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so um welcome everybody uh to  the full  2020 online  uh introduction to the theory of  computing um 18404 6840.  my name is mike sipser  i'm going to be your instructor for the  semester in this class  um  so let me just tell you what the course  is about um  basically it's going to be into two  halves what we're going to be talking  about what are the capabilities and  limitations of computers of computer  algorithms really computation  um  and the two parts of the course are more  or less  divided in half  we're having the first half of the  course is going to talk about a subject  called computability theory  which it really asks what what you can  compute with an algorithm in principle  um  that's a was an active area of research  in the earlier part of the 20th century  it's pretty much closed off as a  research subject these days  mainly because  they answered all of their big questions  um and so uh a mathematical field really  only uh stays  vital when it has problems to solve and  they really solved all of their uh  interesting problems for the most part  not a hundred percent but for the most  part uh it sort of finished off in the  1950s um  just to say a little bit more about what  we're going to talk about there you know  when you're interested to know what  kinds of problems you can solve with an  algorithm  you know there are problems that you  might want to solve that you just can't  solve for example if you want to  you're given a specification for a um  you know a computer problem you want to  solve  whatever whatever that specification  might be say you know your algorithm  actually is a sorting algorithm for  example  and you want to write down that  specification and have an automatic  verifier that's going to check whether a  program meets a specification well  that's just in principle impossible you  cannot meet you cannot make a um  a verifier which is going to uh  answer in all cases whether or not a  program meets a certain specification um  so things like that we will prove uh  this uh this semester you know questions  about mathematical truth what if you're  given a mathematical statement is it  true or is it false uh it'd be great  if you can write a computer program that  would answer that problem well it would  not be great if you're a mathematician  because that would put us all out of  business but if you and you know  you could imagine that might be a nice  thing to have but you can't i mean there  is no algorithm which can answer that  question  um  uh well along the way we're going to  introduce models of computation like  fine automata which we'll see today  touring machines and some model other  models that we'll see along the way  the second half of the course which is  going to be after the midterm we're  going to shift gears and talk about  complexity theory which is instead of  looking at what's computable in  principle you're going to look at what's  computable in practice  so things that you can solve in a  reasonable amount of time  and um for example i'm sure many of you  are aware of the factoring problem which  has connections to uh the um  uh the the rsa crypto system you know  cryptography  um and asks whether you can  factor big numbers quickly um that's a  problem we don't know the answer to um  we just don't know how to factor big  numbers quickly but it's possible that  there are algorithms out there that we  haven't discovered yet that can do so  it's connected with this very famous  problem in the intersection of computer  science and mathematics called the p  versus np problem which many of you may  have heard of we'll talk about that  we'll spend a lot of time on that uh  this uh this term  and along the way we'll talk about  different measures of uh complexity of  computation time and space  time and memory  you know a theorist called memory i like  to call space  that's going to be a big part of the  course  in the complexity theory part introduce  other models of computation such as  probabilistic and interactive  computation  uh talk about the expectations of the  course first of all prerequisites  um  you know there are a bunch of  prerequisites listed you know  6042 18062 or maybe some other subject  as well  the real thing is is that this is a math  class  this is a class where i'm exp you know  and it's not a beginning math class this  is you know a moderate to advanced math  class and i'm expecting people to have  had some prior experience  you know of a substantial nature with um  mathematical theorems and proofs um  you know we'll start off slow but we're  going to ramp up pretty fast so if you  haven't really  you know got the idea or gotten  comfortable with doing proofs coming up  with proofs to mathematical statements  you know that's going to be a concern um  you can you know i would just be  monitoring yourself and seeing how  you're doing  because the homeworks  and the exams are going to count on your  being able to produce proofs um  and so  uh you're gonna stroke be struggling if  if that's gonna be a real uh something  that you haven't had experience with um  so and let me talk a little bit about  the role of theory um  in computer science is this a theory  class  um as you know um  [Music]  so  um  you know before we jump into the  material i just thought it would be  worth for you to give you at least my  perspective on the role of theoretical  computer science within the the  the field uh so you know i've been in in  in  in computer science for a long time you  know i go back i'm sure i'm getting to  be a dinosaur here but i go back to the  days when you had punch cards that's  what we did when i was an undergraduate  and um obviously things are very  different now um  and you can argue that in some you know  that you know computer science as a  discipline has matured and sort of the  the basic stuff has all been solved um  well i would say there's a certain truth  to that but there's a certain way in  which i would say that's not true um i  think we're still at the very beginning  and at least in certain respects of  computer science as a discipline  uh for one thing there are a lot of  things that we do um  uh a lot of things relating to  computation that you know we just don't  know the answer to very fundamental  things you know  you know let's take in this example how  does the brain work  obviously the brain computes in a  certain fashion um  but um  you know and we've made good progress  you can argue with machine learning and  all of those things which that are  really doing very  you know have very uh very powerful and  doing very cool things but i would also  say that  you know at some deeper level we really  have  you know the  the the um the methods that we have so  far  uh don't allow us to understand um  uh creativity um  you know  we're not close to being able to create  a computer program that can do  mathematics or that can do many of the  creative kinds of things that human  beings can do um  you know their i think machine learning  powerful as it is is really successful  only for a very narrow set of tasks um  and so  i think you know  there's a lot  i think there's probably something  deeper and more fundamental going on  that we're missing um that would be my  hunch um now whether something like um  theoretical computer science is gonna  give you an answer there or this kind of  theory or some kind of theory i think  some kind of theory has a as at least a  decent shot at playing a role in helping  us to understand computation in a deeper  way  and the fact that we can't understand  something as basic as can you factor a  big number quickly or not um you know  you can't really say you understand  computation until you can answer  questions like that  so i would argue that we have a really  very primitive understanding of  computation at this stage and that there  is a um a lot that has yet to be  discovered um not just on the  technological side but just on the very  fundamental theoretical side that has a  real shot at playing a role in affecting  the practice of how we use computers  and so i i think for that reason again  i'm not sure what what kind of theory is  going to be the most useful but the the  theory we're going to cover in this  course is a particularly elegant theory  and it has already paid off in many  applications and in terms of our  understanding of computation and i think  you know there is um at least as a  starting point it's a good it's a good  subject to learn um  uh certainly i enjoy it and i've spent a  good you know good chunk of my my career  uh  doing that um  so um  uh let's move on then um and begin with  the um  uh  um  the  subject material um so we're going to  talk about um  uh models of computation as i mentioned  um you know we  we want to try to understand computers  and we want to understand what computers  can do but computers in the real world  are pretty complicated objects and  they're really not nice to talk about  mathematically  so we're going to talk about abstract  models of computers that are much  simpler but really capture just like  models in general capture the important  aspects of the thing we're trying to  understand um  and so uh  we're going to look at several different  kinds of models that vary in their  capabilities and and the way they  approximate the real  computers that we deal with every day  and  to start for starters we're going to  look at a very simple model called the  finite automaton  and um that's going to represent uh you  can think of it as representing a  computer that has a very small  amount of memory um  and a very limited and small amount of  memory and we're going to look at the  capabilities of those kinds of machines  and what's nice about them is that you  can understand them very well  um and so um  uh more powerful models that we're going  to look at later are going to be harder  to understand in in a deeper way um but  for these we can  we can uh develop a very  um  comprehensive theory and so that's what  we're gonna do for the next lecture and  a half  um so this is gonna i'm starting off  with an example  i'm presenting a fine art automaton as a  um  as a diagram we call it a state diagram  that is a bunch of has these circles and  lines and um  uh  labels on the on the lines and and also  on the on these circles so let what's  going on here so this is a finite  automaton i'm giving it the name m1  and it has um  uh  these circles are called states so in  this case there are three states q1 q2  and q3 those are the labels there  there are arrows connecting once states  with each other um so these are we'll  call transitions um  and they're going to tell you how to  compute with this device  and there's going to be a specially  designated starting state which is has a  an arrow coming in from nowhere um  and there are certs  other specially designated states called  accepting states and that's going to  have to do with how the machine computes  but those are the ones that have these  double circles um  and so the talking about the way it  computes  the idea is pretty simple um the input  is going to be some finite string of  zeros and ones in this case we might  have other types of symbols that are  allowed for other  um automata but the example that i have  here it's going to be zeros and ones um  and the way you compute with the thing  is you uh first put your finger which i  can't do um  on zoom so i'm going to use the pointer  you put your pointer on the starting  state the one that has the arrow coming  in from nowhere you put your first you  put your your  pointer there and then  uh you start reading symbols from the  input uh one after the next so let's  take an example here zero one one zero  one  so you start reading  those symbols and you follow those  transitions so you go zero this is and  you go back to the same state then you  go for the next symbol is a one so you  go over to this state from q1 to q2  um now you have another one that comes  in so now you're starting at q2 you have  another one so you follow its associated  transition so  if you notice every state has an  outgoing transition for one and another  outgoing transition for zero so there's  always somewhere to go  every time you read symbols from the  input so now you're at q2 you read that  next um the third symbol which is a one  that's going to take you over to q3  and now you have a zero which loops you  back to the where you were and another  one which looks you back to where you  were and uh because you ended up at an  accept um  that is an ex you you say we accept that  straight um so so that's going to be the  output of this um  a finite attempt on  for each string it's either going to  accept it or reject it so it's just a  binary decision that it's going to be  making sort of like a one or a zero  output but we're calling it accept or  reject  so this one here  because it ended up at the accepting  state is accepted but if you look at the  second example  0 0 1 0 1 so you're going to have 0  0  1  0  one  now we ended up at q2  that's not an accepting state  so therefore we say we reject this input  um  okay very simple um  and now for example one of the questions  you might want to ask given one of these  things is well which are exactly those  strings that the machine accepts um  and  uh  a little bit of thought will  help you understand that and the only  strings which are going to take you over  to q3 are those strings that have a 1-1  appearing somewhere along the way two  consecutive ones and you will end up at  the accepting state um  um you have to i mean i encourage you to  think about that uh for a minute if not  immediately obvious but those are the  strings that are going to be uh accepted  by these this machine  and we call that collection of strings  the language of the machine  um that so that  set a  of those strings that have a 1 1 for  this particular machine is the language  of m1  we also say that m1 recognizes that  language recognizes a  and and in terms of notation we write  that a is l of m1  a is the language of m1 so the language  of a machine is exactly the set of  strings that machine accepts  okay  so one of the first things we're going  to want to be able to do is take a  machine and understand what its language  is what's the set of strings that that  machine accepts another thing we might  want to do is given a language build a  machine which recognizes that language  um  so and then understanding what are the  class of languages can you get any  language from some machine or are there  going to be some languages that you can  do and other languages that you cannot  do so those are the kinds of questions  we're going to be asking about these  finite automata what kinds of things can  those machines do and what can they not  do  um  okay  here's our net here is our uh next  check-in  um  so let me um uh so wake up everybody  who's not paying attention a check-in is  coming um  so we have more questions though i can't  uh  can't keep are these three statements  equivalent what to be statements  um  at the bottom of the slide  yes  those three are equivalent a is a  language those mean the same thing not  only are they equivalent but they  they're just different ways of saying  the same thing um  uh that m1 recognizes the language the  same as saying that's the language of  the machine and that a equals uh that l  of m uh that's that's all  the same way of saying they all six of  one half a dozen over the other it's two  ways of saying the same thing okay so  let's pop up our poll  um  and get that started  uh  whoops  still showing the old one oh here we go  move it to the next  question  okay  okay so  you understand the question here  after reason where do we end up after we  read one zero one what state are we in  do we end up in state q1 q2 or q3  okay  go fast  this is a  okay so i think we got uh  pretty much converged here  i think uh almost everybody got it right  um  uh the answer is indeed um that you  ended up in state q2 because you go one  zero one and that's where you ended up  in state q2 so is the mach is this uh  string accepted no  because you didn't end up in an accept  state so this machine rejects one zero  one  okay um  [Music]  let's keep going  so  now um  [Music]  okay so now we gave it this informal  idea of a finite automaton we're going  to have to try to get a formal  definition now um which is going to be a  more mathematical way of saying it's the  same thing that i just said um  and  the reason for having a formal  definition is for one thing it allows us  to be very precise then we'll know  exactly what we mean by a finite  automaton and it should answer any  questions about what counts and what  doesn't count  it also is um  a way of providing notation um so it'll  help us describe finite automata and  sometimes there might be an automaton  where  you know the picture is just too big so  you might want to be able to describe it  in in in some mathematical terminology  rather than by giving a picture or maybe  you're going to be asked to give a  family of automata where um you know  there is going to be  a parameter n  associated with the uh class of  languages you're trying to describe with  the automaton and then it'll be more  helpful to describe it in this formal  notation rather than as a kind of a  picture because  it might be infinitely many pictures  that are being needed  um so you may maybe examples of that  will come up with now so we'll find out  automaton we call it a five tuple don't  be put off by that a five tuple is just  a list of five things so  uh finite automaton will  in our definition is going to have five  components  um it's going to have uh q  which is going to be a finite set of  states so it's going to be a list you  know finite  set which will designate as the states  of the automaton  sigma  is the alphabet symbols of the automaton  another finite set  delta is the transition function  that tells us how the automaton moves  from state to state those gives it  describes how those  transition arrows those arrows which  connected the states with each other  um it describes them in a mathematical  way instead of in terms of a picture um  and the way i'm doing that is with a  function so delta is a function which  takes two things so you know i'm hoping  you've seen this notation before i'll  help you through it once but um this is  the kind of thing i would expect you to  have seen already so when we have uh q  cross sigma so i'm going to give delta a  state  and an in  a state and an  alphabet symbol so q is states sigma is  alphabet symbols so you're going to get  a state and a and a  alphabet symbol and it's going to give  you back a state  so more  um describing it you know um  [Music]  kind of a  little bit more detail you're going to  you know delta if you give it  state q and symbol a  equals r that means q when you read an a  you go to r  so that's the way this picture gets  translated into a mathematical function  um which describes those transitions  and then now q 0 is going to be the  starting state so there's going to be  that's the one with the in arrow coming  in from nowhere and f is the set of  accepted accepting states so there might  be several there's only going to be one  starting state but there might be  several different or possibly even zero  accepting states that's all legal  when we have a finite automaton um and  so in terms of using the notation going  back to the machine that we just had  from the previous slide which i've given  you here again  let me show you how i would describe  this using this notation that comes out  of the definition so here is m1 again  it's this five tuple  where q now is the set q1 q2 group three  that's the uh the set of states  uh the input alphabet is zero one  it might vary in other automata um and f  is the set q3 which has only the element  q3 because this has just one except  state q3  okay um so i hope that's uh helpful um  oh of course i forgot the transition  function which here i'm describing as a  table  so the transition function says um  you know if you have a q1 and a if you  have a state  and an input alphabet you can look up in  the table where you're supposed to go  under the transition function according  to the um  the uh  the state and the alphabet symbol that  you're you're given so for example you  know if we were in state um  q2  um here getting a zero  then q2 goes back to q1 so that q2 on  zero is q1  but q2 on one here  is q3  okay so that's how that table captures  this uh this picture  okay  and it's just a function it's a rep wave  representing a function a a finite  function um  uh  you know in terms of uh this uh this  table here so i realize for some of you  this may be  uh  slow um we will ramp up on speed but i'm  trying to get us all together in terms  of the language of the course here at  the beginning um  okay uh so now um  let's talk about some more um  uh the computation  uh  a  uh  so strings and languages  a string a string is just a finite  sequence of symbols from the from the  alphabet um so don't  whenever this class  is not going to talk about infinite  strings all of our strings are going to  be finite um  uh there's other mathematical theories  of automata and so on that talk about  infinite inputs and infinite strings  we're not going to talk about that um uh  you know maybe  rarely we'll make it very clear we'll  talk about an infinite string but that's  going to be an exception  um and a language is a set of strings  that's the traditional uh  you know um  a way that uh uh people in this subject  refer to a set of strings they call it a  language really because this subject had  its roots in linguistics actually um and  they were talking about they're trying  to understand languages you know human  languages  um  so this is just a historical fact and  that's the terminology that stuck  okay so two special string a string a  special string in a special language um  the empty string is the string of length  zero this is a totally legitimate string  um that you we're going to run into now  and then and there's the empty language  which is the set with no strings  these are not the same  they're not even of the same type of  object  so don't confuse them with one another  i mean you can have a a set a language  which has just one element which is the  empty string that is not the empty set  that is a set that is not the empty  language that is the a language that has  one element in it with namely the empty  string okay so those are separate things  okay so here's a little bit of a a  mouthful here on the slide um defining  what it means for a an automaton to  accept its input accepts its input  string w  and we can define that formally and you  know it's a little technical looking  it's really not that bad  um  so if you have a your input string w  which you can write as a sequence of  symbols in the alphabet w one w two dot  dot w n so you know like zero one zero  zero one i'm just writing it out symbol  by symbol here  um  so what does it mean for the machine to  accept um that's that input so that  means that there's a sequence of states  in the machine  sequence of states and of members of  queue so these are  a sequence from q these are the states  of the machine  uh that have a certain prod that satisfy  these three properties down here  first of all  and i'm thinking about the sequence is  the the sequence that the machine goes  through  as it's processing  the input w  so  when does it accept w  if that sequence has the feature that it  starts at the start state  each state legally follows the previous  state according to the transition  function  so that says you know the  member of the sequence  uh  is obtained by looking at the previous  one the i minus first member of that  sequence of that the i  the i minus first state in that sequence  and then looking at what happens when  you take the ice input symbol  so as you look at the previous state and  the next input symbol you should get the  next state that's all that this is  saying and this should happen for each  each one of these guys  and lastly for this to be accepted  the very last member here where we ended  up at the end of the input so you only  care about this at the end of the input  you have to be in an accepting state  so you can mathematically capture this  notion of going along this path  um  and uh that's what i'm just trying to  illustrate that you know we could  describe all this very formally i'm not  saying that's the best way to think  about it all the time but that it can be  done  and i think that's something you know  worth appreciating um  okay um  so now in terms of again getting back  we've said this once already but in  terms of the languages that the machine  recognizes it's the collection of  strings that the machine accepts  every machine accepts ex might it might  accept many strings  but it always recognizes one particular  language  even if the machine accepts no strings  then it recognizes the empty language so  a machine always recognizes some one  language  but it may have many many strings that  it's accepting  and we call that language the language  of the machine  and that's the and we say that m  recognizes that language these three  things mean the same thing  okay and now important definition i try  to reserve the most important things or  the highlighted things to be in this  light blue color if you can see that  we say a language is a regular language  if there's some finite automaton that  recognizes it  okay  so there are going to be some languages  that have associated to them finite  automata that actually solve those  languages that recognize those languages  but there might be other languages and  we'll see examples where you just can't  solve them you can't recognize them with  a finite automaton those languages will  not be regular languages the regular  ones are the ones that you can do with a  finite automaton that's the traditional  terminology  okay so let's continue let's go on from  there so look a couple examples here  again is that same getting getting to be  an old friend that language that  automaton m1  uh  remember its language here is the set of  strings that have the substring one one  that is the um  that language a  now what do we know about a  from the previous slide  think with me don't just listen  a is a  regular language now because it's  recognized by some  automaton  so  all of the whenever you find a um  an automaton for a language a finite  automaton for language we know that  language that language is a regular  language  so let's look at a couple of more  examples so if you take the language  let's call this one b  which is the the strings that have an  even number of ones in them  so like the string  one one zero one  would that be in b  no because it has an odd number of ones  so the string one one one one has four  ones in it that's an even number so that  string would be in b the  the zeros don't don't matter for this  language so that string  um  this  of  of of  strings that have an even number of ones  uh that's a regular language  um and the way you would know that is  you would have to make a fine art  automaton that recognizes that language  and i would encourage you to go  and make that automaton you can do it  with two states it's a very simple  automaton but if you haven't had  practice with these i encourage you to  do that and i actually there are lots of  examples that i ask you to  solve at the end of chapter one in the  book and you definitely should spend  some time  playing with it if you're not  if you have not yet seen uh finite  automata before um you need to get  comfortable with these and be able to  make them uh so we're going to start to  we're going to start making some of them  but we're going to be talking about it  at a sort of a more abstract level in a  minute  um  uh basically the reason why you can you  can solve this problem you can you can  make a fine art automaton would  recognize is the language b  is because that finite automaton is  going to keep track  of the parity of the number of ones it's  seen before  this has two states one of them  remembering that it's seen an odd number  one so far or the other one remembering  it's seen an even number of ones before  and that's going to be typical for these  automata finite automata there's going  to be  several different possibilities that you  may have to keep track of  as you're reading the input  and there's going to be a state  associated with each one of those  possibilities  so if you're designing an automaton you  have to think about as you're processing  the input what things you have to keep  track of and you're going to make a  state for each one of those  possibilities  okay so  you need to get comfortable with that um  let's look at another example um  the language c where the inputs have an  equal number of zeros and ones  that turns out to be a not a regular  language  um so what in other words what that  means is there's no way to recognize  that language with a finite automaton  you just can't do it  that's beyond the capabilities of finite  automata  and that's that's a problem that's  that's a statement we will prove later  um  okay um  and our goal over the next uh  lecture or so is to understand the  regular languages which you can do in a  very comprehensive way um so we're going  to start to do that now  so first we're going to introduce these  this concept of regular expressions um  which again these are things you may  have run into in one way or another  before um  so if we have a uh  um  we're going to introduce something  called the regular operations  now uh  you know  you i'm sure you're familiar with the  um  the arithmetical operations like plus  and times  um those apply to numbers  now the operations we're going to talk  about are operations that apply to  languages  so they're going to take  let's say two languages you apply an  operation you can get get back another  language like the union operation for  example  that's one you probably have seen before  um  the union of two languages here is a  collection of strings that are in either  one or the other  um  so  but there are other operations which you  may not have seen before that we're  going to look at um the concatenation  operation for example so that says  you're going to take a string from the  first language and another string from  the second language and stick them  together  and  it's called concatenating them  and you do that in all possible ways and  you're going to get the concatenation  language from these two languages that  you're starting with a and b  the symbol we use for concatenation is  this little circle but sometimes we  don't often we don't we just suppress  that and we write the two languages uh  next to one another with the uh little  circle implied so this also means  concatenation over here  just like this does  and the last uh  last of the regular operations is the  so-called star operation which is a  unary operation that applies to just a  single language  and so what you do is now um you're  going to take to look get a member of  the star language you're going to take a  bunch of strings in in the original  language a  and you stick them together  any number of members of a you stick  them together  and that becomes an element of the star  language i will do an example in a  second if you didn't get that  but one important element is that when  you have the star language you can also  allow to stick zero elements together  and then you get the empty string  so that's always a member of the star  language the empty string  okay so let's look at some examples  let's let let's say a is the language  these are two strings here good bad and  b is the language boy girl  okay  now if we take the union of those two  we get  a  good bad boy girl  that's kind of uh what you'd expect um  and um  if uh  um now let's  take a look at the concatenation  um  now if you concatenate the a and the b  language you're going to get all  possible ways of having an a string  followed by all possible ways having a b  string so you can get good boy good girl  bad boy bad girl  um now uh  looking at the star  um  [Music]  well  that applies to just one language so  let's say it's the good bad language  from uh a  uh and so the  a star that you get from that is all  possible ways of sticking together the  strings from a um so for there are no  using no strings you always get the  empty string that's always guaranteed to  be a member of a  and then just taking one element of a  you get good or another element bad but  now two elements of a you get good good  or good bad and so on um or three  elements of a good good good bit good  good bad  and so in fact a star is going to be an  infinite language  if a itself  is um  uh  contains any  non-empty member so if a is the empty if  a is the empty language or if a contains  just the language empty string  then a star will be  not an infinite language it'll just be  the language empty string  but otherwise it'll be an infinite  language  i'm not even sure  okay uh i'm not  i'm ignoring the chat here i'm hoping  people are getting are you guys getting  your questions answered by rtas  how are we doing thomas uh one question  is are the supplies going to be posted  are the slides going to be posted  well um  uh the whole lecture is going to be  recorded um is it helpful to have the  slide separately i can post the slides  um  sure i'll  remind me if i don't but i'll try to do  that yes it is helpful i will i will do  that  um  yeah um  yeah i will post the slides just  thomas it's your job to remind me okay  um  [Music]  all right good uh so regular so we  talked about the regular operations  let's talk about the regular expressions  so regular expressions are just like you  have the arithmetical operations then  you can get arithmetic expressions like  you know one plus three  um times seven so now we're going to  make expressions out of these operations  first of all you have the more atomic  and things the building blocks of the  expressions which are going to be like  um  uh  elements of sigma  you know elements of the alphabet or the  sigma itself as as an alphabet symbol or  the empty string or the empty language  of the the empty empty language or the  empty string these are going to be the  building blocks for the regular  expressions we'll do an example in a  second  and then you combine those those basic  elements  using the regular operations of union  concatenation and star so those are the  these are the atomic expressions these  are the composite expressions so for  example  um if you look at the expression  zero union one star  so that we can also write that as sigma  star because if sigma is zero  uh and one then sigma star is the same  thing as zero union one is sigma is the  same as zero union one and that just  gives all possible strings over sigma  so this is something you're going to see  frequently sigma star means all pos this  is the language of all strings over the  alphabet we're working with at that  moment  now if you if you take sigma star one  that's gonna you just concatenate one  onto all of the elements of sigma star  and that's going to give you all strings  that end with a one  you know technically you might imagine  writing this with braces around the one  but generally we don't do that we just  single element sets single element  strings we write without the braces  because  it's clear enough without them and it  gets messy with them so sigma star one  is all strings that end with one  or for example you take sigma star  one one sigma star  that is all strings that contain one one  and we already saw that language once  before that's the language of that other  machine that we presented on you know  one or two slides back  okay um  right uh  yes so  um uh yeah but in terms of readings by  the way sorry to i don't know if it's  helpful to you for me to do these  interjections but the readings are  listed also on the homework  so  if you if you  look at the posted homework one it tells  you  which  which chapters you should be reading now  and also if you look at the course  schedule  which is also on the home page it has  the whole  course plan and which readings are uh  for which dates  so it's all it's all there for you  um  and so our goal here this is not an  accident that sigma star one one sigma  star  happens to be the same language as we  saw before from the language of that of  that finite automaton m1  um in fact that's a general phenomenon  anything you can do with a regular  expression you can also do with a finite  automaton and vice versa  they are equivalent in power with  respect to the class of languages they  describe  and that's one we'll prove that  okay  um  so you know it's got you know if you  step back for a second and just let you  let yourself appreciate this it's kind  of an amazing thing because fine at  automata with the states and transitions  and the regular expressions with these  you know operations of union  concatenation and star they look totally  different from one another they look  like they have nothing to do with one  another but in fact they both describe  exactly the regular languages the same  class of languages  um  and so it's kind of a cool fact that you  can prove that these two very different  looking systems actually are equivalent  to one another  um  can we get  empty string from empty set yet  there are a bunch of  uh exotic cases by the way so empty  empty language star  is the language where has just the empty  string  if you don't get that chew on that one  that means but that is a that is that is  true um  uh  okay let's move on  okay let's talk about closure properties  now we're going to start in doing  something that has a little bit more  meat to it in terms of we're going to  have our first theorem of the course  coming here and this is not you know  it's not a baby theorem this is actually  uh there's going to be some meat to this  and you're going to have to uh um  you know  you know it's not totally um  this is not a toy this is we're proving  something that has a real substance  and and the um  uh  the statement of this theorem says that  the regular languages are closed really  the class of regular languages are  closed under union closed under the  union operation  um so what do i mean by that  so you you know  when you say a uh collection of objects  is closed under on some operation that  means applying that operation to those  objects  leaves you in the same class of objects  like the you know the um  the  natural you know the the positive  integers the natural numbers  that's closed under addition because  when you add two you know  positive integers you get back a  positive integer  but they're not closed under subtraction  because you know 2 minus four you get  something which is not a positive  integer  so close means you leave yourself in the  collection  and the fact is that  uh if you look at all the regular  languages these are the languages that  the  fine automata can recognize  they are closed under the union  operation so if you ta you start off  with two regular languages and you apply  the union you get back another regular  language  and that's what the statement of this  theorem is i hope you can i hope that's  clear enough in the way i've written it  if a1 and a2 are regular  then a1 union a2 is also regular that's  what the statement of this is and it's  just simply that um  that's proving that the class of regular  language is closed under union so we're  going to prove that  so how do you prove such a such a thing  so you know the way we  we're going to prove that is you uh  start off with you know what we're  assuming so we're our hypothesis is that  we have two regular languages  and we have to prove our conclusion that  the union is also regular  now the hypothesis that they're regular  you have to unpack that and understand  what it what is that what does that get  you and  them being regular means that there are  finer automata that recognize those  languages so let's give those two  fighter automata names so m1 and m2 are  the two finest automata that recognize  those two languages a1 and a2 that's  what it means that they're regular that  these automata exist  so let's uh have those two atomic m1 and  m2 using the components as we've  described this you know the respective  state sets input alphabet transition  functions the two starting states and  the two collections of stepping states  here i'm assuming that they're over the  same alphabet you could have automata  which operate over different alphabets  it's not interesting to do that  it doesn't add anything the proof would  be exactly the same so let's just not  over complicate our lives and focus on  the more interesting case so assuming  that the two  input alphabets are going to be the same  um and from these two automata we have  to show that this language here the  union is also a regular language and  we're going to do that by  constructing the automaton which  recognizes the union that's really the  only thing that we can do um  so we're going to build an automaton out  of m1 and m2 which recognizes the union  language a1 union a2  and the task of m is that it should  accept its input if either m1 or m2  except  and now what i'd like you to think about  doing that how would how in the world  are we going to come up with this find  out automaton m  um  and uh  uh the way we do that is  to think about  um  how would you  do that union language  you know if i ask you i give you two  automata m1 and m2 and i say here's an  input w  um  are is w in the union language  that's the job that m is supposed to  solve and i suggest you try to figure  out how you would solve it first  i mean this is a good strategy for  solving a lot of the problems in this  course put yourself in the place of the  machine you're trying to build  and so you know if you want to try to  figure out how to do that you know  natural thing is well you take w you  feed it into  m1 and then you feed it into m2 and if  m1 accepts it great and you know it's in  the union and if not you try it in and  try it out in m2 and see if m2 accepts  it um  now you have to be a little careful  because you want to have a strategy that  you can also implement in a fine art  automaton  and  uh  and a fine art automaton only gets one  shot at looking at the input you can't  sort of rewind the input you feed it  first into m1 and then you feed it into  m2  and  uh  operate in a sequential way like that  that's not going to be allowed um in the  way uh a finite automata work  so  you're going to have to be a little bit  more take it to the next level be a  little bit more clever and instead of  feeding it first into m1 and then and  then into m2 you feed them into both in  parallel  so you take m1 and m2 and you run them  both in parallel on the input w  keeping track of which state each of  those two automata are in  and then at the end you see if either  one of those machines is in an accepting  state and then you accept  so that's the strategy we're going to  employ um  in building the finer automaton m out of  m1 and m2 so here's in terms of a  picture here's m1 and m2  here is the automaton we're trying to  build  uh we don't know how it's going to look  like it and uh yeah so kind of getting  ahead of myself but here is the strategy  as i just described for amp  um  we're going to keep track  m is going to keep track of which  state that m1 is in and which state m2  is in at any given moment as we're  reading the symbols of w we're going to  feed it also into m1 and also into m2  and so  the possibilities we have to keep track  of in m  are all the pairs of states that are in  m1 and m2 because you're going to really  be tracking m1 and m2 simultaneously so  you have to remember which state m1 is  in and also which state m2 is in and so  that really corresponds to a pair of  states to remember  one from m1 and one from m2 and that's  why i've indicated it like that so m1 is  in state q m2 is in state r at at some  given point in time and that's kind of  correspond to m being in the pair q  comma r  that's just the label of this particular  state of m that uh we're gonna  apply here  okay  and so um and then  uh am is going to accept if either m1  and m2 so is an accepting state so it's  going to be if either q or r is an  accepting state we're going to make this  into a  an accepting state too  okay  whoops there we go  um  [Music]  so let's describe this formally instead  of by a picture because we can do it  both ways and sometimes it's better to  do it one way and sometimes the other  way  so now if we take uh the components of m  now  are the pairs of states from m1 and m2  again i'm writing this out literally uh  explicitly here but you should be make  sure you're comfortable with this cross  product notation  so this is the collection of pairs of  states q1 and q2 where q1 is in the  state of the first machine q2 is the  state of the second machine  the start state is you start at the two  uh start states of the two machines um  so this is q1 q2 probably i should have  not reused the q notation i should have  called these r's not of them looking at  that but anyway i hope you're not  confused by reusing this q1 and q2 here  are the specific start states of the two  twos of the two machines these are just  two  other states representative states of  those machines  now the transition function for the new  machine is going to be built out of the  transition functions for the old  previous machines so when i have a pair  qr  and i have the symbol a  where we go which new pair do we get  well we just update the state from m1  and update the state from m2 according  to their respective transition functions  and that's what's shown over here  um  now let's take a look at the accepting  states for m  the  natural thing to do is look at the set  of pairs of states  where we have a pair of states  of the  pair of accepting states one from  um the first machine and one from the  second machine  but if but if you're thinking with me  you realize that this is not the right  thing  what is dfas  uh  did i recall them dfa somewhere  oh somebody else is probably doing that  in the chat  uh  the dfa you careful what notation you're  using we haven't introduced dfas yet uh  we'll do that next on thursday uh but  these are dfas these are just fine at  automata deterministic finite automata  that's why they're  why the d anyway so  this is actually not right because if  you think about what this is saying it  says that both  components have to be accepting  and you want either one to be accepting  so  this is not good this is a this would be  the wrong way of defining it  that actually gives the intersection  language and really kind of along the  way it's proving closure under  intersection which we don't care about  but might be useful to have in our back  pocket sometime in the future  um  in order to get closure under reunion  we have to write it this slightly more  complicated looking way which says  that  you know uh the pair what you would want  to have is either the first state is a  an accepting state and then any state  for the second uh element or any state  for the first element and an accepting  state for the second element  that's what it means to have the union  uh to be doing the union okay so let's  let's do uh oh here's a quick check in  um  [Music]  so let's let's do another poll here you  thought we we thought we were done with  these um  again  you know  oh  here we go so i i it was too complicated  to write it out in the poll so i  actually put it up on the slide for you  um  [Music]  so i'm all i'm asking is that m1 has k  states k1 states and m2 has k2 states  how many states does m have  is it the sum  the sum of the squares or the product  okay you have to think about what are  the states of m what do they look like  um  and  uh  come on guys  all right ending the poll  uh  sharing results  yes indeed  it is most of you got it correct it is c  the the product  because when you look at the number of  pairs of states  from m1 and m2 you need all possible  pairs and so it's the number of states  in m1 times the number of states in m2  so make sure you you understand that and  think about that  um to uh  to so that you're you're following um  and and  and get this  all right so um let's move on here  so we have another five minutes or so  let's start thinking about closure under  concatenation  so if we have two regular languages so  is um the concatenation language um  we're gonna try to prove that we won't  finish but we'll at least get our  creative juices going about it um so  we're gonna do the same scheme here  we're gonna  take two machines for a1 and a2 and  build a machine for the concatenation  language out of those two so here are  the two machines for a1 and a2 written  down um  [Music]  and here is now here is  uh  the concatenation language and i'm going  to propose to you a strategy  um which is not going to work but it  still is going to be a good intuition to  have  so what i'm going to do is i'm going to  make a copy of m uh okay let's  understand what m is supposed to do  first  so m should accept its input so think  about this m is doing the concatenation  language so it's given a string and it's  an answer is it in the cat concatenating  the concatenation language  a1a2 or not  so it should accept it if m1 if there's  some way to divide w into two pieces  where m1 accepts the first piece and m2  accepts the second piece  uh  so the here would be the picture  okay and now um we have to try to make a  machine which is going to solve this  intuition so how would you do that  yourself  you're giving you w  and you can simulate m1 and m2  so  the natural thing is you're going to  start out by simulating m1 for a while  and then shift into simulating m2 for a  while  because that's what happens as you're  supposed to happening as you're  processing the input so i'm going to  suggest that in terms of the diagram  like this um  so we have here um  uh m1 and m2 copied here  uh  and what i propose doing is  uh connecting m1 to m2 so that when  m1  has accepted its input we're going to  jump to m2 because that's perhaps you  know the first part of w and now we're  going to have m2 process the second part  of w so the way i'm going to implement  that is by  declassifying the start state of m2  having transition symbols from  the accepting states of m1 to m2  and then removing these guys here as  uh accepting states  and we have to would have to figure out  what sort of labels to apply here but  actually this reasoning doesn't work  it's tempting but flawed  because  what what goes wrong  what happens is that  you know it might be  that  when m1 has accepted an initial part of  w and then it wants m2 to accept the  rest  it might fail because m2 doesn't accept  the rest  and what you might have been better off  doing is waiting longer in m1  because there might have been some other  later place to split w  which is still valid  splitting w in the first place where you  have m1 accepting an initial part may  not be the optimal place to split w you  might want to wait later and then you'll  have a better chance of accepting w  of accepting w  so um  i don't i don't know sure if you quite  follow that but um uh in fact it doesn't  work the question is where to split w  and uh it's challenging because how do  you know where to split w because it  depends upon what you it's a it might it  depends on why and you haven't seen why  yet  so when you're trying to think about it  that way it looks hopeless  but in fact it's still true  and we'll see how to do that  on thursday  so  just to recap what we did today  um  we did our introductory stuff  we defined fine art automata  on regular languages we define the  regular operations and expressions  uh we showed that the regular languages  are closed under union  we start a closure under intersection to  be continued  you

