:15,430

 

1433
00:45:15,430 --> 00:45:16,950

 

1434
00:45:16,950 --> 00:45:19,190

 

1435
00:45:19,190 --> 00:45:20,550

 

1436
00:45:20,550 --> 00:45:21,990

 

1437
00:45:21,990 --> 00:45:22,000

 

1438
00:45:22,000 --> 00:45:22,950


1439
00:45:22,950 --> 00:45:24,630

 

1440
00:45:24,630 --> 00:45:24,640

 

1441
00:45:24,640 --> 00:45:27,430


1442
00:45:27,430 --> 00:45:30,710

 

1443
00:45:30,710 --> 00:45:32,309

 

1444
00:45:32,309 --> 00:45:34,790

 

1445
00:45:34,790 --> 00:45:36,470

 

1446
00:45:36,470 --> 00:45:38,230

 

1447
00:45:38,230 --> 00:45:40,550

 

1448
00:45:40,550 --> 00:45:41,910

 

1449
00:45:41,910 --> 00:45:44,230

 

1450
00:45:44,230 --> 00:45:46,710

 

1451
00:45:46,710 --> 00:45:46,720

 

1452
00:45:46,720 --> 00:45:48,390


1453
00:45:48,390 --> 00:45:50,630

 

1454
00:45:50,630 --> 00:45:50,640

 

1455
00:45:50,640 --> 00:45:52,870


1456
00:45:52,870 --> 00:45:54,309

 

1457
00:45:54,309 --> 00:45:56,069

 

1458
00:45:56,069 --> 00:45:57,829

 

1459
00:45:57,829 --> 00:45:59,109

 

1460
00:45:59,109 --> 00:46:00,550

 

1461
00:46:00,550 --> 00:46:00,560

 

1462
00:46:00,560 --> 00:46:01,910


1463
00:46:01,910 --> 00:46:06,230

 

1464
00:46:06,230 --> 00:46:08,950

 

1465
00:46:08,950 --> 00:46:11,190

 

1466
00:46:11,190 --> 00:46:11,200

 

1467
00:46:11,200 --> 00:46:12,150


1468
00:46:12,150 --> 00:46:15,109

 

1469
00:46:15,109 --> 00:46:17,670

 

1470
00:46:17,670 --> 00:46:20,950

 

1471
00:46:20,950 --> 00:46:23,589

 

1472
00:46:23,589 --> 00:46:25,670

 

1473
00:46:25,670 --> 00:46:27,670

 

1474
00:46:27,670 --> 00:46:29,270

 

1475
00:46:29,270 --> 00:46:31,190

 

1476
00:46:31,190 --> 00:46:33,670

 

1477
00:46:33,670 --> 00:46:35,589

 

1478
00:46:35,589 --> 00:46:38,150

 

1479
00:46:38,150 --> 00:46:40,309

 

1480
00:46:40,309 --> 00:46:40,319

 

1481
00:46:40,319 --> 00:46:44,550


1482
00:46:44,550 --> 00:46:44,560

 

1483
00:46:44,560 --> 00:46:48,470


1484
00:46:48,470 --> 00:46:50,790

 

1485
00:46:50,790 --> 00:46:50,800

 

1486
00:46:50,800 --> 00:46:55,109


1487
00:46:55,109 --> 00:46:56,630

 

1488
00:46:56,630 --> 00:46:58,470

 

1489
00:46:58,470 --> 00:47:00,950

 

1490
00:47:00,950 --> 00:47:02,550

 

1491
00:47:02,550 --> 00:47:04,309

 

1492
00:47:04,309 --> 00:47:06,710

 

1493
00:47:06,710 --> 00:47:08,829

 

1494
00:47:08,829 --> 00:47:11,270

 

1495
00:47:11,270 --> 00:47:13,190

 

1496
00:47:13,190 --> 00:47:13,200

 

1497
00:47:13,200 --> 00:47:13,560


1498
00:47:13,560 --> 00:47:13,570

 

1499
00:47:13,570 --> 00:47:16,470


1500
00:47:16,470 --> 00:47:16,480

 

1501
00:47:16,480 --> 00:47:17,510


1502
00:47:17,510 --> 00:47:19,349

 

1503
00:47:19,349 --> 00:47:21,190

 

1504
00:47:21,190 --> 00:47:22,549

 

1505
00:47:22,549 --> 00:47:26,870

 

1506
00:47:26,870 --> 00:47:29,430

 

1507
00:47:29,430 --> 00:47:32,549

 

1508
00:47:32,549 --> 00:47:34,309

 

1509
00:47:34,309 --> 00:47:36,390

 

1510
00:47:36,390 --> 00:47:39,109

 

1511
00:47:39,109 --> 00:47:41,030

 

1512
00:47:41,030 --> 00:47:43,430

 

1513
00:47:43,430 --> 00:47:45,829

 

1514
00:47:45,829 --> 00:47:47,829

 

1515
00:47:47,829 --> 00:47:47,839

 

1516
00:47:47,839 --> 00:47:48,710


1517
00:47:48,710 --> 00:47:50,630

 

1518
00:47:50,630 --> 00:47:53,430

 

1519
00:47:53,430 --> 00:47:55,750

 

1520
00:47:55,750 --> 00:47:58,390

 

1521
00:47:58,390 --> 00:48:00,150

 

1522
00:48:00,150 --> 00:48:02,710

 

1523
00:48:02,710 --> 00:48:04,549

 

1524
00:48:04,549 --> 00:48:07,109

 

1525
00:48:07,109 --> 00:48:09,750

 

1526
00:48:09,750 --> 00:48:11,829

 

1527
00:48:11,829 --> 00:48:15,030

 

1528
00:48:15,030 --> 00:48:16,630

 

1529
00:48:16,630 --> 00:48:18,549

 

1530
00:48:18,549 --> 00:48:20,870

 

1531
00:48:20,870 --> 00:48:23,270

 

1532
00:48:23,270 --> 00:48:25,190

 

1533
00:48:25,190 --> 00:48:26,549

 

1534
00:48:26,549 --> 00:48:28,630

 

1535
00:48:28,630 --> 00:48:29,670

 

1536
00:48:29,670 --> 00:48:34,309

 

1537
00:48:34,309 --> 00:48:34,319

 

1538
00:48:34,319 --> 00:48:35,190


1539
00:48:35,190 --> 00:48:37,910

 

1540
00:48:37,910 --> 00:48:41,910

 

1541
00:48:41,910 --> 00:48:41,920

 

1542
00:48:41,920 --> 00:48:43,109


1543
00:48:43,109 --> 00:48:47,349

 

1544
00:48:47,349 --> 00:48:47,359

 

1545
00:48:47,359 --> 00:48:48,790


1546
00:48:48,790 --> 00:48:48,800

 

1547
00:48:48,800 --> 00:48:50,470


1548
00:48:50,470 --> 00:48:50,480

 

1549
00:48:50,480 --> 00:48:51,430


1550
00:48:51,430 --> 00:48:54,309

 

1551
00:48:54,309 --> 00:48:57,910

 

1552
00:48:57,910 --> 00:49:00,950

 

1553
00:49:00,950 --> 00:49:02,470

 

1554
00:49:02,470 --> 00:49:04,549

 

1555
00:49:04,549 --> 00:49:06,470

 

1556
00:49:06,470 --> 00:49:08,390

 

1557
00:49:08,390 --> 00:49:08,400

 

1558
00:49:08,400 --> 00:49:09,829


1559
00:49:09,829 --> 00:49:10,870

 

1560
00:49:10,870 --> 00:49:12,390

 

