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why don't we get started  so as i like to do let's just  review where we um have been recently um  which is to  um  discuss  context-free languages we talked about  the context-free grammars and the  push-down automata  as a way of describing the context-free  languages  as you remember  the context-free languages are a larger  class of languages than the regular  languages which is where we started the  languages of the finite automata so when  you add a stack you get more power you  get more languages that you can do  and we're very rapidly going to be  moving on today to  our main model for the semester which is  called the turing machine so let's just  take a look at what we're  going to be covering today  and that is  first we're going to show that a  technique analogous to the one we use  for um  [Music]  proving that languages are not regular  but this time for proving languages are  not context free  so the  push down automate and the grammar still  have their limitations in terms of what  we normally think a computer can do  and with that we're going to use that as  a kind of a lead into our  general purpose model which is the  turing machine  and uh  so we're going to talk about turing  machines  and of and aspects of of that um  and i do i would want to make comment uh  so  i have posted the solutions for the  first problem set i know you're starting  to think about the second problem set  now which i have posted as well  if you want to get a sense of what i'm  looking for in terms of the level of  detail  you can look at the solutions to problem  set one because i consider those to be  model solutions uh that's part of the  reason why i post them just to give you  a sense of the level of detail that i'm  looking for which is not a whole lot but  i do want to make sure you're capturing  the main ideas um  of what  of what's involved in solving the  problem  so have a look at those um  and  for problem set two which i'll talk  about um in a second so i'll just say a  few words if you want to pull that up um  you know you can do that but just to get  you started on a few of the problems if  you're  uh finding some challenges there i i  don't want you to get stuck really  before you even understand what the  problem is saying  um  so for problem number one if you looked  at that so that's a problem where you're  asked to prove um a certain language is  not context-free so that's  um and by the way all of the problems in  this problem set except uh perhaps for  the last one for number six you'll be  able to solve we'll have enough material  at the end of today's lecture to solve  all of them um  i believe that's right uh  yeah so number six you should have  enough as of thursday's lecture to solve  that um  um  uh problem number one um  uh it's proving a language is not  context-free so we're going to introduce  a method for doing that that method is  going to come in handy  for parts b and c uh if you look at the  problem set it has this strange looking  thing sigma sigma sigma and parenthesis  star it's really just a regular  expression that's very simple  um you know  you should just make sure you understand  that that's a way of representing all  strings whose length is a multiple of  three  um  and if i stick a sigma in front of that  it's all strings whose length  is one plus a multiple of three  um  so  um  once you understand that and if you  think about what kinds of strings are in  the language c2 it'll help you want to  understand what happens when you take  those unions um  and um  parts b and c are not intended to be  very hard but you just have to  understand what's going on  uh problem number two is about ambiguous  grammars i touched on that briefly in  lecture um  it's enough to solve the problem um the  book has a little bit more detail about  uh  um  ambiguous languages uh ambiguous  grammars  uh  ambiguous grammars i should say um  and uh  so this is a grammar that's supposed to  represent a fragment of a programming  language um  you know with uh if thens um and if then  else's uh you know i'm sure you're all  familiar with uh those kinds of  constructs in programming languages  um and there is a natural ambiguity that  comes up in a programming language you  know if you have  you know if some condition then  statement one else statement two i  presume you understand what the  semantics of that is what that means um  and the tricky thing is that if you have  a you know those statements can  themselves be if statements and so if  you have the situation where you have  if then  and if then else is the uh what follows  that the question is where does the else  attach is it to the second if foot to  the first if um so that's kind of a big  hint on this problem but that's okay um  i think you uh should uh  um uh  you know you need to take that and  figure out how to get a  uh an actual member of the language  which is ambiguously generated um and  then show that it has that show that it  is by showing two parse trees uh  now or or two  left-most derivations if you read the  book you'll see that's an alternative  way of of representing comparison  so um  and then what you're supposed to do is  give a grammar for the same language  which is unambiguous um you don't have  to prove that it's unambiguous because  that's a bit of a chore but as long as  you understand what's going on you  should be should be able to come up with  an unambiguous grammar which resolves  that ambiguity  and i don't have in mind changing the  language by introducing new uh  programming languages contra constructs  like a begin end uh that's not in the  spirit of this problem because  that's a different it's grammar for a  different language so you need to be  generating the same language um without  any other extraneous things going on  that are going to resolve the ambiguity  the ambiguity needs to be resolved  within the structure of the grammar  itself  so keep that in mind  uh for problem number three about the  key automata um you know that came up  actually as a suggestion um uh last  lecture i believe or two lectures back  what happens if you take a push down  automaton but instead of a push down uh  as a instead of a stack you add a q  um what happens then well actually it  turns out that the model you get is very  powerful and turns out to be equivalent  in power to a turing machine so you'll  see arguments of that kind today how you  show that um  other models are equivalent no not today  that's going to be on um  uh  that's going to so i apologize the key  this is going to be something that  you'll also you know i i i i'm confusing  myself here uh for problems number  problem number three you actually need  thursday thursday's lecture as well uh  to really  at least see examples of how you do that  kind of thing  um  and so uh so thursday  uh  yeah so i i i'll try to send out a note  clarifying this by the end of thursday  you'll be able to do everything except  for problem six and for problem six  you'll need tuesday's lecture a week  from today's lecture to do  um  so uh  um all right uh so  um  problem number four that one you'll be  able to do at the end of today we'll  that's also gonna you know the problem  is i'm i'm working on preparing  thursday's lecture too so i'm getting a  little i'm confusing myself uh problem  number four you'll be able to do after  thursday's lecture maybe we should talk  about that um  next lecture  um  problem number five um you can do today  um  but maybe i'm not gonna say anything  about that and problem number six i  won't say anything about either okay so  why don't we just jump in then um and uh  look at today's material um what about  seven seven is uh  oh seven is an optional problem oh i  should have mentioned that whenever  there's seven is always going to be an  optional i indicate that with a star i  should have made that clear um on the  actual description here but seven is  optional uh it's just like we had for  problem set one  okay let's move let's move on then to uh  what we're going to talk about today um  uh  and just a little bit of review um  so we talked about the equivalence of  context-free grammars and push down  automata as you remember oops let me get  myself out of the picture here um  as we mentioned last time we actually  proved one direction but the other  direction of that you just have to know  it's true but you don't have to know the  proof the proof is a little bit lengthy  i would say it's a nice proof but it's  pretty long  um  and uh  there are two uh important corollaries  to that um if you know what a corollary  is just a simple consequence which  doesn't need much of a proof uh sort of  a very straightforward consequence first  of all uh as i think we pointed out at  last time one one conclusion one  corollary you get is that every regular  language is a context-free language  because um a  finite automaton is a push down  automaton that just happens not to use  its stack  so immediately you get that every  regular language is context-free  and second of all um you also  immediately get that  whenever you have a context-free  language and a  and a regular language and you take  their intersection  you get back a context-free language  so context-free intersect regular is  context-free  okay that that's actually mentioned in  your homework as well as one of the  zero-point x problems um which i give to  try to get you um they're not you don't  have to turn those in but i suggest you  look at them i don't know how many of  you are looking at them but uh this is a  useful fact and and some of those other  facts and in zero point x problems are  useful so i encourage you to look at  them  but anyway intersection of context-free  and regular is context-free you might  ask what about intersection of  context-free and context-free do we have  closure under intersection the answer is  no  we do not have close closure under  intersection we'll talk about that  shortly uh so here's the proof sketch  for um  i should i wanted to say that the  intersection of context free and regular  why do we know that's still context-free  because the push down automaton for a  can be simulating the finite automaton  for b  inside its finite control  inside its finite memory  the problem is if you have two  context-free languages you have two  push-down automata you can't simulate  that with one push-down automaton  because it has only a single stack  so if you're trying to take the  intersection of two context-free  language languages with only a single  stack you're going to be in trouble  because it's hard to um  anyway i mean that's not a proof but at  least it shows you what goes wrong if  you try to do the obvious thing  um  okay  uh so  uh  so if  if then just here's an important point  that i was trying to make before if a  and b are both context-free and you take  the intersection  uh the result may not necessarily be a  context-free language um so the class of  context-free languages is not closed  under the intersection we'll comment  down that uh uh in a bit  um  the context-free languages are closed  under the regular operations however  union intersection  a union concatenation and star so you  should um feel comfortable that you know  how to prove that it's in the again it's  one of the  uh  i think it's problem 0.2 and i think the  solution is even given in the book for  it so you just should know how to prove  that um  uh not very  pretty straightforward um okay  so let's move on then  um to uh  to really basically conclude our um work  on context-free languages and to  understand the limitations of uh content  context-free grammars and what kinds of  languages may not be context-free and  how do you prove that so how do you  prove that for some language there is no  grammar  um  again you know it's not enough just to  say  you know  uh given informal  uh  comment that you will i i couldn't think  of a grammar that or some things of that  kind that's not going to be good enough  we need to have a proof  uh so if we take the language here  0 to the k1 to the k2 to the k so those  are strings which are runs of zeros  followed by an equal number of ones  followed by an equal number of twos so  just for zeros then ones then twos all  the same length  um  uh that's a language which is not going  to be a context-free language and we'll  we'll give a method for proving that  you know if you had a stack you can  match the ones with the zeros but then  once you're done with that the stack is  empty and how do you now make sure that  the number of twos corresponds to the  number of ones that you had so again  that's an informal argument that's not  good enough to uh to be a proof but it  sort of gives an intuition um so we're  going to give a method for proving  non-context-free  languages are not context-free using  again  a pumping lemma  but this is going to be a pumping lemma  that applies to context-free languages  not to regular languages okay it looks  very similar but it has some extra  wrinkles thrown in  because the other older  pumping dilemma was specific to the  regular languages and this is going to  be something that applies to the context  free languages  okay so now um let's just read it and  then we'll um  try to interpret it again it's very  similar in spirit basically it says that  whenever you have a context-free  language all long strings in the  language can be pumped  in some kind of way so it's going to be  a little different kind of pumping than  we had before  and you stay in the language  okay so  before we broke the string into three  pieces where we could repeat that center  piece as many times as as you like um  and you stay in the language here we're  going to end up breaking the um the  string into five pieces so s is going to  be broken up into u v x y z um  uh  and  the  and the way it's gonna work here so here  is a picture so all long strings again  there's gonna be a threshold um for  whenever you have a language there's  going to be some cutoff length  so that all the longer strings in that  language can be pumped and you stay in  the language but the shorter strings  there's no guarantee  um so if you have a long string in the  language length at least this pumping  length p  then you can break it up into five  pieces but now it's that  second and fourth string that are going  to play that special uh pumping roll um  which means that what you can do is you  can repeat those  um  and you stay in the language  and it's important that you repeat them  both that v and that y the same number  of times so you're gonna have a picture  that looks something like this  um and that is going to  you you repeat  uh so you can get you if you repeat the  v and repeat the y you get u  v v x y y z um or if you look at over  here it would be u v squared x y squared  is e  um  and that's going to still be in the  language  um and then we have so that's one  condition we'll have to look at all of  these conditions when we do the proof  but we just understand what the  statement is right now so the second  condition is that v and y together  cannot be empty and really that's  another way of saying they can't both be  the empty string because if they were  both the empty string then repeating  them wouldn't change  um  s  and then of course it would stay in the  language so it would be kind of  meaningless if they were allowed to be  empty  and the last thing is again going to be  there as a matter of convenience for  proving str proving  uh languages are not context-free  because  you have to make sure there's no  possible way of cutting up the string  when you're trying to prove a language  is not context-free you have to show the  pumping fails and then you  it's it's going to be helpful sometimes  to limit the ways in which the string  can be cut up because then you have it's  an easier job for you to work with it um  so here it's a little different than  before but sort of similar that v x y  combined um as a substring so i think i  show that over here  v x y together  is a um  is not too long so the v vxy maybe is  better seen up here is going to be is  going to be at most p  um we'll we'll do an example in a minute  of using this okay so  again here's our pumping limit i just  restated it so um you know we have it in  front of us  and we're going to do a proof  i'm just going to give you the idea of  the proof first and then we'll go  through some of the details the idea is  actually pretty simple  okay  i give it a call proof by picture  again remember what we're trying to do  we're trying to show that we have this  context-free language a  and now  all long strings in a have this pumping  quality that you can break them up into  five pieces  so that the second and the fourth piece  can be repeated and you stay in the  language  so how do we know that that's going to  be true  let's take a look at the proof here and  how why is that going to be true  so first of all  i i i'd like to do it qualitatively  rather than quantitatively so let's just  imagine instead of what thinking we'll  calculate what p is later but just  imagine the s is some really really long  string  that's the way i like to think about it  so s is just really long  what is that going to tell us  it's going to tell us something  important  about  the way the grammar produces s  which is going to be useful in in  getting a way of pumping  pumping s  so if s is really long  um  we're going to look at the parse tree  for s  and we're going to conclude that the  parse tree has to be really tall  because it's impossible for a very  shallow  parse tree to generate a very long  string  and again we'll quantify that in a  second but intuitively i think that's  not too hard to see why that ought to be  true so if you have a long s the pole  the parse tree has to be really tall  because the parse tree can't generate  very many it can't expand by very much  at each level  so we'll look at how much it can expand  but it depends on the grammar  how much expansion can we have at each  level and it's going to be you can't  have just in three levels  you know some small grammar generating a  string of length a million just you know  you'll see that that's just impossible  so once you know that the parse tree is  really tall here  um  then you're actually almost done  because what does it mean to be really  tall it means that there's some path  starting at the start variable e i'm  calling it uh in this parse tree  which goes down to some  um  terminal  symbol  in s  which goes through many steps that's  what it means for the tree to be very  tall  and each one of those steps is a  variable until you get down to the very  end  okay so that's the way parse trees look  you keep expanding variables until you  get to a terminal  so here's that here you get some some  path that's really a long path  and once you have a long path that have  many many variables appearing on here  well  the grammar itself has only some fixed  number of variables in it so you're  going to have to have a repetition  coming among the variables that occur on  that long path  got that  so a long string forces a tall  parse tree forces a repetition  on some path coming out of the start  variable  uh of some of some other variable that  comes out  now that's going to tell us how to cut  up s  because if you look at the subtrees of s  that those two r variables are  generating  shown like this  i'm going to use that  so if you have to follow what i'm saying  here so you know the r here is  generating this  portion of s and the lower r is  generating a smaller portion of s  just  looking at the subtree that you get here  um and that's going to tell us that we  can cut up s  accordingly  so u is that very first part out here  generated by e but not by the first r r  is generated v is generated by the first  r but not by the second r the second r  generates exactly x  and then we have y and z similarly  so that all follows from having a lo a  tall  parse tree  and now we're finished  now we know how to cut up s how do we  know  we can repeat  uh v and y and still be in the language  well i'll actually show your and show  you that you're in the language by  exhibiting a parse tree for the string u  v v x y y z here it is  what i'm gonna get that parse tree by  when i expand this lower r  instead of expanding it to get x i'm  going to follow the same substitutions  that i i had when i expanded the upper r  so it's as if i took this larger subtree  here and i substituted it in for the  smaller uh subtree under under the  second r and so that i get a picture  that looks like this  so here i'm substituting under the  second r the same subtree that i had  originally coming out of the upper r the  first r  and so now this parse tree is generating  the string u v v x y y z  which is what i'm looking for and of  course you can do that again and again  and you're going to keep getting higher  and higher exponents of v and y  um  and in fact you can even get the zero  exponent which means that v and y both  disappear altogether  and for that you do something slightly  different which is that you replace  the larger subtree by the smaller  subtree  okay so here which was originally that  larger subtree generating v x y i stick  instead the smaller subtree i do the  substitutions from the smaller sub tree  and i just get x uh there and so now the  string i generated is u xz which is the  same as u v to the zero x y to the zero  z  okay that's that is the idea of the  proof  now i think  uh you could work out um  [Music]  the quantities that you need in order to  drive this proof um i'm gonna do that  for you i actually hate writing down  lots of inequalities and um  equations and so on on the board because  i think they're just almost  incomprehensible to follow at least they  would be for me but i'm going to put  them up there just for completeness sake  um  so here we're going to give the details  of this proof on the next uh the next  slide here  oh yeah so i just want to give a name to  this i'm going to call this the cutting  and pasting argument because i'm cutting  apart  pieces of this parse tree and i'm  pasting them into other places within  the parse tree to get new strings being  generated  so this is a cutting and pasting  argument so okay let's let's take a look  at the details here um  uh just so we have to understand well  how big does p actually need to be  in order for this thing to kick in  um  well first of all we have to understand  how fast that parse tree can be growing  um as we go level to level and that's  going to be dependent on how big the  right-hand side sides of rules are i  mean that's expand that really tells you  how many what's the fan out you know of  each node  what's the maximum fanout and that's  going to be the maximum uh  length of a right-hand side of any rule  so for example in that other grammar we  had seen last time  for arithmetic expressions we had this e  goes to e plus t and  uh this rule here and in terms of the  parse tree that would look uh like an a  little element like that  and that's actually the longest right  hand side that you can get and so the  parse tree can be growing by a factor of  three each time  now that's going to tell us  how  big the string uh  needs to be that's being generated what  is the value of p in order to get a  high enough parse tree so that you're  going to get a repeated variable  okay um  let's call the height of the parse tree  for sh  okay  so now if you this is just repeating  what i just said if you have a tree of  height h and the maximum branching is b  then you get it most b to the h leaves  because each level you get another  factor of b coming up because that's how  much branching you have  so each node at one level can become b  nodes at the next level down so you're  multiplying by b each time and if you  have h levels you're going to have b to  the h leaves  so the length of s which are really the  leaves here that it is at most b to the  a the reason why i said most and not  exactly is you might be doing  substitutions which are shorter right  hand sides  um  okay so to try to show this as a picture  here uh  pulling that same picture we had before  we want h  the height to be bigger than the number  of variables  to force a repetition  so the number of variables is going to  be written this way v is the variables  uh v with bars around it is going to be  the number of variables and we want that  height to be greater than the number of  variables  so once you know how high you want that  tree to be in order to force the  repetition  then it tells you how big s has to be  so b has to be bigger  than b to the v b to the size of v  um because then  um the the the height that you're going  to get is going to be greater  than uh  uh the size of v which is uh so that's  what you want you want h to be greater  than the size of v  uh so you're gonna set p to be one more  than b to the v  and so if s is at least that length  this whole thing is going to kick in and  you're going to get that repeated  variable um  so we let p to be that value where v is  the number of variables in the grammar  and so if  s  is at least p  which is greater than b to the v  then the length of s is going to be  greater than  um b to the v so h is going to be what  you want to make this thing work  i  you know if you don't follow that those  inequalities i sympathize with you i  would never follow that either in a  lecture so  um but i hope you get the idea um but  you know we're not quite finished yet  because what i want to now circle back  as and look at these three conditions  and make sure  that we've captured them all because  actually it's not totally  uh obvious um  in each of those cases that we've got  them um so there's a few extra things we  need to do um um  okay so this is concluding the argument  you know there are going to be at least  three plus one variables in the longest  path so there's going to be a repetition  so now let's go back here and see now  that we have this picture  with a repeated uh variable  how do we know we can get condition one  well that's just the cutting and pacing  argument from the previous slide  how do we know that v and y are not both  empty  well actually that's not totally obvious  because it's possible  that when you generated v here and you  generated y  um maybe going from this r to that r you  got nothing new  you know it could have been that r got  replaced by t  another variable with nothing new coming  out and then t got replaced by r so you  substituted uh  you know t for r and then r for t and  you've got nothing new coming out and in  that case uh v and y would both be the  empty string and that would violate what  we want  um  the way you  get around that and these are details  here if you're not totally following  these points don't worry uh but i do  want to just try to it's easy they're  easy to describe so i figure let me  present the whole thing in full detail  so if going from this r to that r  doesn't generate anything new you're  getting exactly the same things coming  out v and y are just  the empty string  um how do we avoid that from happening  well what there's a simple way to  address that which is to say  uh if you have  this string s  when you take a parse tree make sure you  take a smallest possible parse tree  you cannot you're not allowed to start  off with an inefficient parse tree that  can be shortened and still generate s i  want the smallest possible parse tree  and that smallest possible parser can't  have an r going to another r which is  generating nothing new because then you  could always have eliminated that step  and you would still have a parse tree  for s but it would be a smaller parse  tree so that would be i want you to  start off with the smallest possible  parser tree and then you're going to be  guaranteed that v or y is going to be  something uh not empty so that takes  care of  uh condition two  condition three  uh  you know how do we know that v x  y together is not very long  and basically it's the same argument all  over again um you just want to make sure  that when you're picking the repetition  are the two r's here you pick the lowest  possible repetitions that occur if you  have many many choices  and those lowest two those lowest  repetitions there's not going to be any  lower repetition here and then by the  same argument um the the uh  since once you have that very first r  there's no more repetitions occurring  below  the v x y can't be very long um because  that would again force another  repetition to occur  um so anyway those are the three  conditions and that's the proof of the  pumping lemma for context-free languages  let's see how we use that  um  okay so let's do an example of proving a  language not context free using the  pumping line  how do you gonna do a go about doing  that because that's the kind of thing  you you know  at the very least you need to know how  to do this in order to do the homework  um  uh i'd like to motivate you that the  stuff is so interesting and fun but it  doesn't work for everybody so  uh for you practical people out there um  pay attention so you can do the homework  okay let's go back to that language we  had a couple of slides back zero to the  g one to the k two to the k it's not a  contact three language we're gonna show  that now using the pumping lemma for  context-free languages  um  uh so it's gonna do  similar to the proofs using for  non-regular non-regul not non-regular  languages um proof by contradiction so  you first you assume the language is  context-free and then we're going to  apply the pumping lemma um and then  we're going to get a contradiction uh  so the pumping limit gives that pumping  length as we subscribed above and now we  just want to pick a longer string in the  language  and show that that  longer string which is supposed to be  pumpable and stay in the language in  fact is not pumpable  um  so the pumping dilemma says that you can  divide it into five pieces satisfying  the three conditions uh condition three  implies  uh that  so now i'm gonna i'm gonna work through  the i'm gonna uh show you get a  contradiction so condition three  implies that you cannot contain both  zeros and let's pull up a picture here  so here's s  uh  zeros ones and then twos all the same  length condition three  uh so if you break it up  condition three says v x y together  cannot be too long well if v x y  together is not too long  uh what you know how could it be that  when you're repeating v and y you stay  in the language  uh for one thing you can't have  zeros ones and twos all occurring within  v x and y  um  some  uh some symbol is going to get left out  so then when you pump up you're going to  have unequal numbers of symbols  and so you're going to be out of the  language  um  okay so no matter how you try to cut it  up uh following condition three which is  one of the things that restricts the  the ways to cut it up you're going to  end up when you pump up  uh going out of the language and so  therefore it's not in  d  d is wrong  uh  b shouldn't say b  can knife can i  nice  supposed to be able to write on this  thing i guess not  i didn't test that  oh well that's supposed to be a b  um  uh so b is a context free language we  close  so that's the assumption that b is a  context-free language that's false and  we conclude that's not a context-free  language okay  let's do uh  oh yeah  i have a check-in here  um  so let's see um what we're going to  we're going to i'm going to ask you to  think about  okay  my head is blocking part of the text  oh that was a while ago um okay uh  yes so with just one question by the way  in in terms of applying the um in in the  pumping dilemma either v or y can be uh  empty but not both but anyway let's get  to this check in here  um  so let's look at these two languages a1  and a2 which look very similar to b  but a little different  um so it's a1 is 0 to the k1 to the gate  2 to the l  where k and l could be any numbers any  positive  you know uh non-negative numbers  so basically what this says saying is  that the number of zeros and ones are  going to be equal but the number of twos  can be anything  whereas a2  similar  but here we're requiring the number of  two ones and twos  to be equal and the number of zeros can  be anything  okay  now you can easily make i hope  uh you should make sure you you can  push down automata that can recognize a1  and a2  um because let's just take a1  the push down automaton can push the  zeros  as it's reading them  pop them as it's reading the ones to  match them off and make sure that  they're the same number of them  and then the twos  it doesn't care how many there are it  just has to make sure that there are no  strings there are no letters coming out  of order but any number of twos is fine  so you can easily make a presentation  recognize a1 similarly for a2  um so what can we conclude from that  here are the three possibilities let me  uh so look at that the class of content  language is not closing intersection  you know you can read it um so i i want  to pull up the uh  poll and launch that  please fill that out  um  10 seconds  again  just  if you don't know the answer just give  any answers so that uh because we're not  counting correctness  um  still a few dribbling in  okay  uh  five seconds  okay end polling uh most of you got that  right  um  i know is it okay to share these things  i don't want to make people who didn't  get the right feel bad  you know but you should understand i  think you know if you're missing  something you should understand  what is what you're missing the pumping  limit shows that a1 union a2 is not a  context-free language no as i mentioned  at the beginning the reg the  context-free languages are closed under  union  so the pumping alum had better not show  that these we already know that these  two languages are context-free because  we get them from push down on top  and we said at the beginning that  contact free language is closed under  union so we know that these two are  context-free so the pumping alignment  better not show it they're not  context-free something would be terribly  gone have gone terribly wrong uh  if that were true and also  um  we know  also from a little bit of further  reasoning that the context-free  languages is not closed under complement  by what we've already what we've already  discussed  uh because they are closed under union  as i pointed out they're not closed  under intersection  and so if they were closed under  complement  de morgan's laws would say that closure  under union and closure under complement  would give you closure under  intersection  but we don't have closer intersections  so in fact they're not closed under  complement  okay uh so in fact this does show us  that the class of context-free language  is not closed under intersection because  the intersection of a1 and a2 two  context three languages is b  and b  is not context-free so it shows that um  this is not the the  um closure under intersection does not  hold  okay um  so let us continue then  we have one more uh example then we'll  take a break  uh so the pumping limit for context-free  languages again uh here's a second  example  uh here's the language f we have  actually seen what's before ww  uh two copies of a string um in in  um  that  two copies of any string um i'm gonna  show that's not a context free language  uh  assume that it is context-free the  pumping limit gives pumping length now  here you have to do a little bit more  often the challenge  in applying  the pumping lemma in either case uh that  we've seen uh involves with choosing  that string that you need to pump that  you're gonna pump  so you have to choose s uh in f which is  longer than p which has to go with so  you might try this one first glance here  isn't string it's in the language  because it's  two copies of the string zero to the p1  zero to and then zero to the p1 that's  in the language  but it's a bad choice  uh let's say let's before i get ahead of  myself let's let's draw a picture of  this which i think is always helpful to  see  um so here is runs of zeros and then a  one runs of zeros and then a one um why  is this a bad choice  because you can pump that string and you  remain in the language um there is a way  to cut that string up  um and you'll stay in the language and  the way to cut it up is to let the x be  just that substring which is just the  one  and the v and the y can be a couple of  zeros or a single zero on either side of  that one  and now that's going to be a small  v x y  but  if you repeat v and y you're going to um  if you repeat v and y  you're going to stay in the language  because you'll just be adding zeros here  you'll be adding same number of zeros  there  and then you're going to have a string  which still looks like ww and you'll  still be in the language  so that  that means of cutting it up doesn't get  you out of the language under pumping  and the fact is that  that's a bad choice for s because there  is that way of cutting it up so you have  to show there's no way you can't you  don't get to pick the way to cut it up  um you have to show that there is no way  to cut it up um  uh in order to um  to violate the uh  the pumping lever so  if instead you use the string 0 to the  p1 to the p0 to the p1 to the p so this  is zeros followed by one followed by  zeros followed by ones all the same  number of them  that can't be pumped satisfying to the  three conditions um  and just going through that  uh  um now if you try to break it up you're  going to lose  or the or the lemon is going to lose  you're going to be happy but the lemon  is not going to be happy because it's  not going it's going to violate the  conditions um  condition 3 says  vxy is not  doesn't span too much and in fact can't  span  uh two runs of zeros or two runs of ones  it's just not big enough because they're  more than p things they're p things  apart  and this one string this string v x y is  only p long  and so therefore if you repeat  of uh v and y  you're going to have two runs of zeros  or two runs of ones that have unequal  length and now  that's not going to be of the form w and  you're going to be out of the language  okay uh so i hope  you know that's uh you've got a little  practice with that um i think we're at  our  uh break and i will see you back here in  uh five uh minutes um  i can get my timer launched here  okay so see you soon  you can this is a good time by the way  to to um  uh message me or the tas and i'll i'll  try to be uh  looking for if you have any questions  in the pumping limit can x  x can be epsilon in the pumping lemma uh  but it's not possible x can be epsilon y  can be epsilon but x and y cannot be  both cannot both be epsilon  because then when you pump you will get  nothing new  technically v and y can include both  zeros and ones  yeah  v and y can conclude both zeros and ones  um  so  let me uh  try to put that back if that's  will  um  [Music]  so v and y can have both zeros and ones  but they can't have zeros from two  different blocks  and you can't have ones from two  different blocks  so what's going to happen is either  you're going to get things out of order  when you repeat like a v has both zeros  and ones in it when you repeat  v you're going to have uh zeros and ones  and zeros and ones and zeros and ones  that's clearly out of the language so  that's so that's no good  your only hope is to have v to be  sticking only inside the zeros and y to  be sticking only inside zeros are only  inside once  but now if you repeat that if just look  at what you're going to get  you're going to have a string which is  going to be  if you try to cut that string in half  it's not going to be of the right form  it's not going to be two copies of the  same string because you know it's going  to have a run of zeros  um followed by  a longer or shorter run of zeros or one  of ones followed by another run of ones  of unequal length so there's no way this  can be two strings  um  two copies of the same string  because that that's what you're required  f has to be two copies of the same  string to be in the language  okay uh let me just see where we're  running out of time here let me just put  my uh  my  my  timer here  we've only got 30 seconds  okay sorry i'm not getting to answer all  the questions here um  okay we are done um  uh  with um  our break  it's going to come back  um and now we're shifting gears in a  major way  because in a sense everything we've done  so far  has been kind of a warm up  these limited  computational models  um  really are kind of  get helping us to set our understanding  of automata and the definitions and the  notation  and they're also going to be helpful in  providing examples later on in the term  um but really in terms of a model of  computation they don't cut it because  uh they cannot do very simple things  that we normally think of a computer as  being able to do  so here we're introducing another model  of computation called the turing machine  and that's really going to be the model  that we're going to stick with for the  rest of the semester because that's  going to be our model of a general  purpose uh  computer the way you normally think  about it  um  so let's uh we'll spend a little time  introducing it and then we will um  uh  you know we'll  we'll continue that discussion next time  um  so uh in terms of a schematic actually  the turing machine model is pretty  simple  um  it has a uh  it has um  it's going to have states and all that  stuff but you know though so there's  going to be a finite control here which  is going to include the states and  transition function as we'll describe in  a minute um the point is is that you  know it's going to have the input  appearing on a tape  uh  the key difference now is that the the  machine is going to be able to change  the symbols on the tape um  and so we think of the machine as being  able to write as well as read the tape  okay so that's really uh the key  uh  feature  um of a turing machine is the ability to  write on the tape everything else in a  sense follows from that  a few other differences  but so the fact that he he head can read  and write  so we can use the tape as  storage  much as we used a stack as storage but  it doesn't it's not limited in the way  we can access it the way a stack is so  that we kind of have very uh  flexible access of the information on  the tape  now  being able to write on the tape doesn't  do any good if you can't go back and  read what you've written later on so  we're going to make the head to be able  to be two-way  so the head can move left to right as  before but it can also move back left  and that's going to be under control of  the transition function so under under  program control essentially  the tape is going to be  oops  sorry the tape is infinite to the right  and  so we're not going to limit how much  storage the machine can have so the tape  is going to think of as having instead  of just having the input on it it's  going to have the input but then the  rest is going to have infinitely many  blanks  blank symbols following the input so the  tape is infinite or  in the right right hand direction  um  and uh so there's infinitely many blanks  i'm going to use that symbol for the  blank to follow the input  um you can accept or reject oh yeah so  that's another thing that's important  normally we think of the machine in the  previous machines  finding a time when i pushed down  automata when you got to the end of the  input that's when you the acceptance of  rejection was decided if you were in  except state at the end of the input  then you accept it but  you have to be in that location um at  the end of the input in order for that  to take effect  that doesn't make any sense anymore  because  the machine might go off beyond that and  still be computing and come back and  read the tape later on so it only really  makes sense to let the machine  accept or reject  upon entering the accept or the reject  state so we're going to have a special  accept state and a special reject state  which is also a little different than  before  and when the machine enters those states  then machine then the action takes  effect the machine those machine holds  and then accepts  or holds and then it rejects so we'll  make that absolutely clear in the formal  definition in a second but just to get  the spirit of it  um  so i'm going to give you an example of  the thing running uh sorry you need to  again my  powerpoint is having issues  um  uh  okay so here is a turing machine  recognizing that language b  actually i switched gears on you instead  of zeros ones and twos i made them a's  b's and c's but same idea  [Music]  so i'm going to show you how the turing  machine operates  and then we'll give a formal  uh definition i hope that's on here  i think it is  in a second but let's this is an  informal discussion of how the machine  is going to operate to do this language  a to the kb to the kc to the k using its  ability to write on the tape as well as  read and move its head in both  directions  okay so let me just first describe  um in english  how this um  [Music]  machine operates and then we will um  see it in action on this little uh  picture i have over here okay so the way  the machine is going to operate is the  very first thing is the head is going to  start here and the head is going to scan  off to the right  uh making sure that the symbols appear  in the correct order  so it's seeing that  that there are a's and then b's and then  c's without checking the quantities just  that the order is correct for that you  don't need to write a finite automaton  can check that the input is of the form  a star b star c star  um  so uh  rioting is not necessary  the machine if it uh detects symbols out  of order it immediately rejects by going  into a special reject state otherwise  it's going to return its head back to  the left end  and um  uh let me just show that here so here is  it  oh no  i better  before i illustrate it over here let's  go through the whole algorithm uh so the  next thing that happens is you're going  to scan right and now you want to do the  counting so you're going to scan right  again but this time you're going to make  a bunch of passes over the input bunch  of scans  and each time you make a scan you're  going to cross  off one symbol of each type so you're  going to cross off an a you'll cross off  a b you'll cross off a c on a single  scan and then repeat that  crossing off the next a the next b the  next c  and  you want to make sure that you've  crossed off all of the symbols on the  same run and not crossing off some  symbols before other symbols uh while  other symbols will remain because that  would mean that the counts were not  equal if you cross them off and they're  all  you know run out on the same scan  same pass  then we know that the numbers have to be  start off being equal sort of i mean  this is sort of baby stuff here i but i  hope you get the idea and we'll kind of  illustrate in in a second so if at least  one of the if you have the last one of  each symbol so what i mean by that is  you you just crossed off the last a the  last b and the last c then you know that  you were originally had an equal number  and so you accept because you're  crossing off one of each on each scan so  if you cross the  on the last scan  each one of them gets crossed off um  then you accept  but if  uh it was the last of some symbol but  not of other symbols  so you know you crossed off the last a  but there um there were several b's  remaining then you started off with an  unequal number of a's b's and c's  and you can reject or  if all symbols still remain after you've  crossed them uh one of each off  then you haven't  done enough passes and you're gonna  repeat  from stage three and do that uh again  another scan  okay so here's a little um animation  which shows this happening on this  diagram so here is  the very first stage where you're  scanning across making sure things are  in the right order  um didn't have to write on the tape and  now you're gonna  reset the the head back to the beginning  this is by the way not the most  efficient procedure for doing this um  so um  now that now we're going to do a scan  crossing off a single a a single b and a  single c  so here i'm going to show that here  single a single b single c  and now as soon as you've crossed out  that last c we can return back to the  beginning um  so scan right cross  uh so if all symbols remain so there are  still symbols remaining of each type  we're going to return to the left and  repeat and repeat  now we'll get another pass single a  single b single c get crossed off  we haven't crossed them all off yet  no there's  of each type there's still our remaining  ones so again we return back to the  beginning now we have a last pass  crossover the last a the last be the  last c the last one of each type was  crossed off so now we know we can accept  because the original string was in the  language  okay so that's  uh to give you at least some idea um  that how the turing machine can operate  you know more like the way you would  think of a computer operating you know  maybe it's very primitive you know you  you could imagine counting also and the  termination can count as well but this  is the simplest uh descript uh procedure  that i can just describe on um you know  uh  for you without getting making too  complicated um  okay so let's do a little checking on  that um  okay so  the way i'm describing this  how do we go how do you think  and you in a sense you don't quite know  enough yet but um  [Music]  uh how do you think we're going to get  this effect  of crossing off with the turing machine  um are we going to get that by  changing the model and adding that  ability to cross off to the model  um are we going to use a tape alphabet  that includes those crossed off symbols  among them  or um we'll just assume that all turing  machines come with the eraser and they  can always cross our stuff  um so what do you think is the  nice way sort of mathematically to  describe  this  this ability to cross things off  yeah again again most of your again i  think are getting this um  uh  so  like 10 laggards here so  please wrap it up um so we can  close the poll  uh  five seconds to go  um  okay  polling ending get your last last call  uh  all right  share the results  so most of you  uh got that right  all turn machines come with an eraser i  don't know that was thrown in there as a  joke but it got came in second so i  don't feel bad if you got it but that's  not what i had in mind  um the the way the turing machine is  going to be riding on the tape is to  write a crossed off symbol uh instead of  the uh  the symbol that was originally there so  we're going to add these new crosstalk  symbols and that's going to be a common  thing for us to do  in um you know when we design turing  machines we're not going to get down to  the implementation level for very long  we're going to very quickly shift to a  higher level of discussion about the  machines but anyway that's how you would  do it if you were going to actually  build a machine  okay so um  let us then look at the formal  definition of the cursing and maybe i  should have done that checking after the  formal definition that might have been  clear but oh well  uh  okay here's the formal definition this  time a turing machine is a seven  tuple um  and  there is a  uh  um  now here  uh we have sigma which is the input  alphabet gamma is the tape alphabet so  now you're it's sort of a little bit  analogous to the stack from before where  gamma was the stack alphabet but these  are the symbols that you're allowed to  write on the tape um or that are allowed  to be on the tape um  so uh obviously all of the input symbols  are among the uh tape uh symbols  because they can appear on the tape so  you have sigma  is a subset of gamma um  one thing i didn't mention here is that  the input alphabet  we don't allow the blank symbol to be uh  in the input alphabet so that  you can actually use the blank symbol as  a delimiter for the end of the input  a marker for the end of the input um so  uh in fact and blank symbol is always  going to be in the tape alphabet uh so  we have um  you know  the this is actually always going to be  a proper subset uh because of the blank  symbol um what we're just allowing  doesn't really matter we're allowing the  tape alphabet to have  other symbols for convenience so for  example these crossed off symbols  now let's look at what the transition  function how that operates so the  transition function remember tells how  the machine is actually doing its  computation  and it says that if you're in a certain  state and the head is looking at a  certain tape symbol  um  then you can go to a new state you write  a new symbol on the uh  at that location on the tape and you can  move the head either left or right  so that's how we get the effect of the  head being able to be bi-directional  and here's the writing on the tape  comes up right here so  you know just an example here we says if  we're in state q and the head is looking  at an a currently on the tape then we  can move to state r we change it a to a  b and we move the head right once one  square  okay um  uh  uh now here this is important  to  on when you're giving a certain input  here to the turing machine it may gr  compute around for a while moving its  head back and forth as we were showing  and it may eventually halt by either  entering the q accept state or the q  reject state which i didn't  uh bring out here but that's important  these are the accepting rejecting  special states of the machine um or the  machine may never enter one of those it  may just go on and on and on and never  halt  we call that looping  a little bit of a misnomer because it's  the looping implies some sort of a  repetition for us looping just means not  holding  um  and um  so therefore m has three possible  outcomes for each um input  w it might accept w  by uh entering uh the except state  it could reject w by entering the reject  state which means it's going to reject  it by holding  or we also say we can reject by looping  you can reject the string by running  forever  um  okay that's just the terminology that's  common in the subject so you either  accept it um by halting and accepting or  rejecting it by either holding and  rejecting or by just going forever  that's also considered to be rejecting  sort of  sort of rejecting the instance by  default if you never actually have  accepted it then it's going to be  rejected  okay uh  checking three here  all right  so now um our last check in for the day  we say this turing machine model is  deterministic  um  i'm just saying that but if you look at  the way we set it up  if you've been following these formal  definitions so far you would understand  why it's deterministic so let's just as  a way of checking that um how would we  change this definition because we will  look on at the next lecture at  non-deterministic turing machines so a  little bit of a lead into that how would  we change  this uh definition to make it a  non-deterministic turing machine which  of those three options would we use  so here i'll launch that poll  uh  got about 10 people left let's give them  another 10 seconds  um  okay i think that's everybody who's  answered it from before  um  here there is so here  i think you pretty much  almost all of you got the right idea uh  it is b in fact uh because when we have  the power set symbol here that means  there might be several there's a subset  of possibilities so that indicates  several different ways to go  um  uh and that's what the that's the  essence of non-determinism  okay so uh i think we're  uh  whoops  um  okay  all right  so look  this is also kind of setting us up for  next lecture  and where we're going to be going with  this um so these are uh  basically two in example two or three  important definitions here um  one is um  we have we talked about the regular  languages  from fine automata we talked about the  context free languages from the grammars  and the push down automata what are the  languages of that the touring machines  can do  those are called in this course anyway  turing recognizable languages is very  orti recognizable um those are the lang  those are the languages that the turing  machine can recognize  and so what  just to make sure we were on the same  page on this  the language of the machine is the  collection of strings that the machine  accepts  so the things that are not in the  language are the things that are  rejected either by looping or by halting  and projecting  so all the only the ones that are  accepted are the light is the language  every machine has just a single language  it's a language of all strings that that  machine accepts um  and we'll say that m recognize that  language if that language is  a collection of such strings that are  accepted  and uh we will call that language a  touring recognizable language  okay if there's some turing machine that  can recognize it  um now uh  this feature of being able to reject by  running forever is a little bit weird  perhaps um  uh  and  uh for from the standpoint of  practicality  uh it's more convenient if the machine  always makes a decision to accept or  reject in a finite time  and doesn't just reject by going forever  and so we're going to  bring out a special class of turing  machines that have that feature  that they always halt they always go to  the holding states by the way maybe  didn't say this explicitly are the q  accept and the q eject states the accept  and reject states are the holding states  so the machine halts that means it ends  up in one of those two so it has made a  decision of accepting or rejecting in in  by the at the point in which it has  halted  um so we'll say a machine is a decider  if it always holds on every input  uh so for every w feed in the machine is  eventually going to come to a q accept  or to reject we call such a machine a  decider  and now we're going to say a language  uh  is that so it will say that the machine  decides a language  if  if it's the language of the machines of  the collection of accepted strings uh  and the machine is a decider so we'll  say that instead of just recognizing the  language we'll say that it decides the  language  um  and a turing decidable language is a  language um  that the machine  of all strings the machine accepts uh  for some turing machine which is is a  decider which is a touring machine that  always holds  so if a turing machine may sometimes  reject by looping then it's only  recognizing this language if the turing  machine is always holding so it's always  rejecting by explicitly coming to a  reject state and halting then we'll say  it's actually deciding the language  so in a sense that's better and we're  going to distinguish between those two  because  um  they're not the same  uh there are some languages which can be  recognized but not decided  and so in fact we're going to get the  following picture here  the deteuring recognizable languages are  a proper subset they include all this  everything that's decidable certainly is  going to be recognizable  because um  uh you know this being a decider is an  additional restriction to impose  additional requirement  so everything that's decidable is going  to be automatically recognizable but  there are things which are recognizable  but you're not decidable as we'll see  um  i'll actually give an example of that  but not prove it next lecture  and just for just to complete out this  picture um  i'm going to uh  also point out we haven't proven this  yet but uh we will prove it um that the  decidable languages also include  all the context-free languages which in  turn include the regular languages as it  was already seen so we haven't shown  this inclusion yet but actually this is  the picture we we get um so there's  actually a hierarchy of containments  here  regular languages are  a subset of the  context-free languages which are in turn  a subset of the decidable languages  which are in turn are a subset of the  recognizer turing recognizable languages  um and so  with that i think we're going to move to  our little bit of a review of what we've  done today  so we proved that pumping lemma  as a tool for showing that languages are  not context-free languages  we define turing machines which is going  to be our model um  that we're going to be focusing on for  the rest of the term uh not forgetting  the other models because they're going  to be useful examples for us  and we define touring deciders turing  machine deciders that halt and all  inputs  okay so i think with that um we have  come to the end of today's lecture  um i will stick around a little bit and  answer questions in the chat um i will  try to share them with everybody um as  i'm answering them so i'm not just  talking to one person  how was the concept of pride applied in  so i'm getting one question about  the practicality of all this uh bunches  of questions are coming in  um  uh  so look uh  is this is this stuff all practical um  uh i would say um uh yes and no um  uh  i i don't know which concept you have in  mind we're going to introduce lots of  concepts in this course and  um uh the you know the concept of  um  uh of you know the fine art automata and  the push down automata and context-free  languages  definitely used in other  subjects in other fields in computer  science and elsewhere these are very  basic and fundamental notions  um and so yes and turing machines well i  mean that's a model of a general  computer if you want to understand  computation you're going to need to  understand some model and a turing  machine is a particularly simple model  and that's why we use it as it turns out  it doesn't really matter what model you  use but we'll talk about that next  next time but yeah i would say there's a  lot of applied material in this course  as time has shown  whether it's  led to things like public key  cryptography which is  used on the internet  or you know understanding various  algorithms  i mean  that's not the reason i study it i'd  study it because i'm more of them coming  at it from more of a mathematical  perspective i just find the material  very uh beautiful and interesting and  challenging um but it does have  applications um  any other questions here i think i'm  going to sign off then so get myself set  up for my office hours which is in a  different on a different zoom like  okay so thank you everybody and see you  uh on thursday bye