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so um welcome everybody uh to  the full  2020 online  uh introduction to the theory of  computing um 18404 6840.  my name is mike sipser  i'm going to be your instructor for the  semester in this class  um  so let me just tell you what the course  is about um  basically it's going to be into two  halves what we're going to be talking  about what are the capabilities and  limitations of computers of computer  algorithms really computation  um  and the two parts of the course are more  or less  divided in half  we're having the first half of the  course is going to talk about a subject  called computability theory  which it really asks what what you can  compute with an algorithm in principle  um  that's a was an active area of research  in the earlier part of the 20th century  it's pretty much closed off as a  research subject these days  mainly because  they answered all of their big questions  um and so uh a mathematical field really  only uh stays  vital when it has problems to solve and  they really solved all of their uh  interesting problems for the most part  not a hundred percent but for the most  part uh it sort of finished off in the  1950s um  just to say a little bit more about what  we're going to talk about there you know  when you're interested to know what  kinds of problems you can solve with an  algorithm  you know there are problems that you  might want to solve that you just can't  solve for example if you want to  you're given a specification for a um  you know a computer problem you want to  solve  whatever whatever that specification  might be say you know your algorithm  actually is a sorting algorithm for  example  and you want to write down that  specification and have an automatic  verifier that's going to check whether a  program meets a specification well  that's just in principle impossible you  cannot meet you cannot make a um  a verifier which is going to uh  answer in all cases whether or not a  program meets a certain specification um  so things like that we will prove uh  this uh this semester you know questions  about mathematical truth what if you're  given a mathematical statement is it  true or is it false uh it'd be great  if you can write a computer program that  would answer that problem well it would  not be great if you're a mathematician  because that would put us all out of  business but if you and you know  you could imagine that might be a nice  thing to have but you can't i mean there  is no algorithm which can answer that  question  um  uh well along the way we're going to  introduce models of computation like  fine automata which we'll see today  touring machines and some model other  models that we'll see along the way  the second half of the course which is  going to be after the midterm we're  going to shift gears and talk about  complexity theory which is instead of  looking at what's computable in  principle you're going to look at what's  computable in practice  so things that you can solve in a  reasonable amount of time  and um for example i'm sure many of you  are aware of the factoring problem which  has connections to uh the um  uh the the rsa crypto system you know  cryptography  um and asks whether you can  factor big numbers quickly um that's a  problem we don't know the answer to um  we just don't know how to factor big  numbers quickly but it's possible that  there are algorithms out there that we  haven't discovered yet that can do so  it's connected with this very famous  problem in the intersection of computer  science and mathematics called the p  versus np problem which many of you may  have heard of we'll talk about that  we'll spend a lot of time on that uh  this uh this term  and along the way we'll talk about  different measures of uh complexity of  computation time and space  time and memory  you know a theorist called memory i like  to call space  that's going to be a big part of the  course  in the complexity theory part introduce  other models of computation such as  probabilistic and interactive  computation  uh talk about the expectations of the  course first of all prerequisites  um  you know there are a bunch of  prerequisites listed you know  6042 18062 or maybe some other subject  as well  the real thing is is that this is a math  class  this is a class where i'm exp you know  and it's not a beginning math class this  is you know a moderate to advanced math  class and i'm expecting people to have  had some prior experience  you know of a substantial nature with um  mathematical theorems and proofs um  you know we'll start off slow but we're  going to ramp up pretty fast so if you  haven't really  you know got the idea or gotten  comfortable with doing proofs coming up  with proofs to mathematical statements  you know that's going to be a concern um  you can you know i would just be  monitoring yourself and seeing how  you're doing  because the homeworks  and the exams are going to count on your  being able to produce proofs um  and so  uh you're gonna stroke be struggling if  if that's gonna be a real uh something  that you haven't had experience with um  so and let me talk a little bit about  the role of theory um  in computer science is this a theory  class  um as you know um  [Music]  so  um  you know before we jump into the  material i just thought it would be  worth for you to give you at least my  perspective on the role of theoretical  computer science within the the  the field uh so you know i've been in in  in  in computer science for a long time you  know i go back i'm sure i'm getting to  be a dinosaur here but i go back to the  days when you had punch cards that's  what we did when i was an undergraduate  and um obviously things are very  different now um  and you can argue that in some you know  that you know computer science as a  discipline has matured and sort of the  the basic stuff has all been solved um  well i would say there's a certain truth  to that but there's a certain way in  which i would say that's not true um i  think we're still at the very beginning  and at least in certain respects of  computer science as a discipline  uh for one thing there are a lot of  things that we do um  uh a lot of things relating to  computation that you know we just don't  know the answer to very fundamental  things you know  you know let's take in this example how  does the brain work  obviously the brain computes in a  certain fashion um  but um  you know and we've made good progress  you can argue with machine learning and  all of those things which that are  really doing very  you know have very uh very powerful and  doing very cool things but i would also  say that  you know at some deeper level we really  have  you know the  the the um the methods that we have so  far  uh don't allow us to understand um  uh creativity um  you know  we're not close to being able to create  a computer program that can do  mathematics or that can do many of the  creative kinds of things that human  beings can do um  you know their i think machine learning  powerful as it is is really successful  only for a very narrow set of tasks um  and so  i think you know  there's a lot  i think there's probably something  deeper and more fundamental going on  that we're missing um that would be my  hunch um now whether something like um  theoretical computer science is gonna  give you an answer there or this kind of  theory or some kind of theory i think  some kind of theory has a as at least a  decent shot at playing a role in helping  us to understand computation in a deeper  way  and the fact that we can't understand  something as basic as can you factor a  big number quickly or not um you know  you can't really say you understand  computation until you can answer  questions like that  so i would argue that we have a really  very primitive understanding of  computation at this stage and that there  is a um a lot that has yet to be  discovered um not just on the  technological side but just on the very  fundamental theoretical side that has a  real shot at playing a role in affecting  the practice of how we use computers  and so i i think for that reason again  i'm not sure what what kind of theory is  going to be the most useful but the the  theory we're going to cover in this  course is a particularly elegant theory  and it has already paid off in many  applications and in terms of our  understanding of computation and i think  you know there is um at least as a  starting point it's a good it's a good  subject to learn um  uh certainly i enjoy it and i've spent a  good you know good chunk of my my career  uh  doing that um  so um  uh let's move on then um and begin with  the um  uh  um  the  subject material um so we're going to  talk about um  uh models of computation as i mentioned  um you know we  we want to try to understand computers  and we want to understand what computers  can do but computers in the real world  are pretty complicated objects and  they're really not nice to talk about  mathematically  so we're going to talk about abstract  models of computers that are much  simpler but really capture just like  models in general capture the important  aspects of the thing we're trying to  understand um  and so uh  we're going to look at several different  kinds of models that vary in their  capabilities and and the way they  approximate the real  computers that we deal with every day  and  to start for starters we're going to  look at a very simple model called the  finite automaton  and um that's going to represent uh you  can think of it as representing a  computer that has a very small  amount of memory um  and a very limited and small amount of  memory and we're going to look at the  capabilities of those kinds of machines  and what's nice about them is that you  can understand them very well  um and so um  uh more powerful models that we're going  to look at later are going to be harder  to understand in in a deeper way um but  for these we can  we can uh develop a very  um  comprehensive theory and so that's what  we're gonna do for the next lecture and  a half  um so this is gonna i'm starting off  with an example  i'm presenting a fine art automaton as a  um  as a diagram we call it a state diagram  that is a bunch of has these circles and  lines and um  uh  labels on the on the lines and and also  on the on these circles so let what's  going on here so this is a finite  automaton i'm giving it the name m1  and it has um  uh  these circles are called states so in  this case there are three states q1 q2  and q3 those are the labels there  there are arrows connecting once states  with each other um so these are we'll  call transitions um  and they're going to tell you how to  compute with this device  and there's going to be a specially  designated starting state which is has a  an arrow coming in from nowhere um  and there are certs  other specially designated states called  accepting states and that's going to  have to do with how the machine computes  but those are the ones that have these  double circles um  and so the talking about the way it  computes  the idea is pretty simple um the input  is going to be some finite string of  zeros and ones in this case we might  have other types of symbols that are  allowed for other  um automata but the example that i have  here it's going to be zeros and ones um  and the way you compute with the thing  is you uh first put your finger which i  can't do um  on zoom so i'm going to use the pointer  you put your pointer on the starting  state the one that has the arrow coming  in from nowhere you put your first you  put your your  pointer there and then  uh you start reading symbols from the  input uh one after the next so let's  take an example here zero one one zero  one  so you start reading  those symbols and you follow those  transitions so you go zero this is and  you go back to the same state then you  go for the next symbol is a one so you  go over to this state from q1 to q2  um now you have another one that comes  in so now you're starting at q2 you have  another one so you follow its associated  transition so  if you notice every state has an  outgoing transition for one and another  outgoing transition for zero so there's  always somewhere to go  every time you read symbols from the  input so now you're at q2 you read that  next um the third symbol which is a one  that's going to take you over to q3  and now you have a zero which loops you  back to the where you were and another  one which looks you back to where you  were and uh because you ended up at an  accept um  that is an ex you you say we accept that  straight um so so that's going to be the  output of this um  a finite attempt on  for each string it's either going to  accept it or reject it so it's just a  binary decision that it's going to be  making sort of like a one or a zero  output but we're calling it accept or  reject  so this one here  because it ended up at the accepting  state is accepted but if you look at the  second example  0 0 1 0 1 so you're going to have 0  0  1  0  one  now we ended up at q2  that's not an accepting state  so therefore we say we reject this input  um  okay very simple um  and now for example one of the questions  you might want to ask given one of these  things is well which are exactly those  strings that the machine accepts um  and  uh  a little bit of thought will  help you understand that and the only  strings which are going to take you over  to q3 are those strings that have a 1-1  appearing somewhere along the way two  consecutive ones and you will end up at  the accepting state um  um you have to i mean i encourage you to  think about that uh for a minute if not  immediately obvious but those are the  strings that are going to be uh accepted  by these this machine  and we call that collection of strings  the language of the machine  um that so that  set a  of those strings that have a 1 1 for  this particular machine is the language  of m1  we also say that m1 recognizes that  language recognizes a  and and in terms of notation we write  that a is l of m1  a is the language of m1 so the language  of a machine is exactly the set of  strings that machine accepts  okay  so one of the first things we're going  to want to be able to do is take a  machine and understand what its language  is what's the set of strings that that  machine accepts another thing we might  want to do is given a language build a  machine which recognizes that language  um  so and then understanding what are the  class of languages can you get any  language from some machine or are there  going to be some languages that you can  do and other languages that you cannot  do so those are the kinds of questions  we're going to be asking about these  finite automata what kinds of things can  those machines do and what can they not  do  um  okay  here's our net here is our uh next  check-in  um  so let me um uh so wake up everybody  who's not paying attention a check-in is  coming um  so we have more questions though i can't  uh  can't keep are these three statements  equivalent what to be statements  um  at the bottom of the slide  yes  those three are equivalent a is a  language those mean the same thing not  only are they equivalent but they  they're just different ways of saying  the same thing um  uh that m1 recognizes the language the  same as saying that's the language of  the machine and that a equals uh that l  of m uh that's that's all  the same way of saying they all six of  one half a dozen over the other it's two  ways of saying the same thing okay so  let's pop up our poll  um  and get that started  uh  whoops  still showing the old one oh here we go  move it to the next  question  okay  okay so  you understand the question here  after reason where do we end up after we  read one zero one what state are we in  do we end up in state q1 q2 or q3  okay  go fast  this is a  okay so i think we got uh  pretty much converged here  i think uh almost everybody got it right  um  uh the answer is indeed um that you  ended up in state q2 because you go one  zero one and that's where you ended up  in state q2 so is the mach is this uh  string accepted no  because you didn't end up in an accept  state so this machine rejects one zero  one  okay um  [Music]  let's keep going  so  now um  [Music]  okay so now we gave it this informal  idea of a finite automaton we're going  to have to try to get a formal  definition now um which is going to be a  more mathematical way of saying it's the  same thing that i just said um  and  the reason for having a formal  definition is for one thing it allows us  to be very precise then we'll know  exactly what we mean by a finite  automaton and it should answer any  questions about what counts and what  doesn't count  it also is um  a way of providing notation um so it'll  help us describe finite automata and  sometimes there might be an automaton  where  you know the picture is just too big so  you might want to be able to describe it  in in in some mathematical terminology  rather than by giving a picture or maybe  you're going to be asked to give a  family of automata where um you know  there is going to be  a parameter n  associated with the uh class of  languages you're trying to describe with  the automaton and then it'll be more  helpful to describe it in this formal  notation rather than as a kind of a  picture because  it might be infinitely many pictures  that are being needed  um so you may maybe examples of that  will come up with now so we'll find out  automaton we call it a five tuple don't  be put off by that a five tuple is just  a list of five things so  uh finite automaton will  in our definition is going to have five  components  um it's going to have uh q  which is going to be a finite set of  states so it's going to be a list you  know finite  set which will designate as the states  of the automaton  sigma  is the alphabet symbols of the automaton  another finite set  delta is the transition function  that tells us how the automaton moves  from state to state those gives it  describes how those  transition arrows those arrows which  connected the states with each other  um it describes them in a mathematical  way instead of in terms of a picture um  and the way i'm doing that is with a  function so delta is a function which  takes two things so you know i'm hoping  you've seen this notation before i'll  help you through it once but um this is  the kind of thing i would expect you to  have seen already so when we have uh q  cross sigma so i'm going to give delta a  state  and an in  a state and an  alphabet symbol so q is states sigma is  alphabet symbols so you're going to get  a state and a and a  alphabet symbol and it's going to give  you back a state  so more  um describing it you know um  [Music]  kind of a  little bit more detail you're going to  you know delta if you give it  state q and symbol a  equals r that means q when you read an a  you go to r  so that's the way this picture gets  translated into a mathematical function  um which describes those transitions  and then now q 0 is going to be the  starting state so there's going to be  that's the one with the in arrow coming  in from nowhere and f is the set of  accepted accepting states so there might  be several there's only going to be one  starting state but there might be  several different or possibly even zero  accepting states that's all legal  when we have a finite automaton um and  so in terms of using the notation going  back to the machine that we just had  from the previous slide which i've given  you here again  let me show you how i would describe  this using this notation that comes out  of the definition so here is m1 again  it's this five tuple  where q now is the set q1 q2 group three  that's the uh the set of states  uh the input alphabet is zero one  it might vary in other automata um and f  is the set q3 which has only the element  q3 because this has just one except  state q3  okay um so i hope that's uh helpful um  oh of course i forgot the transition  function which here i'm describing as a  table  so the transition function says um  you know if you have a q1 and a if you  have a state  and an input alphabet you can look up in  the table where you're supposed to go  under the transition function according  to the um  the uh  the state and the alphabet symbol that  you're you're given so for example you  know if we were in state um  q2  um here getting a zero  then q2 goes back to q1 so that q2 on  zero is q1  but q2 on one here  is q3  okay so that's how that table captures  this uh this picture  okay  and it's just a function it's a rep wave  representing a function a a finite  function um  uh  you know in terms of uh this uh this  table here so i realize for some of you  this may be  uh  slow um we will ramp up on speed but i'm  trying to get us all together in terms  of the language of the course here at  the beginning um  okay uh so now um  let's talk about some more um  uh the computation  uh  a  uh  so strings and languages  a string a string is just a finite  sequence of symbols from the from the  alphabet um so don't  whenever this class  is not going to talk about infinite  strings all of our strings are going to  be finite um  uh there's other mathematical theories  of automata and so on that talk about  infinite inputs and infinite strings  we're not going to talk about that um uh  you know maybe  rarely we'll make it very clear we'll  talk about an infinite string but that's  going to be an exception  um and a language is a set of strings  that's the traditional uh  you know um  a way that uh uh people in this subject  refer to a set of strings they call it a  language really because this subject had  its roots in linguistics actually um and  they were talking about they're trying  to understand languages you know human  languages  um  so this is just a historical fact and  that's the terminology that stuck  okay so two special string a string a  special string in a special language um  the empty string is the string of length  zero this is a totally legitimate string  um that you we're going to run into now  and then and there's the empty language  which is the set with no strings  these are not the same  they're not even of the same type of  object  so don't confuse them with one another  i mean you can have a a set a language  which has just one element which is the  empty string that is not the empty set  that is a set that is not the empty  language that is the a language that has  one element in it with namely the empty  string okay so those are separate things  okay so here's a little bit of a a  mouthful here on the slide um defining  what it means for a an automaton to  accept its input accepts its input  string w  and we can define that formally and you  know it's a little technical looking  it's really not that bad  um  so if you have a your input string w  which you can write as a sequence of  symbols in the alphabet w one w two dot  dot w n so you know like zero one zero  zero one i'm just writing it out symbol  by symbol here  um  so what does it mean for the machine to  accept um that's that input so that  means that there's a sequence of states  in the machine  sequence of states and of members of  queue so these are  a sequence from q these are the states  of the machine  uh that have a certain prod that satisfy  these three properties down here  first of all  and i'm thinking about the sequence is  the the sequence that the machine goes  through  as it's processing  the input w  so  when does it accept w  if that sequence has the feature that it  starts at the start state  each state legally follows the previous  state according to the transition  function  so that says you know the  member of the sequence  uh  is obtained by looking at the previous  one the i minus first member of that  sequence of that the i  the i minus first state in that sequence  and then looking at what happens when  you take the ice input symbol  so as you look at the previous state and  the next input symbol you should get the  next state that's all that this is  saying and this should happen for each  each one of these guys  and lastly for this to be accepted  the very last member here where we ended  up at the end of the input so you only  care about this at the end of the input  you have to be in an accepting state  so you can mathematically capture this  notion of going along this path  um  and uh that's what i'm just trying to  illustrate that you know we could  describe all this very formally i'm not  saying that's the best way to think  about it all the time but that it can be  done  and i think that's something you know  worth appreciating um  okay um  so now in terms of again getting back  we've said this once already but in  terms of the languages that the machine  recognizes it's the collection of  strings that the machine accepts  every machine accepts ex might it might  accept many strings  but it always recognizes one particular  language  even if the machine accepts no strings  then it recognizes the empty language so  a machine always recognizes some one  language  but it may have many many strings that  it's accepting  and we call that language the language  of the machine  and that's the and we say that m  recognizes that language these three  things mean the same thing  okay and now important definition i try  to reserve the most important things or  the highlighted things to be in this  light blue color if you can see that  we say a language is a regular language  if there's some finite automaton that  recognizes it  okay  so there are going to be some languages  that have associated to them finite  automata that actually solve those  languages that recognize those languages  but there might be other languages and  we'll see examples where you just can't  solve them you can't recognize them with  a finite automaton those languages will  not be regular languages the regular  ones are the ones that you can do with a  finite automaton that's the traditional  terminology  okay so let's continue