1561
00:49:12,390 --> 00:49:15,270

 

1562
00:49:15,270 --> 00:49:16,870

 

1563
00:49:16,870 --> 00:49:16,880

 

1564
00:49:16,880 --> 00:49:17,910


1565
00:49:17,910 --> 00:49:20,549

 

1566
00:49:20,549 --> 00:49:23,270

 

1567
00:49:23,270 --> 00:49:25,829

 

1568
00:49:25,829 --> 00:49:25,839

 

1569
00:49:25,839 --> 00:49:27,109


1570
00:49:27,109 --> 00:49:28,470

 

1571
00:49:28,470 --> 00:49:30,870

 

1572
00:49:30,870 --> 00:49:32,870

 

1573
00:49:32,870 --> 00:49:34,950

 

1574
00:49:34,950 --> 00:49:34,960

 

1575
00:49:34,960 --> 00:49:36,950


1576
00:49:36,950 --> 00:49:38,790

 

1577
00:49:38,790 --> 00:49:41,109

 

1578
00:49:41,109 --> 00:49:43,030

 

1579
00:49:43,030 --> 00:49:44,470

 

1580
00:49:44,470 --> 00:49:46,309

 

1581
00:49:46,309 --> 00:49:46,319

 

1582
00:49:46,319 --> 00:49:47,349


1583
00:49:47,349 --> 00:49:47,359

 

1584
00:49:47,359 --> 00:49:48,470


1585
00:49:48,470 --> 00:49:48,480

 

1586
00:49:48,480 --> 00:49:50,150


1587
00:49:50,150 --> 00:49:50,160

 

1588
00:49:50,160 --> 00:49:51,670


1589
00:49:51,670 --> 00:49:53,430

 

1590
00:49:53,430 --> 00:49:54,790

 

1591
00:49:54,790 --> 00:49:56,549

 

1592
00:49:56,549 --> 00:49:58,390

 

1593
00:49:58,390 --> 00:50:01,190

 

1594
00:50:01,190 --> 00:50:03,190

 

1595
00:50:03,190 --> 00:50:04,390

 

1596
00:50:04,390 --> 00:50:06,870

 

1597
00:50:06,870 --> 00:50:09,349

 

1598
00:50:09,349 --> 00:50:11,750

 

1599
00:50:11,750 --> 00:50:13,990

 

1600
00:50:13,990 --> 00:50:15,430

 

1601
00:50:15,430 --> 00:50:17,990

 

1602
00:50:17,990 --> 00:50:18,000

 

1603
00:50:18,000 --> 00:50:18,790


1604
00:50:18,790 --> 00:50:20,790

 

1605
00:50:20,790 --> 00:50:23,109

 

1606
00:50:23,109 --> 00:50:23,119

 

1607
00:50:23,119 --> 00:50:26,069


1608
00:50:26,069 --> 00:50:29,510

 

1609
00:50:29,510 --> 00:50:31,670

 

1610
00:50:31,670 --> 00:50:34,790

 

1611
00:50:34,790 --> 00:50:36,069

 

1612
00:50:36,069 --> 00:50:39,430

 

1613
00:50:39,430 --> 00:50:41,030

 

1614
00:50:41,030 --> 00:50:44,390

 

1615
00:50:44,390 --> 00:50:45,670

 

1616
00:50:45,670 --> 00:50:47,349

 

1617
00:50:47,349 --> 00:50:49,030

 

1618
00:50:49,030 --> 00:50:50,790

 

1619
00:50:50,790 --> 00:50:50,800

 

1620
00:50:50,800 --> 00:50:52,230


1621
00:50:52,230 --> 00:50:52,240

 

1622
00:50:52,240 --> 00:50:53,349


1623
00:50:53,349 --> 00:50:55,829

 

1624
00:50:55,829 --> 00:50:58,710

 

1625
00:50:58,710 --> 00:51:00,069

 

1626
00:51:00,069 --> 00:51:02,870

 

1627
00:51:02,870 --> 00:51:04,870

 

1628
00:51:04,870 --> 00:51:07,109

 

1629
00:51:07,109 --> 00:51:07,119

 

1630
00:51:07,119 --> 00:51:08,470


1631
00:51:08,470 --> 00:51:10,309

 

1632
00:51:10,309 --> 00:51:12,630

 

1633
00:51:12,630 --> 00:51:14,309

 

1634
00:51:14,309 --> 00:51:16,790

 

1635
00:51:16,790 --> 00:51:16,800

 

1636
00:51:16,800 --> 00:51:17,829


1637
00:51:17,829 --> 00:51:17,839

 

1638
00:51:17,839 --> 00:51:19,349


1639
00:51:19,349 --> 00:51:21,510

 

1640
00:51:21,510 --> 00:51:23,510

 

1641
00:51:23,510 --> 00:51:25,030

 

1642
00:51:25,030 --> 00:51:27,829

 

1643
00:51:27,829 --> 00:51:30,230

 

1644
00:51:30,230 --> 00:51:32,870

 

1645
00:51:32,870 --> 00:51:32,880

 

1646
00:51:32,880 --> 00:51:34,230


1647
00:51:34,230 --> 00:51:35,589

 

1648
00:51:35,589 --> 00:51:39,510

 

1649
00:51:39,510 --> 00:51:41,270

 

1650
00:51:41,270 --> 00:51:41,280

 

1651
00:51:41,280 --> 00:51:43,109


1652
00:51:43,109 --> 00:51:43,119

 

1653
00:51:43,119 --> 00:51:45,349


1654
00:51:45,349 --> 00:51:45,359

 

1655
00:51:45,359 --> 00:51:46,470


1656
00:51:46,470 --> 00:51:48,470

 

1657
00:51:48,470 --> 00:51:50,549

 

1658
00:51:50,549 --> 00:51:52,150

 

1659
00:51:52,150 --> 00:51:53,829

 

1660
00:51:53,829 --> 00:51:55,030

 

1661
00:51:55,030 --> 00:51:56,950

 

1662
00:51:56,950 --> 00:51:58,470

 

1663
00:51:58,470 --> 00:52:00,790

 

1664
00:52:00,790 --> 00:52:02,950

 

1665
00:52:02,950 --> 00:52:02,960

 

1666
00:52:02,960 --> 00:52:05,670


1667
00:52:05,670 --> 00:52:06,950

 

1668
00:52:06,950 --> 00:52:06,960

 

1669
00:52:06,960 --> 00:52:07,990


1670
00:52:07,990 --> 00:52:09,829

 

1671
00:52:09,829 --> 00:52:12,230

 

1672
00:52:12,230 --> 00:52:14,549

 

1673
00:52:14,549 --> 00:52:16,390

 

1674
00:52:16,390 --> 00:52:18,470

 

1675
00:52:18,470 --> 00:52:20,309

 

1676
00:52:20,309 --> 00:52:22,790

 

1677
00:52:22,790 --> 00:52:22,800

 

1678
00:52:22,800 --> 00:52:23,589


1679
00:52:23,589 --> 00:52:23,599

 

1680
00:52:23,599 --> 00:52:24,870


1681
00:52:24,870 --> 00:52:25,990

 

1682
00:52:25,990 --> 00:52:28,790

 

1683
00:52:28,790 --> 00:52:30,790

 

1684
00:52:30,790 --> 00:52:33,510

 

1685
00:52:33,510 --> 00:52:35,190

 

1686
00:52:35,190 --> 00:52:36,870

 

1687
00:52:36,870 --> 00:52:38,790

 

1688
00:52:38,790 --> 00:52:39,910

 

1689
00:52:39,910 --> 00:52:41,349

 