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why don't we get started  so as i like to do let's just  review where we um have been recently um  which is to  um  discuss  context-free languages we talked about  the context-free grammars and the  push-down automata  as a way of describing the context-free  languages  as you remember  the context-free languages are a larger  class of languages than the regular  languages which is where we started the  languages of the finite automata so when  you add a stack you get more power you  get more languages that you can do  and we're very rapidly going to be  moving on today to  our main model for the semester which is  called the turing machine so let's just  take a look at what we're  going to be covering today  and that is  first we're going to show that a  technique analogous to the one we use  for um  [Music]  proving that languages are not regular  but this time for proving languages are  not context free  so the  push down automate and the grammar still  have their limitations in terms of what  we normally think a computer can do  and with that we're going to use that as  a kind of a lead into our  general purpose model which is the  turing machine  and uh  so we're going to talk about turing  machines  and of and aspects of of that um  and i do i would want to make comment uh  so  i have posted the solutions for the  first problem set i know you're starting  to think about the second problem set  now which i have posted as well  if you want to get a sense of what i'm  looking for in terms of the level of  detail  you can look at the solutions to problem  set one because i consider those to be  model solutions uh that's part of the  reason why i post them just to give you  a sense of the level of detail that i'm  looking for which is not a whole lot but  i do want to make sure you're capturing  the main ideas um  of what  of what's involved in solving the  problem  so have a look at those um  and  for problem set two which i'll talk  about um in a second so i'll just say a  few words if you want to pull that up um  you know you can do that but just to get  you started on a few of the problems if  you're  uh finding some challenges there i i  don't want you to get stuck really  before you even understand what the  problem is saying  um  so for problem number one if you looked  at that so that's a problem where you're  asked to prove um a certain language is  not context-free so that's  um and by the way all of the problems in  this problem set except uh perhaps for  the last one for number six you'll be  able to solve we'll have enough material  at the end of today's lecture to solve  all of them um  i believe that's right uh  yeah so number six you should have  enough as of thursday's lecture to solve  that um  um  uh problem number one um  uh it's proving a language is not  context-free so we're going to introduce  a method for doing that that method is  going to come in handy  for parts b and c uh if you look at the  problem set it has this strange looking  thing sigma sigma sigma and parenthesis  star it's really just a regular  expression that's very simple  um you know  you should just make sure you understand  that that's a way of representing all  strings whose length is a multiple of  three  um  and if i stick a sigma in front of that  it's all strings whose length  is one plus a multiple of three  um  so  um  once you understand that and if you  think about what kinds of strings are in  the language c2 it'll help you want to  understand what happens when you take  those unions um  and um  parts b and c are not intended to be  very hard but you just have to  understand what's going on  uh problem number two is about ambiguous  grammars i touched on that briefly in  lecture um  it's enough to solve the problem um the  book has a little bit more detail about  uh  um  ambiguous languages uh ambiguous  grammars  uh  ambiguous grammars i should say um  and uh  so this is a grammar that's supposed to  represent a fragment of a programming  language um  you know with uh if thens um and if then  else's uh you know i'm sure you're all  familiar with uh those kinds of  constructs in programming languages  um and there is a natural ambiguity that  comes up in a programming language you  know if you have  you know if some condition then  statement one else statement two i  presume you understand what the  semantics of that is what that means um  and the tricky thing is that if you have  a you know those statements can  themselves be if statements and so if  you have the situation where you have  if then  and if then else is the uh what follows  that the question is where does the else  attach is it to the second if foot to  the first if um so that's kind of a big  hint on this problem but that's okay um  i think you uh should uh  um uh  you know you need to take that and  figure out how to get a  uh an actual member of the language  which is ambiguously generated um and  then show that it has that show that it  is by showing two parse trees uh  now or or two  left-most derivations if you read the  book you'll see that's an alternative  way of of representing comparison  so um  and then what you're supposed to do is  give a grammar for the same language  which is unambiguous um you don't have  to prove that it's unambiguous because  that's a bit of a chore but as long as  you understand what's going on you  should be should be able to come up with  an unambiguous grammar which resolves  that ambiguity  and i don't have in mind changing the  language by introducing new uh  programming languages contra constructs  like a begin end uh that's not in the  spirit of this problem because  that's a different it's grammar for a  different language so you need to be  generating the same language um without  any other extraneous things going on  that are going to resolve the ambiguity  the ambiguity needs to be resolved  within the structure of the grammar  itself  so keep that in mind  uh for problem number three about the  key automata um you know that came up  actually as a suggestion um uh last  lecture i believe or two lectures back  what happens if you take a push down  automaton but instead of a push down uh  as a instead of a stack you add a q  um what happens then well actually it  turns out that the model you get is very  powerful and turns out to be equivalent  in power to a turing machine so you'll  see arguments of that kind today how you  show that um  other models are equivalent no not today  that's going to be on um  uh  that's going to so i apologize the key  this is going to be something that  you'll also you know i i i i'm confusing  myself here uh for problems number  problem number three you actually need  thursday thursday's lecture as well uh  to really  at least see examples of how you do that  kind of thing  um  and so uh so thursday  uh  yeah so i i i'll try to send out a note  clarifying this by the end of thursday  you'll be able to do everything except  for problem six and for problem six  you'll need tuesday's lecture a week  from today's lecture to do  um  so uh  um all right uh so  um  problem number four that one you'll be  able to do at the end of today we'll  that's also gonna you know the problem  is i'm i'm working on preparing  thursday's lecture too so i'm getting a  little i'm confusing myself uh problem  number four you'll be able to do after  thursday's lecture maybe we should talk  about that um  next lecture  um  problem number five um you can do today  um  but maybe i'm not gonna say anything  about that and problem number six i  won't say anything about either okay so  why don't we just jump in then um and uh  look at today's material um what about  seven seven is uh  oh seven is an optional problem oh i  should have mentioned that whenever  there's seven is always going to be an  optional i indicate that with a star i  should have made that clear um on the  actual description here but seven is  optional uh it's just like we had for  problem set one  okay let's move let's move on then to uh  what we're going to talk about today um  uh  and just a little bit of review um  so we talked about the equivalence of  context-free grammars and push down  automata as you remember oops let me get  myself out of the picture here um  as we mentioned last time we actually  proved one direction but the other  direction of that you just have to know  it's true but you don't have to know the  proof the proof is a little bit lengthy  i would say it's a nice proof but it's  pretty long  um  and uh  there are two uh important corollaries  to that um if you know what a corollary  is just a simple consequence which  doesn't need much of a proof uh sort of  a very straightforward consequence first  of all uh as i think we pointed out at  last time one one conclusion one  corollary you get is that every regular  language is a context-free language  because um a  finite automaton is a push down  automaton that just happens not to use  its stack  so immediately you get that every  regular language is context-free  and second of all um you also  immediately get that  whenever you have a context-free  language and a  and a regular language and you take  their intersection  you get back a context-free language  so context-free intersect regular is  context-free  okay that that's actually mentioned in  your homework as well as one of the  zero-point x problems um which i give to  try to get you um they're not you don't  have to turn those in but i suggest you  look at them i don't know how many of  you are looking at them but uh this is a  useful fact and and some of those other  facts and in zero point x problems are  useful so i encourage you to look at  them  but anyway intersection of context-free  and regular is context-free you might  ask what about intersection of  context-free and context-free do we have  closure under intersection the answer is  no  we do not have close closure under  intersection we'll talk about that  shortly uh so here's the proof sketch  for um  i should i wanted to say that the  intersection of context free and regular  why do we know that's still context-free  because the push down automaton for a  can be simulating the finite automaton  for b  inside its finite control  inside its finite memory  the problem is if you have two  context-free languages you have two  push-down automata you can't simulate  that with one push-down automaton  because it has only a single stack  so if you're trying to take the  intersection of two context-free  language languages with only a single  stack you're going to be in trouble  because it's hard to um  anyway i mean that's not a proof but at  least it shows you what goes wrong if  you try to do the obvious thing  um  okay  uh so  uh  so if  if then just here's an important point  that i was trying to make before if a  and b are both context-free and you take  the intersection  uh the result may not necessarily be a  context-free language um so the class of  context-free languages is not closed  under the intersection we'll comment  down that uh uh in a bit  um  the context-free languages are closed  under the regular operations however  union intersection  a union concatenation and star so you  should um feel comfortable that you know  how to prove that it's in the again it's  one of the  uh  i think it's problem 0.2 and i think the  solution is even given in the book for  it so you just should know how to prove  that um  uh not very  pretty straightforward um okay  so let's move on then  um to uh  to really basically conclude our um work  on context-free languages and to  understand the limitations of uh content  context-free grammars and what kinds of  languages may not be context-free and  how do you prove that so how do you  prove that for some language there is no  grammar  um  again you know it's not enough just to  say  you know  uh given informal  uh  comment that you will i i couldn't think  of a grammar that or some things of that  kind that's not going to be good enough  we need to have a proof  uh so if we take the language here  0 to the k1 to the k2 to the k so those  are strings which are runs of zeros  followed by an equal number of ones  followed by an equal number of twos so  just for zeros then ones then twos all  the same length  um  uh that's a language which is not going  to be a context-free language and we'll  we'll give a method for proving that  you know if you had a stack you can  match the ones with the zeros but then  once you're done with that the stack is  empty and how do you now make sure that  the number of twos corresponds to the  number of ones that you had so again  that's an informal argument that's not  good enough to uh to be a proof but it  sort of gives an intuition um so we're  going to give a method for proving  non-context-free  languages are not context-free using  again  a pumping lemma  but this is going to be a pumping lemma  that applies to context-free languages  not to regular languages okay it looks  very similar but it has some extra  wrinkles thrown in  because the other older  pumping dilemma was specific to the  regular languages and this is going to  be something that applies to the context  free languages  okay so now um let's just read it and  then we'll um  try to interpret it again it's very  similar in spirit basically it says that  whenever you have a context-free  language all long strings in the  language can be pumped  in some kind of way so it's going to be  a little different kind of pumping than  we had before  and you stay in the language  okay so  before we broke the string into three  pieces where we could repeat that center  piece as many times as as you like um  and you stay in the language here we're  going to end up breaking the um the  string into five pieces so s is going to  be broken up into u v x y z um  uh  and  the  and the way it's gonna work here so here  is a picture so all long strings again  there's gonna be a threshold um for  whenever you have a language there's  going to be some cutoff length  so that all the longer strings in that  language can be pumped and you stay in  the language but the shorter strings  there's no guarantee  um so if you have a long string in the  language length at least this pumping  length p  then you can break it up into five  pieces but now it's that  second and fourth string that are going  to play that special uh pumping roll um  which means that what you can do is you  can repeat those  um  and you stay in the language  and it's important that you repeat them  both that v and that y the same number  of times so you're gonna have a picture  that looks something like this  um and that is going to  you you repeat  uh so you can get you if you repeat the  v and repeat the y you get u  v v x y y z um or if you look at over  here it would be u v squared x y squared  is e  um  and that's going to still be in the  language  um and then we have so that's one  condition we'll have to look at all of  these conditions when we do the proof  but we just understand what the  statement is right now so the second  condition is that v and y together  cannot be empty and really that's  another way of saying they can't both be  the empty string because if they were  both the empty string then repeating  them wouldn't change  um  s  and then of course it would stay in the  language so it would be kind of  meaningless if they were allowed to be  empty  and the last thing is again going to be  there as a matter of convenience for  proving str proving  uh languages are not context-free  because  you have to make sure there's no  possible way of cutting up the string  when you're trying to prove a language  is not context-free you have to show the  pumping fails and then you  it's it's going to be helpful sometimes  to limit the ways in which the string  can be cut up because then you have it's  an easier job for you to work with it um  so here it's a little different than  before but sort of similar that v x y  combined um as a substring so i think i  show that over here  v x y together  is a um  is not too long so the v vxy maybe is  better seen up here is going to be is  going to be at most p  um we'll we'll do an example in a minute  of using this okay so  again here's our pumping limit i just  restated it so um you know we have it in  front of us  and we're going to do a proof  i'm just going to give you the idea of  the proof first and then we'll go  through some of the details the idea is  actually pretty simple  okay  i give it a call proof by picture  again remember what we're trying to do  we're trying to show that we have this  context-free language a  and now  all long strings in a have this pumping  quality that you can break them up into  five pieces  so that the second and the fourth piece  can be repeated and you stay in the  language  so how do we know that that's going to  be true  let's take a look at the proof here and  how why is that going to be true  so first of all  i i i'd like to do it qualitatively  rather than quantitatively so let's just  imagine instead of what thinking we'll  calculate what p is later but just  imagine the s is some really really long  string  that's the way i like to think about it  so s is just really long  what is that going to tell us  it's going to tell us something  important  about  the way the grammar produces s  which is going to be useful in in  getting a way of pumping  pumping s  so if s is really long  um  we're going to look at the parse tree  for s  and we're going to conclude that the  parse tree has to be really tall  because it's impossible for a very  shallow  parse tree to generate a very long  string  and again we'll quantify that in a  second but intuitively i think that's  not too hard to see why that ought to be  true so if you have a long s the pole  the parse tree has to be really tall  because the parse tree can't generate  very many it can't expand by very much  at each level  so we'll look at how much it can expand  but it depends on the grammar  how much expansion can we have at each  level and it's going to be you can't  have just in three levels  you know some small grammar generating a  string of length a million just you know  you'll see that that's just impossible  so once you know that the parse tree is  really tall here  um  then you're actually almost done  because what does it mean to be really  tall it means that there's some path  starting at the start variable e i'm  calling it uh in this parse tree  which goes down to some  um  terminal  symbol  in s  which goes through many steps that's  what it means for the tree to be very  tall  and each one of those steps is a  variable until you get down to the very  end  okay so that's the way parse trees look  you keep expanding variables until you  get to a terminal  so here's that here you get some some  path that's really a long path  and once you have a long path that have  many many variables appearing on here  well  the grammar itself has only some fixed  number of variables in it so you're  going to have to have a repetition  coming among the variables that occur on  that long path  got that  so a long string forces a tall  parse tree forces a repetition  on some path coming out of the start  variable  uh of some of some other variable that  comes out  now that's going to tell us how to cut  up s  because if you look at the subtrees of s  that those two r variables are  generating  shown like this  i'm going to use that  so if you have to follow what i'm saying  here so you know the r here is  generating this  portion of s and the lower r is  generating a smaller portion of s  just  looking at the subtree that you get here  um and that's going to tell us that we  can cut up s  accordingly  so u is that very first part out here  generated by e but not by the first r r  is generated v is generated by the first  r but not by the second r the second r  generates exactly x  and then we have y and z similarly  so that all follows from having a lo a  tall  parse tree  and now we're finished  now we know how to cut up s how do we  know  we can repeat  uh v and y and still be in the language  well i'll actually show your and show  you that you're in the language by  exhibiting a parse tree for the string u  v v x y y z here it is  what i'm gonna get that parse tree by  when i expand this lower r  instead of expanding it to get x i'm  going to follow the same substitutions  that i i had when i expanded the upper r  so it's as if i took this larger subtree  here and i substituted it in for the  smaller uh subtree under under the  second r and so that i get a picture  that looks like this  so here i'm substituting under the  second r the same subtree that i had  originally coming out of the upper r the  first r  and so now this parse tree is generating  the string u v v x y y z  which is what i'm looking for and of  course you can do that again and again  and you're going to keep getting higher  and higher exponents of v and y  um  and in fact you can even get the zero  exponent which means that v and y both  disappear altogether  and for that you do something slightly  different which is that you replace  the larger subtree by the smaller  subtree  okay so here which was originally that  larger subtree generating v x y i stick  instead the smaller subtree i do the  substitutions from the smaller sub tree  and i just get x uh there and so now the  string i generated is u xz which is the  same as u v to the zero x y to the zero  z  okay that's that is the idea of the  proof  now i think  uh you could work out um  [Music]  the quantities that you need in order to  drive this proof um i'm gonna do that  for you i actually hate writing down  lots of inequalities and um  equations and so on on the board because  i think they're just almost  incomprehensible to follow at least they  would be for me but i'm going to put  them up there just for completeness sake  um  so here we're going to give the details  of this proof on the next uh the next  slide here  oh yeah so i just want to give a name to  this i'm going to call this the cutting  and pasting argument because i'm cutting  apart  pieces of this parse tree and i'm  pasting them into other places within  the parse tree to get new strings being  generated  so this is a cutting and pasting  argument so okay let's let's take a look  at the details here um  uh just so we have to understand well  how big does p actually need to be  in order for this thing to kick in  um  well first of all we have to understand  how fast that parse tree can be growing  um as we go level to level and that's  going to be dependent on how big the  right-hand side sides of rules are i  mean that's expand that really tells you  how many what's the fan out you know of  each node  what's the maximum fanout and that's  going to be the maximum uh  length of a right-hand side of any rule  so for example in that other grammar we  had seen last time  for arithmetic expressions we had this e  goes to e plus t and  uh this rule here and in terms of the  parse tree that would look uh like an a  little element like that  and that's actually the longest right  hand side that you can get and so the  parse tree can be growing by a factor of  three each time  now that's going to tell us  how  big the string uh  needs to be that's being generated what  is the value of p in order to get a  high enough parse tree so that you're  going to get a repeated variable  okay um  let's call the height of the parse tree  for sh  okay  so now if you this is just repeating  what i just said if you have a tree of  height h and the maximum branching is b  then you get it most b to the h leaves  because each level you get another  factor of b coming up because that's how  much branching you have  so each node at one level can become b  nodes at the next level down so you're  multiplying by b each time and if you  have h levels you're going to have b to  the h leaves  so the length of s which are really the  leaves here that it is at most b to the  a the reason why i said most and not  exactly is you might be doing  substitutions which are shorter right  hand sides  um  okay so to try to show this as a picture  here uh  pulling that same picture we had before  we want h  the height to be bigger than the number  of variables  to force a repetition  so the number of variables is going to  be written this way v is the variables  uh v with bars around it is going to be  the number of variables and we want that  height to be greater than the number of  variables  so once you know how high you want that  tree to be in order to force the  repetition  then it tells you how big s has to be  so b has to be bigger  than b to the v b to the size of v  um because then  um the the the height that you're going  to get is going to be greater  than uh  uh the size of v which is uh so that's  what you want you want h to be greater  than the size of v  uh so you're gonna set p to be one more  than b to the v  and so if s is at least that length  this whole thing is going to kick in and  you're going to get that repeated  variable um  so we let p to be that value where v is  the number of variables in the grammar  and so if  s  is at least p  which is greater than b to the v  then the length of s is going to be  greater than  um b to the v so h is going to be what  you want to make this thing work  i  you know if you don't follow that those  inequalities i sympathize with you i  would never follow that either in a  lecture so  um but i hope you get the idea um but  you know we're not quite finished yet  because what i want to now circle back  as and look at these three conditions  and make sure  that we've captured them all because  actually it's not totally  uh obvious um  in each of those cases that we've got  them um so there's a few extra things we  need to do um um  okay so this is concluding the argument  you know there are going to be at least  three plus one variables in the longest  path so there's going to be a repetition  so now let's go back here and see now  that we have this picture  with a repeated uh variable  how do we know we can get condition one  well that's just the cutting and pacing  argument from the previous slide  how do we know that v and y are not both  empty  well actually that's not totally obvious  because it's possible  that when you generated v here and you  generated y  um maybe going from this r to that r you  got nothing new  you know it could have been that r got  replaced by t  another variable with nothing new coming  out and then t got replaced by r so you  substituted uh  you know t for r and then r for t and  you've got nothing new coming out and in  that case uh v and y would both be the  empty string and that would violate what  we want  um  the way you  get around that and these are details  here if you're not totally following  these points don't worry uh but i do  want to just try to it's easy they're  easy to describe so i figure let me  present the whole thing in full detail  so if going from this r to that r  doesn't generate anything new you're  getting exactly the same things coming  out v and y are just  the empty string  um how do we avoid that from happening  well what there's a simple way to  address that which is to say  uh if you have  this string s  when you take a parse tree make sure you  take a smallest possible parse tree  you cannot you're not allowed to start  off with an inefficient parse tree that  can be shortened and still generate s i  want the smallest possible parse tree  and that smallest possible parser can't  have an r going to another r which is  generating nothing new because then you  could always have eliminated that step  and you would still have a parse tree  for s but it would be a smaller parse  tree so that would be i want you to  start off with the smallest possible  parser tree and then you're going to be  guaranteed that v or y is going to be  something uh not empty so that takes  care of  uh condition two  condition three  uh  you know how do we know that v x  y together is not very long  and basically it's the same argument all  over again um you just want to make sure  that when you're picking the repetition  are the two r's here you pick the lowest  possible repetitions that occur if you  have many many choices  and those lowest two those lowest  repetitions there's not going to be any  lower repetition here and then by the  same argument um the the uh  since once you have that very first r  there's no more repetitions occurring  below  the v x y can't be very long um because  that would again force another  repetition to occur  um so anyway those are the three  conditions and that's the proof of the  pumping lemma for context-free languages  let's see how we use that  um  okay so let's do an example of proving a  language not context free using the  pumping line  how do you gonna do a go about doing  that because that's the kind of thing  you you know  at the very least you need to know how  to do this in order to do the homework  um  uh i'd like to motivate you that the  stuff is so interesting and fun but it  doesn't work for everybody so  uh for you practical people out there um  pay attention so you can do the homework  okay let's go back to that language we  had a couple of slides back zero to the  g one to the k two to the k it's not a  contact three language we're gonna show  that now using the pumping lemma for  context-free languages  um  uh so it's gonna do  similar to the proofs using for  non-regular non-regul not non-regular  languages um proof by contradiction so  you first you assume the language is  context-free and then we're going to  apply the pumping lemma um and then  we're going to get a contradiction uh  so the pumping limit gives that pumping  length as we subscribed above and now we  just want to pick a longer string in the  language  and show that that  longer string which is supposed to be  pumpable and stay in the language in  fact is not pumpable  um  so the pumping dilemma says that you can  divide it into five pieces satisfying  the three conditions uh condition three  implies  uh that  so now i'm gonna i'm gonna work through  the i'm gonna uh show you get a  contradiction so condition three  implies that you cannot contain both  zeros and let's pull up a picture here  so here's s  uh  zeros ones and then twos all the same  length condition three  uh so if you break it up  condition three says v x y together  cannot be too long well if v x y  together is not too long  uh what you know how could it be that  when you're repeating v and y you stay  in the language  uh for one thing you can't have  zeros ones and twos all occurring within  v x and y  um  some  uh some symbol is going to get left out  so then when you pump up you're going to  have unequal numbers of symbols  and so you're going to be out of the  language  um  okay so no matter how you try to cut it  up uh following condition three which is  one of the things that restricts the  the ways to cut it up you're going to  end up when you pump up  uh going out of the language and so  therefore it's not in  d  d is wrong  uh  b shouldn't say b  can knife can i  nice  supposed to be able to write on this  thing i guess not  i didn't test that  oh well that's supposed to be a b  um  uh so b is a context free language we  close  so that's the assumption that b is a  