let's go on from  there so look a couple examples here  again is that same getting getting to be  an old friend that language that  automaton m1  uh  remember its language here is the set of  strings that have the substring one one  that is the um  that language a  now what do we know about a  from the previous slide  think with me don't just listen  a is a  regular language now because it's  recognized by some  automaton  so  all of the whenever you find a um  an automaton for a language a finite  automaton for language we know that  language that language is a regular  language  so let's look at a couple of more  examples so if you take the language  let's call this one b  which is the the strings that have an  even number of ones in them  so like the string  one one zero one  would that be in b  no because it has an odd number of ones  so the string one one one one has four  ones in it that's an even number so that  string would be in b the  the zeros don't don't matter for this  language so that string  um  this  of  of of  strings that have an even number of ones  uh that's a regular language  um and the way you would know that is  you would have to make a fine art  automaton that recognizes that language  and i would encourage you to go  and make that automaton you can do it  with two states it's a very simple  automaton but if you haven't had  practice with these i encourage you to  do that and i actually there are lots of  examples that i ask you to  solve at the end of chapter one in the  book and you definitely should spend  some time  playing with it if you're not  if you have not yet seen uh finite  automata before um you need to get  comfortable with these and be able to  make them uh so we're going to start to  we're going to start making some of them  but we're going to be talking about it  at a sort of a more abstract level in a  minute  um  uh basically the reason why you can you  can solve this problem you can you can  make a fine art automaton would  recognize is the language b  is because that finite automaton is  going to keep track  of the parity of the number of ones it's  seen before  this has two states one of them  remembering that it's seen an odd number  one so far or the other one remembering  it's seen an even number of ones before  and that's going to be typical for these  automata finite automata there's going  to be  several different possibilities that you  may have to keep track of  as you're reading the input  and there's going to be a state  associated with each one of those  possibilities  so if you're designing an automaton you  have to think about as you're processing  the input what things you have to keep  track of and you're going to make a  state for each one of those  possibilities  okay so  you need to get comfortable with that um  let's look at another example um  the language c where the inputs have an  equal number of zeros and ones  that turns out to be a not a regular  language  um so what in other words what that  means is there's no way to recognize  that language with a finite automaton  you just can't do it  that's beyond the capabilities of finite  automata  and that's that's a problem that's  that's a statement we will prove later  um  okay um  and our goal over the next uh  lecture or so is to understand the  regular languages which you can do in a  very comprehensive way um so we're going  to start to do that now  so first we're going to introduce these  this concept of regular expressions um  which again these are things you may  have run into in one way or another  before um  so if we have a uh  um  we're going to introduce something  called the regular operations  now uh  you know  you i'm sure you're familiar with the  um  the arithmetical operations like plus  and times  um those apply to numbers  now the operations we're going to talk  about are operations that apply to  languages  so they're going to take  let's say two languages you apply an  operation you can get get back another  language like the union operation for  example  that's one you probably have seen before  um  the union of two languages here is a  collection of strings that are in either  one or the other  um  so  but there are other operations which you  may not have seen before that we're  going to look at um the concatenation  operation for example so that says  you're going to take a string from the  first language and another string from  the second language and stick them  together  and  it's called concatenating them  and you do that in all possible ways and  you're going to get the concatenation  language from these two languages that  you're starting with a and b  the symbol we use for concatenation is  this little circle but sometimes we  don't often we don't we just suppress  that and we write the two languages uh  next to one another with the uh little  circle implied so this also means  concatenation over here  just like this does  and the last uh  last of the regular operations is the  so-called star operation which is a  unary operation that applies to just a  single language  and so what you do is now um you're  going to take to look get a member of  the star language you're going to take a  bunch of strings in in the original  language a  and you stick them together  any number of members of a you stick  them together  and that becomes an element of the star  language i will do an example in a  second if you didn't get that  but one important element is that when  you have the star language you can also  allow to stick zero elements together  and then you get the empty string  so that's always a member of the star  language the empty string  okay so let's look at some examples  let's let let's say a is the language  these are two strings here good bad and  b is the language boy girl  okay  now if we take the union of those two  we get  a  good bad boy girl  that's kind of uh what you'd expect um  and um  if uh  um now let's  take a look at the concatenation  um  now if you concatenate the a and the b  language you're going to get all  possible ways of having an a string  followed by all possible ways having a b  string so you can get good boy good girl  bad boy bad girl  um now uh  looking at the star  um  [Music]  well  that applies to just one language so  let's say it's the good bad language  from uh a  uh and so the  a star that you get from that is all  possible ways of sticking together the  strings from a um so for there are no  using no strings you always get the  empty string that's always guaranteed to  be a member of a  and then just taking one element of a  you get good or another element bad but  now two elements of a you get good good  or good bad and so on um or three  elements of a good good good bit good  good bad  and so in fact a star is going to be an  infinite language  if a itself  is um  uh  contains any  non-empty member so if a is the empty if  a is the empty language or if a contains  just the language empty string  then a star will be  not an infinite language it'll just be  the language empty string  but otherwise it'll be an infinite  language  i'm not even sure  okay uh i'm not  i'm ignoring the chat here i'm hoping  people are getting are you guys getting  your questions answered by rtas  how are we doing thomas uh one question  is are the supplies going to be posted  are the slides going to be posted  well um  uh the whole lecture is going to be  recorded um is it helpful to have the  slide separately i can post the slides  um  sure i'll  remind me if i don't but i'll try to do  that yes it is helpful i will i will do  that  um  yeah um  yeah i will post the slides just  thomas it's your job to remind me okay  um  [Music]  all right good uh so regular so we  talked about the regular operations  let's talk about the regular expressions  so regular expressions are just like you  have the arithmetical operations then  you can get arithmetic expressions like  you know one plus three  um times seven so now we're going to  make expressions out of these operations  first of all you have the more atomic  and things the building blocks of the  expressions which are going to be like  um  uh  elements of sigma  you know elements of the alphabet or the  sigma itself as as an alphabet symbol or  the empty string or the empty language  of the the empty empty language or the  empty string these are going to be the  building blocks for the regular  expressions we'll do an example in a  second  and then you combine those those basic  elements  using the regular operations of union  concatenation and star so those are the  these are the atomic expressions these  are the composite expressions so for  example  um if you look at the expression  zero union one star  so that we can also write that as sigma  star because if sigma is zero  uh and one then sigma star is the same  thing as zero union one is sigma is the  same as zero union one and that just  gives all possible strings over sigma  so this is something you're going to see  frequently sigma star means all pos this  is the language of all strings over the  alphabet we're working with at that  moment  now if you if you take sigma star one  that's gonna you just concatenate one  onto all of the elements of sigma star  and that's going to give you all strings  that end with a one  you know technically you might imagine  writing this with braces around the one  but generally we don't do that we just  single element sets single element  strings we write without the braces  because  it's clear enough without them and it  gets messy with them so sigma star one  is all strings that end with one  or for example you take sigma star  one one sigma star  that is all strings that contain one one  and we already saw that language once  before that's the language of that other  machine that we presented on you know  one or two slides back  okay um  right uh  yes so  um uh yeah but in terms of readings by  the way sorry to i don't know if it's  helpful to you for me to do these  interjections but the readings are  listed also on the homework  so  if you if you  look at the posted homework one it tells  you  which  which chapters you should be reading now  and also if you look at the course  schedule  which is also on the home page it has  the whole  course plan and which readings are uh  for which dates  so it's all it's all there for you  um  and so our goal here this is not an  accident that sigma star one one sigma  star  happens to be the same language as we  saw before from the language of that of  that finite automaton m1  um in fact that's a general phenomenon  anything you can do with a regular  expression you can also do with a finite  automaton and vice versa  they are equivalent in power with  respect to the class of languages they  describe  and that's one we'll prove that  okay  um  so you know it's got you know if you  step back for a second and just let you  let yourself appreciate this it's kind  of an amazing thing because fine at  automata with the states and transitions  and the regular expressions with these  you know operations of union  concatenation and star they look totally  different from one another they look  like they have nothing to do with one  another but in fact they both describe  exactly the regular languages the same  class of languages  um  and so it's kind of a cool fact that you  can prove that these two very different  looking systems actually are equivalent  to one another  um  can we get  empty string from empty set yet  there are a bunch of  uh exotic cases by the way so empty  empty language star  is the language where has just the empty  string  if you don't get that chew on that one  that means but that is a that is that is  true um  uh  okay let's move on  okay let's talk about closure properties  now we're going to start in doing  something that has a little bit more  meat to it in terms of we're going to  have our first theorem of the course  coming here and this is not you know  it's not a baby theorem this is actually  uh there's going to be some meat to this  and you're going to have to uh um  you know  you know it's not totally um  this is not a toy this is we're proving  something that has a real substance  and and the um  uh  the statement of this theorem says that  the regular languages are closed really  the class of regular languages are  closed under union closed under the  union operation  um so what do i mean by that  so you you know  when you say a uh collection of objects  is closed under on some operation that  means applying that operation to those  objects  leaves you in the same class of objects  like the you know the um  the  natural you know the the positive  integers the natural numbers  that's closed under addition because  when you add two you know  positive integers you get back a  positive integer  but they're not closed under subtraction  because you know 2 minus four you get  something which is not a positive  integer  so close means you leave yourself in the  collection  and the fact is that  uh if you look at all the regular  languages these are the languages that  the  fine automata can recognize  they are closed under the union  operation so if you ta you start off  with two regular languages and you apply  the union you get back another regular  language  and that's what the statement of this  theorem is i hope you can i hope that's  clear enough in the way i've written it  if a1 and a2 are regular  then a1 union a2 is also regular that's  what the statement of this is and it's  just simply that um  that's proving that the class of regular  language is closed under union so we're  going to prove that  so how do you prove such a such a thing  so you know the way we  we're going to prove that is you uh  start off with you know what we're  assuming so we're our hypothesis is that  we have two regular languages  and we have to prove our conclusion that  the union is also regular  now the hypothesis that they're regular  you have to unpack that and understand  what it what is that what does that get  you and  them being regular means that there are  finer automata that recognize those  languages so let's give those two  fighter automata names so m1 and m2 are  the two finest automata that recognize  those two languages a1 and a2 that's  what it means that they're regular that  these automata exist  so let's uh have those two atomic m1 and  m2 using the components as we've  described this you know the respective  state sets input alphabet transition  functions the two starting states and  the two collections of stepping states  here i'm assuming that they're over the  same alphabet you could have automata  which operate over different alphabets  it's not interesting to do that  it doesn't add anything the proof would  be exactly the same so let's just not  over complicate our lives and focus on  the more interesting case so assuming  that the two  input alphabets are going to be the same  um and from these two automata we have  to show that this language here the  union is also a regular language and  we're going to do that by  constructing the automaton which  recognizes the union that's really the  only thing that we can do um  so we're going to build an automaton out  of m1 and m2 which recognizes the union  language a1 union a2  and the task of m is that it should  accept its input if either m1 or m2  except  and now what i'd like you to think about  doing that how would how in the world  are we going to come up with this find  out automaton m  um  and uh  uh the way we do that is  to think about  um  how would you  do that union language  you know if i ask you i give you two  automata m1 and m2 and i say here's an  input w  um  are is w in the union language  that's the job that m is supposed to  solve and i suggest you try to figure  out how you would solve it first  i mean this is a good strategy for  solving a lot of the problems in this  course put yourself in the place of the  machine you're trying to build  and so you know if you want to try to  figure out how to do that you know  natural thing is well you take w you  feed it into  m1 and then you feed it into m2 and if  m1 accepts it great and you know it's in  the union and if not you try it in and  try it out in m2 and see if m2 accepts  it um  now you have to be a little careful  because you want to have a strategy that  you can also implement in a fine art  automaton  and  uh  and a fine art automaton only gets one  shot at looking at the input you can't  sort of rewind the input you feed it  first into m1 and then you feed it into  m2  and  uh  operate in a sequential way like that  that's not going to be allowed um in the  way uh a finite automata work  so  you're going to have to be a little bit  more take it to the next level be a  little bit more clever and instead of  feeding it first into m1 and then and  then into m2 you feed them into both in  parallel  so you take m1 and m2 and you run them  both in parallel on the input w  keeping track of which state each of  those two automata are in  and then at the end you see if either  one of those machines is in an accepting  state and then you accept  so that's the strategy we're going to  employ um  in building the finer automaton m out of  m1 and m2 so here's in terms of a  picture here's m1 and m2  here is the automaton we're trying to  build  uh we don't know how it's going to look  like it and uh yeah so kind of getting  ahead of myself but here is the strategy  as i just described for amp  um  we're going to keep track  m is going to keep track of which  state that m1 is in and which state m2  is in at any given moment as we're  reading the symbols of w we're going to  feed it also into m1 and also into m2  and so  the possibilities we have to keep track  of in m  are all the pairs of states that are in  m1 and m2 because you're going to really  be tracking m1 and m2 simultaneously so  you have to remember which state m1 is  in and also which state m2 is in and so  that really corresponds to a pair of  states to remember  one from m1 and one from m2 and that's  why i've indicated it like that so m1 is  in state q m2 is in state r at at some  given point in time and that's kind of  correspond to m being in the pair q  comma r  that's just the label of this particular  state of m that uh we're gonna  apply here  okay  and so um and then  uh am is going to accept if either m1  and m2 so is an accepting state so it's  going to be if either q or r is an  accepting state we're going to make this  into a  an accepting state too  okay  whoops there we go  um  [Music]  so let's describe this formally instead  of by a picture because we can do it  both ways and sometimes it's better to  do it one way and sometimes the other  way  so now if we take uh the components of m  now  are the pairs of states from m1 and m2  again i'm writing this out literally uh  explicitly here but you should be make  sure you're comfortable with this cross  product notation  so this is the collection of pairs of  states q1 and q2 where q1 is in the  state of the first machine q2 is the  state of the second machine  the start state is you start at the two  uh start states of the two machines um  so this is q1 q2 probably i should have  not reused the q notation i should have  called these r's not of them looking at  that but anyway i hope you're not  confused by reusing this q1 and q2 here  are the specific start states of the two  twos of the two machines these are just  two  other states representative states of  those machines  now the transition function for the new  machine is going to be built out of the  transition functions for the old  previous machines so when i have a pair  qr  and i have the symbol a  where we go which new pair do we get  well we just update the state from m1  and update the state from m2 according  to their respective transition functions  and that's what's shown over here  um  now let's take a look at the accepting  states for m  the  natural thing to do is look at the set  of pairs of states  where we have a pair of states  of the  pair of accepting states one from  um the first machine and one from the  second machine  but if but if you're thinking with me  you realize that this is not the right  thing  what is dfas  uh  did i recall them dfa somewhere  oh somebody else is probably doing that  in the chat  uh  the dfa you careful what notation you're  using we haven't introduced dfas yet uh  we'll do that next on thursday uh but  these are dfas these are just fine at  automata deterministic finite automata  that's why they're  why the d anyway so  this is actually not right because if  you think about what this is saying it  says that both  components have to be accepting  and you want either one to be accepting  so  this is not good this is a this would be  the wrong way of defining it  that actually gives the intersection  language and really kind of along the  way it's proving closure under  intersection which we don't care about  but might be useful to have in our back  pocket sometime in the future  um  in order to get closure under reunion  we have to write it this slightly more  complicated looking way which says  that  you know uh the pair what you would want  to have is either the first state is a  an accepting state and then any state  for the second uh element or any state  for the first element and an accepting  state for the second element  that's what it means to have the union  uh to be doing the union okay so let's  let's do uh oh here's a quick check in  um  [Music]  so let's let's do another poll here you  thought we we thought we were done with  these um  again  you know  oh  here we go so i i it was too complicated  to write it out in the poll so i  actually put it up on the slide for you  um  [Music]  so i'm all i'm asking is that m1 has k  states k1 states and m2 has k2 states  how many states does m have  is it the sum  the sum of the squares or the product  okay you have to think about what are  the states of m what do they look like  um  and  uh  come on guys  all right ending the poll  uh  sharing results  yes indeed  it is most of you got it correct it is c  the the product  because when you look at the number of  pairs of states  from m1 and m2 you need all possible  pairs and so it's the number of states  in m1 times the number of states in m2  so make sure you you understand that and  think about that  um to uh  to so that you're you're following um  and and  and get this  all right so um let's move on here  so we have another five minutes or so  let's start thinking about closure under  concatenation  so if we have two regular languages so  is um the concatenation language um  we're gonna try to prove that we won't  finish but we'll at least get our  creative juices going about it um so  we're gonna do the same scheme here  we're gonna  take two machines for a1 and a2 and  build a machine for the concatenation  language out of those two so here are  the two machines for a1 and a2 written  down um  [Music]  and here is now here is  uh  the concatenation language and i'm going  to propose to you a strategy  um which is not going to work but it  still is going to be a good intuition to  have  so what i'm going to do is i'm going to  make a copy of m uh okay let's  understand what m is supposed to do  first  so m should accept its input so think  about this m is doing the concatenation  language so it's given a string and it's  an answer is it in the cat concatenating  the concatenation language  a1a2 or not  so it should accept it if m1 if there's  some way to divide w into two pieces  where m1 accepts the first piece and m2  accepts the second piece  uh  so the here would be the picture  okay and now um we have to try to make a  machine which is going to solve this  intuition so how would you do that  yourself  you're giving you w  and you can simulate m1 and m2  so  the natural thing is you're going to  start out by simulating m1 for a while  and then shift into simulating m2 for a  while  because that's what happens as you're  supposed to happening as you're  processing the input so i'm going to  suggest that in terms of the diagram  like this um  so we have here um  uh m1 and m2 copied here  uh  and what i propose doing is  uh connecting m1 to m2 so that when  m1  has accepted its input we're going to  jump to m2 because that's perhaps you  know the first part of w and now we're  going to have m2 process the second part  of w so the way i'm going to implement  that is by  declassifying the start state of m2  having transition symbols from  the accepting states of m1 to m2  and then removing these guys here as  uh accepting states  and we have to would have to figure out  what sort of labels to apply here but  actually this reasoning doesn't work  it's tempting but flawed  because  what what goes wrong  what happens is that  you know it might be  that  when m1 has accepted an initial part of  w and then it wants m2 to accept the  rest  it might fail because m2 doesn't accept  the rest  and what you might have been better off  doing is waiting longer in m1  because there might have been some other  later place to split w  which is still valid  splitting w in the first place where you  have m1 accepting an initial part may  not be the optimal place to split w you  might want to wait later and then you'll  have a better chance of accepting w  of accepting w  so um  i don't i don't know sure if you quite  follow that but um uh in fact it doesn't  work the question is where to split w  and uh it's challenging because how do  you know where to split w because it  depends upon what you it's a it might it  depends on why and you haven't seen why  yet  so when you're trying to think about it  that way it looks hopeless  but in fact it's still true  and we'll see how to do that  on thursday  so  just to recap what we did today  um  we did our introductory stuff  we defined fine art automata  on regular languages we define the  regular operations and expressions  uh we showed that the regular languages  are closed under union  we start a closure under intersection to  be continued  you
 