1690
00:52:41,349 --> 00:52:43,430

 

1691
00:52:43,430 --> 00:52:45,349

 

1692
00:52:45,349 --> 00:52:47,510

 

1693
00:52:47,510 --> 00:52:48,309

 

1694
00:52:48,309 --> 00:52:50,470

 

1695
00:52:50,470 --> 00:52:50,480

 

1696
00:52:50,480 --> 00:52:51,510


1697
00:52:51,510 --> 00:52:53,270

 

1698
00:52:53,270 --> 00:52:55,270

 

1699
00:52:55,270 --> 00:52:56,950

 

1700
00:52:56,950 --> 00:52:59,829

 

1701
00:52:59,829 --> 00:53:01,510

 

1702
00:53:01,510 --> 00:53:03,750

 

1703
00:53:03,750 --> 00:53:07,109

 

1704
00:53:07,109 --> 00:53:09,670

 

1705
00:53:09,670 --> 00:53:13,750

 

1706
00:53:13,750 --> 00:53:13,760

 

1707
00:53:13,760 --> 00:53:15,030


1708
00:53:15,030 --> 00:53:17,030

 

1709
00:53:17,030 --> 00:53:19,030

 

1710
00:53:19,030 --> 00:53:21,990

 

1711
00:53:21,990 --> 00:53:22,000

 

1712
00:53:22,000 --> 00:53:23,030


1713
00:53:23,030 --> 00:53:26,150

 

1714
00:53:26,150 --> 00:53:28,630

 

1715
00:53:28,630 --> 00:53:30,549

 

1716
00:53:30,549 --> 00:53:30,559

 

1717
00:53:30,559 --> 00:53:31,510


1718
00:53:31,510 --> 00:53:33,589

 

1719
00:53:33,589 --> 00:53:35,109

 

1720
00:53:35,109 --> 00:53:35,119

 

1721
00:53:35,119 --> 00:53:37,270


1722
00:53:37,270 --> 00:53:37,280

 

1723
00:53:37,280 --> 00:53:38,309


1724
00:53:38,309 --> 00:53:39,990

 

1725
00:53:39,990 --> 00:53:41,750

 

1726
00:53:41,750 --> 00:53:43,829

 

1727
00:53:43,829 --> 00:53:46,230

 

1728
00:53:46,230 --> 00:53:46,240

 

1729
00:53:46,240 --> 00:53:48,790


1730
00:53:48,790 --> 00:53:48,800

 

1731
00:53:48,800 --> 00:53:51,270


1732
00:53:51,270 --> 00:53:52,710

 

1733
00:53:52,710 --> 00:53:54,710

 

1734
00:53:54,710 --> 00:53:57,190

 

1735
00:53:57,190 --> 00:53:59,270

 

1736
00:53:59,270 --> 00:54:00,790

 

1737
00:54:00,790 --> 00:54:02,230

 

1738
00:54:02,230 --> 00:54:04,470

 

1739
00:54:04,470 --> 00:54:07,190

 

1740
00:54:07,190 --> 00:54:07,200

 

1741
00:54:07,200 --> 00:54:08,710


1742
00:54:08,710 --> 00:54:13,109

 

1743
00:54:13,109 --> 00:54:14,870

 

1744
00:54:14,870 --> 00:54:17,589

 

1745
00:54:17,589 --> 00:54:19,190

 

1746
00:54:19,190 --> 00:54:21,430

 

1747
00:54:21,430 --> 00:54:25,670

 

1748
00:54:25,670 --> 00:54:42,069

 

1749
00:54:42,069 --> 00:54:48,309

 

1750
00:54:48,309 --> 00:54:51,190

 

1751
00:54:51,190 --> 00:54:53,190

 

1752
00:54:53,190 --> 00:54:55,349

 

1753
00:54:55,349 --> 00:54:57,190

 

1754
00:54:57,190 --> 00:55:01,030

 

1755
00:55:01,030 --> 00:55:03,349

 

1756
00:55:03,349 --> 00:55:06,950

 

1757
00:55:06,950 --> 00:55:10,870

 

1758
00:55:10,870 --> 00:55:12,950

 

1759
00:55:12,950 --> 00:55:15,190

 

1760
00:55:15,190 --> 00:55:16,390

 

1761
00:55:16,390 --> 00:55:18,870

 

1762
00:55:18,870 --> 00:55:21,829

 

1763
00:55:21,829 --> 00:55:23,349

 

1764
00:55:23,349 --> 00:55:23,359

 

1765
00:55:23,359 --> 00:55:25,349


1766
00:55:25,349 --> 00:55:25,359

 

1767
00:55:25,359 --> 00:55:26,950


1768
00:55:26,950 --> 00:55:29,270

 

1769
00:55:29,270 --> 00:55:30,950

 

1770
00:55:30,950 --> 00:55:30,960

 

1771
00:55:30,960 --> 00:55:32,630


1772
00:55:32,630 --> 00:55:35,349

 

1773
00:55:35,349 --> 00:55:38,150

 

1774
00:55:38,150 --> 00:55:40,230

 

1775
00:55:40,230 --> 00:55:42,870

 

1776
00:55:42,870 --> 00:55:44,870

 

1777
00:55:44,870 --> 00:55:47,270

 

1778
00:55:47,270 --> 00:55:48,870

 

1779
00:55:48,870 --> 00:55:51,190

 

1780
00:55:51,190 --> 00:55:53,349

 

1781
00:55:53,349 --> 00:55:54,950

 

1782
00:55:54,950 --> 00:55:56,549

 

1783
00:55:56,549 --> 00:55:59,670

 

1784
00:55:59,670 --> 00:56:01,589

 

1785
00:56:01,589 --> 00:56:01,599

 

1786
00:56:01,599 --> 00:56:02,710


1787
00:56:02,710 --> 00:56:06,069

 

1788
00:56:06,069 --> 00:56:06,079

 

1789
00:56:06,079 --> 00:56:07,270


1790
00:56:07,270 --> 00:56:09,750

 

1791
00:56:09,750 --> 00:56:12,309

 

1792
00:56:12,309 --> 00:56:14,549

 

1793
00:56:14,549 --> 00:56:17,349

 

1794
00:56:17,349 --> 00:56:17,359

 

1795
00:56:17,359 --> 00:56:18,549


1796
00:56:18,549 --> 00:56:20,710

 

1797
00:56:20,710 --> 00:56:23,430

 

1798
00:56:23,430 --> 00:56:25,270

 

1799
00:56:25,270 --> 00:56:27,190

 

1800
00:56:27,190 --> 00:56:29,510

 

1801
00:56:29,510 --> 00:56:31,430

 

1802
00:56:31,430 --> 00:56:37,589

 

1803
00:56:37,589 --> 00:56:39,990

 

1804
00:56:39,990 --> 00:56:42,829

 

1805
00:56:42,829 --> 00:56:49,510

 

1806
00:56:49,510 --> 00:56:49,520

 

1807
00:56:49,520 --> 00:56:53,670


1808
00:56:53,670 --> 00:56:53,680

 

1809
00:56:53,680 --> 00:56:54,710


1810
00:56:54,710 --> 00:56:57,990

 

1811
00:56:57,990 --> 00:56:59,910

 

1812
00:56:59,910 --> 00:57:03,270

 

1813
00:57:03,270 --> 00:57:05,990

 

1814
00:57:05,990 --> 00:57:07,589

 

1815
00:57:07,589 --> 00:57:09,589

 

1816
00:57:09,589 --> 00:57:09,599

 

1817
00:57:09,599 --> 00:57:10,829


1818
00:57:10,829 --> 00:57:10,839

 