context-free language that's false and  we conclude that's not a context-free  language okay  let's do uh  oh yeah  i have a check-in here  um  so let's see um what we're going to  we're going to i'm going to ask you to  think about  okay  my head is blocking part of the text  oh that was a while ago um okay uh  yes so with just one question by the way  in in terms of applying the um in in the  pumping dilemma either v or y can be uh  empty but not both but anyway let's get  to this check in here  um  so let's look at these two languages a1  and a2 which look very similar to b  but a little different  um so it's a1 is 0 to the k1 to the gate  2 to the l  where k and l could be any numbers any  positive  you know uh non-negative numbers  so basically what this says saying is  that the number of zeros and ones are  going to be equal but the number of twos  can be anything  whereas a2  similar  but here we're requiring the number of  two ones and twos  to be equal and the number of zeros can  be anything  okay  now you can easily make i hope  uh you should make sure you you can  push down automata that can recognize a1  and a2  um because let's just take a1  the push down automaton can push the  zeros  as it's reading them  pop them as it's reading the ones to  match them off and make sure that  they're the same number of them  and then the twos  it doesn't care how many there are it  just has to make sure that there are no  strings there are no letters coming out  of order but any number of twos is fine  so you can easily make a presentation  recognize a1 similarly for a2  um so what can we conclude from that  here are the three possibilities let me  uh so look at that the class of content  language is not closing intersection  you know you can read it um so i i want  to pull up the uh  poll and launch that  please fill that out  um  10 seconds  again  just  if you don't know the answer just give  any answers so that uh because we're not  counting correctness  um  still a few dribbling in  okay  uh  five seconds  okay end polling uh most of you got that  right  um  i know is it okay to share these things  i don't want to make people who didn't  get the right feel bad  you know but you should understand i  think you know if you're missing  something you should understand  what is what you're missing the pumping  limit shows that a1 union a2 is not a  context-free language no as i mentioned  at the beginning the reg the  context-free languages are closed under  union  so the pumping alum had better not show  that these we already know that these  two languages are context-free because  we get them from push down on top  and we said at the beginning that  contact free language is closed under  union so we know that these two are  context-free so the pumping alignment  better not show it they're not  context-free something would be terribly  gone have gone terribly wrong uh  if that were true and also  um  we know  also from a little bit of further  reasoning that the context-free  languages is not closed under complement  by what we've already what we've already  discussed  uh because they are closed under union  as i pointed out they're not closed  under intersection  and so if they were closed under  complement  de morgan's laws would say that closure  under union and closure under complement  would give you closure under  intersection  but we don't have closer intersections  so in fact they're not closed under  complement  okay uh so in fact this does show us  that the class of context-free language  is not closed under intersection because  the intersection of a1 and a2 two  context three languages is b  and b  is not context-free so it shows that um  this is not the the  um closure under intersection does not  hold  okay um  so let us continue then  we have one more uh example then we'll  take a break  uh so the pumping limit for context-free  languages again uh here's a second  example  uh here's the language f we have  actually seen what's before ww  uh two copies of a string um in in  um  that  two copies of any string um i'm gonna  show that's not a context free language  uh  assume that it is context-free the  pumping limit gives pumping length now  here you have to do a little bit more  often the challenge  in applying  the pumping lemma in either case uh that  we've seen uh involves with choosing  that string that you need to pump that  you're gonna pump  so you have to choose s uh in f which is  longer than p which has to go with so  you might try this one first glance here  isn't string it's in the language  because it's  two copies of the string zero to the p1  zero to and then zero to the p1 that's  in the language  but it's a bad choice  uh let's say let's before i get ahead of  myself let's let's draw a picture of  this which i think is always helpful to  see  um so here is runs of zeros and then a  one runs of zeros and then a one um why  is this a bad choice  because you can pump that string and you  remain in the language um there is a way  to cut that string up  um and you'll stay in the language and  the way to cut it up is to let the x be  just that substring which is just the  one  and the v and the y can be a couple of  zeros or a single zero on either side of  that one  and now that's going to be a small  v x y  but  if you repeat v and y you're going to um  if you repeat v and y  you're going to stay in the language  because you'll just be adding zeros here  you'll be adding same number of zeros  there  and then you're going to have a string  which still looks like ww and you'll  still be in the language  so that  that means of cutting it up doesn't get  you out of the language under pumping  and the fact is that  that's a bad choice for s because there  is that way of cutting it up so you have  to show there's no way you can't you  don't get to pick the way to cut it up  um you have to show that there is no way  to cut it up um  uh in order to um  to violate the uh  the pumping lever so  if instead you use the string 0 to the  p1 to the p0 to the p1 to the p so this  is zeros followed by one followed by  zeros followed by ones all the same  number of them  that can't be pumped satisfying to the  three conditions um  and just going through that  uh  um now if you try to break it up you're  going to lose  or the or the lemon is going to lose  you're going to be happy but the lemon  is not going to be happy because it's  not going it's going to violate the  conditions um  condition 3 says  vxy is not  doesn't span too much and in fact can't  span  uh two runs of zeros or two runs of ones  it's just not big enough because they're  more than p things they're p things  apart  and this one string this string v x y is  only p long  and so therefore if you repeat  of uh v and y  you're going to have two runs of zeros  or two runs of ones that have unequal  length and now  that's not going to be of the form w and  you're going to be out of the language  okay uh so i hope  you know that's uh you've got a little  practice with that um i think we're at  our  uh break and i will see you back here in  uh five uh minutes um  i can get my timer launched here  okay so see you soon  you can this is a good time by the way  to to um  uh message me or the tas and i'll i'll  try to be uh  looking for if you have any questions  in the pumping limit can x  x can be epsilon in the pumping lemma uh  but it's not possible x can be epsilon y  can be epsilon but x and y cannot be  both cannot both be epsilon  because then when you pump you will get  nothing new  technically v and y can include both  zeros and ones  yeah  v and y can conclude both zeros and ones  um  so  let me uh  try to put that back if that's  will  um  [Music]  so v and y can have both zeros and ones  but they can't have zeros from two  different blocks  and you can't have ones from two  different blocks  so what's going to happen is either  you're going to get things out of order  when you repeat like a v has both zeros  and ones in it when you repeat  v you're going to have uh zeros and ones  and zeros and ones and zeros and ones  that's clearly out of the language so  that's so that's no good  your only hope is to have v to be  sticking only inside the zeros and y to  be sticking only inside zeros are only  inside once  but now if you repeat that if just look  at what you're going to get  you're going to have a string which is  going to be  if you try to cut that string in half  it's not going to be of the right form  it's not going to be two copies of the  same string because you know it's going  to have a run of zeros  um followed by  a longer or shorter run of zeros or one  of ones followed by another run of ones  of unequal length so there's no way this  can be two strings  um  two copies of the same string  because that that's what you're required  f has to be two copies of the same  string to be in the language  okay uh let me just see where we're  running out of time here let me just put  my uh  my  my  timer here  we've only got 30 seconds  okay sorry i'm not getting to answer all  the questions here um  okay we are done um  uh  with um  our break  it's going to come back  um and now we're shifting gears in a  major way  because in a sense everything we've done  so far  has been kind of a warm up  these limited  computational models  um  really are kind of  get helping us to set our understanding  of automata and the definitions and the  notation  and they're also going to be helpful in  providing examples later on in the term  um but really in terms of a model of  computation they don't cut it because  uh they cannot do very simple things  that we normally think of a computer as  being able to do  so here we're introducing another model  of computation called the turing machine  and that's really going to be the model  that we're going to stick with for the  rest of the semester because that's  going to be our model of a general  purpose uh  computer the way you normally think  about it  um  so let's uh we'll spend a little time  introducing it and then we will um  uh  you know we'll  we'll continue that discussion next time  um  so uh in terms of a schematic actually  the turing machine model is pretty  simple  um  it has a uh  it has um  it's going to have states and all that  stuff but you know though so there's  going to be a finite control here which  is going to include the states and  transition function as we'll describe in  a minute um the point is is that you  know it's going to have the input  appearing on a tape  uh  the key difference now is that the the  machine is going to be able to change  the symbols on the tape um  and so we think of the machine as being  able to write as well as read the tape  okay so that's really uh the key  uh  feature  um of a turing machine is the ability to  write on the tape everything else in a  sense follows from that  a few other differences  but so the fact that he he head can read  and write  so we can use the tape as  storage  much as we used a stack as storage but  it doesn't it's not limited in the way  we can access it the way a stack is so  that we kind of have very uh  flexible access of the information on  the tape  now  being able to write on the tape doesn't  do any good if you can't go back and  read what you've written later on so  we're going to make the head to be able  to be two-way  so the head can move left to right as  before but it can also move back left  and that's going to be under control of  the transition function so under under  program control essentially  the tape is going to be  oops  sorry the tape is infinite to the right  and  so we're not going to limit how much  storage the machine can have so the tape  is going to think of as having instead  of just having the input on it it's  going to have the input but then the  rest is going to have infinitely many  blanks  blank symbols following the input so the  tape is infinite or  in the right right hand direction  um  and uh so there's infinitely many blanks  i'm going to use that symbol for the  blank to follow the input  um you can accept or reject oh yeah so  that's another thing that's important  normally we think of the machine in the  previous machines  finding a time when i pushed down  automata when you got to the end of the  input that's when you the acceptance of  rejection was decided if you were in  except state at the end of the input  then you accept it but  you have to be in that location um at  the end of the input in order for that  to take effect  that doesn't make any sense anymore  because  the machine might go off beyond that and  still be computing and come back and  read the tape later on so it only really  makes sense to let the machine  accept or reject  upon entering the accept or the reject  state so we're going to have a special  accept state and a special reject state  which is also a little different than  before  and when the machine enters those states  then machine then the action takes  effect the machine those machine holds  and then accepts  or holds and then it rejects so we'll  make that absolutely clear in the formal  definition in a second but just to get  the spirit of it  um  so i'm going to give you an example of  the thing running uh sorry you need to  again my  powerpoint is having issues  um  uh  okay so here is a turing machine  recognizing that language b  actually i switched gears on you instead  of zeros ones and twos i made them a's  b's and c's but same idea  [Music]  so i'm going to show you how the turing  machine operates  and then we'll give a formal  uh definition i hope that's on here  i think it is  in a second but let's this is an  informal discussion of how the machine  is going to operate to do this language  a to the kb to the kc to the k using its  ability to write on the tape as well as  read and move its head in both  directions  okay so let me just first describe  um in english  how this um  [Music]  machine operates and then we will um  see it in action on this little uh  picture i have over here okay so the way  the machine is going to operate is the  very first thing is the head is going to  start here and the head is going to scan  off to the right  uh making sure that the symbols appear  in the correct order  so it's seeing that  that there are a's and then b's and then  c's without checking the quantities just  that the order is correct for that you  don't need to write a finite automaton  can check that the input is of the form  a star b star c star  um  so uh  rioting is not necessary  the machine if it uh detects symbols out  of order it immediately rejects by going  into a special reject state otherwise  it's going to return its head back to  the left end  and um  uh let me just show that here so here is  it  oh no  i better  before i illustrate it over here let's  go through the whole algorithm uh so the  next thing that happens is you're going  to scan right and now you want to do the  counting so you're going to scan right  again but this time you're going to make  a bunch of passes over the input bunch  of scans  and each time you make a scan you're  going to cross  off one symbol of each type so you're  going to cross off an a you'll cross off  a b you'll cross off a c on a single  scan and then repeat that  crossing off the next a the next b the  next c  and  you want to make sure that you've  crossed off all of the symbols on the  same run and not crossing off some  symbols before other symbols uh while  other symbols will remain because that  would mean that the counts were not  equal if you cross them off and they're  all  you know run out on the same scan  same pass  then we know that the numbers have to be  start off being equal sort of i mean  this is sort of baby stuff here i but i  hope you get the idea and we'll kind of  illustrate in in a second so if at least  one of the if you have the last one of  each symbol so what i mean by that is  you you just crossed off the last a the  last b and the last c then you know that  you were originally had an equal number  and so you accept because you're  crossing off one of each on each scan so  if you cross the  on the last scan  each one of them gets crossed off um  then you accept  but if  uh it was the last of some symbol but  not of other symbols  so you know you crossed off the last a  but there um there were several b's  remaining then you started off with an  unequal number of a's b's and c's  and you can reject or  if all symbols still remain after you've  crossed them uh one of each off  then you haven't  done enough passes and you're gonna  repeat  from stage three and do that uh again  another scan  okay so here's a little um animation  which shows this happening on this  diagram so here is  the very first stage where you're  scanning across making sure things are  in the right order  um didn't have to write on the tape and  now you're gonna  reset the the head back to the beginning  this is by the way not the most  efficient procedure for doing this um  so um  now that now we're going to do a scan  crossing off a single a a single b and a  single c  so here i'm going to show that here  single a single b single c  and now as soon as you've crossed out  that last c we can return back to the  beginning um  so scan right cross  uh so if all symbols remain so there are  still symbols remaining of each type  we're going to return to the left and  repeat and repeat  now we'll get another pass single a  single b single c get crossed off  we haven't crossed them all off yet  no there's  of each type there's still our remaining  ones so again we return back to the  beginning now we have a last pass  crossover the last a the last be the  last c the last one of each type was  crossed off so now we know we can accept  because the original string was in the  language  okay so that's  uh to give you at least some idea um  that how the turing machine can operate  you know more like the way you would  think of a computer operating you know  maybe it's very primitive you know you  you could imagine counting also and the  termination can count as well but this  is the simplest uh descript uh procedure  that i can just describe on um you know  uh  for you without getting making too  complicated um  okay so let's do a little checking on  that um  okay so  the way i'm describing this  how do we go how do you think  and you in a sense you don't quite know  enough yet but um  [Music]  uh how do you think we're going to get  this effect  of crossing off with the turing machine  um are we going to get that by  changing the model and adding that  ability to cross off to the model  um are we going to use a tape alphabet  that includes those crossed off symbols  among them  or um we'll just assume that all turing  machines come with the eraser and they  can always cross our stuff  um so what do you think is the  nice way sort of mathematically to  describe  this  this ability to cross things off  yeah again again most of your again i  think are getting this um  uh  so  like 10 laggards here so  please wrap it up um so we can  close the poll  uh  five seconds to go  um  okay  polling ending get your last last call  uh  all right  share the results  so most of you  uh got that right  all turn machines come with an eraser i  don't know that was thrown in there as a  joke but it got came in second so i  don't feel bad if you got it but that's  not what i had in mind  um the the way the turing machine is  going to be riding on the tape is to  write a crossed off symbol uh instead of  the uh  the symbol that was originally there so  we're going to add these new crosstalk  symbols and that's going to be a common  thing for us to do  in um you know when we design turing  machines we're not going to get down to  the implementation level for very long  we're going to very quickly shift to a  higher level of discussion about the  machines but anyway that's how you would  do it if you were going to actually  build a machine  okay so um  let us then look at the formal  definition of the cursing and maybe i  should have done that checking after the  formal definition that might have been  clear but oh well  uh  okay here's the formal definition this  time a turing machine is a seven  tuple um  and  there is a  uh  um  now here  uh we have sigma which is the input  alphabet gamma is the tape alphabet so  now you're it's sort of a little bit  analogous to the stack from before where  gamma was the stack alphabet but these  are the symbols that you're allowed to  write on the tape um or that are allowed  to be on the tape um  so uh obviously all of the input symbols  are among the uh tape uh symbols  because they can appear on the tape so  you have sigma  is a subset of gamma um  one thing i didn't mention here is that  the input alphabet  we don't allow the blank symbol to be uh  in the input alphabet so that  you can actually use the blank symbol as  a delimiter for the end of the input  a marker for the end of the input um so  uh in fact and blank symbol is always  going to be in the tape alphabet uh so  we have um  you know  the this is actually always going to be  a proper subset uh because of the blank  symbol um what we're just allowing  doesn't really matter we're allowing the  tape alphabet to have  other symbols for convenience so for  example these crossed off symbols  now let's look at what the transition  function how that operates so the  transition function remember tells how  the machine is actually doing its  computation  and it says that if you're in a certain  state and the head is looking at a  certain tape symbol  um  then you can go to a new state you write  a new symbol on the uh  at that location on the tape and you can  move the head either left or right  so that's how we get the effect of the  head being able to be bi-directional  and here's the writing on the tape  comes up right here so  you know just an example here we says if  we're in state q and the head is looking  at an a currently on the tape then we  can move to state r we change it a to a  b and we move the head right once one  square  okay um  uh  uh now here this is important  to  on when you're giving a certain input  here to the turing machine it may gr  compute around for a while moving its  head back and forth as we were showing  and it may eventually halt by either  entering the q accept state or the q  reject state which i didn't  uh bring out here but that's important  these are the accepting rejecting  special states of the machine um or the  machine may never enter one of those it  may just go on and on and on and never  halt  we call that looping  a little bit of a misnomer because it's  the looping implies some sort of a  repetition for us looping just means not  holding  um  and um  so therefore m has three possible  outcomes for each um input  w it might accept w  by uh entering uh the except state  it could reject w by entering the reject  state which means it's going to reject  it by holding  or we also say we can reject by looping  you can reject the string by running  forever  um  okay that's just the terminology that's  common in the subject so you either  accept it um by halting and accepting or  rejecting it by either holding and  rejecting or by just going forever  that's also considered to be rejecting  sort of  sort of rejecting the instance by  default if you never actually have  accepted it then it's going to be  rejected  okay uh  checking three here  all right  so now um our last check in for the day  we say this turing machine model is  deterministic  um  i'm just saying that but if you look at  the way we set it up  if you've been following these formal  definitions so far you would understand  why it's deterministic so let's just as  a way of checking that um how would we  change this definition because we will  look on at the next lecture at  non-deterministic turing machines so a  little bit of a lead into that how would  we change  this uh definition to make it a  non-deterministic turing machine which  of those three options would we use  so here i'll launch that poll  uh  got about 10 people left let's give them  another 10 seconds  um  okay i think that's everybody who's  answered it from before  um  here there is so here  i think you pretty much  almost all of you got the right idea uh  it is b in fact uh because when we have  the power set symbol here that means  there might be several there's a subset  of possibilities so that indicates  several different ways to go  um  uh and that's what the that's the  essence of non-determinism  okay so uh i think we're  uh  whoops  um  okay  all right  so look  this is also kind of setting us up for  next lecture  and where we're going to be going with  this um so these are uh  basically two in example two or three  important definitions here um  one is um  we have we talked about the regular  languages  from fine automata we talked about the  context free languages from the grammars  and the push down automata what are the  languages of that the touring machines  can do  those are called in this course anyway  turing recognizable languages is very  orti recognizable um those are the lang  those are the languages that the turing  machine can recognize  and so what  just to make sure we were on the same  page on this  the language of the machine is the  collection of strings that the machine  accepts  so the things that are not in the  language are the things that are  rejected either by looping or by halting  and projecting  so all the only the ones that are  accepted are the light is the language  every machine has just a single language  it's a language of all strings that that  machine accepts um  and we'll say that m recognize that  language if that language is  a collection of such strings that are  accepted  and uh we will call that language a  touring recognizable language  okay if there's some turing machine that  can recognize it  um now uh  this feature of being able to reject by  running forever is a little bit weird  perhaps um  uh  and  uh for from the standpoint of  practicality  uh it's more convenient if the machine  always makes a decision to accept or  reject in a finite time  and doesn't just reject by going forever  and so we're going to  bring out a special class of turing  machines that have that feature  that they always halt they always go to  the holding states by the way maybe  didn't say this explicitly are the q  accept and the q eject states the accept  and reject states are the holding states  so the machine halts that means it ends  up in one of those two so it has made a  decision of accepting or rejecting in in  by the at the point in which it has  halted  um so we'll say a machine is a decider  if it always holds on every input  uh so for every w feed in the machine is  eventually going to come to a q accept  or to reject we call such a machine a  decider  and now we're going to say a language  uh  is that so it will say that the machine  decides a language  if  if it's the language of the machines of  the collection of accepted strings uh  and the machine is a decider so we'll  say that instead of just recognizing the  language we'll say that it decides the  language  um  and a turing decidable language is a  language um  that the machine  of all strings the machine accepts uh  for some turing machine which is is a  decider which is a touring machine that  always holds  so if a turing machine may sometimes  reject by looping then it's only  recognizing this language if the turing  machine is always holding so it's always  rejecting by explicitly coming to a  reject state and halting then we'll say  it's actually deciding the language  so in a sense that's better and we're  going to distinguish between those two  because  um  they're not the same  uh there are some languages which can be  recognized but not decided  and so in fact we're going to get the  following picture here  the deteuring recognizable languages are  a proper subset they include all this  everything that's decidable certainly is  going to be recognizable  because um  uh you know this being a decider is an  additional restriction to impose  additional requirement  so everything that's decidable is going  to be automatically recognizable but  there are things which are recognizable  but you're not decidable as we'll see  um  i'll actually give an example of that  but not prove it next lecture  and just for just to complete out this  picture um  i'm going to uh  also point out we haven't proven this  yet but uh we will prove it um that the  decidable languages also include  all the context-free languages which in  turn include the regular languages as it  was already seen so we haven't shown  this inclusion yet but actually this is  the picture we we get um so there's  actually a hierarchy of containments  here  regular languages are  a subset of the  context-free languages which are in turn  a subset of the decidable languages  which are in turn are a subset of the  recognizer turing recognizable languages  um and so  with that i think we're going to move to  our little bit of a review of what we've  done today  so we proved that pumping lemma  as a tool for showing that languages are  not context-free languages  we define turing machines which is going  to be our model um  that we're going to be focusing on for  the rest of the term uh not forgetting  the other models because they're going  to be useful examples for us  and we define touring deciders turing  machine deciders that halt and all  inputs  okay so i think with that um we have  come to the end of today's lecture  um i will stick around a little bit and  answer questions in the chat um i will  try to share them with everybody um as  i'm answering them so i'm not just  talking to one person  how was the concept of pride applied in  so i'm getting one question about  the practicality of all this uh bunches  of questions are coming in  um  uh  so look uh  is this is this stuff all practical um  uh i would say um uh yes and no um  uh  i i don't know which concept you have in  mind we're going to introduce lots of  concepts in this course and  um uh the you know the concept of  um  uh of you know the fine art automata and  the push down automata and context-free  languages  definitely used in other  subjects in other fields in computer  science and elsewhere these are very  basic and fundamental notions  um and so yes and turing machines well i  mean that's a model of a general  computer if you want to understand  computation you're going to need to  understand some model and a turing  machine is a particularly simple model  and that's why we use it as it turns out  it doesn't really matter what model you  use but we'll talk about that next  next time but yeah i would say there's a  lot of applied material in this course  as time has shown  whether it's  led to things like public key  cryptography which is  used on the internet  or you know understanding various  algorithms  i mean  that's not the reason i study it i'd  study it because i'm more of them coming  at it from more of a mathematical  perspective i just find the material  very uh beautiful and interesting and  challenging um but it does have  applications um  any other questions here i think i'm  going to sign off then so get myself set  up for my office hours which is in a  different on a different zoom like  okay so thank you everybody and see you  uh on thursday bye
 