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250
00:07:51,510 --> 00:07:55,189

 

251
00:07:55,189 --> 00:07:56,790

 

252
00:07:56,790 --> 00:07:58,629

 

253
00:07:58,629 --> 00:08:01,189

 

254
00:08:01,189 --> 00:08:03,430

 

255
00:08:03,430 --> 00:08:05,990

 

256
00:08:05,990 --> 00:08:07,990

 

257
00:08:07,990 --> 00:08:08,000

 

258
00:08:08,000 --> 00:08:09,189


259
00:08:09,189 --> 00:08:10,950

 

260
00:08:10,950 --> 00:08:13,270

 

261
00:08:13,270 --> 00:08:16,309

 

262
00:08:16,309 --> 00:08:17,670

 

263
00:08:17,670 --> 00:08:19,350

 

264
00:08:19,350 --> 00:08:21,510

 

265
00:08:21,510 --> 00:08:23,749

 

266
00:08:23,749 --> 00:08:25,189

 

267
00:08:25,189 --> 00:08:28,390

 

268
00:08:28,390 --> 00:08:31,110

 

269
00:08:31,110 --> 00:08:32,949

 

270
00:08:32,949 --> 00:08:34,870

 

271
00:08:34,870 --> 00:08:37,029

 

272
00:08:37,029 --> 00:08:39,110

 

273
00:08:39,110 --> 00:08:42,550

 

274
00:08:42,550 --> 00:08:44,470

 

275
00:08:44,470 --> 00:08:46,150

 

276
00:08:46,150 --> 00:08:47,910

 

277
00:08:47,910 --> 00:08:49,030

 

278
00:08:49,030 --> 00:08:51,190

 

279
00:08:51,190 --> 00:08:53,269

 

280
00:08:53,269 --> 00:08:55,190

 

281
00:08:55,190 --> 00:08:57,190

 

282
00:08:57,190 --> 00:08:59,030

 

283
00:08:59,030 --> 00:09:00,550

 

284
00:09:00,550 --> 00:09:03,430

 

285
00:09:03,430 --> 00:09:05,670

 

286
00:09:05,670 --> 00:09:07,829

 

287
00:09:07,829 --> 00:09:07,839

 

288
00:09:07,839 --> 00:09:08,550


289
00:09:08,550 --> 00:09:11,110

 

290
00:09:11,110 --> 00:09:12,790

 

291
00:09:12,790 --> 00:09:16,070

 

292
00:09:16,070 --> 00:09:18,710

 

293
00:09:18,710 --> 00:09:18,720

 

294
00:09:18,720 --> 00:09:20,470


295
00:09:20,470 --> 00:09:20,480

 

296
00:09:20,480 --> 00:09:21,750


297
00:09:21,750 --> 00:09:21,760

 

298
00:09:21,760 --> 00:09:23,030


299
00:09:23,030 --> 00:09:25,750

 

300
00:09:25,750 --> 00:09:27,670

 

301
00:09:27,670 --> 00:09:30,710

 

302
00:09:30,710 --> 00:09:32,310

 

303
00:09:32,310 --> 00:09:34,230

 

304
00:09:34,230 --> 00:09:35,670

 

305
00:09:35,670 --> 00:09:37,590

 

306
00:09:37,590 --> 00:09:39,030

 

307
00:09:39,030 --> 00:09:40,389

 

308
00:09:40,389 --> 00:09:40,399

 

309
00:09:40,399 --> 00:09:41,750


310
00:09:41,750 --> 00:09:43,430

 

311
00:09:43,430 --> 00:09:45,269

 

312
00:09:45,269 --> 00:09:47,750

 

313
00:09:47,750 --> 00:09:50,230

 

314
00:09:50,230 --> 00:09:51,670

 

315
00:09:51,670 --> 00:09:53,590

 

316
00:09:53,590 --> 00:09:55,670

 

317
00:09:55,670 --> 00:09:56,949

 

318
00:09:56,949 --> 00:09:58,790

 

319
00:09:58,790 --> 00:10:00,550

 

320
00:10:00,550 --> 00:10:03,190

 

321
00:10:03,190 --> 00:10:05,670

 

322
00:10:05,670 --> 00:10:05,680

 

323
00:10:05,680 --> 00:10:06,550


324
00:10:06,550 --> 00:10:08,470

 

325
00:10:08,470 --> 00:10:10,310

 

326
00:10:10,310 --> 00:10:11,990

 

327
00:10:11,990 --> 00:10:15,269

 

328
00:10:15,269 --> 00:10:16,550

 

329
00:10:16,550 --> 00:10:18,470

 

330
00:10:18,470 --> 00:10:20,630

 

331
00:10:20,630 --> 00:10:22,550

 

332
00:10:22,550 --> 00:10:24,069

 

333
00:10:24,069 --> 00:10:26,470

 

334
00:10:26,470 --> 00:10:27,990

 

335
00:10:27,990 --> 00:10:29,910

 

336
00:10:29,910 --> 00:10:32,550

 

337
00:10:32,550 --> 00:10:34,630

 

338
00:10:34,630 --> 00:10:36,230

 

339
00:10:36,230 --> 00:10:39,269

 

340
00:10:39,269 --> 00:10:40,550

 

341
00:10:40,550 --> 00:10:43,110

 

342
00:10:43,110 --> 00:10:43,120

 

343
00:10:43,120 --> 00:10:44,870


344
00:10:44,870 --> 00:10:46,550

 

345
00:10:46,550 --> 00:10:48,310

 

346
00:10:48,310 --> 00:10:49,590

 

347
00:10:49,590 --> 00:10:51,750

 

348
00:10:51,750 --> 00:10:53,110

 

349
00:10:53,110 --> 00:10:56,230

 

350
00:10:56,230 --> 00:10:56,240

 

351
00:10:56,240 --> 00:10:57,350


352
00:10:57,350 --> 00:10:59,910

 

353
00:10:59,910 --> 00:11:01,910

 

354
00:11:01,910 --> 00:11:04,069

 

355
00:11:04,069 --> 00:11:04,079

 

356
00:11:04,079 --> 00:11:05,430


357
00:11:05,430 --> 00:11:08,310

 

358
00:11:08,310 --> 00:11:10,310

 

359
00:11:10,310 --> 00:11:12,389

 

360
00:11:12,389 --> 00:11:15,030

 

361
00:11:15,030 --> 00:11:18,389

 

362
00:11:18,389 --> 00:11:18,399

 

363
00:11:18,399 --> 00:11:19,910


364
00:11:19,910 --> 00:11:22,230

 

365
00:11:22,230 --> 00:11:24,470

 

366
00:11:24,470 --> 00:11:26,870

 

367
00:11:26,870 --> 00:11:30,150

 

368
00:11:30,150 --> 00:11:32,310

 

369
00:11:32,310 --> 00:11:34,790

 

370
00:11:34,790 --> 00:11:36,389

 

371
00:11:36,389 --> 00:11:39,030

 

372
00:11:39,030 --> 00:11:40,870

 

373
00:11:40,870 --> 00:11:44,310

 

374
00:11:44,310 --> 00:11:47,350

 

375
00:11:47,350 --> 00:11:48,870

 

376
00:11:48,870 --> 00:11:50,949

 

377
00:11:50,949 --> 00:11:52,389

 

378
00:11:52,389 --> 00:11:54,470

 

379
00:11:54,470 --> 00:11:55,590

 

380
00:11:55,590 --> 00:11:57,590

 

381
00:11:57,590 --> 00:11:59,509

 

382
00:11:59,509 --> 00:11:59,519

 

383
00:11:59,519 --> 00:12:00,550


384
00:12:00,550 --> 00:12:03,350

 

385
00:12:03,350 --> 00:12:04,949

 

386
00:12:04,949 --> 00:12:07,110

 

387
00:12:07,110 --> 00:12:09,110

 

388
00:12:09,110 --> 00:12:10,470

 

389
00:12:10,470 --> 00:12:13,030

 

390
00:12:13,030 --> 00:12:15,829

 

391
00:12:15,829 --> 00:12:17,670

 

392
00:12:17,670 --> 00:12:20,790

 

393
00:12:20,790 --> 00:12:22,790

 

394
00:12:22,790 --> 00:12:24,550

 

395
00:12:24,550 --> 00:12:26,790

 

396
00:12:26,790 --> 00:12:28,470

 

397
00:12:28,470 --> 00:12:30,629

 

398
00:12:30,629 --> 00:12:32,710

 

399
00:12:32,710 --> 00:12:35,590

 

400
00:12:35,590 --> 00:12:37,750

 

401
00:12:37,750 --> 00:12:40,150

 

402
00:12:40,150 --> 00:12:43,190

 

403
00:12:43,190 --> 00:12:43,200

 

404
00:12:43,200 --> 00:12:44,069


405
00:12:44,069 --> 00:12:45,829

 

406
00:12:45,829 --> 00:12:47,670

 

407
00:12:47,670 --> 00:12:50,870

 

408
00:12:50,870 --> 00:12:52,790

 

409
00:12:52,790 --> 00:12:54,949

 

410
00:12:54,949 --> 00:12:58,069

 

411
00:12:58,069 --> 00:13:00,230

 

412
00:13:00,230 --> 00:13:03,110

 

413
00:13:03,110 --> 00:13:05,030

 

414
00:13:05,030 --> 00:13:07,030

 

415
00:13:07,030 --> 00:13:09,750

 

416
00:13:09,750 --> 00:13:12,470

 

417
00:13:12,470 --> 00:13:14,629

 

418
00:13:14,629 --> 00:13:16,470

 

419
00:13:16,470 --> 00:13:17,750

 

420
00:13:17,750 --> 00:13:20,710

 

421
00:13:20,710 --> 00:13:23,269

 

422
00:13:23,269 --> 00:13:25,670

 

423
00:13:25,670 --> 00:13:27,829

 

424
00:13:27,829 --> 00:13:31,750

 

425
00:13:31,750 --> 00:13:33,190

 

426
00:13:33,190 --> 00:13:36,389

 

427
00:13:36,389 --> 00:13:38,629

 

428
00:13:38,629 --> 00:13:42,310

 

429
00:13:42,310 --> 00:13:45,269

 

430
00:13:45,269 --> 00:13:47,269

 

431
00:13:47,269 --> 00:13:49,030

 

432
00:13:49,030 --> 00:13:50,790

 