1819
00:57:10,839 --> 00:57:12,390


1820
00:57:12,390 --> 00:57:12,400

 

1821
00:57:12,400 --> 00:57:13,750


1822
00:57:13,750 --> 00:57:13,760

 

1823
00:57:13,760 --> 00:57:16,390


1824
00:57:16,390 --> 00:57:18,630

 

1825
00:57:18,630 --> 00:57:18,640

 

1826
00:57:18,640 --> 00:57:21,030


1827
00:57:21,030 --> 00:57:21,040

 

1828
00:57:21,040 --> 00:57:21,829


1829
00:57:21,829 --> 00:57:21,839

 

1830
00:57:21,839 --> 00:57:23,270


1831
00:57:23,270 --> 00:57:25,109

 

1832
00:57:25,109 --> 00:57:27,990

 

1833
00:57:27,990 --> 00:57:30,470

 

1834
00:57:30,470 --> 00:57:33,190

 

1835
00:57:33,190 --> 00:57:34,630

 

1836
00:57:34,630 --> 00:57:36,309

 

1837
00:57:36,309 --> 00:57:39,910

 

1838
00:57:39,910 --> 00:57:42,549

 

1839
00:57:42,549 --> 00:57:44,870

 

1840
00:57:44,870 --> 00:57:46,789

 

1841
00:57:46,789 --> 00:57:48,230

 

1842
00:57:48,230 --> 00:57:49,990

 

1843
00:57:49,990 --> 00:57:56,549

 

1844
00:57:56,549 --> 00:57:56,559

 

1845
00:57:56,559 --> 00:57:59,030


1846
00:57:59,030 --> 00:57:59,040

 

1847
00:57:59,040 --> 00:58:03,829


1848
00:58:03,829 --> 00:58:03,839

 

1849
00:58:03,839 --> 00:58:04,789


1850
00:58:04,789 --> 00:58:08,950

 

1851
00:58:08,950 --> 00:58:11,430

 

1852
00:58:11,430 --> 00:58:15,190

 

1853
00:58:15,190 --> 00:58:20,309

 

1854
00:58:20,309 --> 00:58:22,470

 

1855
00:58:22,470 --> 00:58:24,549

 

1856
00:58:24,549 --> 00:58:24,559

 

1857
00:58:24,559 --> 00:58:26,470


1858
00:58:26,470 --> 00:58:30,870

 

1859
00:58:30,870 --> 00:58:32,789

 

1860
00:58:32,789 --> 00:58:32,799

 

1861
00:58:32,799 --> 00:58:33,589


1862
00:58:33,589 --> 00:58:36,710

 

1863
00:58:36,710 --> 00:58:38,390

 

1864
00:58:38,390 --> 00:58:39,829

 

1865
00:58:39,829 --> 00:58:41,990

 

1866
00:58:41,990 --> 00:58:43,750

 

1867
00:58:43,750 --> 00:58:46,230

 

1868
00:58:46,230 --> 00:58:47,990

 

1869
00:58:47,990 --> 00:58:49,270

 

1870
00:58:49,270 --> 00:58:49,280

 

1871
00:58:49,280 --> 00:58:53,589


1872
00:58:53,589 --> 00:59:00,710

 

1873
00:59:00,710 --> 00:59:00,720

 

1874
00:59:00,720 --> 00:59:03,270


1875
00:59:03,270 --> 00:59:05,510

 

1876
00:59:05,510 --> 00:59:07,109

 

1877
00:59:07,109 --> 00:59:09,990

 

1878
00:59:09,990 --> 00:59:12,390

 

1879
00:59:12,390 --> 00:59:14,789

 

1880
00:59:14,789 --> 00:59:15,829

 

1881
00:59:15,829 --> 00:59:18,309

 

1882
00:59:18,309 --> 00:59:20,309

 

1883
00:59:20,309 --> 00:59:22,309

 

1884
00:59:22,309 --> 00:59:22,319

 

1885
00:59:22,319 --> 00:59:23,270


1886
00:59:23,270 --> 00:59:24,710

 

1887
00:59:24,710 --> 00:59:26,390

 

1888
00:59:26,390 --> 00:59:27,829

 

1889
00:59:27,829 --> 00:59:30,829

 

1890
00:59:30,829 --> 00:59:30,839

 

1891
00:59:30,839 --> 00:59:32,630


1892
00:59:32,630 --> 00:59:34,069

 

1893
00:59:34,069 --> 00:59:35,670

 

1894
00:59:35,670 --> 00:59:37,349

 

1895
00:59:37,349 --> 00:59:38,870

 

1896
00:59:38,870 --> 00:59:40,870

 

1897
00:59:40,870 --> 00:59:40,880

 

1898
00:59:40,880 --> 00:59:41,990


1899
00:59:41,990 --> 00:59:43,510

 

1900
00:59:43,510 --> 00:59:43,520

 

1901
00:59:43,520 --> 00:59:44,950


1902
00:59:44,950 --> 00:59:47,270

 

1903
00:59:47,270 --> 00:59:49,430

 

1904
00:59:49,430 --> 00:59:51,430

 

1905
00:59:51,430 --> 00:59:53,910

 

1906
00:59:53,910 --> 00:59:56,150

 

1907
00:59:56,150 --> 00:59:58,150

 

1908
00:59:58,150 --> 01:00:00,549

 

1909
01:00:00,549 --> 01:00:00,559

 

1910
01:00:00,559 --> 01:00:01,670


1911
01:00:01,670 --> 01:00:04,150

 

1912
01:00:04,150 --> 01:00:07,349

 

1913
01:00:07,349 --> 01:00:11,829

 

1914
01:00:11,829 --> 01:00:14,549

 

1915
01:00:14,549 --> 01:00:16,549

 

1916
01:00:16,549 --> 01:00:18,549

 

1917
01:00:18,549 --> 01:00:21,750

 

1918
01:00:21,750 --> 01:00:24,069

 

1919
01:00:24,069 --> 01:00:25,589

 

1920
01:00:25,589 --> 01:00:28,549

 

1921
01:00:28,549 --> 01:00:29,750

 

1922
01:00:29,750 --> 01:00:29,760

 

1923
01:00:29,760 --> 01:00:30,789


1924
01:00:30,789 --> 01:00:32,870

 

1925
01:00:32,870 --> 01:00:34,390

 

1926
01:00:34,390 --> 01:00:36,230

 

1927
01:00:36,230 --> 01:00:36,240

 

1928
01:00:36,240 --> 01:00:37,270


1929
01:00:37,270 --> 01:00:39,109

 

1930
01:00:39,109 --> 01:00:41,510

 

1931
01:00:41,510 --> 01:00:44,549

 

1932
01:00:44,549 --> 01:00:46,230

 

1933
01:00:46,230 --> 01:00:49,030

 

1934
01:00:49,030 --> 01:00:51,349

 

1935
01:00:51,349 --> 01:00:51,359

 

1936
01:00:51,359 --> 01:00:54,230


1937
01:00:54,230 --> 01:00:57,270

 

1938
01:00:57,270 --> 01:00:58,950

 

1939
01:00:58,950 --> 01:01:01,109

 

1940
01:01:01,109 --> 01:01:02,470

 

1941
01:01:02,470 --> 01:01:04,390

 

1942
01:01:04,390 --> 01:01:06,549

 

1943
01:01:06,549 --> 01:01:08,069

 

1944
01:01:08,069 --> 01:01:10,470

 

1945
01:01:10,470 --> 01:01:13,430

 

1946
01:01:13,430 --> 01:01:16,470

 

1947
01:01:16,470 --> 01:01:18,789

 