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349
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350
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351
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352
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353
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354
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355
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356
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359
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360
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361
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367
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369
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370
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371
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379
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380
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381
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382
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384
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385
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386
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387
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388
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389
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390
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391
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392
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394
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395
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396
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397
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398
00:12:36,550 --> 00:12:38,790

 

399
00:12:38,790 --> 00:12:40,470

 

400
00:12:40,470 --> 00:12:43,269

 

401
00:12:43,269 --> 00:12:45,269

 

402
00:12:45,269 --> 00:12:48,069

 

403
00:12:48,069 --> 00:12:49,269

 

404
00:12:49,269 --> 00:12:49,279

 

405
00:12:49,279 --> 00:12:50,069


406
00:12:50,069 --> 00:12:52,790

 

407
00:12:52,790 --> 00:12:54,389

 

408
00:12:54,389 --> 00:12:55,910

 

409
00:12:55,910 --> 00:12:57,430

 

410
00:12:57,430 --> 00:12:58,790

 

411
00:12:58,790 --> 00:13:01,430

 

412
00:13:01,430 --> 00:13:03,829

 

413
00:13:03,829 --> 00:13:06,629

 

414
00:13:06,629 --> 00:13:10,069

 

415
00:13:10,069 --> 00:13:12,310

 

416
00:13:12,310 --> 00:13:16,150

 

417
00:13:16,150 --> 00:13:19,190

 

418
00:13:19,190 --> 00:13:21,269

 

419
00:13:21,269 --> 00:13:22,629

 

420
00:13:22,629 --> 00:13:24,470

 

421
00:13:24,470 --> 00:13:24,480

 

422
00:13:24,480 --> 00:13:25,590


423
00:13:25,590 --> 00:13:25,600

 

424
00:13:25,600 --> 00:13:26,470


425
00:13:26,470 --> 00:13:28,230

 

426
00:13:28,230 --> 00:13:28,240

 

427
00:13:28,240 --> 00:13:29,110


428
00:13:29,110 --> 00:13:30,230

 

429
00:13:30,230 --> 00:13:33,829

 

430
00:13:33,829 --> 00:13:33,839

 

431
00:13:33,839 --> 00:13:34,870


432
00:13:34,870 --> 00:13:36,790

 

433
00:13:36,790 --> 00:13:39,269

 

434
00:13:39,269 --> 00:13:40,310

 

435
00:13:40,310 --> 00:13:41,990

 

436
00:13:41,990 --> 00:13:44,949

 

437
00:13:44,949 --> 00:13:47,670

 

438
00:13:47,670 --> 00:13:49,590

 

439
00:13:49,590 --> 00:13:51,110

 

440
00:13:51,110 --> 00:13:53,269

 

441
00:13:53,269 --> 00:13:55,670

 

442
00:13:55,670 --> 00:13:57,030

 

443
00:13:57,030 --> 00:13:57,040

 

444
00:13:57,040 --> 00:13:58,230


445
00:13:58,230 --> 00:14:00,870

 

446
00:14:00,870 --> 00:14:02,629

 

447
00:14:02,629 --> 00:14:05,110

 

448
00:14:05,110 --> 00:14:07,829

 

449
00:14:07,829 --> 00:14:10,629

 

450
00:14:10,629 --> 00:14:12,230

 

451
00:14:12,230 --> 00:14:14,310

 

452
00:14:14,310 --> 00:14:17,030

 

453
00:14:17,030 --> 00:14:19,670

 

454
00:14:19,670 --> 00:14:21,350

 

455
00:14:21,350 --> 00:14:24,150

 

456
00:14:24,150 --> 00:14:27,110

 

457
00:14:27,110 --> 00:14:30,230

 

458
00:14:30,230 --> 00:14:30,240

 

459
00:14:30,240 --> 00:14:32,470


460
00:14:32,470 --> 00:14:34,870

 

461
00:14:34,870 --> 00:14:34,880

 

462
00:14:34,880 --> 00:14:36,230


463
00:14:36,230 --> 00:14:38,069

 

464
00:14:38,069 --> 00:14:39,910

 

465
00:14:39,910 --> 00:14:41,670

 

466
00:14:41,670 --> 00:14:43,829

 

467
00:14:43,829 --> 00:14:45,750

 

468
00:14:45,750 --> 00:14:47,509

 

469
00:14:47,509 --> 00:14:49,670

 

470
00:14:49,670 --> 00:14:51,110

 

471
00:14:51,110 --> 00:14:52,629

 

472
00:14:52,629 --> 00:14:54,550

 

473
00:14:54,550 --> 00:14:56,710

 

474
00:14:56,710 --> 00:14:59,990

 

475
00:14:59,990 --> 00:15:01,829

 

476
00:15:01,829 --> 00:15:03,590

 

477
00:15:03,590 --> 00:15:05,910

 

478
00:15:05,910 --> 00:15:07,269

 

479
00:15:07,269 --> 00:15:10,069

 

480
00:15:10,069 --> 00:15:12,470

 

481
00:15:12,470 --> 00:15:14,230

 

482
00:15:14,230 --> 00:15:15,509

 

483
00:15:15,509 --> 00:15:17,829

 

484
00:15:17,829 --> 00:15:20,069

 

485
00:15:20,069 --> 00:15:22,710

 

486
00:15:22,710 --> 00:15:24,710

 

487
00:15:24,710 --> 00:15:27,269

 

488
00:15:27,269 --> 00:15:29,749

 

489
00:15:29,749 --> 00:15:31,670

 

490
00:15:31,670 --> 00:15:33,829

 

491
00:15:33,829 --> 00:15:36,710

 

492
00:15:36,710 --> 00:15:41,670

 

493
00:15:41,670 --> 00:15:41,680

 

494
00:15:41,680 --> 00:15:42,870


495
00:15:42,870 --> 00:15:42,880

 

496
00:15:42,880 --> 00:15:43,670


497
00:15:43,670 --> 00:15:43,680

 

498
00:15:43,680 --> 00:15:45,670


499
00:15:45,670 --> 00:15:47,910

 

500
00:15:47,910 --> 00:15:50,310

 

501
00:15:50,310 --> 00:15:52,069

 

502
00:15:52,069 --> 00:15:53,189

 

503
00:15:53,189 --> 00:15:55,430

 

504
00:15:55,430 --> 00:15:57,110

 

505
00:15:57,110 --> 00:15:59,030

 

506
00:15:59,030 --> 00:16:00,710

 

507
00:16:00,710 --> 00:16:02,230

 

508
00:16:02,230 --> 00:16:04,389

 

509
00:16:04,389 --> 00:16:05,910

 

510
00:16:05,910 --> 00:16:07,350

 

511
00:16:07,350 --> 00:16:08,790

 

512
00:16:08,790 --> 00:16:12,230

 

513
00:16:12,230 --> 00:16:14,790

 

514
00:16:14,790 --> 00:16:18,550

 

515
00:16:18,550 --> 00:16:20,790

 

516
00:16:20,790 --> 00:16:23,430

 

517
00:16:23,430 --> 00:16:23,440

 

518
00:16:23,440 --> 00:16:24,230


519
00:16:24,230 --> 00:16:26,629

 

520
00:16:26,629 --> 00:16:28,310

 

521
00:16:28,310 --> 00:16:31,110

 

522
00:16:31,110 --> 00:16:32,870

 

523
00:16:32,870 --> 00:16:34,870

 

524
00:16:34,870 --> 00:16:38,470

 

525
00:16:38,470 --> 00:16:40,790

 

526
00:16:40,790 --> 00:16:42,870

 

527
00:16:42,870 --> 00:16:45,269

 

528
00:16:45,269 --> 00:16:49,910

 

529
00:16:49,910 --> 00:16:53,189

 

530
00:16:53,189 --> 00:16:53,990

 

531
00:16:53,990 --> 00:16:54,000

 

532
00:16:54,000 --> 00:16:54,870


533
00:16:54,870 --> 00:16:56,470

 

534
00:16:56,470 --> 00:16:56,480

 

535
00:16:56,480 --> 00:16:57,990


536
00:16:57,990 --> 00:17:00,389

 

537
00:17:00,389 --> 00:17:01,829

 

538
00:17:01,829 --> 00:17:03,749

 

539
00:17:03,749 --> 00:17:05,029

 

540
00:17:05,029 --> 00:17:07,110

 

541
00:17:07,110 --> 00:17:09,750

 

542
00:17:09,750 --> 00:17:11,829

 

543
00:17:11,829 --> 00:17:13,829

 

544
00:17:13,829 --> 00:17:15,909

 

545
00:17:15,909 --> 00:17:17,669

 

546
00:17:17,669 --> 00:17:19,429

 

547
00:17:19,429 --> 00:17:19,439

 

548
00:17:19,439 --> 00:17:20,549


549
00:17:20,549 --> 00:17:20,559

 

550
00:17:20,559 --> 00:17:21,590


551
00:17:21,590 --> 00:17:22,789

 

552
00:17:22,789 --> 00:17:23,990

 

553
00:17:23,990 --> 00:17:25,750

 

554
00:17:25,750 --> 00:17:25,760

 

555
00:17:25,760 --> 00:17:26,789


556
00:17:26,789 --> 00:17:28,870

 

557
00:17:28,870 --> 00:17:31,029

 

558
00:17:31,029 --> 00:17:33,510

 

559
00:17:33,510 --> 00:17:36,070

 

560
00:17:36,070 --> 00:17:36,080

 

561
00:17:36,080 --> 00:17:37,430


562
00:17:37,430 --> 00:17:38,870

 

563
00:17:38,870 --> 00:17:41,590

 