433
00:13:50,790 --> 00:13:53,030

 

434
00:13:53,030 --> 00:13:54,870

 

435
00:13:54,870 --> 00:13:56,629

 

436
00:13:56,629 --> 00:13:58,550

 

437
00:13:58,550 --> 00:13:58,560

 

438
00:13:58,560 --> 00:13:59,750


439
00:13:59,750 --> 00:14:01,910

 

440
00:14:01,910 --> 00:14:03,750

 

441
00:14:03,750 --> 00:14:05,990

 

442
00:14:05,990 --> 00:14:07,910

 

443
00:14:07,910 --> 00:14:11,750

 

444
00:14:11,750 --> 00:14:11,760

 

445
00:14:11,760 --> 00:14:13,030


446
00:14:13,030 --> 00:14:13,040

 

447
00:14:13,040 --> 00:14:14,230


448
00:14:14,230 --> 00:14:14,240

 

449
00:14:14,240 --> 00:14:15,189


450
00:14:15,189 --> 00:14:15,199

 

451
00:14:15,199 --> 00:14:16,310


452
00:14:16,310 --> 00:14:18,310

 

453
00:14:18,310 --> 00:14:20,629

 

454
00:14:20,629 --> 00:14:23,910

 

455
00:14:23,910 --> 00:14:23,920

 

456
00:14:23,920 --> 00:14:25,350


457
00:14:25,350 --> 00:14:27,350

 

458
00:14:27,350 --> 00:14:29,189

 

459
00:14:29,189 --> 00:14:30,790

 

460
00:14:30,790 --> 00:14:33,350

 

461
00:14:33,350 --> 00:14:36,550

 

462
00:14:36,550 --> 00:14:36,560

 

463
00:14:36,560 --> 00:14:38,310


464
00:14:38,310 --> 00:14:38,320

 

465
00:14:38,320 --> 00:14:39,269


466
00:14:39,269 --> 00:14:42,310

 

467
00:14:42,310 --> 00:14:45,269

 

468
00:14:45,269 --> 00:14:46,790

 

469
00:14:46,790 --> 00:14:50,069

 

470
00:14:50,069 --> 00:14:52,310

 

471
00:14:52,310 --> 00:14:54,629

 

472
00:14:54,629 --> 00:14:57,189

 

473
00:14:57,189 --> 00:14:59,189

 

474
00:14:59,189 --> 00:15:01,509

 

475
00:15:01,509 --> 00:15:03,910

 

476
00:15:03,910 --> 00:15:06,470

 

477
00:15:06,470 --> 00:15:08,470

 

478
00:15:08,470 --> 00:15:11,670

 

479
00:15:11,670 --> 00:15:14,310

 

480
00:15:14,310 --> 00:15:17,430

 

481
00:15:17,430 --> 00:15:18,550

 

482
00:15:18,550 --> 00:15:20,870

 

483
00:15:20,870 --> 00:15:23,350

 

484
00:15:23,350 --> 00:15:24,629

 

485
00:15:24,629 --> 00:15:27,509

 

486
00:15:27,509 --> 00:15:29,749

 

487
00:15:29,749 --> 00:15:31,990

 

488
00:15:31,990 --> 00:15:34,310

 

489
00:15:34,310 --> 00:15:36,949

 

490
00:15:36,949 --> 00:15:38,870

 

491
00:15:38,870 --> 00:15:42,550

 

492
00:15:42,550 --> 00:15:42,560

 

493
00:15:42,560 --> 00:15:43,990


494
00:15:43,990 --> 00:15:45,269

 

495
00:15:45,269 --> 00:15:47,189

 

496
00:15:47,189 --> 00:15:49,110

 

497
00:15:49,110 --> 00:15:50,629

 

498
00:15:50,629 --> 00:15:52,629

 

499
00:15:52,629 --> 00:15:54,470

 

500
00:15:54,470 --> 00:15:57,509

 

501
00:15:57,509 --> 00:15:57,519

 

502
00:15:57,519 --> 00:15:58,550


503
00:15:58,550 --> 00:16:00,790

 

504
00:16:00,790 --> 00:16:02,629

 

505
00:16:02,629 --> 00:16:04,310

 

506
00:16:04,310 --> 00:16:05,749

 

507
00:16:05,749 --> 00:16:07,590

 

508
00:16:07,590 --> 00:16:09,430

 

509
00:16:09,430 --> 00:16:11,189

 

510
00:16:11,189 --> 00:16:13,590

 

511
00:16:13,590 --> 00:16:15,829

 

512
00:16:15,829 --> 00:16:15,839

 

513
00:16:15,839 --> 00:16:16,710


514
00:16:16,710 --> 00:16:16,720

 

515
00:16:16,720 --> 00:16:18,069


516
00:16:18,069 --> 00:16:18,079

 

517
00:16:18,079 --> 00:16:20,310


518
00:16:20,310 --> 00:16:23,189

 

519
00:16:23,189 --> 00:16:23,199

 

520
00:16:23,199 --> 00:16:24,150


521
00:16:24,150 --> 00:16:24,160

 

522
00:16:24,160 --> 00:16:25,189


523
00:16:25,189 --> 00:16:28,550

 

524
00:16:28,550 --> 00:16:31,509

 

525
00:16:31,509 --> 00:16:33,189

 

526
00:16:33,189 --> 00:16:36,310

 

527
00:16:36,310 --> 00:16:36,320

 

528
00:16:36,320 --> 00:16:37,430


529
00:16:37,430 --> 00:16:38,949

 

530
00:16:38,949 --> 00:16:41,350

 

531
00:16:41,350 --> 00:16:41,360

 

532
00:16:41,360 --> 00:16:42,550


533
00:16:42,550 --> 00:16:46,310

 

534
00:16:46,310 --> 00:16:46,320

 

535
00:16:46,320 --> 00:16:47,269


536
00:16:47,269 --> 00:16:49,829

 

537
00:16:49,829 --> 00:16:51,990

 

538
00:16:51,990 --> 00:16:53,749

 

539
00:16:53,749 --> 00:16:55,030

 

540
00:16:55,030 --> 00:16:57,189

 

541
00:16:57,189 --> 00:16:59,590

 

542
00:16:59,590 --> 00:17:01,030

 

543
00:17:01,030 --> 00:17:03,829

 

544
00:17:03,829 --> 00:17:06,390

 

545
00:17:06,390 --> 00:17:08,309

 

546
00:17:08,309 --> 00:17:10,390

 

547
00:17:10,390 --> 00:17:12,549

 

548
00:17:12,549 --> 00:17:14,630

 

549
00:17:14,630 --> 00:17:14,640

 

550
00:17:14,640 --> 00:17:15,750


551
00:17:15,750 --> 00:17:17,590

 

552
00:17:17,590 --> 00:17:17,600

 

553
00:17:17,600 --> 00:17:18,390


554
00:17:18,390 --> 00:17:18,400

 

555
00:17:18,400 --> 00:17:21,189


556
00:17:21,189 --> 00:17:23,590

 

557
00:17:23,590 --> 00:17:24,949

 

558
00:17:24,949 --> 00:17:24,959

 

559
00:17:24,959 --> 00:17:26,069


560
00:17:26,069 --> 00:17:26,079

 

561
00:17:26,079 --> 00:17:31,590


562
00:17:31,590 --> 00:17:34,390

 

563
00:17:34,390 --> 00:17:36,150

 

564
00:17:36,150 --> 00:17:38,870

 

565
00:17:38,870 --> 00:17:45,990

 

566
00:17:45,990 --> 00:17:52,070

 

567
00:17:52,070 --> 00:17:52,080

 

568
00:17:52,080 --> 00:17:54,789


569
00:17:54,789 --> 00:17:56,870

 

570
00:17:56,870 --> 00:18:01,350

 

571
00:18:01,350 --> 00:18:05,669

 

572
00:18:05,669 --> 00:18:07,909

 

573
00:18:07,909 --> 00:18:10,390

 

574
00:18:10,390 --> 00:18:10,400

 

575
00:18:10,400 --> 00:18:11,750


576
00:18:11,750 --> 00:18:15,029

 

577
00:18:15,029 --> 00:18:17,990

 

578
00:18:17,990 --> 00:18:20,390

 

579
00:18:20,390 --> 00:18:23,029

 

580
00:18:23,029 --> 00:18:25,110

 

581
00:18:25,110 --> 00:18:26,549

 

582
00:18:26,549 --> 00:18:28,870

 

583
00:18:28,870 --> 00:18:28,880

 

584
00:18:28,880 --> 00:18:30,310


585
00:18:30,310 --> 00:18:31,730

 

586
00:18:31,730 --> 00:18:31,740

 

587
00:18:31,740 --> 00:18:33,270


588
00:18:33,270 --> 00:18:35,029

 

589
00:18:35,029 --> 00:18:35,039

 

590
00:18:35,039 --> 00:18:35,830


591
00:18:35,830 --> 00:18:37,000

 

592
00:18:37,000 --> 00:18:37,010

 

593
00:18:37,010 --> 00:18:41,510


594
00:18:41,510 --> 00:18:44,150

 

595
00:18:44,150 --> 00:18:46,470

 

596
00:18:46,470 --> 00:18:47,669

 

597
00:18:47,669 --> 00:18:49,909

 

598
00:18:49,909 --> 00:18:52,549

 

599
00:18:52,549 --> 00:18:56,230

 

600
00:18:56,230 --> 00:18:56,240

 

601
00:18:56,240 --> 00:18:57,029


602
00:18:57,029 --> 00:18:58,390

 

603
00:18:58,390 --> 00:19:00,950

 

604
00:19:00,950 --> 00:19:02,710

 

605
00:19:02,710 --> 00:19:04,310

 

606
00:19:04,310 --> 00:19:06,230

 

607
00:19:06,230 --> 00:19:07,510

 

608
00:19:07,510 --> 00:19:08,630

 

609
00:19:08,630 --> 00:19:10,710

 

610
00:19:10,710 --> 00:19:13,270

 

611
00:19:13,270 --> 00:19:15,590

 

612
00:19:15,590 --> 00:19:17,750

 

613
00:19:17,750 --> 00:19:17,760

 

614
00:19:17,760 --> 00:19:18,710


615
00:19:18,710 --> 00:19:20,789

 

616
00:19:20,789 --> 00:19:22,390

 

617
00:19:22,390 --> 00:19:25,110

 

618
00:19:25,110 --> 00:19:26,870

 

619
00:19:26,870 --> 00:19:27,909

 

620
00:19:27,909 --> 00:19:31,190

 

621
00:19:31,190 --> 00:19:32,870

 

622
00:19:32,870 --> 00:19:34,549

 

623
00:19:34,549 --> 00:19:37,029

 

624
00:19:37,029 --> 00:19:38,710

 

625
00:19:38,710 --> 00:19:41,110

 

626
00:19:41,110 --> 00:19:43,430

 

627
00:19:43,430 --> 00:19:46,310

 

628
00:19:46,310 --> 00:19:47,750

 

629
00:19:47,750 --> 00:19:49,669

 

630
00:19:49,669 --> 00:19:51,110

 

631
00:19:51,110 --> 00:19:53,830

 

632
00:19:53,830 --> 00:19:55,750

 

633
00:19:55,750 --> 00:19:58,549

 

634
00:19:58,549 --> 00:20:00,950

 

635
00:20:00,950 --> 00:20:03,270

 

636
00:20:03,270 --> 00:20:05,270

 

637
00:20:05,270 --> 00:20:07,110

 

638
00:20:07,110 --> 00:20:07,120

 

639
00:20:07,120 --> 00:20:08,470


640
00:20:08,470 --> 00:20:11,270

 

641
00:20:11,270 --> 00:20:12,630

 

642
00:20:12,630 --> 00:20:14,630

 

643
00:20:14,630 --> 00:20:15,909

 

644
00:20:15,909 --> 00:20:17,830

 

645
00:20:17,830 --> 00:20:19,830

 

646
00:20:19,830 --> 00:20:19,840

 

647
00:20:19,840 --> 00:20:20,789


648
00:20:20,789 --> 00:20:22,870

 

649
00:20:22,870 --> 00:20:26,789

 

650
00:20:26,789 --> 00:20:29,350

 

651
00:20:29,350 --> 00:20:31,750

 

652
00:20:31,750 --> 00:20:33,909

 

653
00:20:33,909 --> 00:20:35,990

 

654
00:20:35,990 --> 00:20:38,390

 

655
00:20:38,390 --> 00:20:40,470

 

656
00:20:40,470 --> 00:20:42,789

 

657
00:20:42,789 --> 00:20:45,510

 

658
00:20:45,510 --> 00:20:46,870

 

659
00:20:46,870 --> 00:20:50,310

 

660
00:20:50,310 --> 00:20:52,310

 

661
00:20:52,310 --> 00:20:53,750

 

662
00:20:53,750 --> 00:20:56,470

 

663
00:20:56,470 --> 00:20:58,070

 

664
00:20:58,070 --> 00:21:00,630

 

665
00:21:00,630 --> 00:21:03,750

 

666
00:21:03,750 --> 00:21:03,760

 

667
00:21:03,760 --> 00:21:04,710


668
00:21:04,710 --> 00:21:05,990

 

669
00:21:05,990 --> 00:21:07,990

 

670
00:21:07,990 --> 00:21:10,789

 

671
00:21:10,789 --> 00:21:12,310

 

672
00:21:12,310 --> 00:21:14,390

 

673
00:21:14,390 --> 00:21:15,990

 

674
00:21:15,990 --> 00:21:17,990

 

675
00:21:17,990 --> 00:21:19,430

 

676
00:21:19,430 --> 00:21:21,820

 

677
00:21:21,820 --> 00:21:21,830

 

678
00:21:21,830 --> 00:21:23,270


679
00:21:23,270 --> 00:21:24,630

 

680
00:21:24,630 --> 00:21:26,870

 

681
00:21:26,870 --> 00:21:29,430

 

682
00:21:29,430 --> 00:21:32,230

 

683
00:21:32,230 --> 00:21:35,669

 

684
00:21:35,669 --> 00:21:37,830

 

685
00:21:37,830 --> 00:21:39,909

 

686
00:21:39,909 --> 00:21:43,270

 

687
00:21:43,270 --> 00:21:47,270

 

688
00:21:47,270 --> 00:21:49,110

 

689
00:21:49,110 --> 00:21:50,390

 

690
00:21:50,390 --> 00:21:51,909

 

691
00:21:51,909 --> 00:21:54,310

 

692
00:21:54,310 --> 00:21:56,870

 

693
00:21:56,870 --> 00:21:58,549

 

694
00:21:58,549 --> 00:21:59,830

 

695
00:21:59,830 --> 00:22:02,310

 

696
00:22:02,310 --> 00:22:04,710

 

697
00:22:04,710 --> 00:22:07,590

 

698
00:22:07,590 --> 00:22:09,909

 

699
00:22:09,909 --> 00:22:11,190

 

700
00:22:11,190 --> 00:22:12,630

 

701
00:22:12,630 --> 00:22:14,070

 

702
00:22:14,070 --> 00:22:15,510

 

703
00:22:15,510 --> 00:22:17,669

 

704
00:22:17,669 --> 00:22:20,870

 

705
00:22:20,870 --> 00:22:23,029

 

706
00:22:23,029 --> 00:22:26,230

 

707
00:22:26,230 --> 00:22:29,110

 

708
00:22:29,110 --> 00:22:31,990

 

709
00:22:31,990 --> 00:22:35,350

 

710
00:22:35,350 --> 00:22:37,990

 

711
00:22:37,990 --> 00:22:40,149

 

712
00:22:40,149 --> 00:22:41,750

 

713
00:22:41,750 --> 00:22:45,750

 

714
00:22:45,750 --> 00:22:48,549

 

715
00:22:48,549 --> 00:22:50,390

 

716
00:22:50,390 --> 00:22:50,400

 

717
00:22:50,400 --> 00:22:51,669


718
00:22:51,669 --> 00:22:55,110

 

719
00:22:55,110 --> 00:22:58,230

 

720
00:22:58,230 --> 00:22:59,750

 

721
00:22:59,750 --> 00:23:02,549

 

722
00:23:02,549 --> 00:23:05,029

 

723
00:23:05,029 --> 00:23:06,950

 

724
00:23:06,950 --> 00:23:08,549

 

725
00:23:08,549 --> 00:23:09,750

 

726
00:23:09,750 --> 00:23:12,470

 

727
00:23:12,470 --> 00:23:15,270

 

728
00:23:15,270 --> 00:23:17,830

 

729
00:23:17,830 --> 00:23:17,840

 

730
00:23:17,840 --> 00:23:19,669


731
00:23:19,669 --> 00:23:22,630

 

732
00:23:22,630 --> 00:23:25,750

 

733
00:23:25,750 --> 00:23:27,510

 

734
00:23:27,510 --> 00:23:29,909

 

735
00:23:29,909 --> 00:23:32,070

 

736
00:23:32,070 --> 00:23:34,390

 

737
00:23:34,390 --> 00:23:36,789

 

738
00:23:36,789 --> 00:23:36,799

 

739
00:23:36,799 --> 00:23:37,590


740
00:23:37,590 --> 00:23:39,110

 

741
00:23:39,110 --> 00:23:41,430

 

742
00:23:41,430 --> 00:23:43,029

 

743
00:23:43,029 --> 00:23:43,039

 

744
00:23:43,039 --> 00:23:45,110


745
00:23:45,110 --> 00:23:47,350

 

746
00:23:47,350 --> 00:23:49,990

 

747
00:23:49,990 --> 00:23:51,430

 

748
00:23:51,430 --> 00:23:51,440

 

749
00:23:51,440 --> 00:23:52,310


750
00:23:52,310 --> 00:23:55,269

 

751
00:23:55,269 --> 00:23:57,590

 

752
00:23:57,590 --> 00:23:59,510

 

753
00:23:59,510 --> 00:24:01,430

 

754
00:24:01,430 --> 00:24:04,630

 

755
00:24:04,630 --> 00:24:07,430

 

756
00:24:07,430 --> 00:24:09,110

 

757
00:24:09,110 --> 00:24:09,120

 

758
00:24:09,120 --> 00:24:10,149


759
00:24:10,149 --> 00:24:10,159

 

760
00:24:10,159 --> 00:24:11,830


761
00:24:11,830 --> 00:24:11,840

 

762
00:24:11,840 --> 00:24:12,630


763
00:24:12,630 --> 00:24:14,390

 

764
00:24:14,390 --> 00:24:16,070

 

765
00:24:16,070 --> 00:24:17,990

 

766
00:24:17,990 --> 00:24:20,870

 

767
00:24:20,870 --> 00:24:23,029

 

768
00:24:23,029 --> 00:24:24,310

 

769
00:24:24,310 --> 00:24:25,750

 

770
00:24:25,750 --> 00:24:27,669

 

771
00:24:27,669 --> 00:24:30,390

 

772
00:24:30,390 --> 00:24:32,149

 

773
00:24:32,149 --> 00:24:34,070

 

774
00:24:34,070 --> 00:24:39,110

 

775
00:24:39,110 --> 00:24:40,310

 

776
00:24:40,310 --> 00:24:42,070

 

777
00:24:42,070 --> 00:24:43,430

 

778
00:24:43,430 --> 00:24:44,830

 

779
00:24:44,830 --> 00:24:48,470

 

780
00:24:48,470 --> 00:24:51,110

 

781
00:24:51,110 --> 00:24:52,710

 

782
00:24:52,710 --> 00:24:55,990

 

783
00:24:55,990 --> 00:24:57,830

 

784
00:24:57,830 --> 00:25:00,230

 

785
00:25:00,230 --> 00:25:03,350

 

786
00:25:03,350 --> 00:25:04,470

 

787
00:25:04,470 --> 00:25:06,230

 

788
00:25:06,230 --> 00:25:06,240

 

789
00:25:06,240 --> 00:25:07,350


790
00:25:07,350 --> 00:25:07,360

 

791
00:25:07,360 --> 00:25:08,470


792
00:25:08,470 --> 00:25:10,630

 

793
00:25:10,630 --> 00:25:13,029

 

794
00:25:13,029 --> 00:25:15,510

 

795
00:25:15,510 --> 00:25:17,750

 

796
00:25:17,750 --> 00:25:19,430

 

797
00:25:19,430 --> 00:25:22,230

 

798
00:25:22,230 --> 00:25:24,549

 

799
00:25:24,549 --> 00:25:26,789

 

800
00:25:26,789 --> 00:25:29,590

 

801
00:25:29,590 --> 00:25:31,029

 

802
00:25:31,029 --> 00:25:32,789

 

803
00:25:32,789 --> 00:25:32,799

 

804
00:25:32,799 --> 00:25:33,830


805
00:25:33,830 --> 00:25:36,789

 

806
00:25:36,789 --> 00:25:39,830

 

807
00:25:39,830 --> 00:25:41,669

 

808
00:25:41,669 --> 00:25:44,549

 

809
00:25:44,549 --> 00:25:46,230

 

810
00:25:46,230 --> 00:25:48,149

 

811
00:25:48,149 --> 00:25:49,830

 

812
00:25:49,830 --> 00:25:53,830

 

813
00:25:53,830 --> 00:25:56,070

 

814
00:25:56,070 --> 00:25:58,789

 

815
00:25:58,789 --> 00:26:01,669

 

816
00:26:01,669 --> 00:26:04,549

 

817
00:26:04,549 --> 00:26:06,710

 

818
00:26:06,710 --> 00:26:08,830

 