1948
01:01:18,789 --> 01:01:21,030

 

1949
01:01:21,030 --> 01:01:23,349

 

1950
01:01:23,349 --> 01:01:25,910

 

1951
01:01:25,910 --> 01:01:28,630

 

1952
01:01:28,630 --> 01:01:30,230

 

1953
01:01:30,230 --> 01:01:36,150

 

1954
01:01:36,150 --> 01:01:36,160

 

1955
01:01:36,160 --> 01:01:37,430


1956
01:01:37,430 --> 01:01:39,430

 

1957
01:01:39,430 --> 01:01:39,440

 

1958
01:01:39,440 --> 01:01:40,630


1959
01:01:40,630 --> 01:01:40,640

 

1960
01:01:40,640 --> 01:01:41,750


1961
01:01:41,750 --> 01:01:41,760

 

1962
01:01:41,760 --> 01:01:43,190


1963
01:01:43,190 --> 01:01:44,549

 

1964
01:01:44,549 --> 01:01:46,630

 

1965
01:01:46,630 --> 01:01:46,640

 

1966
01:01:46,640 --> 01:01:47,670


1967
01:01:47,670 --> 01:01:50,470

 

1968
01:01:50,470 --> 01:01:52,549

 

1969
01:01:52,549 --> 01:01:54,789

 

1970
01:01:54,789 --> 01:01:57,270

 

1971
01:01:57,270 --> 01:01:59,670

 

1972
01:01:59,670 --> 01:01:59,680

 

1973
01:01:59,680 --> 01:02:01,190


1974
01:02:01,190 --> 01:02:02,950

 

1975
01:02:02,950 --> 01:02:05,029

 

1976
01:02:05,029 --> 01:02:07,029

 

1977
01:02:07,029 --> 01:02:07,039

 

1978
01:02:07,039 --> 01:02:08,309


1979
01:02:08,309 --> 01:02:09,829

 

1980
01:02:09,829 --> 01:02:11,750

 

1981
01:02:11,750 --> 01:02:13,589

 

1982
01:02:13,589 --> 01:02:15,510

 

1983
01:02:15,510 --> 01:02:17,589

 

1984
01:02:17,589 --> 01:02:20,710

 

1985
01:02:20,710 --> 01:02:23,589

 

1986
01:02:23,589 --> 01:02:25,910

 

1987
01:02:25,910 --> 01:02:27,829

 

1988
01:02:27,829 --> 01:02:29,750

 

1989
01:02:29,750 --> 01:02:31,510

 

1990
01:02:31,510 --> 01:02:35,430

 

1991
01:02:35,430 --> 01:02:37,990

 

1992
01:02:37,990 --> 01:02:40,390

 

1993
01:02:40,390 --> 01:02:42,870

 

1994
01:02:42,870 --> 01:02:46,150

 

1995
01:02:46,150 --> 01:02:48,069

 

1996
01:02:48,069 --> 01:02:50,069

 

1997
01:02:50,069 --> 01:02:52,390

 

1998
01:02:52,390 --> 01:02:52,400

 

1999
01:02:52,400 --> 01:02:53,510


2000
01:02:53,510 --> 01:02:55,430

 

2001
01:02:55,430 --> 01:02:57,190

 

2002
01:02:57,190 --> 01:03:00,309

 

2003
01:03:00,309 --> 01:03:02,470

 

2004
01:03:02,470 --> 01:03:03,910

 

2005
01:03:03,910 --> 01:03:06,710

 

2006
01:03:06,710 --> 01:03:10,230

 

2007
01:03:10,230 --> 01:03:11,750

 

2008
01:03:11,750 --> 01:03:14,309

 

2009
01:03:14,309 --> 01:03:15,589

 

2010
01:03:15,589 --> 01:03:17,910

 

2011
01:03:17,910 --> 01:03:19,589

 

2012
01:03:19,589 --> 01:03:21,510

 

2013
01:03:21,510 --> 01:03:24,950

 

2014
01:03:24,950 --> 01:03:26,390

 

2015
01:03:26,390 --> 01:03:28,870

 

2016
01:03:28,870 --> 01:03:28,880

 

2017
01:03:28,880 --> 01:03:31,430


2018
01:03:31,430 --> 01:03:33,270

 

2019
01:03:33,270 --> 01:03:36,230

 

2020
01:03:36,230 --> 01:03:39,910

 

2021
01:03:39,910 --> 01:03:42,230

 

2022
01:03:42,230 --> 01:03:44,789

 

2023
01:03:44,789 --> 01:03:44,799

 

2024
01:03:44,799 --> 01:03:45,910


2025
01:03:45,910 --> 01:03:48,230

 

2026
01:03:48,230 --> 01:03:50,230

 

2027
01:03:50,230 --> 01:03:53,109

 

2028
01:03:53,109 --> 01:03:53,119

 

2029
01:03:53,119 --> 01:03:53,990


2030
01:03:53,990 --> 01:03:56,309

 

2031
01:03:56,309 --> 01:04:01,029

 

2032
01:04:01,029 --> 01:04:02,950

 

2033
01:04:02,950 --> 01:04:05,750

 

2034
01:04:05,750 --> 01:04:07,430

 

2035
01:04:07,430 --> 01:04:09,990

 

2036
01:04:09,990 --> 01:04:13,430

 

2037
01:04:13,430 --> 01:04:15,430

 

2038
01:04:15,430 --> 01:04:17,430

 

2039
01:04:17,430 --> 01:04:19,589

 

2040
01:04:19,589 --> 01:04:21,670

 

2041
01:04:21,670 --> 01:04:24,870

 

2042
01:04:24,870 --> 01:04:24,880

 

2043
01:04:24,880 --> 01:04:26,549


2044
01:04:26,549 --> 01:04:30,390

 

2045
01:04:30,390 --> 01:04:33,750

 

2046
01:04:33,750 --> 01:04:35,589

 

2047
01:04:35,589 --> 01:04:37,750

 

2048
01:04:37,750 --> 01:04:39,750

 

2049
01:04:39,750 --> 01:04:41,510

 

2050
01:04:41,510 --> 01:04:42,870

 

2051
01:04:42,870 --> 01:04:44,549

 

2052
01:04:44,549 --> 01:04:46,870

 

2053
01:04:46,870 --> 01:04:48,870

 

2054
01:04:48,870 --> 01:04:50,470

 

2055
01:04:50,470 --> 01:04:52,470

 

2056
01:04:52,470 --> 01:04:55,109

 

2057
01:04:55,109 --> 01:04:57,349

 

2058
01:04:57,349 --> 01:04:59,829

 

2059
01:04:59,829 --> 01:05:01,589

 

2060
01:05:01,589 --> 01:05:02,870

 

2061
01:05:02,870 --> 01:05:05,190

 

2062
01:05:05,190 --> 01:05:07,510

 

2063
01:05:07,510 --> 01:05:10,549

 

2064
01:05:10,549 --> 01:05:13,109

 

2065
01:05:13,109 --> 01:05:15,109

 

2066
01:05:15,109 --> 01:05:16,630

 

2067
01:05:16,630 --> 01:05:19,109

 

2068
01:05:19,109 --> 01:05:19,119

 

2069
01:05:19,119 --> 01:05:19,990


2070
01:05:19,990 --> 01:05:21,589

 

2071
01:05:21,589 --> 01:05:28,789

 

2072
01:05:28,789 --> 01:05:30,950

 

2073
01:05:30,950 --> 01:05:33,750

 

2074
01:05:33,750 --> 01:05:37,349

 

2075
01:05:37,349 --> 01:05:38,950

 