564
00:17:41,590 --> 00:17:42,789

 

565
00:17:42,789 --> 00:17:44,390

 

566
00:17:44,390 --> 00:17:48,549

 

567
00:17:48,549 --> 00:17:51,110

 

568
00:17:51,110 --> 00:17:53,029

 

569
00:17:53,029 --> 00:17:55,510

 

570
00:17:55,510 --> 00:17:58,549

 

571
00:17:58,549 --> 00:18:00,150

 

572
00:18:00,150 --> 00:18:03,510

 

573
00:18:03,510 --> 00:18:06,310

 

574
00:18:06,310 --> 00:18:07,669

 

575
00:18:07,669 --> 00:18:09,590

 

576
00:18:09,590 --> 00:18:11,590

 

577
00:18:11,590 --> 00:18:14,230

 

578
00:18:14,230 --> 00:18:16,150

 

579
00:18:16,150 --> 00:18:17,590

 

580
00:18:17,590 --> 00:18:19,590

 

581
00:18:19,590 --> 00:18:22,150

 

582
00:18:22,150 --> 00:18:23,750

 

583
00:18:23,750 --> 00:18:26,870

 

584
00:18:26,870 --> 00:18:29,029

 

585
00:18:29,029 --> 00:18:31,029

 

586
00:18:31,029 --> 00:18:32,390

 

587
00:18:32,390 --> 00:18:33,669

 

588
00:18:33,669 --> 00:18:35,909

 

589
00:18:35,909 --> 00:18:38,310

 

590
00:18:38,310 --> 00:18:38,320

 

591
00:18:38,320 --> 00:18:39,190


592
00:18:39,190 --> 00:18:43,909

 

593
00:18:43,909 --> 00:18:45,270

 

594
00:18:45,270 --> 00:18:47,430

 

595
00:18:47,430 --> 00:18:49,750

 

596
00:18:49,750 --> 00:18:50,950

 

597
00:18:50,950 --> 00:18:54,470

 

598
00:18:54,470 --> 00:18:56,710

 

599
00:18:56,710 --> 00:18:58,310

 

600
00:18:58,310 --> 00:19:00,630

 

601
00:19:00,630 --> 00:19:02,390

 

602
00:19:02,390 --> 00:19:02,400

 

603
00:19:02,400 --> 00:19:04,549


604
00:19:04,549 --> 00:19:06,150

 

605
00:19:06,150 --> 00:19:07,430

 

606
00:19:07,430 --> 00:19:09,110

 

607
00:19:09,110 --> 00:19:11,110

 

608
00:19:11,110 --> 00:19:13,270

 

609
00:19:13,270 --> 00:19:15,669

 

610
00:19:15,669 --> 00:19:17,669

 

611
00:19:17,669 --> 00:19:20,070

 

612
00:19:20,070 --> 00:19:21,990

 

613
00:19:21,990 --> 00:19:23,990

 

614
00:19:23,990 --> 00:19:24,000

 

615
00:19:24,000 --> 00:19:25,510


616
00:19:25,510 --> 00:19:26,950

 

617
00:19:26,950 --> 00:19:29,029

 

618
00:19:29,029 --> 00:19:31,190

 

619
00:19:31,190 --> 00:19:32,390

 

620
00:19:32,390 --> 00:19:32,400

 

621
00:19:32,400 --> 00:19:33,430


622
00:19:33,430 --> 00:19:33,440

 

623
00:19:33,440 --> 00:19:36,470


624
00:19:36,470 --> 00:19:39,909

 

625
00:19:39,909 --> 00:19:41,430

 

626
00:19:41,430 --> 00:19:43,669

 

627
00:19:43,669 --> 00:19:45,110

 

628
00:19:45,110 --> 00:19:47,270

 

629
00:19:47,270 --> 00:19:47,280

 

630
00:19:47,280 --> 00:19:48,630


631
00:19:48,630 --> 00:19:49,830

 

632
00:19:49,830 --> 00:19:50,870

 

633
00:19:50,870 --> 00:19:52,630

 

634
00:19:52,630 --> 00:19:55,590

 

635
00:19:55,590 --> 00:19:58,230

 

636
00:19:58,230 --> 00:19:58,240

 

637
00:19:58,240 --> 00:19:59,590


638
00:19:59,590 --> 00:20:02,710

 

639
00:20:02,710 --> 00:20:02,720

 

640
00:20:02,720 --> 00:20:03,830


641
00:20:03,830 --> 00:20:05,350

 

642
00:20:05,350 --> 00:20:07,430

 

643
00:20:07,430 --> 00:20:09,669

 

644
00:20:09,669 --> 00:20:12,549

 

645
00:20:12,549 --> 00:20:14,470

 

646
00:20:14,470 --> 00:20:16,390

 

647
00:20:16,390 --> 00:20:18,870

 

648
00:20:18,870 --> 00:20:20,390

 

649
00:20:20,390 --> 00:20:22,070

 

650
00:20:22,070 --> 00:20:24,230

 

651
00:20:24,230 --> 00:20:25,909

 

652
00:20:25,909 --> 00:20:27,669

 

653
00:20:27,669 --> 00:20:30,070

 

654
00:20:30,070 --> 00:20:31,909

 

655
00:20:31,909 --> 00:20:34,149

 

656
00:20:34,149 --> 00:20:37,270

 

657
00:20:37,270 --> 00:20:40,630

 

658
00:20:40,630 --> 00:20:42,870

 

659
00:20:42,870 --> 00:20:42,880

 

660
00:20:42,880 --> 00:20:43,669


661
00:20:43,669 --> 00:20:46,630

 

662
00:20:46,630 --> 00:20:48,710

 

663
00:20:48,710 --> 00:20:50,950

 

664
00:20:50,950 --> 00:20:53,270

 

665
00:20:53,270 --> 00:20:56,149

 

666
00:20:56,149 --> 00:20:58,230

 

667
00:20:58,230 --> 00:20:58,240

 

668
00:20:58,240 --> 00:20:59,190


669
00:20:59,190 --> 00:20:59,200

 

670
00:20:59,200 --> 00:21:00,390


671
00:21:00,390 --> 00:21:00,400

 

672
00:21:00,400 --> 00:21:01,430


673
00:21:01,430 --> 00:21:02,630

 

674
00:21:02,630 --> 00:21:04,549

 

675
00:21:04,549 --> 00:21:06,070

 

676
00:21:06,070 --> 00:21:06,080

 

677
00:21:06,080 --> 00:21:06,870


678
00:21:06,870 --> 00:21:09,029

 

679
00:21:09,029 --> 00:21:11,029

 

680
00:21:11,029 --> 00:21:11,039

 

681
00:21:11,039 --> 00:21:13,669


682
00:21:13,669 --> 00:21:15,750

 

683
00:21:15,750 --> 00:21:17,750

 

684
00:21:17,750 --> 00:21:19,350

 

685
00:21:19,350 --> 00:21:21,510

 

686
00:21:21,510 --> 00:21:25,029

 

687
00:21:25,029 --> 00:21:27,590

 

688
00:21:27,590 --> 00:21:30,149

 

689
00:21:30,149 --> 00:21:30,159

 

690
00:21:30,159 --> 00:21:30,950


691
00:21:30,950 --> 00:21:32,870

 

692
00:21:32,870 --> 00:21:34,789

 

693
00:21:34,789 --> 00:21:37,750

 

694
00:21:37,750 --> 00:21:40,230

 

695
00:21:40,230 --> 00:21:44,390

 

696
00:21:44,390 --> 00:21:45,669

 

697
00:21:45,669 --> 00:21:48,549

 

698
00:21:48,549 --> 00:21:51,430

 

699
00:21:51,430 --> 00:21:53,990

 

700
00:21:53,990 --> 00:21:54,000

 

701
00:21:54,000 --> 00:21:55,510


702
00:21:55,510 --> 00:21:58,310

 

703
00:21:58,310 --> 00:22:00,070

 

704
00:22:00,070 --> 00:22:02,470

 

705
00:22:02,470 --> 00:22:03,750

 

706
00:22:03,750 --> 00:22:06,470

 

707
00:22:06,470 --> 00:22:08,230

 

708
00:22:08,230 --> 00:22:08,240

 

709
00:22:08,240 --> 00:22:10,390


710
00:22:10,390 --> 00:22:13,270

 

711
00:22:13,270 --> 00:22:15,029

 

712
00:22:15,029 --> 00:22:16,710

 

713
00:22:16,710 --> 00:22:18,549

 

714
00:22:18,549 --> 00:22:19,909

 

715
00:22:19,909 --> 00:22:21,909

 

716
00:22:21,909 --> 00:22:24,230

 

717
00:22:24,230 --> 00:22:24,240

 

718
00:22:24,240 --> 00:22:24,950


719
00:22:24,950 --> 00:22:27,350

 

720
00:22:27,350 --> 00:22:29,029

 

721
00:22:29,029 --> 00:22:31,990

 

722
00:22:31,990 --> 00:22:32,000

 

723
00:22:32,000 --> 00:22:33,990


724
00:22:33,990 --> 00:22:37,669

 

725
00:22:37,669 --> 00:22:40,549

 

726
00:22:40,549 --> 00:22:42,950

 

727
00:22:42,950 --> 00:22:45,430

 

728
00:22:45,430 --> 00:22:47,270

 

729
00:22:47,270 --> 00:22:51,909

 

730
00:22:51,909 --> 00:22:54,789

 

731
00:22:54,789 --> 00:22:54,799

 

732
00:22:54,799 --> 00:22:55,830


733
00:22:55,830 --> 00:22:57,110

 

734
00:22:57,110 --> 00:22:58,630

 

735
00:22:58,630 --> 00:23:00,789

 

736
00:23:00,789 --> 00:23:00,799

 

737
00:23:00,799 --> 00:23:01,590


738
00:23:01,590 --> 00:23:03,430

 

739
00:23:03,430 --> 00:23:06,070

 

740
00:23:06,070 --> 00:23:07,909

 

741
00:23:07,909 --> 00:23:09,029

 

742
00:23:09,029 --> 00:23:11,990

 

743
00:23:11,990 --> 00:23:17,350

 

744
00:23:17,350 --> 00:23:20,230

 

745
00:23:20,230 --> 00:23:22,789

 

746
00:23:22,789 --> 00:23:25,110

 

747
00:23:25,110 --> 00:23:27,350

 

748
00:23:27,350 --> 00:23:30,630

 

749
00:23:30,630 --> 00:23:33,830

 

750
00:23:33,830 --> 00:23:36,390

 

751
00:23:36,390 --> 00:23:39,190

 

752
00:23:39,190 --> 00:23:41,110

 

753
00:23:41,110 --> 00:23:45,190

 

754
00:23:45,190 --> 00:23:47,590

 

755
00:23:47,590 --> 00:23:50,070

 

756
00:23:50,070 --> 00:23:52,230

 

757
00:23:52,230 --> 00:23:54,070

 

758
00:23:54,070 --> 00:23:56,310

 

759
00:23:56,310 --> 00:23:59,590

 

760
00:23:59,590 --> 00:24:00,950

 

761
00:24:00,950 --> 00:24:03,110

 

762
00:24:03,110 --> 00:24:04,870

 

763
00:24:04,870 --> 00:24:08,070

 

764
00:24:08,070 --> 00:24:08,080

 

765
00:24:08,080 --> 00:24:09,029


766
00:24:09,029 --> 00:24:11,029

 

767
00:24:11,029 --> 00:24:13,750

 

768
00:24:13,750 --> 00:24:15,430

 

769
00:24:15,430 --> 00:24:17,029

 

770
00:24:17,029 --> 00:24:20,390

 

771
00:24:20,390 --> 00:24:22,149

 

772
00:24:22,149 --> 00:24:22,159

 

773
00:24:22,159 --> 00:24:25,190


774
00:24:25,190 --> 00:24:27,590

 

775
00:24:27,590 --> 00:24:30,710

 

776
00:24:30,710 --> 00:24:32,710

 

777
00:24:32,710 --> 00:24:34,710

 

778
00:24:34,710 --> 00:24:37,510

 

779
00:24:37,510 --> 00:24:40,149

 

780
00:24:40,149 --> 00:24:42,630

 

781
00:24:42,630 --> 00:24:42,640

 

782
00:24:42,640 --> 00:24:44,230


783
00:24:44,230 --> 00:24:47,350

 

784
00:24:47,350 --> 00:24:47,360

 

785
00:24:47,360 --> 00:24:48,470


786
00:24:48,470 --> 00:24:49,909

 

787
00:24:49,909 --> 00:24:52,150

 

788
00:24:52,150 --> 00:24:52,160

 

789
00:24:52,160 --> 00:24:54,149


790
00:24:54,149 --> 00:24:56,390

 

791
00:24:56,390 --> 00:24:58,710

 

792
00:24:58,710 --> 00:25:01,269

 

793
00:25:01,269 --> 00:25:05,269

 

794
00:25:05,269 --> 00:25:06,950

 

795
00:25:06,950 --> 00:25:08,149

 

796
00:25:08,149 --> 00:25:09,830

 

797
00:25:09,830 --> 00:25:11,269

 

798
00:25:11,269 --> 00:25:13,750

 

799
00:25:13,750 --> 00:25:13,760

 

800
00:25:13,760 --> 00:25:15,190


801
00:25:15,190 --> 00:25:16,549

 

802
00:25:16,549 --> 00:25:18,310

 

803
00:25:18,310 --> 00:25:19,590

 

804
00:25:19,590 --> 00:25:21,190

 

805
00:25:21,190 --> 00:25:22,630

 

806
00:25:22,630 --> 00:25:24,630

 

807
00:25:24,630 --> 00:25:24,640

 

808
00:25:24,640 --> 00:25:25,669


809
00:25:25,669 --> 00:25:28,070

 

810
00:25:28,070 --> 00:25:30,070

 

811
00:25:30,070 --> 00:25:32,230

 

812
00:25:32,230 --> 00:25:32,240

 

813
00:25:32,240 --> 00:25:34,630


814
00:25:34,630 --> 00:25:35,830

 

815
00:25:35,830 --> 00:25:37,590

 

816
00:25:37,590 --> 00:25:39,909

 

817
00:25:39,909 --> 00:25:41,990

 

818
00:25:41,990 --> 00:25:44,950

 

819
00:25:44,950 --> 00:25:47,909

 

820
00:25:47,909 --> 00:25:47,919

 

821
00:25:47,919 --> 00:25:49,669


822
00:25:49,669 --> 00:25:51,909

 

823
00:25:51,909 --> 00:25:55,029

 

824
00:25:55,029 --> 00:25:57,350

 

825
00:25:57,350 --> 00:25:59,909

 

826
00:25:59,909 --> 00:26:02,710

 

827
00:26:02,710 --> 00:26:05,110

 

828
00:26:05,110 --> 00:26:07,669

 

829
00:26:07,669 --> 00:26:09,350

 

830
00:26:09,350 --> 00:26:11,110

 

831
00:26:11,110 --> 00:26:13,510

 

832
00:26:13,510 --> 00:26:17,110

 

833
00:26:17,110 --> 00:26:19,029

 

834
00:26:19,029 --> 00:26:21,190

 

835
00:26:21,190 --> 00:26:23,590

 

836
00:26:23,590 --> 00:26:25,669

 

837
00:26:25,669 --> 00:26:27,750

 

838
00:26:27,750 --> 00:26:30,789

 

839
00:26:30,789 --> 00:26:32,549

 

840
00:26:32,549 --> 00:26:34,549

 

841
00:26:34,549 --> 00:26:37,110

 

842
00:26:37,110 --> 00:26:39,269

 

843
00:26:39,269 --> 00:26:41,269

 

844
00:26:41,269 --> 00:26:42,950

 

845
00:26:42,950 --> 00:26:42,960

 

846
00:26:42,960 --> 00:26:43,830


847
00:26:43,830 --> 00:26:45,909

 

848
00:26:45,909 --> 00:26:47,909

 

849
00:26:47,909 --> 00:26:50,870

 

850
00:26:50,870 --> 00:26:52,870

 

851
00:26:52,870 --> 00:26:55,669

 

852
00:26:55,669 --> 00:26:57,269

 

853
00:26:57,269 --> 00:26:58,950

 

854
00:26:58,950 --> 00:27:00,630

 

855
00:27:00,630 --> 00:27:00,640

 

856
00:27:00,640 --> 00:27:01,830


857
00:27:01,830 --> 00:27:03,430

 

858
00:27:03,430 --> 00:27:05,269

 

859
00:27:05,269 --> 00:27:08,149

 

860
00:27:08,149 --> 00:27:10,070

 

861
00:27:10,070 --> 00:27:12,549

 

862
00:27:12,549 --> 00:27:14,870

 

863
00:27:14,870 --> 00:27:17,350

 

864
00:27:17,350 --> 00:27:19,830

 

865
00:27:19,830 --> 00:27:21,590

 

866
00:27:21,590 --> 00:27:23,590

 

867
00:27:23,590 --> 00:27:25,190

 

868
00:27:25,190 --> 00:27:28,149

 

869
00:27:28,149 --> 00:27:30,230

 

870
00:27:30,230 --> 00:27:32,789

 

871
00:27:32,789 --> 00:27:34,310

 

872
00:27:34,310 --> 00:27:35,750

 

873
00:27:35,750 --> 00:27:37,990

 

874
00:27:37,990 --> 00:27:39,190

 

875
00:27:39,190 --> 00:27:39,200

 

876
00:27:39,200 --> 00:27:40,149


877
00:27:40,149 --> 00:27:42,310

 

878
00:27:42,310 --> 00:27:43,909

 

879
00:27:43,909 --> 00:27:46,230

 

880
00:27:46,230 --> 00:27:49,269

 

881
00:27:49,269 --> 00:27:51,510

 

882
00:27:51,510 --> 00:27:53,269

 

883
00:27:53,269 --> 00:27:55,909

 

884
00:27:55,909 --> 00:27:57,590

 

885
00:27:57,590 --> 00:28:00,870

 

886
00:28:00,870 --> 00:28:03,110

 

887
00:28:03,110 --> 00:28:04,950

 

888
00:28:04,950 --> 00:28:06,389

 

889
00:28:06,389 --> 00:28:06,399

 

890
00:28:06,399 --> 00:28:07,669


891
00:28:07,669 --> 00:28:09,750

 

892
00:28:09,750 --> 00:28:11,110

 

893
00:28:11,110 --> 00:28:11,120

 

894
00:28:11,120 --> 00:28:12,470


895
00:28:12,470 --> 00:28:15,190

 

896
00:28:15,190 --> 00:28:17,669

 

897
00:28:17,669 --> 00:28:21,830

 

898
00:28:21,830 --> 00:28:23,750

 

899
00:28:23,750 --> 00:28:26,149

 

900
00:28:26,149 --> 00:28:28,470

 

901
00:28:28,470 --> 00:28:30,710

 

902
00:28:30,710 --> 00:28:33,830

 

903
00:28:33,830 --> 00:28:35,269

 

904
00:28:35,269 --> 00:28:36,549

 

905
00:28:36,549 --> 00:28:39,510

 

906
00:28:39,510 --> 00:28:41,190

 

907
00:28:41,190 --> 00:28:43,669

 

908
00:28:43,669 --> 00:28:45,110

 

909
00:28:45,110 --> 00:28:46,230

 

910
00:28:46,230 --> 00:28:48,230

 

911
00:28:48,230 --> 00:28:51,190

 

912
00:28:51,190 --> 00:28:53,110

 

913
00:28:53,110 --> 00:28:54,870

 

914
00:28:54,870 --> 00:28:54,880

 

915
00:28:54,880 --> 00:28:55,909


916
00:28:55,909 --> 00:28:57,350

 

917
00:28:57,350 --> 00:28:59,430

 

918
00:28:59,430 --> 00:29:01,990

 

919
00:29:01,990 --> 00:29:03,269

 

920
00:29:03,269 --> 00:29:05,510

 

921
00:29:05,510 --> 00:29:08,389

 

922
00:29:08,389 --> 00:29:08,399

 

923
00:29:08,399 --> 00:29:09,269


924
00:29:09,269 --> 00:29:11,190

 

925
00:29:11,190 --> 00:29:13,510

 

926
00:29:13,510 --> 00:29:14,870

 

927
00:29:14,870 --> 00:29:16,549

 

928
00:29:16,549 --> 00:29:19,909

 

929
00:29:19,909 --> 00:29:21,510

 

930
00:29:21,510 --> 00:29:23,590

 

931
00:29:23,590 --> 00:29:25,909

 

932
00:29:25,909 --> 00:29:27,830

 

933
00:29:27,830 --> 00:29:29,750

 

934
00:29:29,750 --> 00:29:32,230

 

935
00:29:32,230 --> 00:29:34,470

 

936
00:29:34,470 --> 00:29:35,830

 

937
00:29:35,830 --> 00:29:38,310

 

938
00:29:38,310 --> 00:29:41,269

 

939
00:29:41,269 --> 00:29:43,830

 

940
00:29:43,830 --> 00:29:44,789

 

941
00:29:44,789 --> 00:29:46,149

 

942
00:29:46,149 --> 00:29:47,990

 

943
00:29:47,990 --> 00:29:50,230

 

944
00:29:50,230 --> 00:29:53,190

 

945
00:29:53,190 --> 00:29:56,070

 

946
00:29:56,070 --> 00:29:58,789

 

947
00:29:58,789 --> 00:30:00,310

 

948
00:30:00,310 --> 00:30:04,950

 

949
00:30:04,950 --> 00:30:07,269

 

950
00:30:07,269 --> 00:30:07,279

 

951
00:30:07,279 --> 00:30:08,630


952
00:30:08,630 --> 00:30:11,750

 

953
00:30:11,750 --> 00:30:13,990

 

954
00:30:13,990 --> 00:30:16,230

 

955
00:30:16,230 --> 00:30:18,070

 

956
00:30:18,070 --> 00:30:20,950

 

957
00:30:20,950 --> 00:30:23,590

 

958
00:30:23,590 --> 00:30:25,110

 

959
00:30:25,110 --> 00:30:26,950

 

960
00:30:26,950 --> 00:30:28,789

 

961
00:30:28,789 --> 00:30:31,669

 

962
00:30:31,669 --> 00:30:33,190

 

963
00:30:33,190 --> 00:30:35,110

 

964
00:30:35,110 --> 00:30:36,950

 

965
00:30:36,950 --> 00:30:39,590

 

966
00:30:39,590 --> 00:30:41,350

 

967
00:30:41,350 --> 00:30:42,710

 

968
00:30:42,710 --> 00:30:42,720

 

969
00:30:42,720 --> 00:30:43,990


970
00:30:43,990 --> 00:30:45,269

 

971
00:30:45,269 --> 00:30:47,269

 

972
00:30:47,269 --> 00:30:48,389

 

973
00:30:48,389 --> 00:30:50,950

 

974
00:30:50,950 --> 00:30:52,470

 

975
00:30:52,470 --> 00:30:54,310

 

976
00:30:54,310 --> 00:30:59,590

 

977
00:30:59,590 --> 00:31:01,830

 

978
00:31:01,830 --> 00:31:03,990

 

979
00:31:03,990 --> 00:31:06,070

 

980
00:31:06,070 --> 00:31:07,990

 

981
00:31:07,990 --> 00:31:09,269

 

982
00:31:09,269 --> 00:31:11,830

 

983
00:31:11,830 --> 00:31:14,630

 

984
00:31:14,630 --> 00:31:16,870

 

985
00:31:16,870 --> 00:31:20,310

 

986
00:31:20,310 --> 00:31:22,070

 

987
00:31:22,070 --> 00:31:24,389

 

988
00:31:24,389 --> 00:31:27,110

 

989
00:31:27,110 --> 00:31:28,950

 

990
00:31:28,950 --> 00:31:31,750

 

991
00:31:31,750 --> 00:31:34,310

 

992
00:31:34,310 --> 00:31:36,549

 

993
00:31:36,549 --> 00:31:38,950

 

994
00:31:38,950 --> 00:31:41,590

 

995
00:31:41,590 --> 00:31:43,269

 

996
00:31:43,269 --> 00:31:45,590

 

997
00:31:45,590 --> 00:31:47,269

 

998
00:31:47,269 --> 00:31:49,029

 

999
00:31:49,029 --> 00:31:50,789

 

1000
00:31:50,789 --> 00:31:53,110

 

1001
00:31:53,110 --> 00:31:54,789

 

1002
00:31:54,789 --> 00:31:57,430

 

1003
00:31:57,430 --> 00:32:00,789

 

1004
00:32:00,789 --> 00:32:03,590

 

1005
00:32:03,590 --> 00:32:05,590

 

1006
00:32:05,590 --> 00:32:07,269

 

1007
00:32:07,269 --> 00:32:07,279

 

1008
00:32:07,279 --> 00:32:08,549


1009
00:32:08,549 --> 00:32:11,909

 

1010
00:32:11,909 --> 00:32:15,190

 

1011
00:32:15,190 --> 00:32:17,269

 

1012
00:32:17,269 --> 00:32:19,590

 

1013
00:32:19,590 --> 00:32:21,750

 

1014
00:32:21,750 --> 00:32:24,310

 

1015
00:32:24,310 --> 00:32:26,149

 

1016
00:32:26,149 --> 00:32:28,389

 

1017
00:32:28,389 --> 00:32:30,070

 

1018
00:32:30,070 --> 00:32:31,350

 

1019
00:32:31,350 --> 00:32:33,669

 