819
00:26:08,830 --> 00:26:11,029

 

820
00:26:11,029 --> 00:26:13,110

 

821
00:26:13,110 --> 00:26:13,120

 

822
00:26:13,120 --> 00:26:14,070


823
00:26:14,070 --> 00:26:17,590

 

824
00:26:17,590 --> 00:26:19,269

 

825
00:26:19,269 --> 00:26:22,549

 

826
00:26:22,549 --> 00:26:25,430

 

827
00:26:25,430 --> 00:26:27,830

 

828
00:26:27,830 --> 00:26:29,830

 

829
00:26:29,830 --> 00:26:29,840

 

830
00:26:29,840 --> 00:26:30,710


831
00:26:30,710 --> 00:26:32,230

 

832
00:26:32,230 --> 00:26:35,430

 

833
00:26:35,430 --> 00:26:37,990

 

834
00:26:37,990 --> 00:26:39,669

 

835
00:26:39,669 --> 00:26:41,990

 

836
00:26:41,990 --> 00:26:43,909

 

837
00:26:43,909 --> 00:26:45,510

 

838
00:26:45,510 --> 00:26:46,870

 

839
00:26:46,870 --> 00:26:49,110

 

840
00:26:49,110 --> 00:26:51,190

 

841
00:26:51,190 --> 00:26:52,390

 

842
00:26:52,390 --> 00:26:54,070

 

843
00:26:54,070 --> 00:26:56,470

 

844
00:26:56,470 --> 00:26:56,480

 

845
00:26:56,480 --> 00:26:57,669


846
00:26:57,669 --> 00:26:59,669

 

847
00:26:59,669 --> 00:27:01,990

 

848
00:27:01,990 --> 00:27:02,000

 

849
00:27:02,000 --> 00:27:02,710


850
00:27:02,710 --> 00:27:04,630

 

851
00:27:04,630 --> 00:27:07,110

 

852
00:27:07,110 --> 00:27:10,310

 

853
00:27:10,310 --> 00:27:13,190

 

854
00:27:13,190 --> 00:27:14,950

 

855
00:27:14,950 --> 00:27:14,960

 

856
00:27:14,960 --> 00:27:17,190


857
00:27:17,190 --> 00:27:19,830

 

858
00:27:19,830 --> 00:27:21,590

 

859
00:27:21,590 --> 00:27:21,600

 

860
00:27:21,600 --> 00:27:22,549


861
00:27:22,549 --> 00:27:24,870

 

862
00:27:24,870 --> 00:27:26,710

 

863
00:27:26,710 --> 00:27:28,549

 

864
00:27:28,549 --> 00:27:31,110

 

865
00:27:31,110 --> 00:27:32,549

 

866
00:27:32,549 --> 00:27:36,070

 

867
00:27:36,070 --> 00:27:38,389

 

868
00:27:38,389 --> 00:27:40,310

 

869
00:27:40,310 --> 00:27:41,909

 

870
00:27:41,909 --> 00:27:43,830

 

871
00:27:43,830 --> 00:27:45,590

 

872
00:27:45,590 --> 00:27:48,389

 

873
00:27:48,389 --> 00:27:50,630

 

874
00:27:50,630 --> 00:27:53,269

 

875
00:27:53,269 --> 00:27:55,110

 

876
00:27:55,110 --> 00:27:58,470

 

877
00:27:58,470 --> 00:28:00,470

 

878
00:28:00,470 --> 00:28:03,510

 

879
00:28:03,510 --> 00:28:03,520

 

880
00:28:03,520 --> 00:28:05,430


881
00:28:05,430 --> 00:28:07,750

 

882
00:28:07,750 --> 00:28:09,430

 

883
00:28:09,430 --> 00:28:11,029

 

884
00:28:11,029 --> 00:28:12,149

 

885
00:28:12,149 --> 00:28:14,230

 

886
00:28:14,230 --> 00:28:14,240

 

887
00:28:14,240 --> 00:28:15,110


888
00:28:15,110 --> 00:28:16,470

 

889
00:28:16,470 --> 00:28:19,029

 

890
00:28:19,029 --> 00:28:20,789

 

891
00:28:20,789 --> 00:28:22,710

 

892
00:28:22,710 --> 00:28:24,149

 

893
00:28:24,149 --> 00:28:25,990

 

894
00:28:25,990 --> 00:28:27,909

 

895
00:28:27,909 --> 00:28:30,549

 

896
00:28:30,549 --> 00:28:33,350

 

897
00:28:33,350 --> 00:28:35,110

 

898
00:28:35,110 --> 00:28:37,750

 

899
00:28:37,750 --> 00:28:37,760

 

900
00:28:37,760 --> 00:28:39,590


901
00:28:39,590 --> 00:28:41,590

 

902
00:28:41,590 --> 00:28:44,230

 

903
00:28:44,230 --> 00:28:47,190

 

904
00:28:47,190 --> 00:28:47,200

 

905
00:28:47,200 --> 00:28:48,310


906
00:28:48,310 --> 00:28:50,230

 

907
00:28:50,230 --> 00:28:51,830

 

908
00:28:51,830 --> 00:28:53,510

 

909
00:28:53,510 --> 00:28:54,710

 

910
00:28:54,710 --> 00:28:56,470

 

911
00:28:56,470 --> 00:28:58,230

 

912
00:28:58,230 --> 00:29:02,070

 

913
00:29:02,070 --> 00:29:04,230

 

914
00:29:04,230 --> 00:29:06,389

 

915
00:29:06,389 --> 00:29:08,149

 

916
00:29:08,149 --> 00:29:10,470

 

917
00:29:10,470 --> 00:29:14,230

 

918
00:29:14,230 --> 00:29:16,549

 

919
00:29:16,549 --> 00:29:19,029

 

920
00:29:19,029 --> 00:29:19,039

 

921
00:29:19,039 --> 00:29:19,990


922
00:29:19,990 --> 00:29:21,750

 

923
00:29:21,750 --> 00:29:23,909

 

924
00:29:23,909 --> 00:29:25,510

 

925
00:29:25,510 --> 00:29:27,590

 

926
00:29:27,590 --> 00:29:29,029

 

927
00:29:29,029 --> 00:29:30,870

 

928
00:29:30,870 --> 00:29:32,470

 

929
00:29:32,470 --> 00:29:34,789

 

930
00:29:34,789 --> 00:29:36,870

 

931
00:29:36,870 --> 00:29:38,630

 

932
00:29:38,630 --> 00:29:41,029

 

933
00:29:41,029 --> 00:29:41,039

 

934
00:29:41,039 --> 00:29:43,590


935
00:29:43,590 --> 00:29:45,909

 

936
00:29:45,909 --> 00:29:48,549

 

937
00:29:48,549 --> 00:29:50,789

 

938
00:29:50,789 --> 00:29:52,470

 

939
00:29:52,470 --> 00:29:54,470

 

940
00:29:54,470 --> 00:29:54,480

 

941
00:29:54,480 --> 00:29:56,070


942
00:29:56,070 --> 00:29:59,269

 

943
00:29:59,269 --> 00:30:02,470

 

944
00:30:02,470 --> 00:30:05,590

 

945
00:30:05,590 --> 00:30:07,190

 

946
00:30:07,190 --> 00:30:09,430

 

947
00:30:09,430 --> 00:30:12,389

 

948
00:30:12,389 --> 00:30:15,110

 

949
00:30:15,110 --> 00:30:16,789

 

950
00:30:16,789 --> 00:30:18,789

 

951
00:30:18,789 --> 00:30:20,830

 

952
00:30:20,830 --> 00:30:20,840

 

953
00:30:20,840 --> 00:30:22,630


954
00:30:22,630 --> 00:30:22,640

 

955
00:30:22,640 --> 00:30:23,590


956
00:30:23,590 --> 00:30:26,950

 

957
00:30:26,950 --> 00:30:28,950

 

958
00:30:28,950 --> 00:30:30,389

 

959
00:30:30,389 --> 00:30:31,830

 

960
00:30:31,830 --> 00:30:31,840

 

961
00:30:31,840 --> 00:30:34,310


962
00:30:34,310 --> 00:30:35,430

 

963
00:30:35,430 --> 00:30:37,830

 

964
00:30:37,830 --> 00:30:39,590

 

965
00:30:39,590 --> 00:30:42,389

 

966
00:30:42,389 --> 00:30:44,870

 

967
00:30:44,870 --> 00:30:46,710

 

968
00:30:46,710 --> 00:30:49,029

 

969
00:30:49,029 --> 00:30:51,190

 

970
00:30:51,190 --> 00:30:54,070

 

971
00:30:54,070 --> 00:30:57,110

 

972
00:30:57,110 --> 00:30:58,789

 

973
00:30:58,789 --> 00:31:00,549

 

974
00:31:00,549 --> 00:31:02,710

 

975
00:31:02,710 --> 00:31:05,909

 

976
00:31:05,909 --> 00:31:05,919

 

977
00:31:05,919 --> 00:31:07,190


978
00:31:07,190 --> 00:31:07,200

 

979
00:31:07,200 --> 00:31:07,909


980
00:31:07,909 --> 00:31:07,919

 

981
00:31:07,919 --> 00:31:08,870


982
00:31:08,870 --> 00:31:10,310

 

983
00:31:10,310 --> 00:31:13,190

 

984
00:31:13,190 --> 00:31:15,750

 

985
00:31:15,750 --> 00:31:17,830

 

986
00:31:17,830 --> 00:31:19,269

 

987
00:31:19,269 --> 00:31:21,430

 

988
00:31:21,430 --> 00:31:23,830

 

989
00:31:23,830 --> 00:31:25,750

 

990
00:31:25,750 --> 00:31:27,269

 

991
00:31:27,269 --> 00:31:29,269

 

992
00:31:29,269 --> 00:31:31,190

 

993
00:31:31,190 --> 00:31:33,190

 

994
00:31:33,190 --> 00:31:36,070

 

995
00:31:36,070 --> 00:31:38,230

 

996
00:31:38,230 --> 00:31:40,230

 

997
00:31:40,230 --> 00:31:41,269

 

998
00:31:41,269 --> 00:31:43,509

 

999
00:31:43,509 --> 00:31:45,509

 

1000
00:31:45,509 --> 00:31:47,909

 

1001
00:31:47,909 --> 00:31:49,669

 

1002
00:31:49,669 --> 00:31:51,590

 

1003
00:31:51,590 --> 00:31:53,110

 

1004
00:31:53,110 --> 00:31:54,549

 

1005
00:31:54,549 --> 00:31:56,230

 

1006
00:31:56,230 --> 00:31:56,240

 

1007
00:31:56,240 --> 00:31:57,110


1008
00:31:57,110 --> 00:31:57,120

 

1009
00:31:57,120 --> 00:31:59,110


1010
00:31:59,110 --> 00:32:01,430

 

1011
00:32:01,430 --> 00:32:03,669

 

1012
00:32:03,669 --> 00:32:05,990

 

1013
00:32:05,990 --> 00:32:08,070

 

1014
00:32:08,070 --> 00:32:09,830

 

1015
00:32:09,830 --> 00:32:12,389

 

1016
00:32:12,389 --> 00:32:14,549

 

1017
00:32:14,549 --> 00:32:15,830

 

1018
00:32:15,830 --> 00:32:17,750

 

1019
00:32:17,750 --> 00:32:19,350

 

1020
00:32:19,350 --> 00:32:20,950

 

1021
00:32:20,950 --> 00:32:23,830

 

1022
00:32:23,830 --> 00:32:25,430

 

1023
00:32:25,430 --> 00:32:27,669

 

1024
00:32:27,669 --> 00:32:28,389

 

1025
00:32:28,389 --> 00:32:30,230

 

1026
00:32:30,230 --> 00:32:32,149

 

1027
00:32:32,149 --> 00:32:33,990

 

1028
00:32:33,990 --> 00:32:35,509

 

1029
00:32:35,509 --> 00:32:36,950

 

1030
00:32:36,950 --> 00:32:36,960

 

1031
00:32:36,960 --> 00:32:38,470


1032
00:32:38,470 --> 00:32:40,789

 

1033
00:32:40,789 --> 00:32:43,110

 

1034
00:32:43,110 --> 00:32:45,110

 

1035
00:32:45,110 --> 00:32:46,149

 

1036
00:32:46,149 --> 00:32:47,509

 

1037
00:32:47,509 --> 00:32:47,519

 

1038
00:32:47,519 --> 00:32:49,509


1039
00:32:49,509 --> 00:32:50,950

 

1040
00:32:50,950 --> 00:32:54,070

 

1041
00:32:54,070 --> 00:32:56,870

 

1042
00:32:56,870 --> 00:32:59,190

 

1043
00:32:59,190 --> 00:33:02,470

 

1044
00:33:02,470 --> 00:33:04,950

 

1045
00:33:04,950 --> 00:33:04,960

 

1046
00:33:04,960 --> 00:33:06,149


1047
00:33:06,149 --> 00:33:08,630

 

1048
00:33:08,630 --> 00:33:11,590

 

1049
00:33:11,590 --> 00:33:13,430

 

1050
00:33:13,430 --> 00:33:14,789

 

1051
00:33:14,789 --> 00:33:16,710

 

1052
00:33:16,710 --> 00:33:16,720

 

1053
00:33:16,720 --> 00:33:17,750


1054
00:33:17,750 --> 00:33:19,350

 

1055
00:33:19,350 --> 00:33:21,590

 

1056
00:33:21,590 --> 00:33:21,600

 

1057
00:33:21,600 --> 00:33:23,269


1058
00:33:23,269 --> 00:33:24,870

 

1059
00:33:24,870 --> 00:33:27,990

 

1060
00:33:27,990 --> 00:33:29,990

 

1061
00:33:29,990 --> 00:33:32,389

 

1062
00:33:32,389 --> 00:33:34,789

 

1063
00:33:34,789 --> 00:33:37,750

 

1064
00:33:37,750 --> 00:33:39,269

 

1065
00:33:39,269 --> 00:33:42,549

 

1066
00:33:42,549 --> 00:33:43,990

 

1067
00:33:43,990 --> 00:33:45,350

 

1068
00:33:45,350 --> 00:33:47,029

 

1069
00:33:47,029 --> 00:33:50,310

 

1070
00:33:50,310 --> 00:33:50,320

 

1071
00:33:50,320 --> 00:33:51,509


1072
00:33:51,509 --> 00:33:52,549

 

1073
00:33:52,549 --> 00:33:54,549

 

1074
00:33:54,549 --> 00:33:56,389

 

1075
00:33:56,389 --> 00:33:57,350

 

1076
00:33:57,350 --> 00:33:59,750

 

1077
00:33:59,750 --> 00:33:59,760

 

1078
00:33:59,760 --> 00:34:00,630


1079
00:34:00,630 --> 00:34:02,710

 

1080
00:34:02,710 --> 00:34:03,830

 

1081
00:34:03,830 --> 00:34:07,350

 

1082
00:34:07,350 --> 00:34:09,349

 

1083
00:34:09,349 --> 00:34:11,190

 

1084
00:34:11,190 --> 00:34:11,200

 

1085
00:34:11,200 --> 00:34:12,550


1086
00:34:12,550 --> 00:34:14,310

 

1087
00:34:14,310 --> 00:34:16,470

 

1088
00:34:16,470 --> 00:34:18,310

 

1089
00:34:18,310 --> 00:34:20,310

 

1090
00:34:20,310 --> 00:34:20,320

 

1091
00:34:20,320 --> 00:34:21,510


1092
00:34:21,510 --> 00:34:24,069

 

1093
00:34:24,069 --> 00:34:24,079

 

1094
00:34:24,079 --> 00:34:24,950


1095
00:34:24,950 --> 00:34:26,950

 

1096
00:34:26,950 --> 00:34:28,950

 

1097
00:34:28,950 --> 00:34:30,629

 

1098
00:34:30,629 --> 00:34:30,639

 

1099
00:34:30,639 --> 00:34:32,069


1100
00:34:32,069 --> 00:34:32,079

 

1101
00:34:32,079 --> 00:34:32,869


1102
00:34:32,869 --> 00:34:34,550

 

1103
00:34:34,550 --> 00:34:35,750

 

1104
00:34:35,750 --> 00:34:38,310

 

1105
00:34:38,310 --> 00:34:40,869

 

1106
00:34:40,869 --> 00:34:42,869

 

1107
00:34:42,869 --> 00:34:45,030

 

1108
00:34:45,030 --> 00:34:46,629

 

1109
00:34:46,629 --> 00:34:46,639

 

1110
00:34:46,639 --> 00:34:49,270


1111
00:34:49,270 --> 00:34:49,280

 

1112
00:34:49,280 --> 00:34:50,230


1113
00:34:50,230 --> 00:34:52,310

 

1114
00:34:52,310 --> 00:34:54,629

 

1115
00:34:54,629 --> 00:34:56,310

 

1116
00:34:56,310 --> 00:34:58,069

 

1117
00:34:58,069 --> 00:35:01,030

 

1118
00:35:01,030 --> 00:35:02,870

 

1119
00:35:02,870 --> 00:35:04,950

 

1120
00:35:04,950 --> 00:35:06,870

 

1121
00:35:06,870 --> 00:35:09,270

 

1122
00:35:09,270 --> 00:35:11,510

 

1123
00:35:11,510 --> 00:35:13,109

 

1124
00:35:13,109 --> 00:35:14,870

 

1125
00:35:14,870 --> 00:35:17,349

 

1126
00:35:17,349 --> 00:35:19,030

 

1127
00:35:19,030 --> 00:35:20,870

 

1128
00:35:20,870 --> 00:35:23,430

 

1129
00:35:23,430 --> 00:35:25,349

 

1130
00:35:25,349 --> 00:35:27,030

 

1131
00:35:27,030 --> 00:35:30,470

 

1132
00:35:30,470 --> 00:35:33,030

 

1133
00:35:33,030 --> 00:35:35,270

 

1134
00:35:35,270 --> 00:35:37,109

 

1135
00:35:37,109 --> 00:35:38,390

 

1136
00:35:38,390 --> 00:35:39,910

 

1137
00:35:39,910 --> 00:35:42,150

 

1138
00:35:42,150 --> 00:35:43,430

 

1139
00:35:43,430 --> 00:35:45,670

 

1140
00:35:45,670 --> 00:35:47,510

 

1141
00:35:47,510 --> 00:35:49,349

 

1142
00:35:49,349 --> 00:35:51,270

 

1143
00:35:51,270 --> 00:35:53,589

 

1144
00:35:53,589 --> 00:35:57,349

 

1145
00:35:57,349 --> 00:35:59,829

 

1146
00:35:59,829 --> 00:36:02,630

 

1147
00:36:02,630 --> 00:36:05,510

 

1148
00:36:05,510 --> 00:36:07,670

 

1149
00:36:07,670 --> 00:36:10,150

 

1150
00:36:10,150 --> 00:36:12,550

 

1151
00:36:12,550 --> 00:36:15,670

 

1152
00:36:15,670 --> 00:36:15,680

 

1153
00:36:15,680 --> 00:36:17,670


1154
00:36:17,670 --> 00:36:21,270

 

1155
00:36:21,270 --> 00:36:22,310

 

1156
00:36:22,310 --> 00:36:22,320

 

1157
00:36:22,320 --> 00:36:23,109


1158
00:36:23,109 --> 00:36:25,510

 

1159
00:36:25,510 --> 00:36:29,270

 

1160
00:36:29,270 --> 00:36:31,510

 

1161
00:36:31,510 --> 00:36:33,030

 

1162
00:36:33,030 --> 00:36:34,630

 

1163
00:36:34,630 --> 00:36:37,109

 

1164
00:36:37,109 --> 00:36:37,119

 

1165
00:36:37,119 --> 00:36:38,069


1166
00:36:38,069 --> 00:36:40,390

 

1167
00:36:40,390 --> 00:36:42,069

 

1168
00:36:42,069 --> 00:36:44,470

 

1169
00:36:44,470 --> 00:36:46,230

 

1170
00:36:46,230 --> 00:36:48,950

 

1171
00:36:48,950 --> 00:36:52,829

 

1172
00:36:52,829 --> 00:36:55,750

 

1173
00:36:55,750 --> 00:36:57,750

 

1174
00:36:57,750 --> 00:36:57,760

 

1175
00:36:57,760 --> 00:36:58,140


1176
00:36:58,140 --> 00:36:58,150

 

1177
00:36:58,150 --> 00:36:59,589


1178
00:36:59,589 --> 00:36:59,599

 

1179
00:36:59,599 --> 00:37:00,790


1180
00:37:00,790 --> 00:37:03,030

 

1181
00:37:03,030 --> 00:37:05,430

 

1182
00:37:05,430 --> 00:37:07,109

 

1183
00:37:07,109 --> 00:37:08,470

 

1184
00:37:08,470 --> 00:37:10,630

 

1185
00:37:10,630 --> 00:37:12,870

 

1186
00:37:12,870 --> 00:37:16,069

 

1187
00:37:16,069 --> 00:37:17,430

 

1188
00:37:17,430 --> 00:37:19,030

 

1189
00:37:19,030 --> 00:37:20,390

 

1190
00:37:20,390 --> 00:37:22,550

 

1191
00:37:22,550 --> 00:37:24,790

 

1192
00:37:24,790 --> 00:37:26,870

 

1193
00:37:26,870 --> 00:37:29,750

 

1194
00:37:29,750 --> 00:37:31,829

 

1195
00:37:31,829 --> 00:37:32,870

 

1196
00:37:32,870 --> 00:37:35,510

 