2076
01:05:38,950 --> 01:05:39,910

 

2077
01:05:39,910 --> 01:05:41,349

 

2078
01:05:41,349 --> 01:05:43,109

 

2079
01:05:43,109 --> 01:05:45,829

 

2080
01:05:45,829 --> 01:05:47,990

 

2081
01:05:47,990 --> 01:05:50,470

 

2082
01:05:50,470 --> 01:05:51,829

 

2083
01:05:51,829 --> 01:05:54,789

 

2084
01:05:54,789 --> 01:05:57,829

 

2085
01:05:57,829 --> 01:06:00,470

 

2086
01:06:00,470 --> 01:06:03,670

 

2087
01:06:03,670 --> 01:06:03,680

 

2088
01:06:03,680 --> 01:06:04,950


2089
01:06:04,950 --> 01:06:07,990

 

2090
01:06:07,990 --> 01:06:09,349

 

2091
01:06:09,349 --> 01:06:11,430

 

2092
01:06:11,430 --> 01:06:12,789

 

2093
01:06:12,789 --> 01:06:14,870

 

2094
01:06:14,870 --> 01:06:15,910

 

2095
01:06:15,910 --> 01:06:18,870

 

2096
01:06:18,870 --> 01:06:21,589

 

2097
01:06:21,589 --> 01:06:25,029

 

2098
01:06:25,029 --> 01:06:25,039

 

2099
01:06:25,039 --> 01:06:26,789


2100
01:06:26,789 --> 01:06:29,430

 

2101
01:06:29,430 --> 01:06:29,440

 

2102
01:06:29,440 --> 01:06:30,710


2103
01:06:30,710 --> 01:06:33,029

 

2104
01:06:33,029 --> 01:06:35,109

 

2105
01:06:35,109 --> 01:06:35,119

 

2106
01:06:35,119 --> 01:06:36,150


2107
01:06:36,150 --> 01:06:36,160

 

2108
01:06:36,160 --> 01:06:38,630


2109
01:06:38,630 --> 01:06:40,549

 

2110
01:06:40,549 --> 01:06:42,549

 

2111
01:06:42,549 --> 01:06:44,630

 

2112
01:06:44,630 --> 01:06:46,390

 

2113
01:06:46,390 --> 01:06:47,750

 

2114
01:06:47,750 --> 01:06:49,670

 

2115
01:06:49,670 --> 01:06:51,589

 

2116
01:06:51,589 --> 01:06:53,109

 

2117
01:06:53,109 --> 01:06:54,549

 

2118
01:06:54,549 --> 01:06:57,029

 

2119
01:06:57,029 --> 01:06:59,109

 

2120
01:06:59,109 --> 01:07:00,710

 

2121
01:07:00,710 --> 01:07:02,390

 

2122
01:07:02,390 --> 01:07:02,400

 

2123
01:07:02,400 --> 01:07:03,270


2124
01:07:03,270 --> 01:07:05,270

 

2125
01:07:05,270 --> 01:07:07,109

 

2126
01:07:07,109 --> 01:07:09,589

 

2127
01:07:09,589 --> 01:07:11,829

 

2128
01:07:11,829 --> 01:07:14,069

 

2129
01:07:14,069 --> 01:07:16,069

 

2130
01:07:16,069 --> 01:07:21,270

 

2131
01:07:21,270 --> 01:07:23,589

 

2132
01:07:23,589 --> 01:07:25,829

 

2133
01:07:25,829 --> 01:07:26,710

 

2134
01:07:26,710 --> 01:07:29,750

 

2135
01:07:29,750 --> 01:07:31,349

 

2136
01:07:31,349 --> 01:07:32,870

 

2137
01:07:32,870 --> 01:07:35,750

 

2138
01:07:35,750 --> 01:07:40,069

 

2139
01:07:40,069 --> 01:07:42,309

 

2140
01:07:42,309 --> 01:07:44,309

 

2141
01:07:44,309 --> 01:07:46,390

 

2142
01:07:46,390 --> 01:07:48,309

 

2143
01:07:48,309 --> 01:07:48,319

 

2144
01:07:48,319 --> 01:07:49,510


2145
01:07:49,510 --> 01:07:51,670

 

2146
01:07:51,670 --> 01:07:53,510

 

2147
01:07:53,510 --> 01:07:55,029

 

2148
01:07:55,029 --> 01:07:57,750

 

2149
01:07:57,750 --> 01:07:59,750

 

2150
01:07:59,750 --> 01:08:01,589

 

2151
01:08:01,589 --> 01:08:02,710

 

2152
01:08:02,710 --> 01:08:03,829

 

2153
01:08:03,829 --> 01:08:05,109

 

2154
01:08:05,109 --> 01:08:08,309

 

2155
01:08:08,309 --> 01:08:10,150

 

2156
01:08:10,150 --> 01:08:10,160

 

2157
01:08:10,160 --> 01:08:13,430


2158
01:08:13,430 --> 01:08:15,190

 

2159
01:08:15,190 --> 01:08:16,390

 

2160
01:08:16,390 --> 01:08:18,390

 

2161
01:08:18,390 --> 01:08:20,709

 

2162
01:08:20,709 --> 01:08:22,829

 

2163
01:08:22,829 --> 01:08:25,269

 

2164
01:08:25,269 --> 01:08:25,279

 

2165
01:08:25,279 --> 01:08:26,229


2166
01:08:26,229 --> 01:08:28,149

 

2167
01:08:28,149 --> 01:08:29,829

 

2168
01:08:29,829 --> 01:08:29,839

 

2169
01:08:29,839 --> 01:08:31,590


2170
01:08:31,590 --> 01:08:31,600

 

2171
01:08:31,600 --> 01:08:34,149


2172
01:08:34,149 --> 01:08:36,149

 

2173
01:08:36,149 --> 01:08:36,159

 

2174
01:08:36,159 --> 01:08:37,030


2175
01:08:37,030 --> 01:08:39,269

 

2176
01:08:39,269 --> 01:08:41,910

 

2177
01:08:41,910 --> 01:08:43,669

 

2178
01:08:43,669 --> 01:08:43,679

 

2179
01:08:43,679 --> 01:08:45,030


2180
01:08:45,030 --> 01:08:46,870

 

2181
01:08:46,870 --> 01:08:48,309

 

2182
01:08:48,309 --> 01:08:50,149

 

2183
01:08:50,149 --> 01:08:52,550

 

2184
01:08:52,550 --> 01:08:53,749

 

2185
01:08:53,749 --> 01:08:55,189

 

2186
01:08:55,189 --> 01:08:57,269

 

2187
01:08:57,269 --> 01:08:59,510

 

2188
01:08:59,510 --> 01:09:01,030

 

2189
01:09:01,030 --> 01:09:03,990

 

2190
01:09:03,990 --> 01:09:04,000

 

2191
01:09:04,000 --> 01:09:04,870


2192
01:09:04,870 --> 01:09:07,669

 

2193
01:09:07,669 --> 01:09:09,749

 

2194
01:09:09,749 --> 01:09:12,149

 

2195
01:09:12,149 --> 01:09:14,229

 

2196
01:09:14,229 --> 01:09:14,239

 

2197
01:09:14,239 --> 01:09:15,189


2198
01:09:15,189 --> 01:09:18,070

 

2199
01:09:18,070 --> 01:09:18,080

 

2200
01:09:18,080 --> 01:09:18,950


2201
01:09:18,950 --> 01:09:21,829

 

2202
01:09:21,829 --> 01:09:24,390

 

2203
01:09:24,390 --> 01:09:24,400

 