1020
00:32:33,669 --> 00:32:37,590

 

1021
00:32:37,590 --> 00:32:39,350

 

1022
00:32:39,350 --> 00:32:41,029

 

1023
00:32:41,029 --> 00:32:41,039

 

1024
00:32:41,039 --> 00:32:42,710


1025
00:32:42,710 --> 00:32:46,950

 

1026
00:32:46,950 --> 00:32:48,549

 

1027
00:32:48,549 --> 00:32:50,950

 

1028
00:32:50,950 --> 00:32:53,029

 

1029
00:32:53,029 --> 00:32:55,750

 

1030
00:32:55,750 --> 00:32:57,590

 

1031
00:32:57,590 --> 00:32:59,269

 

1032
00:32:59,269 --> 00:32:59,279

 

1033
00:32:59,279 --> 00:33:00,789


1034
00:33:00,789 --> 00:33:02,630

 

1035
00:33:02,630 --> 00:33:04,549

 

1036
00:33:04,549 --> 00:33:05,909

 

1037
00:33:05,909 --> 00:33:07,269

 

1038
00:33:07,269 --> 00:33:08,310

 

1039
00:33:08,310 --> 00:33:09,590

 

1040
00:33:09,590 --> 00:33:11,509

 

1041
00:33:11,509 --> 00:33:13,350

 

1042
00:33:13,350 --> 00:33:13,360

 

1043
00:33:13,360 --> 00:33:14,870


1044
00:33:14,870 --> 00:33:16,950

 

1045
00:33:16,950 --> 00:33:19,430

 

1046
00:33:19,430 --> 00:33:21,669

 

1047
00:33:21,669 --> 00:33:25,029

 

1048
00:33:25,029 --> 00:33:28,310

 

1049
00:33:28,310 --> 00:33:29,830

 

1050
00:33:29,830 --> 00:33:32,310

 

1051
00:33:32,310 --> 00:33:34,070

 

1052
00:33:34,070 --> 00:33:36,230

 

1053
00:33:36,230 --> 00:33:38,230

 

1054
00:33:38,230 --> 00:33:39,830

 

1055
00:33:39,830 --> 00:33:39,840

 

1056
00:33:39,840 --> 00:33:41,269


1057
00:33:41,269 --> 00:33:43,990

 

1058
00:33:43,990 --> 00:33:46,149

 

1059
00:33:46,149 --> 00:33:49,029

 

1060
00:33:49,029 --> 00:33:51,830

 

1061
00:33:51,830 --> 00:33:54,389

 

1062
00:33:54,389 --> 00:33:56,389

 

1063
00:33:56,389 --> 00:33:59,029

 

1064
00:33:59,029 --> 00:34:02,310

 

1065
00:34:02,310 --> 00:34:04,070

 

1066
00:34:04,070 --> 00:34:06,470

 

1067
00:34:06,470 --> 00:34:08,470

 

1068
00:34:08,470 --> 00:34:08,480

 

1069
00:34:08,480 --> 00:34:09,510


1070
00:34:09,510 --> 00:34:11,349

 

1071
00:34:11,349 --> 00:34:13,349

 

1072
00:34:13,349 --> 00:34:15,270

 

1073
00:34:15,270 --> 00:34:17,109

 

1074
00:34:17,109 --> 00:34:17,119

 

1075
00:34:17,119 --> 00:34:19,349


1076
00:34:19,349 --> 00:34:21,030

 

1077
00:34:21,030 --> 00:34:23,430

 

1078
00:34:23,430 --> 00:34:25,750

 

1079
00:34:25,750 --> 00:34:25,760

 

1080
00:34:25,760 --> 00:34:26,950


1081
00:34:26,950 --> 00:34:28,230

 

1082
00:34:28,230 --> 00:34:29,750

 

1083
00:34:29,750 --> 00:34:31,829

 

1084
00:34:31,829 --> 00:34:34,790

 

1085
00:34:34,790 --> 00:34:37,109

 

1086
00:34:37,109 --> 00:34:39,109

 

1087
00:34:39,109 --> 00:34:41,030

 

1088
00:34:41,030 --> 00:34:41,040

 

1089
00:34:41,040 --> 00:34:42,230


1090
00:34:42,230 --> 00:34:44,230

 

1091
00:34:44,230 --> 00:34:46,829

 

1092
00:34:46,829 --> 00:34:50,950

 

1093
00:34:50,950 --> 00:34:53,750

 

1094
00:34:53,750 --> 00:34:56,230

 

1095
00:34:56,230 --> 00:34:57,990

 

1096
00:34:57,990 --> 00:35:00,710

 

1097
00:35:00,710 --> 00:35:02,870

 

1098
00:35:02,870 --> 00:35:04,870

 

1099
00:35:04,870 --> 00:35:07,990

 

1100
00:35:07,990 --> 00:35:10,950

 

1101
00:35:10,950 --> 00:35:12,630

 

1102
00:35:12,630 --> 00:35:12,640

 

1103
00:35:12,640 --> 00:35:13,589


1104
00:35:13,589 --> 00:35:13,599

 

1105
00:35:13,599 --> 00:35:14,829


1106
00:35:14,829 --> 00:35:18,390

 

1107
00:35:18,390 --> 00:35:20,310

 

1108
00:35:20,310 --> 00:35:22,630

 

1109
00:35:22,630 --> 00:35:23,670

 

1110
00:35:23,670 --> 00:35:23,680

 

1111
00:35:23,680 --> 00:35:24,950


1112
00:35:24,950 --> 00:35:24,960

 

1113
00:35:24,960 --> 00:35:27,109


1114
00:35:27,109 --> 00:35:28,950

 

1115
00:35:28,950 --> 00:35:32,550

 

1116
00:35:32,550 --> 00:35:35,270

 

1117
00:35:35,270 --> 00:35:37,109

 

1118
00:35:37,109 --> 00:35:39,430

 

1119
00:35:39,430 --> 00:35:42,950

 

1120
00:35:42,950 --> 00:35:45,510

 

1121
00:35:45,510 --> 00:35:45,520

 

1122
00:35:45,520 --> 00:35:46,230


1123
00:35:46,230 --> 00:35:47,430

 

1124
00:35:47,430 --> 00:35:47,440

 

1125
00:35:47,440 --> 00:35:48,790


1126
00:35:48,790 --> 00:35:51,349

 

1127
00:35:51,349 --> 00:35:53,030

 

1128
00:35:53,030 --> 00:35:53,040

 

1129
00:35:53,040 --> 00:35:56,870


1130
00:35:56,870 --> 00:35:58,069

 

1131
00:35:58,069 --> 00:36:00,069

 

1132
00:36:00,069 --> 00:36:02,069

 

1133
00:36:02,069 --> 00:36:04,390

 

1134
00:36:04,390 --> 00:36:04,400

 

1135
00:36:04,400 --> 00:36:07,829


1136
00:36:07,829 --> 00:36:09,750

 

1137
00:36:09,750 --> 00:36:09,760

 

1138
00:36:09,760 --> 00:36:10,710


1139
00:36:10,710 --> 00:36:12,470

 

1140
00:36:12,470 --> 00:36:13,829

 

1141
00:36:13,829 --> 00:36:15,430

 

1142
00:36:15,430 --> 00:36:16,950

 

1143
00:36:16,950 --> 00:36:18,630

 

1144
00:36:18,630 --> 00:36:19,589

 

1145
00:36:19,589 --> 00:36:21,430

 

1146
00:36:21,430 --> 00:36:21,440

 

1147
00:36:21,440 --> 00:36:22,790


1148
00:36:22,790 --> 00:36:26,470

 

1149
00:36:26,470 --> 00:36:27,750

 

1150
00:36:27,750 --> 00:36:31,990

 

1151
00:36:31,990 --> 00:36:32,000

 

1152
00:36:32,000 --> 00:36:34,230


1153
00:36:34,230 --> 00:36:37,510

 

1154
00:36:37,510 --> 00:36:43,510

 

1155
00:36:43,510 --> 00:36:45,990

 

1156
00:36:45,990 --> 00:36:48,790

 

1157
00:36:48,790 --> 00:36:51,910

 

1158
00:36:51,910 --> 00:36:54,150

 

1159
00:36:54,150 --> 00:36:57,430

 

1160
00:36:57,430 --> 00:36:57,440

 

1161
00:36:57,440 --> 00:36:58,310


1162
00:36:58,310 --> 00:37:00,630

 

1163
00:37:00,630 --> 00:37:02,950

 

1164
00:37:02,950 --> 00:37:04,470

 

1165
00:37:04,470 --> 00:37:08,390

 

1166
00:37:08,390 --> 00:37:11,109

 

1167
00:37:11,109 --> 00:37:13,190

 

1168
00:37:13,190 --> 00:37:13,200

 

1169
00:37:13,200 --> 00:37:14,150


1170
00:37:14,150 --> 00:37:16,630

 

1171
00:37:16,630 --> 00:37:18,630

 

1172
00:37:18,630 --> 00:37:20,310

 

1173
00:37:20,310 --> 00:37:21,910

 

1174
00:37:21,910 --> 00:37:23,750

 

1175
00:37:23,750 --> 00:37:25,349

 

1176
00:37:25,349 --> 00:37:25,359

 

1177
00:37:25,359 --> 00:37:26,710


1178
00:37:26,710 --> 00:37:28,470

 

1179
00:37:28,470 --> 00:37:30,470

 

1180
00:37:30,470 --> 00:37:32,710

 

1181
00:37:32,710 --> 00:37:34,870

 

1182
00:37:34,870 --> 00:37:34,880

 

1183
00:37:34,880 --> 00:37:36,710


1184
00:37:36,710 --> 00:37:39,589

 

1185
00:37:39,589 --> 00:37:41,910

 

1186
00:37:41,910 --> 00:37:45,190

 

1187
00:37:45,190 --> 00:37:46,790

 

1188
00:37:46,790 --> 00:37:49,510

 

1189
00:37:49,510 --> 00:37:52,069

 

1190
00:37:52,069 --> 00:37:52,079

 

1191
00:37:52,079 --> 00:37:53,670


1192
00:37:53,670 --> 00:37:55,430

 

1193
00:37:55,430 --> 00:37:57,109

 

1194
00:37:57,109 --> 00:37:58,470

 

1195
00:37:58,470 --> 00:38:00,630

 

1196
00:38:00,630 --> 00:38:02,790

 

1197
00:38:02,790 --> 00:38:04,310

 

1198
00:38:04,310 --> 00:38:05,589

 

1199
00:38:05,589 --> 00:38:07,109

 

1200
00:38:07,109 --> 00:38:09,829

 

1201
00:38:09,829 --> 00:38:11,510

 

1202
00:38:11,510 --> 00:38:14,470

 

1203
00:38:14,470 --> 00:38:19,589

 

1204
00:38:19,589 --> 00:38:21,190

 

1205
00:38:21,190 --> 00:38:23,750

 

1206
00:38:23,750 --> 00:38:26,550

 

1207
00:38:26,550 --> 00:38:28,870

 

1208
00:38:28,870 --> 00:38:30,829

 

1209
00:38:30,829 --> 00:38:38,069

 

1210
00:38:38,069 --> 00:38:41,910

 

1211
00:38:41,910 --> 00:38:41,920

 

1212
00:38:41,920 --> 00:38:43,190


1213
00:38:43,190 --> 00:38:46,710

 

1214
00:38:46,710 --> 00:38:46,720

 

1215
00:38:46,720 --> 00:38:48,150


1216
00:38:48,150 --> 00:38:48,160

 

1217
00:38:48,160 --> 00:38:48,870


1218
00:38:48,870 --> 00:38:50,310

 

1219
00:38:50,310 --> 00:38:52,230

 

1220
00:38:52,230 --> 00:38:54,630

 

1221
00:38:54,630 --> 00:38:54,640

 

1222
00:38:54,640 --> 00:38:56,950


1223
00:38:56,950 --> 00:38:58,710

 

1224
00:38:58,710 --> 00:38:58,720

 

1225
00:38:58,720 --> 00:39:01,109


1226
00:39:01,109 --> 00:39:01,119

 

1227
00:39:01,119 --> 00:39:02,870


1228
00:39:02,870 --> 00:39:04,550

 

1229
00:39:04,550 --> 00:39:07,190

 

1230
00:39:07,190 --> 00:39:07,200

 

1231
00:39:07,200 --> 00:39:08,230


1232
00:39:08,230 --> 00:39:08,240

 

1233
00:39:08,240 --> 00:39:10,710


1234
00:39:10,710 --> 00:39:12,069

 

1235
00:39:12,069 --> 00:39:13,109

 

1236
00:39:13,109 --> 00:39:15,670

 

1237
00:39:15,670 --> 00:39:17,030

 

1238
00:39:17,030 --> 00:39:18,230

 

1239
00:39:18,230 --> 00:39:19,910

 

1240
00:39:19,910 --> 00:39:21,990

 

1241
00:39:21,990 --> 00:39:24,150

 

1242
00:39:24,150 --> 00:39:26,310

 

1243
00:39:26,310 --> 00:39:28,390

 

1244
00:39:28,390 --> 00:39:30,069

 

1245
00:39:30,069 --> 00:39:30,079

 

1246
00:39:30,079 --> 00:39:31,349


1247
00:39:31,349 --> 00:39:33,829

 

1248
00:39:33,829 --> 00:39:35,670

 

1249
00:39:35,670 --> 00:39:37,190

 

1250
00:39:37,190 --> 00:39:39,990

 

1251
00:39:39,990 --> 00:39:41,670

 

1252
00:39:41,670 --> 00:39:43,030

 

1253
00:39:43,030 --> 00:39:44,950

 

1254
00:39:44,950 --> 00:39:46,390

 

1255
00:39:46,390 --> 00:39:47,510

 

1256
00:39:47,510 --> 00:39:49,190

 

1257
00:39:49,190 --> 00:39:52,310

 

1258
00:39:52,310 --> 00:39:55,510

 

1259
00:39:55,510 --> 00:39:55,520

 

1260
00:39:55,520 --> 00:39:56,870


1261
00:39:56,870 --> 00:39:57,910

 

1262
00:39:57,910 --> 00:39:59,270

 

1263
00:39:59,270 --> 00:40:00,550

 

1264
00:40:00,550 --> 00:40:02,470

 

1265
00:40:02,470 --> 00:40:04,470

 

1266
00:40:04,470 --> 00:40:04,480

 

1267
00:40:04,480 --> 00:40:05,750


1268
00:40:05,750 --> 00:40:08,390

 

1269
00:40:08,390 --> 00:40:10,069

 

1270
00:40:10,069 --> 00:40:12,230

 

1271
00:40:12,230 --> 00:40:13,829

 

1272
00:40:13,829 --> 00:40:13,839

 

1273
00:40:13,839 --> 00:40:14,870


1274
00:40:14,870 --> 00:40:17,109

 

1275
00:40:17,109 --> 00:40:18,870

 

1276
00:40:18,870 --> 00:40:19,910

 

1277
00:40:19,910 --> 00:40:19,920

 

1278
00:40:19,920 --> 00:40:21,430


1279
00:40:21,430 --> 00:40:23,190

 

1280
00:40:23,190 --> 00:40:24,870

 

1281
00:40:24,870 --> 00:40:24,880

 

1282
00:40:24,880 --> 00:40:27,430


1283
00:40:27,430 --> 00:40:30,069

 

1284
00:40:30,069 --> 00:40:31,510

 

1285
00:40:31,510 --> 00:40:33,510

 

1286
00:40:33,510 --> 00:40:35,589

 

1287
00:40:35,589 --> 00:40:38,630

 

1288
00:40:38,630 --> 00:40:39,589

 

1289
00:40:39,589 --> 00:40:43,270

 

1290
00:40:43,270 --> 00:40:44,950

 

1291
00:40:44,950 --> 00:40:46,790

 

1292
00:40:46,790 --> 00:40:46,800

 

1293
00:40:46,800 --> 00:40:48,150


1294
00:40:48,150 --> 00:40:49,829

 

1295
00:40:49,829 --> 00:40:53,510

 

1296
00:40:53,510 --> 00:40:55,750

 

1297
00:40:55,750 --> 00:40:58,069

 

1298
00:40:58,069 --> 00:41:00,069

 

1299
00:41:00,069 --> 00:41:02,309

 

1300
00:41:02,309 --> 00:41:02,319

 

1301
00:41:02,319 --> 00:41:03,349


1302
00:41:03,349 --> 00:41:05,670

 

1303
00:41:05,670 --> 00:41:08,710

 

1304
00:41:08,710 --> 00:41:12,630

 

1305
00:41:12,630 --> 00:41:12,640

 

1306
00:41:12,640 --> 00:41:13,430


1307
00:41:13,430 --> 00:41:13,440

 

1308
00:41:13,440 --> 00:41:14,390


1309
00:41:14,390 --> 00:41:16,790

 

1310
00:41:16,790 --> 00:41:19,430

 

1311
00:41:19,430 --> 00:41:19,440

 

1312
00:41:19,440 --> 00:41:21,990


1313
00:41:21,990 --> 00:41:24,230

 

1314
00:41:24,230 --> 00:41:26,710

 

1315
00:41:26,710 --> 00:41:28,230

 

1316
00:41:28,230 --> 00:41:30,150

 

1317
00:41:30,150 --> 00:41:31,910

 

1318
00:41:31,910 --> 00:41:34,550

 

1319
00:41:34,550 --> 00:41:36,950

 

1320
00:41:36,950 --> 00:41:38,790

 

1321
00:41:38,790 --> 00:41:39,990

 

1322
00:41:39,990 --> 00:41:42,950

 

1323
00:41:42,950 --> 00:41:45,510

 

1324
00:41:45,510 --> 00:41:48,230

 

1325
00:41:48,230 --> 00:41:49,510

 

1326
00:41:49,510 --> 00:41:50,710

 

1327
00:41:50,710 --> 00:41:53,589

 

1328
00:41:53,589 --> 00:41:55,349

 

1329
00:41:55,349 --> 00:41:56,790

 

1330
00:41:56,790 --> 00:41:59,190

 

1331
00:41:59,190 --> 00:42:01,190

 

1332
00:42:01,190 --> 00:42:02,710

 

1333
00:42:02,710 --> 00:42:04,470

 

1334
00:42:04,470 --> 00:42:04,480

 

1335
00:42:04,480 --> 00:42:05,190


1336
00:42:05,190 --> 00:42:08,069

 

1337
00:42:08,069 --> 00:42:10,550

 

1338
00:42:10,550 --> 00:42:12,069

 

1339
00:42:12,069 --> 00:42:14,550

 

1340
00:42:14,550 --> 00:42:17,349

 

1341
00:42:17,349 --> 00:42:19,349

 

1342
00:42:19,349 --> 00:42:21,030

 

1343
00:42:21,030 --> 00:42:24,390

 

1344
00:42:24,390 --> 00:42:26,790

 

1345
00:42:26,790 --> 00:42:26,800

 

1346
00:42:26,800 --> 00:42:27,990


1347
00:42:27,990 --> 00:42:30,790

 

1348
00:42:30,790 --> 00:42:33,190

 

1349
00:42:33,190 --> 00:42:34,230

 

1350
00:42:34,230 --> 00:42:37,109

 

1351
00:42:37,109 --> 00:42:38,870

 

1352
00:42:38,870 --> 00:42:38,880

 

1353
00:42:38,880 --> 00:42:39,589


1354
00:42:39,589 --> 00:42:46,550

 

1355
00:42:46,550 --> 00:42:48,069

 

1356
00:42:48,069 --> 00:42:49,750

 

1357
00:42:49,750 --> 00:42:51,670

 

1358
00:42:51,670 --> 00:42:53,270

 

1359
00:42:53,270 --> 00:42:53,280

 

1360
00:42:53,280 --> 00:42:54,150


1361
00:42:54,150 --> 00:42:56,870

 

1362
00:42:56,870 --> 00:42:58,710

 

1363
00:42:58,710 --> 00:43:00,309

 

1364
00:43:00,309 --> 00:43:01,430

 

1365
00:43:01,430 --> 00:43:03,109

 

1366
00:43:03,109 --> 00:43:06,390

 

1367
00:43:06,390 --> 00:43:08,390

 

1368
00:43:08,390 --> 00:43:10,309

 

1369
00:43:10,309 --> 00:43:11,829

 

1370
00:43:11,829 --> 00:43:14,150

 

1371
00:43:14,150 --> 00:43:16,230

 

1372
00:43:16,230 --> 00:43:17,990

 

1373
00:43:17,990 --> 00:43:20,230

 

1374
00:43:20,230 --> 00:43:22,309

 

1375
00:43:22,309 --> 00:43:24,390

 

1376
00:43:24,390 --> 00:43:27,190

 

1377
00:43:27,190 --> 00:43:29,030

 

1378
00:43:29,030 --> 00:43:31,510

 

1379
00:43:31,510 --> 00:43:33,030

 

1380
00:43:33,030 --> 00:43:35,190

 

1381
00:43:35,190 --> 00:43:36,390

 

1382
00:43:36,390 --> 00:43:38,950

 

1383
00:43:38,950 --> 00:43:40,870

 

1384
00:43:40,870 --> 00:43:43,030

 

1385
00:43:43,030 --> 00:43:43,040

 

1386
00:43:43,040 --> 00:43:45,430


1387
00:43:45,430 --> 00:43:48,069

 

1388
00:43:48,069 --> 00:43:49,109

 

1389
00:43:49,109 --> 00:43:51,270

 

1390
00:43:51,270 --> 00:43:52,550

 

1391
00:43:52,550 --> 00:43:54,390

 

1392
00:43:54,390 --> 00:43:55,750

 

1393
00:43:55,750 --> 00:43:58,150

 

1394
00:43:58,150 --> 00:44:00,790

 

1395
00:44:00,790 --> 00:44:03,109

 

1396
00:44:03,109 --> 00:44:06,150

 

1397
00:44:06,150 --> 00:44:06,160

 

1398
00:44:06,160 --> 00:44:07,190


1399
00:44:07,190 --> 00:44:11,190

 

1400
00:44:11,190 --> 00:44:13,349

 

1401
00:44:13,349 --> 00:44:14,870

 

1402
00:44:14,870 --> 00:44:14,880

 

1403
00:44:14,880 --> 00:44:15,990


1404
00:44:15,990 --> 00:44:18,390

 

1405
00:44:18,390 --> 00:44:20,150

 

1406
00:44:20,150 --> 00:44:22,710

 

1407
00:44:22,710 --> 00:44:25,349

 

1408
00:44:25,349 --> 00:44:26,790

 

1409
00:44:26,790 --> 00:44:28,309

 

1410
00:44:28,309 --> 00:44:29,990

 

1411
00:44:29,990 --> 00:44:31,910

 

1412
00:44:31,910 --> 00:44:34,069

 

1413
00:44:34,069 --> 00:44:36,470

 

1414
00:44:36,470 --> 00:44:38,790

 

1415
00:44:38,790 --> 00:44:41,190

 

1416
00:44:41,190 --> 00:44:41,200

 

1417
00:44:41,200 --> 00:44:42,150


1418
00:44:42,150 --> 00:44:45,510

 

1419
00:44:45,510 --> 00:44:48,550

 

1420
00:44:48,550 --> 00:44:53,430

 

1421
00:44:53,430 --> 00:44:57,670

 

1422
00:44:57,670 --> 00:44:58,950

 

1423
00:44:58,950 --> 00:45:01,510

 

1424
00:45:01,510 --> 00:45:04,150

 

1425
00:45:04,150 --> 00:45:05,670

 

1426
00:45:05,670 --> 00:45:08,550

 

1427
00:45:08,550 --> 00:45:10,710

 

1428
00:45:10,710 --> 00:45:14,150

 

1429
00:45:14,150 --> 00:45:16,630

 

1430
00:45:16,630 --> 00:45:19,510

 

1431
00:45:19,510 --> 00:45:22,230

 

1432
00:45:22,230 --> 00:45:23,670

 

1433
00:45:23,670 --> 00:45:25,910

 

1434
00:45:25,910 --> 00:45:27,670

 

1435
00:45:27,670 --> 00:45:30,150

 

1436
00:45:30,150 --> 00:45:30,160

 

1437
00:45:30,160 --> 00:45:31,109


1438
00:45:31,109 --> 00:45:33,990

 

1439
00:45:33,990 --> 00:45:34,000

 

1440
00:45:34,000 --> 00:45:38,470


1441
00:45:38,470 --> 00:45:38,480

 

1442
00:45:38,480 --> 00:45:41,190


1443
00:45:41,190 --> 00:45:43,349

 

1444
00:45:43,349 --> 00:45:45,829

 

1445
00:45:45,829 --> 00:45:45,839

 

1446
00:45:45,839 --> 00:45:46,950


1447
00:45:46,950 --> 00:45:46,960

 

1448
00:45:46,960 --> 00:45:47,230


1449
00:45:47,230 --> 00:45:47,240

 

1450
00:45:47,240 --> 00:45:50,870


1451
00:45:50,870 --> 00:45:54,470

 

1452
00:45:54,470 --> 00:45:57,030

 

1453
00:45:57,030 --> 00:46:00,870

 

1454
00:46:00,870 --> 00:46:02,150

 

1455
00:46:02,150 --> 00:46:03,430

 

1456
00:46:03,430 --> 00:46:05,349

 

1457
00:46:05,349 --> 00:46:06,790

 

1458
00:46:06,790 --> 00:46:09,270

 

1459
00:46:09,270 --> 00:46:11,510

 