1197
00:37:35,510 --> 00:37:37,030

 

1198
00:37:37,030 --> 00:37:39,190

 

1199
00:37:39,190 --> 00:37:41,910

 

1200
00:37:41,910 --> 00:37:41,920

 

1201
00:37:41,920 --> 00:37:43,030


1202
00:37:43,030 --> 00:37:44,550

 

1203
00:37:44,550 --> 00:37:47,990

 

1204
00:37:47,990 --> 00:37:50,310

 

1205
00:37:50,310 --> 00:37:52,790

 

1206
00:37:52,790 --> 00:37:55,190

 

1207
00:37:55,190 --> 00:37:57,030

 

1208
00:37:57,030 --> 00:37:59,030

 

1209
00:37:59,030 --> 00:38:00,550

 

1210
00:38:00,550 --> 00:38:00,560

 

1211
00:38:00,560 --> 00:38:01,750


1212
00:38:01,750 --> 00:38:07,589

 

1213
00:38:07,589 --> 00:38:09,670

 

1214
00:38:09,670 --> 00:38:11,430

 

1215
00:38:11,430 --> 00:38:12,870

 

1216
00:38:12,870 --> 00:38:15,510

 

1217
00:38:15,510 --> 00:38:17,910

 

1218
00:38:17,910 --> 00:38:21,190

 

1219
00:38:21,190 --> 00:38:23,270

 

1220
00:38:23,270 --> 00:38:24,550

 

1221
00:38:24,550 --> 00:38:26,630

 

1222
00:38:26,630 --> 00:38:29,109

 

1223
00:38:29,109 --> 00:38:31,990

 

1224
00:38:31,990 --> 00:38:32,000

 

1225
00:38:32,000 --> 00:38:32,950


1226
00:38:32,950 --> 00:38:34,230

 

1227
00:38:34,230 --> 00:38:36,069

 

1228
00:38:36,069 --> 00:38:38,069

 

1229
00:38:38,069 --> 00:38:38,079

 

1230
00:38:38,079 --> 00:38:39,030


1231
00:38:39,030 --> 00:38:39,040

 

1232
00:38:39,040 --> 00:38:41,349


1233
00:38:41,349 --> 00:38:43,190

 

1234
00:38:43,190 --> 00:38:45,349

 

1235
00:38:45,349 --> 00:38:49,670

 

1236
00:38:49,670 --> 00:38:49,680

 

1237
00:38:49,680 --> 00:38:50,240


1238
00:38:50,240 --> 00:38:50,250

 

1239
00:38:50,250 --> 00:38:51,589


1240
00:38:51,589 --> 00:38:53,910

 

1241
00:38:53,910 --> 00:38:55,270

 

1242
00:38:55,270 --> 00:38:57,589

 

1243
00:38:57,589 --> 00:38:59,750

 

1244
00:38:59,750 --> 00:39:01,510

 

1245
00:39:01,510 --> 00:39:03,510

 

1246
00:39:03,510 --> 00:39:05,349

 

1247
00:39:05,349 --> 00:39:07,829

 

1248
00:39:07,829 --> 00:39:10,150

 

1249
00:39:10,150 --> 00:39:12,150

 

1250
00:39:12,150 --> 00:39:14,150

 

1251
00:39:14,150 --> 00:39:16,390

 

1252
00:39:16,390 --> 00:39:16,400

 

1253
00:39:16,400 --> 00:39:18,069


1254
00:39:18,069 --> 00:39:18,079

 

1255
00:39:18,079 --> 00:39:19,510


1256
00:39:19,510 --> 00:39:21,190

 

1257
00:39:21,190 --> 00:39:24,390

 

1258
00:39:24,390 --> 00:39:27,910

 

1259
00:39:27,910 --> 00:39:30,470

 

1260
00:39:30,470 --> 00:39:32,710

 

1261
00:39:32,710 --> 00:39:34,310

 

1262
00:39:34,310 --> 00:39:35,750

 

1263
00:39:35,750 --> 00:39:37,589

 

1264
00:39:37,589 --> 00:39:37,599

 

1265
00:39:37,599 --> 00:39:38,470


1266
00:39:38,470 --> 00:39:41,589

 

1267
00:39:41,589 --> 00:39:41,599

 

1268
00:39:41,599 --> 00:39:42,470


1269
00:39:42,470 --> 00:39:44,390

 

1270
00:39:44,390 --> 00:39:46,550

 

1271
00:39:46,550 --> 00:39:48,310

 

1272
00:39:48,310 --> 00:39:50,390

 

1273
00:39:50,390 --> 00:39:50,400

 

1274
00:39:50,400 --> 00:39:51,510


1275
00:39:51,510 --> 00:39:54,150

 

1276
00:39:54,150 --> 00:39:57,750

 

1277
00:39:57,750 --> 00:39:59,589

 

1278
00:39:59,589 --> 00:40:02,230

 

1279
00:40:02,230 --> 00:40:05,030

 

1280
00:40:05,030 --> 00:40:06,870

 

1281
00:40:06,870 --> 00:40:08,870

 

1282
00:40:08,870 --> 00:40:11,589

 

1283
00:40:11,589 --> 00:40:13,109

 

1284
00:40:13,109 --> 00:40:15,910

 

1285
00:40:15,910 --> 00:40:18,069

 

1286
00:40:18,069 --> 00:40:19,430

 

1287
00:40:19,430 --> 00:40:19,440

 

1288
00:40:19,440 --> 00:40:22,230


1289
00:40:22,230 --> 00:40:25,190

 

1290
00:40:25,190 --> 00:40:27,829

 

1291
00:40:27,829 --> 00:40:30,230

 

1292
00:40:30,230 --> 00:40:31,750

 

1293
00:40:31,750 --> 00:40:33,589

 

1294
00:40:33,589 --> 00:40:35,430

 

1295
00:40:35,430 --> 00:40:38,470

 

1296
00:40:38,470 --> 00:40:42,710

 

1297
00:40:42,710 --> 00:40:44,870

 

1298
00:40:44,870 --> 00:40:46,390

 

1299
00:40:46,390 --> 00:40:46,400

 

1300
00:40:46,400 --> 00:40:47,349


1301
00:40:47,349 --> 00:40:49,430

 

1302
00:40:49,430 --> 00:40:51,670

 

1303
00:40:51,670 --> 00:40:53,670

 

1304
00:40:53,670 --> 00:40:57,829

 

1305
00:40:57,829 --> 00:40:59,990

 

1306
00:40:59,990 --> 00:41:02,790

 

1307
00:41:02,790 --> 00:41:04,150

 

1308
00:41:04,150 --> 00:41:06,230

 

1309
00:41:06,230 --> 00:41:07,990

 

1310
00:41:07,990 --> 00:41:11,030

 

1311
00:41:11,030 --> 00:41:15,030

 

1312
00:41:15,030 --> 00:41:16,550

 

1313
00:41:16,550 --> 00:41:18,829

 

1314
00:41:18,829 --> 00:41:21,750

 

1315
00:41:21,750 --> 00:41:22,950

 

1316
00:41:22,950 --> 00:41:24,230

 

1317
00:41:24,230 --> 00:41:26,870

 

1318
00:41:26,870 --> 00:41:29,190

 

1319
00:41:29,190 --> 00:41:29,200

 

1320
00:41:29,200 --> 00:41:29,910


1321
00:41:29,910 --> 00:41:31,270

 

1322
00:41:31,270 --> 00:41:34,069

 

1323
00:41:34,069 --> 00:41:34,079

 

1324
00:41:34,079 --> 00:41:34,870


1325
00:41:34,870 --> 00:41:34,880

 

1326
00:41:34,880 --> 00:41:36,870


1327
00:41:36,870 --> 00:41:38,710

 

1328
00:41:38,710 --> 00:41:40,069

 

1329
00:41:40,069 --> 00:41:40,079

 

1330
00:41:40,079 --> 00:41:41,670


1331
00:41:41,670 --> 00:41:44,309

 

1332
00:41:44,309 --> 00:41:45,990

 

1333
00:41:45,990 --> 00:41:48,390

 

1334
00:41:48,390 --> 00:41:49,750

 

1335
00:41:49,750 --> 00:41:52,069

 

1336
00:41:52,069 --> 00:41:52,079

 

1337
00:41:52,079 --> 00:41:54,309


1338
00:41:54,309 --> 00:41:56,150

 

1339
00:41:56,150 --> 00:41:58,870

 

1340
00:41:58,870 --> 00:41:58,880

 

1341
00:41:58,880 --> 00:41:59,750


1342
00:41:59,750 --> 00:42:02,150

 

1343
00:42:02,150 --> 00:42:04,309

 

1344
00:42:04,309 --> 00:42:06,470

 

1345
00:42:06,470 --> 00:42:10,309

 

1346
00:42:10,309 --> 00:42:11,990

 

1347
00:42:11,990 --> 00:42:13,910

 

1348
00:42:13,910 --> 00:42:16,550

 

1349
00:42:16,550 --> 00:42:19,030

 

1350
00:42:19,030 --> 00:42:20,950

 

1351
00:42:20,950 --> 00:42:20,960

 

1352
00:42:20,960 --> 00:42:22,710


1353
00:42:22,710 --> 00:42:26,630

 

1354
00:42:26,630 --> 00:42:26,640

 

1355
00:42:26,640 --> 00:42:29,030


1356
00:42:29,030 --> 00:42:29,040

 

1357
00:42:29,040 --> 00:42:30,550


1358
00:42:30,550 --> 00:42:32,390

 

1359
00:42:32,390 --> 00:42:34,630

 

1360
00:42:34,630 --> 00:42:36,710

 

1361
00:42:36,710 --> 00:42:38,710

 

1362
00:42:38,710 --> 00:42:41,109

 

1363
00:42:41,109 --> 00:42:43,270

 

1364
00:42:43,270 --> 00:42:44,790

 

1365
00:42:44,790 --> 00:42:47,190

 

1366
00:42:47,190 --> 00:42:48,710

 

1367
00:42:48,710 --> 00:42:50,069

 

1368
00:42:50,069 --> 00:42:52,150

 

1369
00:42:52,150 --> 00:42:53,990

 

1370
00:42:53,990 --> 00:42:55,430

 

1371
00:42:55,430 --> 00:42:55,440

 

1372
00:42:55,440 --> 00:42:56,230


1373
00:42:56,230 --> 00:42:58,069

 

1374
00:42:58,069 --> 00:43:00,309

 

1375
00:43:00,309 --> 00:43:02,390

 

1376
00:43:02,390 --> 00:43:04,230

 

1377
00:43:04,230 --> 00:43:04,240

 

1378
00:43:04,240 --> 00:43:07,109


1379
00:43:07,109 --> 00:43:09,190

 

1380
00:43:09,190 --> 00:43:11,589

 

1381
00:43:11,589 --> 00:43:13,109

 

1382
00:43:13,109 --> 00:43:16,470

 

1383
00:43:16,470 --> 00:43:18,309

 

1384
00:43:18,309 --> 00:43:19,990

 

1385
00:43:19,990 --> 00:43:20,000

 

1386
00:43:20,000 --> 00:43:21,030


1387
00:43:21,030 --> 00:43:23,109

 

1388
00:43:23,109 --> 00:43:24,870

 

1389
00:43:24,870 --> 00:43:26,630

 

1390
00:43:26,630 --> 00:43:26,640

 

1391
00:43:26,640 --> 00:43:28,069


1392
00:43:28,069 --> 00:43:30,630

 

1393
00:43:30,630 --> 00:43:32,630

 

1394
00:43:32,630 --> 00:43:34,069

 

1395
00:43:34,069 --> 00:43:35,349

 

1396
00:43:35,349 --> 00:43:37,589

 

1397
00:43:37,589 --> 00:43:40,069

 

1398
00:43:40,069 --> 00:43:41,910

 

1399
00:43:41,910 --> 00:43:44,390

 

1400
00:43:44,390 --> 00:43:45,829

 

1401
00:43:45,829 --> 00:43:48,230

 

1402
00:43:48,230 --> 00:43:49,190

 

1403
00:43:49,190 --> 00:43:52,950

 

1404
00:43:52,950 --> 00:43:54,550

 

1405
00:43:54,550 --> 00:43:57,829

 

1406
00:43:57,829 --> 00:43:59,990

 

1407
00:43:59,990 --> 00:44:00,000

 

1408
00:44:00,000 --> 00:44:00,870


1409
00:44:00,870 --> 00:44:04,630

 

1410
00:44:04,630 --> 00:44:07,910

 

1411
00:44:07,910 --> 00:44:09,910

 

1412
00:44:09,910 --> 00:44:11,589

 

1413
00:44:11,589 --> 00:44:13,270

 

1414
00:44:13,270 --> 00:44:15,270

 

1415
00:44:15,270 --> 00:44:17,670

 

1416
00:44:17,670 --> 00:44:20,710

 

1417
00:44:20,710 --> 00:44:23,270

 

1418
00:44:23,270 --> 00:44:25,270

 

1419
00:44:25,270 --> 00:44:25,280

 

1420
00:44:25,280 --> 00:44:26,470


1421
00:44:26,470 --> 00:44:29,750

 

1422
00:44:29,750 --> 00:44:32,950

 

1423
00:44:32,950 --> 00:44:32,960

 

1424
00:44:32,960 --> 00:44:33,829


1425
00:44:33,829 --> 00:44:35,510

 

1426
00:44:35,510 --> 00:44:37,670

 

1427
00:44:37,670 --> 00:44:39,750

 

1428
00:44:39,750 --> 00:44:41,829

 

1429
00:44:41,829 --> 00:44:43,270

 

1430
00:44:43,270 --> 00:44:44,790

 

1431
00:44:44,790 --> 00:44:47,270

 

1432
00:44:47,270 --> 00:44:50,309

 

1433
00:44:50,309 --> 00:44:51,589

 

1434
00:44:51,589 --> 00:44:51,599

 

1435
00:44:51,599 --> 00:44:52,710


1436
00:44:52,710 --> 00:44:55,030

 

1437
00:44:55,030 --> 00:44:55,040

 

1438
00:44:55,040 --> 00:44:56,390


1439
00:44:56,390 --> 00:44:58,150

 

1440
00:44:58,150 --> 00:44:59,829

 

1441
00:44:59,829 --> 00:45:01,349

 

1442
00:45:01,349 --> 00:45:01,359

 

1443
00:45:01,359 --> 00:45:02,309


1444
00:45:02,309 --> 00:45:05,030

 

1445
00:45:05,030 --> 00:45:06,630

 

1446
00:45:06,630 --> 00:45:08,710

 

1447
00:45:08,710 --> 00:45:10,309

 

1448
00:45:10,309 --> 00:45:12,470

 

1449
00:45:12,470 --> 00:45:12,480

 

1450
00:45:12,480 --> 00:45:13,829


1451
00:45:13,829 --> 00:45:14,950

 

1452
00:45:14,950 --> 00:45:16,870

 

1453
00:45:16,870 --> 00:45:18,630

 

1454
00:45:18,630 --> 00:45:20,870

 

1455
00:45:20,870 --> 00:45:23,990

 

1456
00:45:23,990 --> 00:45:25,750

 

1457
00:45:25,750 --> 00:45:27,990

 

1458
00:45:27,990 --> 00:45:29,750

 

1459
00:45:29,750 --> 00:45:31,270

 

1460
00:45:31,270 --> 00:45:33,829

 

1461
00:45:33,829 --> 00:45:36,390

 

1462
00:45:36,390 --> 00:45:38,390

 

1463
00:45:38,390 --> 00:45:40,710

 

1464
00:45:40,710 --> 00:45:42,309

 

1465
00:45:42,309 --> 00:45:44,710

 

1466
00:45:44,710 --> 00:45:47,270

 

1467
00:45:47,270 --> 00:45:49,589

 

1468
00:45:49,589 --> 00:45:52,230

 

1469
00:45:52,230 --> 00:45:54,870

 

1470
00:45:54,870 --> 00:45:57,190

 

1471
00:45:57,190 --> 00:45:58,870

 

1472
00:45:58,870 --> 00:46:00,870

 

1473
00:46:00,870 --> 00:46:02,870

 

1474
00:46:02,870 --> 00:46:04,790

 

1475
00:46:04,790 --> 00:46:06,630

 

1476
00:46:06,630 --> 00:46:09,990

 

1477
00:46:09,990 --> 00:46:12,630

 

1478
00:46:12,630 --> 00:46:14,470

 

1479
00:46:14,470 --> 00:46:15,750

 

1480
00:46:15,750 --> 00:46:18,710

 

1481
00:46:18,710 --> 00:46:21,349

 

1482
00:46:21,349 --> 00:46:23,430

 

1483
00:46:23,430 --> 00:46:25,589

 

1484
00:46:25,589 --> 00:46:28,309

 

1485
00:46:28,309 --> 00:46:30,069

 

1486
00:46:30,069 --> 00:46:32,230

 

1487
00:46:32,230 --> 00:46:33,829

 

1488
00:46:33,829 --> 00:46:36,150

 

1489
00:46:36,150 --> 00:46:38,069

 

1490
00:46:38,069 --> 00:46:39,990

 

1491
00:46:39,990 --> 00:46:41,589

 

1492
00:46:41,589 --> 00:46:44,069

 

1493
00:46:44,069 --> 00:46:45,990

 

1494
00:46:45,990 --> 00:46:47,589

 

1495
00:46:47,589 --> 00:46:48,790

 

1496
00:46:48,790 --> 00:46:51,190

 

1497
00:46:51,190 --> 00:46:53,670

 

1498
00:46:53,670 --> 00:46:55,910

 

1499
00:46:55,910 --> 00:46:58,150

 

1500
00:46:58,150 --> 00:47:00,390

 

1501
00:47:00,390 --> 00:47:02,470

 

1502
00:47:02,470 --> 00:47:04,470

 

1503
00:47:04,470 --> 00:47:07,190

 

1504
00:47:07,190 --> 00:47:09,510

 

1505
00:47:09,510 --> 00:47:12,470

 

1506
00:47:12,470 --> 00:47:15,190

 

1507
00:47:15,190 --> 00:47:18,150

 

1508
00:47:18,150 --> 00:47:20,870

 

1509
00:47:20,870 --> 00:47:20,880

 

1510
00:47:20,880 --> 00:47:22,790


1511
00:47:22,790 --> 00:47:24,470

 

1512
00:47:24,470 --> 00:47:26,150

 

1513
00:47:26,150 --> 00:47:27,510

 

1514
00:47:27,510 --> 00:47:29,030

 

1515
00:47:29,030 --> 00:47:29,040

 

1516
00:47:29,040 --> 00:47:30,390


1517
00:47:30,390 --> 00:47:32,950

 

1518
00:47:32,950 --> 00:47:36,790

 

1519
00:47:36,790 --> 00:47:38,710

 

1520
00:47:38,710 --> 00:47:38,720

 

1521
00:47:38,720 --> 00:47:41,990


1522
00:47:41,990 --> 00:47:43,349

 

1523
00:47:43,349 --> 00:47:45,910

 

1524
00:47:45,910 --> 00:47:47,910

 

1525
00:47:47,910 --> 00:47:51,190

 

1526
00:47:51,190 --> 00:47:53,190

 

1527
00:47:53,190 --> 00:47:53,200

 

1528
00:47:53,200 --> 00:47:54,390


1529
00:47:54,390 --> 00:47:56,950

 

1530
00:47:56,950 --> 00:47:59,109

 

1531
00:47:59,109 --> 00:48:01,190

 

1532
00:48:01,190 --> 00:48:03,589

 

1533
00:48:03,589 --> 00:48:05,589

 

1534
00:48:05,589 --> 00:48:06,950

 

1535
00:48:06,950 --> 00:48:08,630

 

1536
00:48:08,630 --> 00:48:10,630

 

1537
00:48:10,630 --> 00:48:13,190

 

1538
00:48:13,190 --> 00:48:14,309

 

1539
00:48:14,309 --> 00:48:16,309

 

1540
00:48:16,309 --> 00:48:17,510

 

1541
00:48:17,510 --> 00:48:20,870

 

1542
00:48:20,870 --> 00:48:22,950

 

1543
00:48:22,950 --> 00:48:24,950

 

1544
00:48:24,950 --> 00:48:27,109

 

1545
00:48:27,109 --> 00:48:28,710

 

1546
00:48:28,710 --> 00:48:30,069

 

1547
00:48:30,069 --> 00:48:31,829

 

1548
00:48:31,829 --> 00:48:34,870

 

1549
00:48:34,870 --> 00:48:34,880

 

1550
00:48:34,880 --> 00:48:37,349


1551
00:48:37,349 --> 00:48:37,359

 

1552
00:48:37,359 --> 00:48:38,309


1553
00:48:38,309 --> 00:48:38,319

 

1554
00:48:38,319 --> 00:48:39,109


1555
00:48:39,109 --> 00:48:41,430

 

1556
00:48:41,430 --> 00:48:43,510

 

1557
00:48:43,510 --> 00:48:45,910

 

1558
00:48:45,910 --> 00:48:48,069

 

1559
00:48:48,069 --> 00:48:48,079

 

1560
00:48:48,079 --> 00:48:49,030


1561
00:48:49,030 --> 00:48:49,040

 

1562
00:48:49,040 --> 00:48:49,829


1563
00:48:49,829 --> 00:48:49,839

 

1564
00:48:49,839 --> 00:48:50,549


1565
00:48:50,549 --> 00:48:52,870

 

1566
00:48:52,870 --> 00:48:55,349

 