2204
01:09:24,400 --> 01:09:26,070


2205
01:09:26,070 --> 01:09:28,309

 

2206
01:09:28,309 --> 01:09:30,309

 

2207
01:09:30,309 --> 01:09:32,630

 

2208
01:09:32,630 --> 01:09:35,590

 

2209
01:09:35,590 --> 01:09:38,309

 

2210
01:09:38,309 --> 01:09:40,950

 

2211
01:09:40,950 --> 01:09:42,870

 

2212
01:09:42,870 --> 01:09:45,669

 

2213
01:09:45,669 --> 01:09:47,990

 

2214
01:09:47,990 --> 01:09:50,709

 

2215
01:09:50,709 --> 01:09:52,789

 

2216
01:09:52,789 --> 01:09:52,799

 

2217
01:09:52,799 --> 01:09:53,829


2218
01:09:53,829 --> 01:09:55,430

 

2219
01:09:55,430 --> 01:09:56,950

 

2220
01:09:56,950 --> 01:09:56,960

 

2221
01:09:56,960 --> 01:09:57,750


2222
01:09:57,750 --> 01:10:00,390

 

2223
01:10:00,390 --> 01:10:02,229

 

2224
01:10:02,229 --> 01:10:03,669

 

2225
01:10:03,669 --> 01:10:03,679

 

2226
01:10:03,679 --> 01:10:05,910


2227
01:10:05,910 --> 01:10:08,630

 

2228
01:10:08,630 --> 01:10:09,910

 

2229
01:10:09,910 --> 01:10:12,550

 

2230
01:10:12,550 --> 01:10:15,910

 

2231
01:10:15,910 --> 01:10:18,149

 

2232
01:10:18,149 --> 01:10:21,110

 

2233
01:10:21,110 --> 01:10:23,189

 

2234
01:10:23,189 --> 01:10:26,790

 

2235
01:10:26,790 --> 01:10:29,350

 

2236
01:10:29,350 --> 01:10:29,360

 

2237
01:10:29,360 --> 01:10:30,630


2238
01:10:30,630 --> 01:10:32,390

 

2239
01:10:32,390 --> 01:10:34,630

 

2240
01:10:34,630 --> 01:10:38,229

 

2241
01:10:38,229 --> 01:10:38,239

 

2242
01:10:38,239 --> 01:10:39,270


2243
01:10:39,270 --> 01:10:41,189

 

2244
01:10:41,189 --> 01:10:42,630

 

2245
01:10:42,630 --> 01:10:44,550

 

2246
01:10:44,550 --> 01:10:45,990

 

2247
01:10:45,990 --> 01:10:48,070

 

2248
01:10:48,070 --> 01:10:50,070

 

2249
01:10:50,070 --> 01:10:51,910

 

2250
01:10:51,910 --> 01:10:53,990

 

2251
01:10:53,990 --> 01:10:57,590

 

2252
01:10:57,590 --> 01:10:59,189

 

2253
01:10:59,189 --> 01:10:59,199

 

2254
01:10:59,199 --> 01:11:00,229


2255
01:11:00,229 --> 01:11:02,229

 

2256
01:11:02,229 --> 01:11:03,830

 

2257
01:11:03,830 --> 01:11:05,270

 

2258
01:11:05,270 --> 01:11:06,790

 

2259
01:11:06,790 --> 01:11:08,070

 

2260
01:11:08,070 --> 01:11:10,070

 

2261
01:11:10,070 --> 01:11:11,990

 

2262
01:11:11,990 --> 01:11:14,229

 

2263
01:11:14,229 --> 01:11:16,950

 

2264
01:11:16,950 --> 01:11:18,470

 

2265
01:11:18,470 --> 01:11:18,480

 

2266
01:11:18,480 --> 01:11:20,870


2267
01:11:20,870 --> 01:11:20,880

 

2268
01:11:20,880 --> 01:11:21,830


2269
01:11:21,830 --> 01:11:24,390

 

2270
01:11:24,390 --> 01:11:26,870

 

2271
01:11:26,870 --> 01:11:29,430

 

2272
01:11:29,430 --> 01:11:31,270

 

2273
01:11:31,270 --> 01:11:34,070

 

2274
01:11:34,070 --> 01:11:34,080

 

2275
01:11:34,080 --> 01:11:35,430


2276
01:11:35,430 --> 01:11:38,149

 

2277
01:11:38,149 --> 01:11:39,430

 

2278
01:11:39,430 --> 01:11:41,270

 

2279
01:11:41,270 --> 01:11:41,280

 

2280
01:11:41,280 --> 01:11:42,630


2281
01:11:42,630 --> 01:11:44,390

 

2282
01:11:44,390 --> 01:11:46,070

 

2283
01:11:46,070 --> 01:11:46,080

 

2284
01:11:46,080 --> 01:11:47,270


2285
01:11:47,270 --> 01:11:48,950

 

2286
01:11:48,950 --> 01:11:50,550

 

2287
01:11:50,550 --> 01:11:52,550

 

2288
01:11:52,550 --> 01:11:55,030

 

2289
01:11:55,030 --> 01:11:56,709

 

2290
01:11:56,709 --> 01:11:58,550

 

2291
01:11:58,550 --> 01:12:00,790

 

2292
01:12:00,790 --> 01:12:02,390

 

2293
01:12:02,390 --> 01:12:03,990

 

2294
01:12:03,990 --> 01:12:04,000

 

2295
01:12:04,000 --> 01:12:05,030


2296
01:12:05,030 --> 01:12:07,270

 

2297
01:12:07,270 --> 01:12:09,430

 

2298
01:12:09,430 --> 01:12:11,110

 

2299
01:12:11,110 --> 01:12:12,310

 

2300
01:12:12,310 --> 01:12:13,830

 

2301
01:12:13,830 --> 01:12:15,430

 

2302
01:12:15,430 --> 01:12:17,669

 

2303
01:12:17,669 --> 01:12:17,679

 

2304
01:12:17,679 --> 01:12:21,430


2305
01:12:21,430 --> 01:12:22,550

 

2306
01:12:22,550 --> 01:12:32,790

 

2307
01:12:32,790 --> 01:12:35,270

 

2308
01:12:35,270 --> 01:12:35,280

 

2309
01:12:35,280 --> 01:12:38,470


2310
01:12:38,470 --> 01:12:40,709

 

2311
01:12:40,709 --> 01:12:44,310

 

2312
01:12:44,310 --> 01:12:47,669

 

2313
01:12:47,669 --> 01:12:48,709

 

2314
01:12:48,709 --> 01:12:50,390

 

2315
01:12:50,390 --> 01:12:52,470

 

2316
01:12:52,470 --> 01:12:54,470

 

2317
01:12:54,470 --> 01:12:57,830

 

2318
01:12:57,830 --> 01:13:00,709

 

2319
01:13:00,709 --> 01:13:02,790

 

2320
01:13:02,790 --> 01:13:04,950

 

2321
01:13:04,950 --> 01:13:07,110

 

2322
01:13:07,110 --> 01:13:08,790

 

2323
01:13:08,790 --> 01:13:08,800

 

2324
01:13:08,800 --> 01:13:09,830


2325
01:13:09,830 --> 01:13:11,350

 

2326
01:13:11,350 --> 01:13:14,470

 

2327
01:13:14,470 --> 01:13:16,070

 

2328
01:13:16,070 --> 01:13:18,950

 

2329
01:13:18,950 --> 01:13:20,630

 

2330
01:13:20,630 --> 01:13:21,669

 

2331
01:13:21,669 --> 01:13:21,679

 