1460
00:46:11,510 --> 00:46:15,030

 

1461
00:46:15,030 --> 00:46:16,550

 

1462
00:46:16,550 --> 00:46:17,910

 

1463
00:46:17,910 --> 00:46:19,910

 

1464
00:46:19,910 --> 00:46:22,150

 

1465
00:46:22,150 --> 00:46:24,950

 

1466
00:46:24,950 --> 00:46:26,710

 

1467
00:46:26,710 --> 00:46:28,069

 

1468
00:46:28,069 --> 00:46:29,670

 

1469
00:46:29,670 --> 00:46:31,109

 

1470
00:46:31,109 --> 00:46:32,710

 

1471
00:46:32,710 --> 00:46:34,790

 

1472
00:46:34,790 --> 00:46:36,950

 

1473
00:46:36,950 --> 00:46:38,630

 

1474
00:46:38,630 --> 00:46:39,990

 

1475
00:46:39,990 --> 00:46:41,670

 

1476
00:46:41,670 --> 00:46:44,069

 

1477
00:46:44,069 --> 00:46:46,069

 

1478
00:46:46,069 --> 00:46:48,710

 

1479
00:46:48,710 --> 00:46:50,470

 

1480
00:46:50,470 --> 00:46:52,630

 

1481
00:46:52,630 --> 00:46:54,710

 

1482
00:46:54,710 --> 00:46:54,720

 

1483
00:46:54,720 --> 00:46:56,230


1484
00:46:56,230 --> 00:46:58,710

 

1485
00:46:58,710 --> 00:47:00,230

 

1486
00:47:00,230 --> 00:47:01,829

 

1487
00:47:01,829 --> 00:47:05,430

 

1488
00:47:05,430 --> 00:47:07,270

 

1489
00:47:07,270 --> 00:47:09,349

 

1490
00:47:09,349 --> 00:47:11,990

 

1491
00:47:11,990 --> 00:47:12,000

 

1492
00:47:12,000 --> 00:47:14,150


1493
00:47:14,150 --> 00:47:14,160

 

1494
00:47:14,160 --> 00:47:15,270


1495
00:47:15,270 --> 00:47:17,030

 

1496
00:47:17,030 --> 00:47:24,870

 

1497
00:47:24,870 --> 00:47:26,390

 

1498
00:47:26,390 --> 00:47:29,430

 

1499
00:47:29,430 --> 00:47:32,630

 

1500
00:47:32,630 --> 00:47:32,640

 

1501
00:47:32,640 --> 00:47:33,510


1502
00:47:33,510 --> 00:47:35,190

 

1503
00:47:35,190 --> 00:47:37,430

 

1504
00:47:37,430 --> 00:47:40,710

 

1505
00:47:40,710 --> 00:47:43,589

 

1506
00:47:43,589 --> 00:47:45,430

 

1507
00:47:45,430 --> 00:47:48,470

 

1508
00:47:48,470 --> 00:47:49,670

 

1509
00:47:49,670 --> 00:47:53,030

 

1510
00:47:53,030 --> 00:47:54,549

 

1511
00:47:54,549 --> 00:47:56,390

 

1512
00:47:56,390 --> 00:47:56,400

 

1513
00:47:56,400 --> 00:47:57,670


1514
00:47:57,670 --> 00:47:59,030

 

1515
00:47:59,030 --> 00:48:01,589

 

1516
00:48:01,589 --> 00:48:03,750

 

1517
00:48:03,750 --> 00:48:03,760

 

1518
00:48:03,760 --> 00:48:05,349


1519
00:48:05,349 --> 00:48:06,870

 

1520
00:48:06,870 --> 00:48:09,510

 

1521
00:48:09,510 --> 00:48:11,670

 

1522
00:48:11,670 --> 00:48:14,950

 

1523
00:48:14,950 --> 00:48:17,670

 

1524
00:48:17,670 --> 00:48:19,030

 

1525
00:48:19,030 --> 00:48:20,790

 

1526
00:48:20,790 --> 00:48:23,670

 

1527
00:48:23,670 --> 00:48:25,750

 

1528
00:48:25,750 --> 00:48:27,349

 

1529
00:48:27,349 --> 00:48:28,549

 

1530
00:48:28,549 --> 00:48:30,470

 

1531
00:48:30,470 --> 00:48:32,870

 

1532
00:48:32,870 --> 00:48:34,630

 

1533
00:48:34,630 --> 00:48:35,990

 

1534
00:48:35,990 --> 00:48:37,750

 

1535
00:48:37,750 --> 00:48:37,760

 

1536
00:48:37,760 --> 00:48:39,349


1537
00:48:39,349 --> 00:48:41,109

 

1538
00:48:41,109 --> 00:48:43,990

 

1539
00:48:43,990 --> 00:48:44,000

 

1540
00:48:44,000 --> 00:48:45,349


1541
00:48:45,349 --> 00:48:46,630

 

1542
00:48:46,630 --> 00:48:48,950

 

1543
00:48:48,950 --> 00:48:48,960

 

1544
00:48:48,960 --> 00:48:49,990


1545
00:48:49,990 --> 00:48:52,309

 

1546
00:48:52,309 --> 00:48:53,750

 

1547
00:48:53,750 --> 00:48:53,760

 

1548
00:48:53,760 --> 00:48:54,710


1549
00:48:54,710 --> 00:48:54,720

 

1550
00:48:54,720 --> 00:48:56,630


1551
00:48:56,630 --> 00:48:59,990

 

1552
00:48:59,990 --> 00:49:01,589

 

1553
00:49:01,589 --> 00:49:03,109

 

1554
00:49:03,109 --> 00:49:04,710

 

1555
00:49:04,710 --> 00:49:07,030

 

1556
00:49:07,030 --> 00:49:08,150

 

1557
00:49:08,150 --> 00:49:09,750

 

1558
00:49:09,750 --> 00:49:12,549

 

1559
00:49:12,549 --> 00:49:14,710

 

1560
00:49:14,710 --> 00:49:16,470

 

1561
00:49:16,470 --> 00:49:16,480

 

1562
00:49:16,480 --> 00:49:17,270


1563
00:49:17,270 --> 00:49:19,750

 

1564
00:49:19,750 --> 00:49:21,990

 

1565
00:49:21,990 --> 00:49:24,470

 

1566
00:49:24,470 --> 00:49:26,390

 

1567
00:49:26,390 --> 00:49:29,030

 

1568
00:49:29,030 --> 00:49:32,150

 

1569
00:49:32,150 --> 00:49:32,160

 

1570
00:49:32,160 --> 00:49:33,510


1571
00:49:33,510 --> 00:49:33,520

 

1572
00:49:33,520 --> 00:49:34,790


1573
00:49:34,790 --> 00:49:37,589

 

1574
00:49:37,589 --> 00:49:39,349

 

1575
00:49:39,349 --> 00:49:41,270

 

1576
00:49:41,270 --> 00:49:43,670

 

1577
00:49:43,670 --> 00:49:46,549

 

1578
00:49:46,549 --> 00:49:48,549

 

1579
00:49:48,549 --> 00:49:51,430

 

1580
00:49:51,430 --> 00:49:51,440

 

1581
00:49:51,440 --> 00:49:52,470


1582
00:49:52,470 --> 00:49:54,630

 

1583
00:49:54,630 --> 00:49:56,230

 

1584
00:49:56,230 --> 00:49:59,030

 

1585
00:49:59,030 --> 00:50:01,510

 

1586
00:50:01,510 --> 00:50:03,670

 

1587
00:50:03,670 --> 00:50:05,109

 

1588
00:50:05,109 --> 00:50:05,119

 

1589
00:50:05,119 --> 00:50:06,309


1590
00:50:06,309 --> 00:50:08,230

 

1591
00:50:08,230 --> 00:50:10,309

 

1592
00:50:10,309 --> 00:50:12,470

 

1593
00:50:12,470 --> 00:50:13,910

 

1594
00:50:13,910 --> 00:50:15,190

 

1595
00:50:15,190 --> 00:50:16,870

 

1596
00:50:16,870 --> 00:50:19,510

 

1597
00:50:19,510 --> 00:50:20,950

 

1598
00:50:20,950 --> 00:50:23,270

 

1599
00:50:23,270 --> 00:50:26,069

 

1600
00:50:26,069 --> 00:50:27,910

 

1601
00:50:27,910 --> 00:50:27,920

 

1602
00:50:27,920 --> 00:50:29,589


1603
00:50:29,589 --> 00:50:34,230

 

1604
00:50:34,230 --> 00:50:34,240

 

1605
00:50:34,240 --> 00:50:35,510


1606
00:50:35,510 --> 00:50:37,589

 

1607
00:50:37,589 --> 00:50:39,829

 

1608
00:50:39,829 --> 00:50:41,349

 

1609
00:50:41,349 --> 00:50:43,190

 

1610
00:50:43,190 --> 00:50:44,950

 

1611
00:50:44,950 --> 00:50:46,630

 

1612
00:50:46,630 --> 00:50:46,640

 

1613
00:50:46,640 --> 00:50:47,750


1614
00:50:47,750 --> 00:50:50,309

 

1615
00:50:50,309 --> 00:50:53,589

 

1616
00:50:53,589 --> 00:50:56,150

 

1617
00:50:56,150 --> 00:50:56,160

 

1618
00:50:56,160 --> 00:50:57,349


1619
00:50:57,349 --> 00:51:00,309

 

1620
00:51:00,309 --> 00:51:01,589

 

1621
00:51:01,589 --> 00:51:03,270

 

1622
00:51:03,270 --> 00:51:05,750

 

1623
00:51:05,750 --> 00:51:07,829

 

1624
00:51:07,829 --> 00:51:09,990

 

1625
00:51:09,990 --> 00:51:11,990

 

1626
00:51:11,990 --> 00:51:13,030

 

1627
00:51:13,030 --> 00:51:14,870

 

1628
00:51:14,870 --> 00:51:17,190

 

1629
00:51:17,190 --> 00:51:19,190

 

1630
00:51:19,190 --> 00:51:20,549

 

1631
00:51:20,549 --> 00:51:22,630

 

1632
00:51:22,630 --> 00:51:25,030

 

1633
00:51:25,030 --> 00:51:26,390

 

1634
00:51:26,390 --> 00:51:27,589

 

1635
00:51:27,589 --> 00:51:29,030

 

1636
00:51:29,030 --> 00:51:29,040

 

1637
00:51:29,040 --> 00:51:30,390


1638
00:51:30,390 --> 00:51:32,630

 

1639
00:51:32,630 --> 00:51:34,390

 

1640
00:51:34,390 --> 00:51:37,030

 

1641
00:51:37,030 --> 00:51:39,270

 

1642
00:51:39,270 --> 00:51:42,390

 

1643
00:51:42,390 --> 00:51:44,470

 

1644
00:51:44,470 --> 00:51:46,390

 

1645
00:51:46,390 --> 00:51:48,790

 

1646
00:51:48,790 --> 00:51:50,150

 

1647
00:51:50,150 --> 00:51:50,160

 

1648
00:51:50,160 --> 00:51:51,349


1649
00:51:51,349 --> 00:51:53,910

 

1650
00:51:53,910 --> 00:51:55,910

 

1651
00:51:55,910 --> 00:51:59,030

 

1652
00:51:59,030 --> 00:52:00,470

 

1653
00:52:00,470 --> 00:52:03,270

 

1654
00:52:03,270 --> 00:52:05,270

 

1655
00:52:05,270 --> 00:52:07,190

 

1656
00:52:07,190 --> 00:52:08,790

 

1657
00:52:08,790 --> 00:52:08,800

 

1658
00:52:08,800 --> 00:52:09,910


1659
00:52:09,910 --> 00:52:12,150

 

1660
00:52:12,150 --> 00:52:14,870

 

1661
00:52:14,870 --> 00:52:16,470

 

1662
00:52:16,470 --> 00:52:18,549

 

1663
00:52:18,549 --> 00:52:18,559

 

1664
00:52:18,559 --> 00:52:20,630


1665
00:52:20,630 --> 00:52:20,640

 

1666
00:52:20,640 --> 00:52:21,829


1667
00:52:21,829 --> 00:52:24,069

 

1668
00:52:24,069 --> 00:52:26,390

 

1669
00:52:26,390 --> 00:52:28,309

 

1670
00:52:28,309 --> 00:52:29,990

 

1671
00:52:29,990 --> 00:52:32,910

 

1672
00:52:32,910 --> 00:52:32,920

 

1673
00:52:32,920 --> 00:52:35,589


1674
00:52:35,589 --> 00:52:37,030

 

1675
00:52:37,030 --> 00:52:38,470

 

1676
00:52:38,470 --> 00:52:40,390

 

1677
00:52:40,390 --> 00:52:43,510

 

1678
00:52:43,510 --> 00:52:45,349

 

1679
00:52:45,349 --> 00:52:46,790

 

1680
00:52:46,790 --> 00:52:48,309

 

1681
00:52:48,309 --> 00:52:50,710

 

1682
00:52:50,710 --> 00:52:53,190

 

1683
00:52:53,190 --> 00:52:55,430

 

1684
00:52:55,430 --> 00:52:56,470

 

1685
00:52:56,470 --> 00:52:56,480

 

1686
00:52:56,480 --> 00:52:57,750


1687
00:52:57,750 --> 00:53:00,790

 

1688
00:53:00,790 --> 00:53:02,710

 

1689
00:53:02,710 --> 00:53:04,280

 

1690
00:53:04,280 --> 00:53:04,290

 

1691
00:53:04,290 --> 00:53:05,670


1692
00:53:05,670 --> 00:53:09,430

 

1693
00:53:09,430 --> 00:53:12,309

 

1694
00:53:12,309 --> 00:53:14,790

 

1695
00:53:14,790 --> 00:53:16,309

 

1696
00:53:16,309 --> 00:53:18,309

 

1697
00:53:18,309 --> 00:53:20,230

 

1698
00:53:20,230 --> 00:53:21,990

 

1699
00:53:21,990 --> 00:53:24,150

 

1700
00:53:24,150 --> 00:53:25,750

 

1701
00:53:25,750 --> 00:53:27,349

 

1702
00:53:27,349 --> 00:53:29,109

 

1703
00:53:29,109 --> 00:53:31,589

 

1704
00:53:31,589 --> 00:53:33,990

 

1705
00:53:33,990 --> 00:53:36,230

 

1706
00:53:36,230 --> 00:53:38,230

 

1707
00:53:38,230 --> 00:53:40,150

 

1708
00:53:40,150 --> 00:53:40,160

 

1709
00:53:40,160 --> 00:53:40,950


1710
00:53:40,950 --> 00:53:43,270

 

1711
00:53:43,270 --> 00:53:45,430

 

1712
00:53:45,430 --> 00:53:48,069

 

1713
00:53:48,069 --> 00:53:50,390

 

1714
00:53:50,390 --> 00:53:52,710

 

1715
00:53:52,710 --> 00:53:53,990

 

1716
00:53:53,990 --> 00:53:55,589

 

1717
00:53:55,589 --> 00:53:57,589

 

1718
00:53:57,589 --> 00:54:00,069

 

1719
00:54:00,069 --> 00:54:00,079

 

1720
00:54:00,079 --> 00:54:00,790


1721
00:54:00,790 --> 00:54:02,630

 

1722
00:54:02,630 --> 00:54:04,870

 

1723
00:54:04,870 --> 00:54:06,230

 

1724
00:54:06,230 --> 00:54:08,390

 

1725
00:54:08,390 --> 00:54:09,589

 

1726
00:54:09,589 --> 00:54:11,589

 

1727
00:54:11,589 --> 00:54:13,510

 

1728
00:54:13,510 --> 00:54:15,910

 

1729
00:54:15,910 --> 00:54:18,390

 

1730
00:54:18,390 --> 00:54:19,589

 

1731
00:54:19,589 --> 00:54:21,430

 

1732
00:54:21,430 --> 00:54:22,829

 

1733
00:54:22,829 --> 00:54:25,430

 

1734
00:54:25,430 --> 00:54:27,670

 

1735
00:54:27,670 --> 00:54:30,150

 

1736
00:54:30,150 --> 00:54:32,470

 

1737
00:54:32,470 --> 00:54:34,230

 

1738
00:54:34,230 --> 00:54:35,190

 

1739
00:54:35,190 --> 00:54:35,200

 

1740
00:54:35,200 --> 00:54:36,150


1741
00:54:36,150 --> 00:54:37,510

 

1742
00:54:37,510 --> 00:54:39,990

 

1743
00:54:39,990 --> 00:54:42,069

 

1744
00:54:42,069 --> 00:54:44,069

 

1745
00:54:44,069 --> 00:54:46,069

 

1746
00:54:46,069 --> 00:54:47,430

 

1747
00:54:47,430 --> 00:54:49,190

 

1748
00:54:49,190 --> 00:54:49,200

 

1749
00:54:49,200 --> 00:54:50,230


1750
00:54:50,230 --> 00:54:53,190

 

1751
00:54:53,190 --> 00:54:54,309

 

1752
00:54:54,309 --> 00:54:56,549

 

1753
00:54:56,549 --> 00:54:57,910

 

1754
00:54:57,910 --> 00:55:00,309

 

1755
00:55:00,309 --> 00:55:02,150

 

1756
00:55:02,150 --> 00:55:04,549

 

1757
00:55:04,549 --> 00:55:07,270

 

1758
00:55:07,270 --> 00:55:09,430

 

1759
00:55:09,430 --> 00:55:11,829

 

1760
00:55:11,829 --> 00:55:14,710

 

1761
00:55:14,710 --> 00:55:17,430

 

1762
00:55:17,430 --> 00:55:18,789

 

1763
00:55:18,789 --> 00:55:21,430

 

1764
00:55:21,430 --> 00:55:22,870

 

1765
00:55:22,870 --> 00:55:24,230

 

1766
00:55:24,230 --> 00:55:26,789

 

1767
00:55:26,789 --> 00:55:28,870

 

1768
00:55:28,870 --> 00:55:29,829

 

1769
00:55:29,829 --> 00:55:31,670

 

1770
00:55:31,670 --> 00:55:33,430

 

1771
00:55:33,430 --> 00:55:35,109

 

1772
00:55:35,109 --> 00:55:37,430

 

1773
00:55:37,430 --> 00:55:39,430

 

1774
00:55:39,430 --> 00:55:42,309

 

1775
00:55:42,309 --> 00:55:44,710

 

1776
00:55:44,710 --> 00:55:47,030

 

1777
00:55:47,030 --> 00:55:49,670

 

1778
00:55:49,670 --> 00:55:51,270

 

1779
00:55:51,270 --> 00:55:53,349

 

1780
00:55:53,349 --> 00:55:53,359

 

1781
00:55:53,359 --> 00:55:54,390


1782
00:55:54,390 --> 00:55:57,829

 

1783
00:55:57,829 --> 00:55:59,109

 

1784
00:55:59,109 --> 00:56:02,230

 

1785
00:56:02,230 --> 00:56:03,829

 

1786
00:56:03,829 --> 00:56:06,230

 

1787
00:56:06,230 --> 00:56:07,670

 

1788
00:56:07,670 --> 00:56:09,349

 

1789
00:56:09,349 --> 00:56:10,870

 

1790
00:56:10,870 --> 00:56:12,710

 

1791
00:56:12,710 --> 00:56:13,750

 

1792
00:56:13,750 --> 00:56:18,870

 

1793
00:56:18,870 --> 00:56:20,390

 

1794
00:56:20,390 --> 00:56:23,670

 

1795
00:56:23,670 --> 00:56:28,710

 

1796
00:56:28,710 --> 00:56:30,789

 

1797
00:56:30,789 --> 00:56:33,670

 

1798
00:56:33,670 --> 00:56:34,950

 

1799
00:56:34,950 --> 00:56:36,630

 

1800
00:56:36,630 --> 00:56:39,589

 

1801
00:56:39,589 --> 00:56:40,870

 

1802
00:56:40,870 --> 00:56:42,829

 

1803
00:56:42,829 --> 00:56:44,630

 

1804
00:56:44,630 --> 00:56:48,309

 

1805
00:56:48,309 --> 00:56:50,710

 

1806
00:56:50,710 --> 00:56:52,470

 

1807
00:56:52,470 --> 00:56:53,829

 

1808
00:56:53,829 --> 00:56:57,270

 

1809
00:56:57,270 --> 00:57:00,150

 

1810
00:57:00,150 --> 00:57:02,870

 

1811
00:57:02,870 --> 00:57:04,870

 

1812
00:57:04,870 --> 00:57:06,710

 

1813
00:57:06,710 --> 00:57:08,470

 

1814
00:57:08,470 --> 00:57:10,630

 

1815
00:57:10,630 --> 00:57:13,349

 

1816
00:57:13,349 --> 00:57:15,190

 

1817
00:57:15,190 --> 00:57:17,430

 

1818
00:57:17,430 --> 00:57:20,150

 

1819
00:57:20,150 --> 00:57:22,309

 

1820
00:57:22,309 --> 00:57:22,319

 

1821
00:57:22,319 --> 00:57:23,510


1822
00:57:23,510 --> 00:57:25,349

 

1823
00:57:25,349 --> 00:57:27,910

 

1824
00:57:27,910 --> 00:57:30,150

 

1825
00:57:30,150 --> 00:57:31,670

 

1826
00:57:31,670 --> 00:57:33,750

 

1827
00:57:33,750 --> 00:57:35,510

 

1828
00:57:35,510 --> 00:57:37,670

 

1829
00:57:37,670 --> 00:57:39,750

 

1830
00:57:39,750 --> 00:57:43,030

 

1831
00:57:43,030 --> 00:57:45,750

 

1832
00:57:45,750 --> 00:57:45,760

 

1833
00:57:45,760 --> 00:57:48,309


1834
00:57:48,309 --> 00:57:49,510

 

1835
00:57:49,510 --> 00:57:51,589

 

1836
00:57:51,589 --> 00:57:53,349

 

1837
00:57:53,349 --> 00:57:55,349

 

1838
00:57:55,349 --> 00:57:56,870

 

1839
00:57:56,870 --> 00:57:59,109

 

1840
00:57:59,109 --> 00:58:01,030

 

1841
00:58:01,030 --> 00:58:02,549

 

1842
00:58:02,549 --> 00:58:04,080

 

1843
00:58:04,080 --> 00:58:04,090

 

1844
00:58:04,090 --> 00:58:06,150


1845
00:58:06,150 --> 00:58:07,510

 

1846
00:58:07,510 --> 00:58:08,710

 

1847
00:58:08,710 --> 00:58:11,670

 

1848
00:58:11,670 --> 00:58:14,549

 

1849
00:58:14,549 --> 00:58:16,069

 

1850
00:58:16,069 --> 00:58:18,470

 

1851
00:58:18,470 --> 00:58:20,630

 

1852
00:58:20,630 --> 00:58:23,030

 

1853
00:58:23,030 --> 00:58:24,309

 

1854
00:58:24,309 --> 00:58:27,030

 

1855
00:58:27,030 --> 00:58:28,470

 

1856
00:58:28,470 --> 00:58:30,150

 

1857
00:58:30,150 --> 00:58:32,710

 

1858
00:58:32,710 --> 00:58:34,789

 

1859
00:58:34,789 --> 00:58:34,799

 

1860
00:58:34,799 --> 00:58:36,870


1861
00:58:36,870 --> 00:58:36,880

 

1862
00:58:36,880 --> 00:58:43,430


1863
00:58:43,430 --> 00:58:55,910

 

1864
00:58:55,910 --> 00:58:58,230

 

1865
00:58:58,230 --> 00:59:00,950

 

1866
00:59:00,950 --> 00:59:00,960

 

1867
00:59:00,960 --> 00:59:01,750


1868
00:59:01,750 --> 00:59:01,760

 

1869
00:59:01,760 --> 00:59:05,670


1870
00:59:05,670 --> 00:59:08,150

 

1871
00:59:08,150 --> 00:59:11,190

 

1872
00:59:11,190 --> 00:59:12,630

 

1873
00:59:12,630 --> 00:59:12,640

 

1874
00:59:12,640 --> 00:59:14,069


1875
00:59:14,069 --> 00:59:16,829

 

1876
00:59:16,829 --> 00:59:16,839

 

1877
00:59:16,839 --> 00:59:18,630


1878
00:59:18,630 --> 00:59:18,640

 

1879
00:59:18,640 --> 00:59:20,390


1880
00:59:20,390 --> 00:59:23,829

 

1881
00:59:23,829 --> 00:59:23,839

 

1882
00:59:23,839 --> 00:59:25,349


1883
00:59:25,349 --> 00:59:26,390

 

1884
00:59:26,390 --> 00:59:28,150

 

1885
00:59:28,150 --> 00:59:29,589

 

1886
00:59:29,589 --> 00:59:31,510

 

1887
00:59:31,510 --> 00:59:33,030

 

1888
00:59:33,030 --> 00:59:34,230

 

1889
00:59:34,230 --> 00:59:37,270

 

1890
00:59:37,270 --> 00:59:38,789

 

1891
00:59:38,789 --> 00:59:40,309

 

1892
00:59:40,309 --> 00:59:42,390

 

1893
00:59:42,390 --> 00:59:44,710

 

1894
00:59:44,710 --> 00:59:47,670

 

1895
00:59:47,670 --> 00:59:49,190

 

1896
00:59:49,190 --> 00:59:51,589

 

1897
00:59:51,589 --> 00:59:53,109

 