1567
00:48:55,349 --> 00:48:58,390

 

1568
00:48:58,390 --> 00:48:58,400

 

1569
00:48:58,400 --> 00:48:59,349


1570
00:48:59,349 --> 00:49:00,470

 

1571
00:49:00,470 --> 00:49:01,990

 

1572
00:49:01,990 --> 00:49:04,309

 

1573
00:49:04,309 --> 00:49:06,230

 

1574
00:49:06,230 --> 00:49:08,630

 

1575
00:49:08,630 --> 00:49:08,640

 

1576
00:49:08,640 --> 00:49:11,270


1577
00:49:11,270 --> 00:49:13,829

 

1578
00:49:13,829 --> 00:49:17,510

 

1579
00:49:17,510 --> 00:49:20,549

 

1580
00:49:20,549 --> 00:49:24,309

 

1581
00:49:24,309 --> 00:49:26,870

 

1582
00:49:26,870 --> 00:49:28,710

 

1583
00:49:28,710 --> 00:49:31,030

 

1584
00:49:31,030 --> 00:49:32,630

 

1585
00:49:32,630 --> 00:49:34,309

 

1586
00:49:34,309 --> 00:49:37,030

 

1587
00:49:37,030 --> 00:49:39,430

 

1588
00:49:39,430 --> 00:49:43,030

 

1589
00:49:43,030 --> 00:49:44,549

 

1590
00:49:44,549 --> 00:49:44,559

 

1591
00:49:44,559 --> 00:49:45,430


1592
00:49:45,430 --> 00:49:46,790

 

1593
00:49:46,790 --> 00:49:50,470

 

1594
00:49:50,470 --> 00:49:52,470

 

1595
00:49:52,470 --> 00:49:54,710

 

1596
00:49:54,710 --> 00:49:54,720

 

1597
00:49:54,720 --> 00:49:58,309


1598
00:49:58,309 --> 00:50:00,549

 

1599
00:50:00,549 --> 00:50:03,670

 

1600
00:50:03,670 --> 00:50:06,870

 

1601
00:50:06,870 --> 00:50:09,270

 

1602
00:50:09,270 --> 00:50:11,349

 

1603
00:50:11,349 --> 00:50:16,150

 

1604
00:50:16,150 --> 00:50:17,430

 

1605
00:50:17,430 --> 00:50:19,910

 

1606
00:50:19,910 --> 00:50:21,270

 

1607
00:50:21,270 --> 00:50:24,309

 

1608
00:50:24,309 --> 00:50:26,230

 

1609
00:50:26,230 --> 00:50:29,109

 

1610
00:50:29,109 --> 00:50:30,710

 

1611
00:50:30,710 --> 00:50:33,109

 

1612
00:50:33,109 --> 00:50:34,710

 

1613
00:50:34,710 --> 00:50:36,150

 

1614
00:50:36,150 --> 00:50:38,150

 

1615
00:50:38,150 --> 00:50:40,710

 

1616
00:50:40,710 --> 00:50:43,349

 

1617
00:50:43,349 --> 00:50:45,430

 

1618
00:50:45,430 --> 00:50:47,670

 

1619
00:50:47,670 --> 00:50:48,790

 

1620
00:50:48,790 --> 00:50:51,109

 

1621
00:50:51,109 --> 00:50:53,990

 

1622
00:50:53,990 --> 00:50:55,990

 

1623
00:50:55,990 --> 00:50:56,000

 

1624
00:50:56,000 --> 00:50:57,270


1625
00:50:57,270 --> 00:51:00,069

 

1626
00:51:00,069 --> 00:51:03,349

 

1627
00:51:03,349 --> 00:51:06,150

 

1628
00:51:06,150 --> 00:51:08,870

 

1629
00:51:08,870 --> 00:51:10,630

 

1630
00:51:10,630 --> 00:51:11,589

 

1631
00:51:11,589 --> 00:51:13,670

 

1632
00:51:13,670 --> 00:51:13,680

 

1633
00:51:13,680 --> 00:51:15,270


1634
00:51:15,270 --> 00:51:16,870

 

1635
00:51:16,870 --> 00:51:16,880

 

1636
00:51:16,880 --> 00:51:17,380


1637
00:51:17,380 --> 00:51:17,390

 

1638
00:51:17,390 --> 00:51:19,430


1639
00:51:19,430 --> 00:51:21,190

 

1640
00:51:21,190 --> 00:51:22,950

 

1641
00:51:22,950 --> 00:51:24,790

 

1642
00:51:24,790 --> 00:51:26,230

 

1643
00:51:26,230 --> 00:51:26,240

 

1644
00:51:26,240 --> 00:51:27,589


1645
00:51:27,589 --> 00:51:30,069

 

1646
00:51:30,069 --> 00:51:30,079

 

1647
00:51:30,079 --> 00:51:30,870


1648
00:51:30,870 --> 00:51:34,549

 

1649
00:51:34,549 --> 00:51:37,030

 

1650
00:51:37,030 --> 00:51:39,270

 

1651
00:51:39,270 --> 00:51:40,790

 

1652
00:51:40,790 --> 00:51:42,150

 

1653
00:51:42,150 --> 00:51:43,670

 

1654
00:51:43,670 --> 00:51:46,390

 

1655
00:51:46,390 --> 00:51:47,990

 

1656
00:51:47,990 --> 00:51:50,390

 

1657
00:51:50,390 --> 00:51:52,870

 

1658
00:51:52,870 --> 00:51:55,670

 

1659
00:51:55,670 --> 00:51:58,870

 

1660
00:51:58,870 --> 00:52:01,349

 

1661
00:52:01,349 --> 00:52:03,270

 

1662
00:52:03,270 --> 00:52:04,549

 

1663
00:52:04,549 --> 00:52:07,430

 

1664
00:52:07,430 --> 00:52:10,390

 

1665
00:52:10,390 --> 00:52:12,630

 

1666
00:52:12,630 --> 00:52:12,640

 

1667
00:52:12,640 --> 00:52:13,829


1668
00:52:13,829 --> 00:52:15,589

 

1669
00:52:15,589 --> 00:52:17,750

 

1670
00:52:17,750 --> 00:52:20,150

 

1671
00:52:20,150 --> 00:52:21,670

 

1672
00:52:21,670 --> 00:52:23,349

 

1673
00:52:23,349 --> 00:52:25,910

 

1674
00:52:25,910 --> 00:52:25,920

 

1675
00:52:25,920 --> 00:52:28,470


1676
00:52:28,470 --> 00:52:30,470

 

1677
00:52:30,470 --> 00:52:33,109

 

1678
00:52:33,109 --> 00:52:35,750

 

1679
00:52:35,750 --> 00:52:38,309

 

1680
00:52:38,309 --> 00:52:40,230

 

1681
00:52:40,230 --> 00:52:44,390

 

1682
00:52:44,390 --> 00:52:44,400

 

1683
00:52:44,400 --> 00:52:45,270


1684
00:52:45,270 --> 00:52:47,910

 

1685
00:52:47,910 --> 00:52:49,910

 

1686
00:52:49,910 --> 00:52:49,920

 

1687
00:52:49,920 --> 00:52:50,630


1688
00:52:50,630 --> 00:52:52,710

 

1689
00:52:52,710 --> 00:52:54,470

 

1690
00:52:54,470 --> 00:52:57,109

 

1691
00:52:57,109 --> 00:52:58,470

 

1692
00:52:58,470 --> 00:53:01,349

 

1693
00:53:01,349 --> 00:53:03,109

 

1694
00:53:03,109 --> 00:53:05,190

 

1695
00:53:05,190 --> 00:53:07,109

 

1696
00:53:07,109 --> 00:53:09,190

 

1697
00:53:09,190 --> 00:53:09,200

 

1698
00:53:09,200 --> 00:53:12,230


1699
00:53:12,230 --> 00:53:15,030

 

1700
00:53:15,030 --> 00:53:15,040

 

1701
00:53:15,040 --> 00:53:16,309


1702
00:53:16,309 --> 00:53:18,470

 

1703
00:53:18,470 --> 00:53:19,990

 

1704
00:53:19,990 --> 00:53:21,109

 

1705
00:53:21,109 --> 00:53:21,119

 

1706
00:53:21,119 --> 00:53:22,549


1707
00:53:22,549 --> 00:53:25,030

 

1708
00:53:25,030 --> 00:53:28,069

 

1709
00:53:28,069 --> 00:53:30,309

 

1710
00:53:30,309 --> 00:53:31,910

 

1711
00:53:31,910 --> 00:53:33,829

 

1712
00:53:33,829 --> 00:53:34,950

 

1713
00:53:34,950 --> 00:53:36,950

 

1714
00:53:36,950 --> 00:53:38,710

 

1715
00:53:38,710 --> 00:53:40,309

 

1716
00:53:40,309 --> 00:53:41,829

 

1717
00:53:41,829 --> 00:53:44,150

 

1718
00:53:44,150 --> 00:53:47,670

 

1719
00:53:47,670 --> 00:53:47,680

 

1720
00:53:47,680 --> 00:53:48,630


1721
00:53:48,630 --> 00:53:51,349

 

1722
00:53:51,349 --> 00:53:53,349

 

1723
00:53:53,349 --> 00:53:54,870

 

1724
00:53:54,870 --> 00:53:56,470

 

1725
00:53:56,470 --> 00:53:57,829

 

1726
00:53:57,829 --> 00:53:59,670

 

1727
00:53:59,670 --> 00:54:01,190

 

1728
00:54:01,190 --> 00:54:03,030

 

1729
00:54:03,030 --> 00:54:03,040

 

1730
00:54:03,040 --> 00:54:04,470


1731
00:54:04,470 --> 00:54:06,870

 

1732
00:54:06,870 --> 00:54:08,390

 

1733
00:54:08,390 --> 00:54:10,549

 

1734
00:54:10,549 --> 00:54:10,559

 

1735
00:54:10,559 --> 00:54:11,589


1736
00:54:11,589 --> 00:54:13,829

 

1737
00:54:13,829 --> 00:54:17,510

 

1738
00:54:17,510 --> 00:54:19,349

 

1739
00:54:19,349 --> 00:54:22,549

 

1740
00:54:22,549 --> 00:54:24,309

 

1741
00:54:24,309 --> 00:54:26,150

 

1742
00:54:26,150 --> 00:54:28,950

 

1743
00:54:28,950 --> 00:54:32,390

 

1744
00:54:32,390 --> 00:54:35,349

 

1745
00:54:35,349 --> 00:54:35,359

 

1746
00:54:35,359 --> 00:54:35,770


1747
00:54:35,770 --> 00:54:35,780

 

1748
00:54:35,780 --> 00:54:37,589


1749
00:54:37,589 --> 00:54:39,910

 

1750
00:54:39,910 --> 00:54:41,510

 

1751
00:54:41,510 --> 00:54:44,470

 

1752
00:54:44,470 --> 00:54:44,480

 

1753
00:54:44,480 --> 00:54:45,510


1754
00:54:45,510 --> 00:54:47,349

 

1755
00:54:47,349 --> 00:54:47,359

 

1756
00:54:47,359 --> 00:54:49,510


1757
00:54:49,510 --> 00:54:52,069

 

1758
00:54:52,069 --> 00:54:53,510

 

1759
00:54:53,510 --> 00:54:55,670

 

1760
00:54:55,670 --> 00:54:55,680

 

1761
00:54:55,680 --> 00:54:56,280


1762
00:54:56,280 --> 00:54:56,290

 

1763
00:54:56,290 --> 00:54:58,150


1764
00:54:58,150 --> 00:55:01,030

 

1765
00:55:01,030 --> 00:55:03,750

 

1766
00:55:03,750 --> 00:55:08,549

 

1767
00:55:08,549 --> 00:55:10,470

 

1768
00:55:10,470 --> 00:55:17,190

 

1769
00:55:17,190 --> 00:55:18,950

 

1770
00:55:18,950 --> 00:55:21,510

 

1771
00:55:21,510 --> 00:55:21,520

 

1772
00:55:21,520 --> 00:55:22,870


1773
00:55:22,870 --> 00:55:22,880

 

1774
00:55:22,880 --> 00:55:23,990


1775
00:55:23,990 --> 00:55:24,000

 

1776
00:55:24,000 --> 00:55:25,910


1777
00:55:25,910 --> 00:55:29,109

 

1778
00:55:29,109 --> 00:55:31,109

 

1779
00:55:31,109 --> 00:55:31,119

 

1780
00:55:31,119 --> 00:55:31,910


1781
00:55:31,910 --> 00:55:33,430

 

1782
00:55:33,430 --> 00:55:35,430

 

1783
00:55:35,430 --> 00:55:38,230

 

1784
00:55:38,230 --> 00:55:39,990

 

1785
00:55:39,990 --> 00:55:41,349

 

1786
00:55:41,349 --> 00:55:42,789

 

1787
00:55:42,789 --> 00:55:45,349

 

1788
00:55:45,349 --> 00:55:47,430

 

1789
00:55:47,430 --> 00:55:49,670

 

1790
00:55:49,670 --> 00:55:51,589

 

1791
00:55:51,589 --> 00:55:52,789

 

1792
00:55:52,789 --> 00:55:54,950

 

1793
00:55:54,950 --> 00:55:57,430

 

1794
00:55:57,430 --> 00:55:58,309

 

1795
00:55:58,309 --> 00:55:59,589

 

1796
00:55:59,589 --> 00:56:04,069

 

1797
00:56:04,069 --> 00:56:05,910

 

1798
00:56:05,910 --> 00:56:08,470

 

1799
00:56:08,470 --> 00:56:08,480

 

1800
00:56:08,480 --> 00:56:11,510


1801
00:56:11,510 --> 00:56:13,670

 

1802
00:56:13,670 --> 00:56:17,190

 

1803
00:56:17,190 --> 00:56:19,109

 

1804
00:56:19,109 --> 00:56:20,789

 

1805
00:56:20,789 --> 00:56:23,589

 

1806
00:56:23,589 --> 00:56:25,430

 

1807
00:56:25,430 --> 00:56:26,950

 

1808
00:56:26,950 --> 00:56:29,510

 

1809
00:56:29,510 --> 00:56:31,190

 

1810
00:56:31,190 --> 00:56:33,349

 

1811
00:56:33,349 --> 00:56:35,190

 

1812
00:56:35,190 --> 00:56:36,330

 

1813
00:56:36,330 --> 00:56:36,340

 

1814
00:56:36,340 --> 00:56:38,150


1815
00:56:38,150 --> 00:56:40,630

 

1816
00:56:40,630 --> 00:56:40,640

 

1817
00:56:40,640 --> 00:56:41,829


1818
00:56:41,829 --> 00:56:43,510

 

1819
00:56:43,510 --> 00:56:45,510

 

1820
00:56:45,510 --> 00:56:47,750

 

1821
00:56:47,750 --> 00:56:49,270

 

1822
00:56:49,270 --> 00:56:49,280

 

1823
00:56:49,280 --> 00:56:50,390


1824
00:56:50,390 --> 00:56:51,829

 

1825
00:56:51,829 --> 00:56:54,069

 

1826
00:56:54,069 --> 00:56:55,510

 

1827
00:56:55,510 --> 00:56:55,520

 

1828
00:56:55,520 --> 00:56:56,309


1829
00:56:56,309 --> 00:56:58,390

 

1830
00:56:58,390 --> 00:57:00,309

 

1831
00:57:00,309 --> 00:57:02,870

 

1832
00:57:02,870 --> 00:57:05,270

 

1833
00:57:05,270 --> 00:57:07,190

 

1834
00:57:07,190 --> 00:57:08,950

 

1835
00:57:08,950 --> 00:57:11,829

 

1836
00:57:11,829 --> 00:57:14,710

 

1837
00:57:14,710 --> 00:57:16,710

 

1838
00:57:16,710 --> 00:57:18,710

 

1839
00:57:18,710 --> 00:57:18,720

 

1840
00:57:18,720 --> 00:57:20,390


1841
00:57:20,390 --> 00:57:23,109

 

1842
00:57:23,109 --> 00:57:26,390

 

1843
00:57:26,390 --> 00:57:28,390

 

1844
00:57:28,390 --> 00:57:30,230

 

1845
00:57:30,230 --> 00:57:30,240

 

1846
00:57:30,240 --> 00:57:32,549


1847
00:57:32,549 --> 00:57:34,470

 

1848
00:57:34,470 --> 00:57:37,349

 

1849
00:57:37,349 --> 00:57:37,359

 

1850
00:57:37,359 --> 00:57:38,150


1851
00:57:38,150 --> 00:57:39,349

 

1852
00:57:39,349 --> 00:57:41,589

 

1853
00:57:41,589 --> 00:57:43,510

 

1854
00:57:43,510 --> 00:57:43,520

 

1855
00:57:43,520 --> 00:57:44,549


1856
00:57:44,549 --> 00:57:46,870

 

1857
00:57:46,870 --> 00:57:47,990

 

1858
00:57:47,990 --> 00:57:49,750

 

1859
00:57:49,750 --> 00:57:51,349

 

1860
00:57:51,349 --> 00:57:54,390

 

1861
00:57:54,390 --> 00:57:56,789

 

1862
00:57:56,789 --> 00:58:00,309

 

1863
00:58:00,309 --> 00:58:00,319

 

1864
00:58:00,319 --> 00:58:01,589


1865
00:58:01,589 --> 00:58:06,230

 

1866
00:58:06,230 --> 00:58:09,910

 

1867
00:58:09,910 --> 00:58:09,920

 

1868
00:58:09,920 --> 00:58:11,109


1869
00:58:11,109 --> 00:58:13,510

 

1870
00:58:13,510 --> 00:58:15,990

 

1871
00:58:15,990 --> 00:58:18,150

 

1872
00:58:18,150 --> 00:58:20,069

 

1873
00:58:20,069 --> 00:58:21,670

 

1874
00:58:21,670 --> 00:58:23,190

 

1875
00:58:23,190 --> 00:58:25,990

 

1876
00:58:25,990 --> 00:58:28,309

 

1877
00:58:28,309 --> 00:58:30,950

 

1878
00:58:30,950 --> 00:58:35,190

 

1879
00:58:35,190 --> 00:58:37,349

 

1880
00:58:37,349 --> 00:58:38,470

 

1881
00:58:38,470 --> 00:58:42,230

 

1882
00:58:42,230 --> 00:58:44,789

 

1883
00:58:44,789 --> 00:58:47,430

 

1884
00:58:47,430 --> 00:58:47,440

 

1885
00:58:47,440 --> 00:58:48,630


1886
00:58:48,630 --> 00:58:50,470

 

1887
00:58:50,470 --> 00:58:52,069

 

1888
00:58:52,069 --> 00:58:53,510

 

1889
00:58:53,510 --> 00:58:53,520

 

1890
00:58:53,520 --> 00:58:54,950


1891
00:58:54,950 --> 00:58:58,390

 

1892
00:58:58,390 --> 00:59:01,430

 

1893
00:59:01,430 --> 00:59:01,440

 

1894
00:59:01,440 --> 00:59:03,829


1895
00:59:03,829 --> 00:59:06,630

 

1896
00:59:06,630 --> 00:59:07,910

 

1897
00:59:07,910 --> 00:59:09,589

 

1898
00:59:09,589 --> 00:59:12,710

 

1899
00:59:12,710 --> 00:59:14,390

 

1900
00:59:14,390 --> 00:59:17,109

 

1901
00:59:17,109 --> 00:59:19,829

 

1902
00:59:19,829 --> 00:59:22,630

 

1903
00:59:22,630 --> 00:59:25,190

 

1904
00:59:25,190 --> 00:59:27,910

 

1905
00:59:27,910 --> 00:59:29,670

 

1906
00:59:29,670 --> 00:59:32,710

 

1907
00:59:32,710 --> 00:59:34,309

 

1908
00:59:34,309 --> 00:59:35,990

 

1909
00:59:35,990 --> 00:59:37,510

 

1910
00:59:37,510 --> 00:59:40,390

 

1911
00:59:40,390 --> 00:59:42,789

 

1912
00:59:42,789 --> 00:59:45,750

 

1913
00:59:45,750 --> 00:59:47,510

 

1914
00:59:47,510 --> 00:59:50,470

 

1915
00:59:50,470 --> 00:59:52,870

 

1916
00:59:52,870 --> 00:59:52,880

 

1917
00:59:52,880 --> 00:59:54,230


1918
00:59:54,230 --> 00:59:55,430

 

1919
00:59:55,430 --> 00:59:57,750

 

1920
00:59:57,750 --> 00:59:59,750

 

1921
00:59:59,750 --> 01:00:01,430

 

1922
01:00:01,430 --> 01:00:03,109

 

1923
01:00:03,109 --> 01:00:03,119

 

1924
01:00:03,119 --> 01:00:04,069


1925
01:00:04,069 --> 01:00:06,390

 

1926
01:00:06,390 --> 01:00:06,400

 

1927
01:00:06,400 --> 01:00:07,829


1928
01:00:07,829 --> 01:00:10,069

 

1929
01:00:10,069 --> 01:00:12,549

 

1930
01:00:12,549 --> 01:00:14,470

 

1931
01:00:14,470 --> 01:00:17,270

 

1932
01:00:17,270 --> 01:00:18,950

 

1933
01:00:18,950 --> 01:00:20,789

 

1934
01:00:20,789 --> 01:00:22,470

 

1935
01:00:22,470 --> 01:00:33,430

 

1936
01:00:33,430 --> 01:00:33,440

 

1937
01:00:33,440 --> 01:00:35,520