2332
01:13:21,679 --> 01:13:23,750


2333
01:13:23,750 --> 01:13:29,750

 

2334
01:13:29,750 --> 01:13:29,760

 

2335
01:13:29,760 --> 01:13:30,790


2336
01:13:30,790 --> 01:13:32,790

 

2337
01:13:32,790 --> 01:13:34,070

 

2338
01:13:34,070 --> 01:13:35,750

 

2339
01:13:35,750 --> 01:13:42,149

 

2340
01:13:42,149 --> 01:13:42,159

 

2341
01:13:42,159 --> 01:13:45,030


2342
01:13:45,030 --> 01:13:46,950

 

2343
01:13:46,950 --> 01:13:49,189

 

2344
01:13:49,189 --> 01:13:51,189

 

2345
01:13:51,189 --> 01:13:53,030

 

2346
01:13:53,030 --> 01:13:54,229

 

2347
01:13:54,229 --> 01:13:56,709

 

2348
01:13:56,709 --> 01:13:58,870

 

2349
01:13:58,870 --> 01:14:01,510

 

2350
01:14:01,510 --> 01:14:03,590

 

2351
01:14:03,590 --> 01:14:05,350

 

2352
01:14:05,350 --> 01:14:06,870

 

2353
01:14:06,870 --> 01:14:08,550

 

2354
01:14:08,550 --> 01:14:10,790

 

2355
01:14:10,790 --> 01:14:12,950

 

2356
01:14:12,950 --> 01:14:14,870

 

2357
01:14:14,870 --> 01:14:16,470

 

2358
01:14:16,470 --> 01:14:18,550

 

2359
01:14:18,550 --> 01:14:20,709

 

2360
01:14:20,709 --> 01:14:22,070

 

2361
01:14:22,070 --> 01:14:24,950

 

2362
01:14:24,950 --> 01:14:26,950

 

2363
01:14:26,950 --> 01:14:28,790

 

2364
01:14:28,790 --> 01:14:28,800

 

2365
01:14:28,800 --> 01:14:29,990


2366
01:14:29,990 --> 01:14:33,030

 

2367
01:14:33,030 --> 01:14:36,149

 

2368
01:14:36,149 --> 01:14:36,159

 

2369
01:14:36,159 --> 01:14:37,110


2370
01:14:37,110 --> 01:14:39,189

 

2371
01:14:39,189 --> 01:14:42,390

 

2372
01:14:42,390 --> 01:14:44,950

 

2373
01:14:44,950 --> 01:14:44,960

 

2374
01:14:44,960 --> 01:14:52,229


2375
01:14:52,229 --> 01:14:53,990

 

2376
01:14:53,990 --> 01:14:56,070

 

2377
01:14:56,070 --> 01:14:57,990

 

2378
01:14:57,990 --> 01:14:59,910

 

2379
01:14:59,910 --> 01:15:02,149

 

2380
01:15:02,149 --> 01:15:03,350

 

2381
01:15:03,350 --> 01:15:05,510

 

2382
01:15:05,510 --> 01:15:07,510

 

2383
01:15:07,510 --> 01:15:09,110

 

2384
01:15:09,110 --> 01:15:11,270

 

2385
01:15:11,270 --> 01:15:14,630

 

2386
01:15:14,630 --> 01:15:15,750

 

2387
01:15:15,750 --> 01:15:17,189

 

2388
01:15:17,189 --> 01:15:18,630

 

2389
01:15:18,630 --> 01:15:21,669

 

2390
01:15:21,669 --> 01:15:23,910

 

2391
01:15:23,910 --> 01:15:25,990

 

2392
01:15:25,990 --> 01:15:27,669

 

2393
01:15:27,669 --> 01:15:29,510

 

2394
01:15:29,510 --> 01:15:32,310

 

2395
01:15:32,310 --> 01:15:32,320

 

2396
01:15:32,320 --> 01:15:35,030


2397
01:15:35,030 --> 01:15:37,350

 

2398
01:15:37,350 --> 01:15:37,360

 

2399
01:15:37,360 --> 01:15:38,950


2400
01:15:38,950 --> 01:15:38,960

 

2401
01:15:38,960 --> 01:15:39,910


2402
01:15:39,910 --> 01:15:42,070

 

2403
01:15:42,070 --> 01:15:42,080

 

2404
01:15:42,080 --> 01:15:43,270


2405
01:15:43,270 --> 01:15:45,669

 

2406
01:15:45,669 --> 01:15:47,430

 

2407
01:15:47,430 --> 01:15:49,990

 

2408
01:15:49,990 --> 01:15:51,110

 

2409
01:15:51,110 --> 01:15:53,750

 

2410
01:15:53,750 --> 01:15:55,350

 

2411
01:15:55,350 --> 01:15:57,830

 

2412
01:15:57,830 --> 01:15:59,270

 

2413
01:15:59,270 --> 01:15:59,280

 

2414
01:15:59,280 --> 01:16:00,070


2415
01:16:00,070 --> 01:16:01,030

 

2416
01:16:01,030 --> 01:16:02,870

 

2417
01:16:02,870 --> 01:16:04,830

 

2418
01:16:04,830 --> 01:16:07,430

 

2419
01:16:07,430 --> 01:16:07,440

 

2420
01:16:07,440 --> 01:16:08,390


2421
01:16:08,390 --> 01:16:10,790

 

2422
01:16:10,790 --> 01:16:13,350

 

2423
01:16:13,350 --> 01:16:16,149

 

2424
01:16:16,149 --> 01:16:16,159

 

2425
01:16:16,159 --> 01:16:17,189


2426
01:16:17,189 --> 01:16:18,310

 

2427
01:16:18,310 --> 01:16:19,669

 

2428
01:16:19,669 --> 01:16:22,070

 

2429
01:16:22,070 --> 01:16:24,310

 

2430
01:16:24,310 --> 01:16:26,229

 

2431
01:16:26,229 --> 01:16:28,310

 

2432
01:16:28,310 --> 01:16:30,149

 

2433
01:16:30,149 --> 01:16:31,910

 

2434
01:16:31,910 --> 01:16:34,390

 

2435
01:16:34,390 --> 01:16:36,390

 

2436
01:16:36,390 --> 01:16:38,950

 

2437
01:16:38,950 --> 01:16:41,030

 

2438
01:16:41,030 --> 01:16:42,390

 

2439
01:16:42,390 --> 01:16:43,590

 

2440
01:16:43,590 --> 01:16:45,750

 

2441
01:16:45,750 --> 01:16:45,760

 

2442
01:16:45,760 --> 01:16:47,270


2443
01:16:47,270 --> 01:16:49,350

 

2444
01:16:49,350 --> 01:16:51,030

 

2445
01:16:51,030 --> 01:16:52,790

 

2446
01:16:52,790 --> 01:16:52,800

 

2447
01:16:52,800 --> 01:16:53,990


2448
01:16:53,990 --> 01:16:55,750

 

2449
01:16:55,750 --> 01:16:58,870

 

2450
01:16:58,870 --> 01:17:00,550

 

2451
01:17:00,550 --> 01:17:02,950

 

2452
01:17:02,950 --> 01:17:02,960

 

2453
01:17:02,960 --> 01:17:05,510


2454
01:17:05,510 --> 01:17:08,229

 

2455
01:17:08,229 --> 01:17:09,510

 

2456
01:17:09,510 --> 01:17:13,110

 

2457
01:17:13,110 --> 01:17:15,430

 

2458
01:17:15,430 --> 01:17:31,750

 

2459
01:17:31,750 --> 01:17:31,760

 

2460
01:17:31,760 --> 01:17:33,840