1898
00:59:53,109 --> 00:59:54,950

 

1899
00:59:54,950 --> 00:59:56,470

 

1900
00:59:56,470 --> 00:59:59,270

 

1901
00:59:59,270 --> 01:00:00,789

 

1902
01:00:00,789 --> 01:00:02,390

 

1903
01:00:02,390 --> 01:00:04,069

 

1904
01:00:04,069 --> 01:00:06,950

 

1905
01:00:06,950 --> 01:00:08,710

 

1906
01:00:08,710 --> 01:00:09,750

 

1907
01:00:09,750 --> 01:00:11,190

 

1908
01:00:11,190 --> 01:00:13,030

 

1909
01:00:13,030 --> 01:00:15,030

 

1910
01:00:15,030 --> 01:00:16,549

 

1911
01:00:16,549 --> 01:00:17,990

 

1912
01:00:17,990 --> 01:00:19,430

 

1913
01:00:19,430 --> 01:00:21,190

 

1914
01:00:21,190 --> 01:00:21,200

 

1915
01:00:21,200 --> 01:00:22,390


1916
01:00:22,390 --> 01:00:24,230

 

1917
01:00:24,230 --> 01:00:26,829

 

1918
01:00:26,829 --> 01:00:29,030

 

1919
01:00:29,030 --> 01:00:29,040

 

1920
01:00:29,040 --> 01:00:30,069


1921
01:00:30,069 --> 01:00:31,990

 

1922
01:00:31,990 --> 01:00:32,000

 

1923
01:00:32,000 --> 01:00:34,470


1924
01:00:34,470 --> 01:00:34,480

 

1925
01:00:34,480 --> 01:00:35,270


1926
01:00:35,270 --> 01:00:36,710

 

1927
01:00:36,710 --> 01:00:38,630

 

1928
01:00:38,630 --> 01:00:42,230

 

1929
01:00:42,230 --> 01:00:44,390

 

1930
01:00:44,390 --> 01:00:46,470

 

1931
01:00:46,470 --> 01:00:48,230

 

1932
01:00:48,230 --> 01:00:49,990

 

1933
01:00:49,990 --> 01:00:53,030

 

1934
01:00:53,030 --> 01:00:55,109

 

1935
01:00:55,109 --> 01:00:58,549

 

1936
01:00:58,549 --> 01:01:01,829

 

1937
01:01:01,829 --> 01:01:03,589

 

1938
01:01:03,589 --> 01:01:05,109

 

1939
01:01:05,109 --> 01:01:07,750

 

1940
01:01:07,750 --> 01:01:09,349

 

1941
01:01:09,349 --> 01:01:11,750

 

1942
01:01:11,750 --> 01:01:14,710

 

1943
01:01:14,710 --> 01:01:17,510

 

1944
01:01:17,510 --> 01:01:19,190

 

1945
01:01:19,190 --> 01:01:23,190

 

1946
01:01:23,190 --> 01:01:26,309

 

1947
01:01:26,309 --> 01:01:28,150

 

1948
01:01:28,150 --> 01:01:30,549

 

1949
01:01:30,549 --> 01:01:32,230

 

1950
01:01:32,230 --> 01:01:33,109

 

1951
01:01:33,109 --> 01:01:34,549

 

1952
01:01:34,549 --> 01:01:36,950

 

1953
01:01:36,950 --> 01:01:39,349

 

1954
01:01:39,349 --> 01:01:41,430

 

1955
01:01:41,430 --> 01:01:43,109

 

1956
01:01:43,109 --> 01:01:45,030

 

1957
01:01:45,030 --> 01:01:47,349

 

1958
01:01:47,349 --> 01:01:48,789

 

1959
01:01:48,789 --> 01:01:50,870

 

1960
01:01:50,870 --> 01:01:52,470

 

1961
01:01:52,470 --> 01:01:53,750

 

1962
01:01:53,750 --> 01:01:53,760

 

1963
01:01:53,760 --> 01:01:55,109


1964
01:01:55,109 --> 01:01:57,750

 

1965
01:01:57,750 --> 01:01:59,990

 

1966
01:01:59,990 --> 01:02:01,589

 

1967
01:02:01,589 --> 01:02:01,599

 

1968
01:02:01,599 --> 01:02:03,589


1969
01:02:03,589 --> 01:02:06,549

 

1970
01:02:06,549 --> 01:02:08,630

 

1971
01:02:08,630 --> 01:02:10,549

 

1972
01:02:10,549 --> 01:02:13,589

 

1973
01:02:13,589 --> 01:02:15,029

 

1974
01:02:15,029 --> 01:02:19,109

 

1975
01:02:19,109 --> 01:02:21,430

 

1976
01:02:21,430 --> 01:02:23,349

 

1977
01:02:23,349 --> 01:02:26,549

 

1978
01:02:26,549 --> 01:02:29,029

 

1979
01:02:29,029 --> 01:02:31,109

 

1980
01:02:31,109 --> 01:02:33,029

 

1981
01:02:33,029 --> 01:02:34,950

 

1982
01:02:34,950 --> 01:02:34,960

 

1983
01:02:34,960 --> 01:02:36,069


1984
01:02:36,069 --> 01:02:38,069

 

1985
01:02:38,069 --> 01:02:38,079

 

1986
01:02:38,079 --> 01:02:39,430


1987
01:02:39,430 --> 01:02:43,270

 

1988
01:02:43,270 --> 01:02:43,280

 

1989
01:02:43,280 --> 01:02:44,069


1990
01:02:44,069 --> 01:02:46,309

 

1991
01:02:46,309 --> 01:02:49,109

 

1992
01:02:49,109 --> 01:02:50,950

 

1993
01:02:50,950 --> 01:02:53,190

 

1994
01:02:53,190 --> 01:02:55,109

 

1995
01:02:55,109 --> 01:02:57,430

 

1996
01:02:57,430 --> 01:02:59,829

 

1997
01:02:59,829 --> 01:03:01,910

 

1998
01:03:01,910 --> 01:03:03,750

 

1999
01:03:03,750 --> 01:03:06,470

 

2000
01:03:06,470 --> 01:03:08,150

 

2001
01:03:08,150 --> 01:03:09,670

 

2002
01:03:09,670 --> 01:03:09,680

 

2003
01:03:09,680 --> 01:03:11,029


2004
01:03:11,029 --> 01:03:13,029

 

2005
01:03:13,029 --> 01:03:14,950

 

2006
01:03:14,950 --> 01:03:16,870

 

2007
01:03:16,870 --> 01:03:19,349

 

2008
01:03:19,349 --> 01:03:19,359

 

2009
01:03:19,359 --> 01:03:21,349


2010
01:03:21,349 --> 01:03:21,359

 

2011
01:03:21,359 --> 01:03:22,390


2012
01:03:22,390 --> 01:03:25,109

 

2013
01:03:25,109 --> 01:03:27,029

 

2014
01:03:27,029 --> 01:03:31,670

 

2015
01:03:31,670 --> 01:03:34,549

 

2016
01:03:34,549 --> 01:03:38,069

 

2017
01:03:38,069 --> 01:03:40,789

 

2018
01:03:40,789 --> 01:03:42,390

 

2019
01:03:42,390 --> 01:03:44,390

 

2020
01:03:44,390 --> 01:03:47,670

 

2021
01:03:47,670 --> 01:03:49,190

 

2022
01:03:49,190 --> 01:03:49,200

 

2023
01:03:49,200 --> 01:03:50,390


2024
01:03:50,390 --> 01:03:50,400

 

2025
01:03:50,400 --> 01:03:51,430


2026
01:03:51,430 --> 01:03:53,190

 

2027
01:03:53,190 --> 01:03:55,670

 

2028
01:03:55,670 --> 01:03:59,670

 

2029
01:03:59,670 --> 01:04:01,589

 

2030
01:04:01,589 --> 01:04:03,990

 

2031
01:04:03,990 --> 01:04:06,150

 

2032
01:04:06,150 --> 01:04:07,109

 

2033
01:04:07,109 --> 01:04:08,710

 

2034
01:04:08,710 --> 01:04:10,230

 

2035
01:04:10,230 --> 01:04:11,990

 

2036
01:04:11,990 --> 01:04:12,000

 

2037
01:04:12,000 --> 01:04:15,349


2038
01:04:15,349 --> 01:04:16,870

 

2039
01:04:16,870 --> 01:04:18,470

 

2040
01:04:18,470 --> 01:04:19,829

 

2041
01:04:19,829 --> 01:04:22,950

 

2042
01:04:22,950 --> 01:04:24,789

 

2043
01:04:24,789 --> 01:04:24,799

 

2044
01:04:24,799 --> 01:04:26,789


2045
01:04:26,789 --> 01:04:26,799

 

2046
01:04:26,799 --> 01:04:27,829


2047
01:04:27,829 --> 01:04:31,190

 

2048
01:04:31,190 --> 01:04:32,870

 

2049
01:04:32,870 --> 01:04:34,630

 

2050
01:04:34,630 --> 01:04:36,710

 

2051
01:04:36,710 --> 01:04:38,710

 

2052
01:04:38,710 --> 01:04:41,670

 

2053
01:04:41,670 --> 01:04:43,349

 

2054
01:04:43,349 --> 01:04:45,430

 

2055
01:04:45,430 --> 01:04:47,109

 

2056
01:04:47,109 --> 01:04:48,950

 

2057
01:04:48,950 --> 01:04:50,069

 

2058
01:04:50,069 --> 01:04:52,470

 

2059
01:04:52,470 --> 01:04:54,549

 

2060
01:04:54,549 --> 01:04:58,630

 

2061
01:04:58,630 --> 01:05:00,789

 

2062
01:05:00,789 --> 01:05:00,799

 

2063
01:05:00,799 --> 01:05:07,990


2064
01:05:07,990 --> 01:05:09,910

 

2065
01:05:09,910 --> 01:05:12,630

 

2066
01:05:12,630 --> 01:05:12,640

 

2067
01:05:12,640 --> 01:05:17,510


2068
01:05:17,510 --> 01:05:19,829

 

2069
01:05:19,829 --> 01:05:21,430

 

2070
01:05:21,430 --> 01:05:21,440

 

2071
01:05:21,440 --> 01:05:22,950


2072
01:05:22,950 --> 01:05:24,390

 

2073
01:05:24,390 --> 01:05:25,910

 

2074
01:05:25,910 --> 01:05:28,069

 

2075
01:05:28,069 --> 01:05:30,950

 

2076
01:05:30,950 --> 01:05:32,789

 

2077
01:05:32,789 --> 01:05:34,789

 

2078
01:05:34,789 --> 01:05:37,029

 

2079
01:05:37,029 --> 01:05:39,589

 

2080
01:05:39,589 --> 01:05:39,599

 

2081
01:05:39,599 --> 01:05:40,789


2082
01:05:40,789 --> 01:05:42,390

 

2083
01:05:42,390 --> 01:05:44,230

 

2084
01:05:44,230 --> 01:05:46,630

 

2085
01:05:46,630 --> 01:05:46,640

 

2086
01:05:46,640 --> 01:05:48,069


2087
01:05:48,069 --> 01:05:48,079

 

2088
01:05:48,079 --> 01:05:49,109


2089
01:05:49,109 --> 01:05:49,119

 

2090
01:05:49,119 --> 01:05:50,549


2091
01:05:50,549 --> 01:05:50,559

 

2092
01:05:50,559 --> 01:05:53,750


2093
01:05:53,750 --> 01:05:55,589

 

2094
01:05:55,589 --> 01:05:56,789

 

2095
01:05:56,789 --> 01:05:59,029

 

2096
01:05:59,029 --> 01:06:01,270

 

2097
01:06:01,270 --> 01:06:02,390

 

2098
01:06:02,390 --> 01:06:06,069

 

2099
01:06:06,069 --> 01:06:08,470

 

2100
01:06:08,470 --> 01:06:11,430

 

2101
01:06:11,430 --> 01:06:13,430

 

2102
01:06:13,430 --> 01:06:15,349

 

2103
01:06:15,349 --> 01:06:15,359

 

2104
01:06:15,359 --> 01:06:16,549


2105
01:06:16,549 --> 01:06:19,430

 

2106
01:06:19,430 --> 01:06:21,670

 

2107
01:06:21,670 --> 01:06:23,510

 

2108
01:06:23,510 --> 01:06:26,150

 

2109
01:06:26,150 --> 01:06:28,069

 

2110
01:06:28,069 --> 01:06:30,789

 

2111
01:06:30,789 --> 01:06:34,069

 

2112
01:06:34,069 --> 01:06:37,270

 

2113
01:06:37,270 --> 01:06:39,029

 

2114
01:06:39,029 --> 01:06:40,789

 

2115
01:06:40,789 --> 01:06:42,069

 

2116
01:06:42,069 --> 01:06:43,910

 

2117
01:06:43,910 --> 01:06:46,549

 

2118
01:06:46,549 --> 01:06:48,549

 

2119
01:06:48,549 --> 01:06:50,549

 

2120
01:06:50,549 --> 01:06:50,559

 

2121
01:06:50,559 --> 01:06:52,950


2122
01:06:52,950 --> 01:06:54,230

 

2123
01:06:54,230 --> 01:06:55,190

 

2124
01:06:55,190 --> 01:06:58,309

 

2125
01:06:58,309 --> 01:06:59,670

 

2126
01:06:59,670 --> 01:07:01,270

 

2127
01:07:01,270 --> 01:07:03,109

 

2128
01:07:03,109 --> 01:07:05,270

 

2129
01:07:05,270 --> 01:07:06,870

 

2130
01:07:06,870 --> 01:07:08,789

 

2131
01:07:08,789 --> 01:07:10,549

 

2132
01:07:10,549 --> 01:07:12,710

 

2133
01:07:12,710 --> 01:07:14,470

 

2134
01:07:14,470 --> 01:07:14,480

 

2135
01:07:14,480 --> 01:07:15,829


2136
01:07:15,829 --> 01:07:18,069

 

2137
01:07:18,069 --> 01:07:21,910

 

2138
01:07:21,910 --> 01:07:23,270

 

2139
01:07:23,270 --> 01:07:26,829

 

2140
01:07:26,829 --> 01:07:32,230

 

2141
01:07:32,230 --> 01:07:34,390

 

2142
01:07:34,390 --> 01:07:36,789

 

2143
01:07:36,789 --> 01:07:38,950

 

2144
01:07:38,950 --> 01:07:38,960

 

2145
01:07:38,960 --> 01:07:39,990


2146
01:07:39,990 --> 01:07:40,000

 

2147
01:07:40,000 --> 01:07:41,190


2148
01:07:41,190 --> 01:07:44,549

 

2149
01:07:44,549 --> 01:07:44,559

 

2150
01:07:44,559 --> 01:07:45,990


2151
01:07:45,990 --> 01:07:48,230

 

2152
01:07:48,230 --> 01:07:50,630

 

2153
01:07:50,630 --> 01:07:52,710

 

2154
01:07:52,710 --> 01:07:55,430

 

2155
01:07:55,430 --> 01:07:57,349

 

2156
01:07:57,349 --> 01:07:59,510

 

2157
01:07:59,510 --> 01:08:01,750

 

2158
01:08:01,750 --> 01:08:04,950

 

2159
01:08:04,950 --> 01:08:07,029

 

2160
01:08:07,029 --> 01:08:09,029

 

2161
01:08:09,029 --> 01:08:11,190

 

2162
01:08:11,190 --> 01:08:13,510

 

2163
01:08:13,510 --> 01:08:16,070

 

2164
01:08:16,070 --> 01:08:17,749

 

2165
01:08:17,749 --> 01:08:21,510

 

2166
01:08:21,510 --> 01:08:23,349

 

2167
01:08:23,349 --> 01:08:23,359

 

2168
01:08:23,359 --> 01:08:24,630


2169
01:08:24,630 --> 01:08:28,149

 

2170
01:08:28,149 --> 01:08:31,590

 

2171
01:08:31,590 --> 01:08:34,470

 

2172
01:08:34,470 --> 01:08:36,870

 

2173
01:08:36,870 --> 01:08:39,430

 

2174
01:08:39,430 --> 01:08:39,440

 

2175
01:08:39,440 --> 01:08:40,950


2176
01:08:40,950 --> 01:08:44,789

 

2177
01:08:44,789 --> 01:08:44,799

 

2178
01:08:44,799 --> 01:08:45,590


2179
01:08:45,590 --> 01:08:47,510

 

2180
01:08:47,510 --> 01:08:49,510

 

2181
01:08:49,510 --> 01:08:49,520

 

2182
01:08:49,520 --> 01:08:50,309


2183
01:08:50,309 --> 01:08:51,749

 

2184
01:08:51,749 --> 01:08:53,910

 

2185
01:08:53,910 --> 01:08:55,749

 

2186
01:08:55,749 --> 01:08:58,390

 

2187
01:08:58,390 --> 01:09:00,149

 

2188
01:09:00,149 --> 01:09:00,159

 

2189
01:09:00,159 --> 01:09:01,669


2190
01:09:01,669 --> 01:09:01,679

 

2191
01:09:01,679 --> 01:09:02,630


2192
01:09:02,630 --> 01:09:04,829

 

2193
01:09:04,829 --> 01:09:06,789

 

2194
01:09:06,789 --> 01:09:08,470

 

2195
01:09:08,470 --> 01:09:10,709

 

2196
01:09:10,709 --> 01:09:12,390

 

2197
01:09:12,390 --> 01:09:13,910

 

2198
01:09:13,910 --> 01:09:15,669

 

2199
01:09:15,669 --> 01:09:18,070

 

2200
01:09:18,070 --> 01:09:19,829

 

2201
01:09:19,829 --> 01:09:21,829

 

2202
01:09:21,829 --> 01:09:24,390

 

2203
01:09:24,390 --> 01:09:27,269

 

2204
01:09:27,269 --> 01:09:29,749

 

2205
01:09:29,749 --> 01:09:32,070

 

2206
01:09:32,070 --> 01:09:34,630

 

2207
01:09:34,630 --> 01:09:36,149

 

2208
01:09:36,149 --> 01:09:36,159

 

2209
01:09:36,159 --> 01:09:37,269


2210
01:09:37,269 --> 01:09:37,279

 

2211
01:09:37,279 --> 01:09:38,309


2212
01:09:38,309 --> 01:09:39,829

 

2213
01:09:39,829 --> 01:09:41,669

 

2214
01:09:41,669 --> 01:09:44,070

 

2215
01:09:44,070 --> 01:09:45,590

 

2216
01:09:45,590 --> 01:09:47,269

 

2217
01:09:47,269 --> 01:09:49,510

 

2218
01:09:49,510 --> 01:09:52,070

 

2219
01:09:52,070 --> 01:09:53,590

 

2220
01:09:53,590 --> 01:09:55,990

 

2221
01:09:55,990 --> 01:09:58,310

 

2222
01:09:58,310 --> 01:10:00,630

 

2223
01:10:00,630 --> 01:10:02,709

 

2224
01:10:02,709 --> 01:10:04,550

 

2225
01:10:04,550 --> 01:10:05,910

 

2226
01:10:05,910 --> 01:10:07,990

 

2227
01:10:07,990 --> 01:10:09,430

 

2228
01:10:09,430 --> 01:10:11,990

 

2229
01:10:11,990 --> 01:10:12,000

 

2230
01:10:12,000 --> 01:10:13,110


2231
01:10:13,110 --> 01:10:14,390

 

2232
01:10:14,390 --> 01:10:16,630

 

2233
01:10:16,630 --> 01:10:18,950

 

2234
01:10:18,950 --> 01:10:20,550

 

2235
01:10:20,550 --> 01:10:23,510

 

2236
01:10:23,510 --> 01:10:25,270

 

2237
01:10:25,270 --> 01:10:29,350

 

2238
01:10:29,350 --> 01:10:32,070

 

2239
01:10:32,070 --> 01:10:34,550

 

2240
01:10:34,550 --> 01:10:36,149

 

2241
01:10:36,149 --> 01:10:37,750

 

2242
01:10:37,750 --> 01:10:40,229

 

2243
01:10:40,229 --> 01:10:42,709

 

2244
01:10:42,709 --> 01:10:44,709

 

2245
01:10:44,709 --> 01:10:44,719

 

2246
01:10:44,719 --> 01:10:45,510


2247
01:10:45,510 --> 01:10:47,350

 

2248
01:10:47,350 --> 01:10:48,950

 

2249
01:10:48,950 --> 01:10:50,550

 

2250
01:10:50,550 --> 01:10:52,470

 

2251
01:10:52,470 --> 01:10:54,470

 

2252
01:10:54,470 --> 01:10:56,830

 

2253
01:10:56,830 --> 01:10:58,790

 

2254
01:10:58,790 --> 01:11:00,470

 

2255
01:11:00,470 --> 01:11:02,070

 

2256
01:11:02,070 --> 01:11:03,990

 

2257
01:11:03,990 --> 01:11:05,990

 

2258
01:11:05,990 --> 01:11:07,669

 

2259
01:11:07,669 --> 01:11:10,709

 

2260
01:11:10,709 --> 01:11:12,550

 

2261
01:11:12,550 --> 01:11:15,830

 

2262
01:11:15,830 --> 01:11:17,590

 

2263
01:11:17,590 --> 01:11:19,910

 

2264
01:11:19,910 --> 01:11:20,950

 

2265
01:11:20,950 --> 01:11:23,990

 

2266
01:11:23,990 --> 01:11:26,390

 

2267
01:11:26,390 --> 01:11:28,229

 

2268
01:11:28,229 --> 01:11:28,239

 

2269
01:11:28,239 --> 01:11:29,590


2270
01:11:29,590 --> 01:11:32,630

 

2271
01:11:32,630 --> 01:11:34,870

 

2272
01:11:34,870 --> 01:11:36,630

 

2273
01:11:36,630 --> 01:11:38,950

 

2274
01:11:38,950 --> 01:11:41,110

 

2275
01:11:41,110 --> 01:11:42,229

 

2276
01:11:42,229 --> 01:11:46,310

 

2277
01:11:46,310 --> 01:11:49,270

 

2278
01:11:49,270 --> 01:11:51,510

 

2279
01:11:51,510 --> 01:11:54,550

 

2280
01:11:54,550 --> 01:11:56,149

 

2281
01:11:56,149 --> 01:11:56,159

 

2282
01:11:56,159 --> 01:11:58,870


2283
01:11:58,870 --> 01:11:58,880

 

2284
01:11:58,880 --> 01:12:00,310


2285
01:12:00,310 --> 01:12:02,229

 

2286
01:12:02,229 --> 01:12:05,590

 

2287
01:12:05,590 --> 01:12:13,270

 

2288
01:12:13,270 --> 01:12:13,280

 

2289
01:12:13,280 --> 01:12:15,189


2290
01:12:15,189 --> 01:12:16,709

 

2291
01:12:16,709 --> 01:12:18,390

 

2292
01:12:18,390 --> 01:12:20,830

 

2293
01:12:20,830 --> 01:12:24,870

 

2294
01:12:24,870 --> 01:12:24,880

 

2295
01:12:24,880 --> 01:12:26,630


2296
01:12:26,630 --> 01:12:29,750

 

2297
01:12:29,750 --> 01:12:32,709

 

2298
01:12:32,709 --> 01:12:32,719

 

2299
01:12:32,719 --> 01:12:33,830


2300
01:12:33,830 --> 01:12:36,630

 

2301
01:12:36,630 --> 01:12:38,790

 

2302
01:12:38,790 --> 01:12:40,870

 

2303
01:12:40,870 --> 01:12:42,870

 

2304
01:12:42,870 --> 01:12:46,229

 

2305
01:12:46,229 --> 01:12:47,430

 

2306
01:12:47,430 --> 01:12:48,870

 

2307
01:12:48,870 --> 01:12:50,950

 

2308
01:12:50,950 --> 01:12:52,630

 

2309
01:12:52,630 --> 01:12:54,070

 

2310
01:12:54,070 --> 01:12:56,070

 

2311
01:12:56,070 --> 01:12:57,270

 

2312
01:12:57,270 --> 01:12:59,750

 

2313
01:12:59,750 --> 01:13:01,830

 

2314
01:13:01,830 --> 01:13:03,830

 

2315
01:13:03,830 --> 01:13:06,310

 

2316
01:13:06,310 --> 01:13:07,510

 

2317
01:13:07,510 --> 01:13:09,270

 

2318
01:13:09,270 --> 01:13:11,110

 

2319
01:13:11,110 --> 01:13:13,590

 

2320
01:13:13,590 --> 01:13:15,590

 

2321
01:13:15,590 --> 01:13:15,600

 

2322
01:13:15,600 --> 01:13:16,870


2323
01:13:16,870 --> 01:13:17,750

 

2324
01:13:17,750 --> 01:13:19,510

 

2325
01:13:19,510 --> 01:13:21,510

 

2326
01:13:21,510 --> 01:13:22,790

 

2327
01:13:22,790 --> 01:13:24,709

 

2328
01:13:24,709 --> 01:13:26,709

 

2329
01:13:26,709 --> 01:13:29,350

 

2330
01:13:29,350 --> 01:13:32,709

 

2331
01:13:32,709 --> 01:13:34,310

 

2332
01:13:34,310 --> 01:13:36,550

 

2333
01:13:36,550 --> 01:13:38,229

 

2334
01:13:38,229 --> 01:13:40,390

 

2335
01:13:40,390 --> 01:13:43,189

 

2336
01:13:43,189 --> 01:13:47,480