transcripts/3PzuSPQPEU4.json

[{"text": "[SQUEAKING]\n[RUSTLING] [CLICKING]", "start": 0.0, "duration": 5.434}, {"text": "PROFESSOR: OK, why\ndon't I get started?", "start": 25.51, "duration": 2.59}, {"text": "So OK, what have we been doing?", "start": 28.1, "duration": 6.68}, {"text": "So last time, we\nconsidered a bunch", "start": 34.78, "duration": 2.16}, {"text": "of procedures for testing\nproperties of various automata", "start": 36.94, "duration": 4.77}, {"text": "and grammars, the acceptance\nproblem for DFAs, for NFAs,", "start": 41.71, "duration": 8.75}, {"text": "the acceptance problem,\nwhich is really", "start": 50.46, "duration": 2.04}, {"text": "degeneration problem for\ncontext-free grammars,", "start": 52.5, "duration": 3.03}, {"text": "and emptiness problems for\nDFAs and context-free grammars.", "start": 55.53, "duration": 4.89}, {"text": "And also, we showed that\nA,TM is Turing-recognizable.", "start": 60.42, "duration": 6.138}, {"text": "There's a question here in\nthe chat already about that.", "start": 66.558, "duration": 2.292}, {"text": "Yes, we did show that.", "start": 68.85, "duration": 1.32}, {"text": "That was the universal\nTuring machine", "start": 70.17, "duration": 1.56}, {"text": "that we presented at the end.", "start": 71.73, "duration": 1.27}, {"text": "That's what shows that A,TM\nis Turing-recognizable.", "start": 73.0, "duration": 4.01}, {"text": "We mentioned that\nA,TM is not decidable,", "start": 77.01, "duration": 4.57}, {"text": "which we promised\nwe will prove today.", "start": 81.58, "duration": 1.88}, {"text": "And we will do so.", "start": 83.46, "duration": 1.98}, {"text": "OK, so that is the plan--", "start": 85.44, "duration": 2.64}, {"text": "proving A,TM is undecidable.", "start": 88.08, "duration": 2.1}, {"text": "And we'll introduce the\nmethod for doing that called", "start": 90.18, "duration": 2.28}, {"text": "the diagonalization method.", "start": 92.46, "duration": 2.01}, {"text": "I will also show the\ncomplement of A,TM is", "start": 94.47, "duration": 4.2}, {"text": "Turing-unrecognizable.", "start": 98.67, "duration": 2.46}, {"text": "Even though A,TM\nitself is recognizable,", "start": 101.13, "duration": 2.49}, {"text": "the complement is not.", "start": 103.62, "duration": 2.04}, {"text": "And then we will introduce\nanother method called", "start": 105.66, "duration": 3.81}, {"text": "the reducibility method for\nshowing other problems are not", "start": 109.47, "duration": 3.45}, {"text": "decidable and give one example\nof another problem, which", "start": 112.92, "duration": 3.892}, {"text": "is not decidable, assuming\nwe have time to get to it.", "start": 116.812, "duration": 2.208}, {"text": "If not, we will delay that part\nuntil next Tuesday's lecture.", "start": 119.02, "duration": 6.47}, {"text": "All right, acceptance\nproblem for Turing Machines.", "start": 125.49, "duration": 3.25}, {"text": "So just as I mentioned,\nwe showed that A,TM,", "start": 128.74, "duration": 4.82}, {"text": "which is the language of Turing\nmachines and inputs where", "start": 133.56, "duration": 3.34}, {"text": "the machine accepts the input.", "start": 136.9, "duration": 2.9}, {"text": "We showed that was recognizable.", "start": 139.8, "duration": 1.68}, {"text": "We claimed it was decidable.", "start": 141.48, "duration": 1.23}, {"text": "Today, we're going to\nprove it's not decidable.", "start": 142.71, "duration": 2.52}, {"text": "All right, now the method\nthat we're going to use,", "start": 145.23, "duration": 5.29}, {"text": "which is really the\nonly method out there", "start": 150.52, "duration": 2.42}, {"text": "for proving a problem\nas undecidable", "start": 152.94, "duration": 6.3}, {"text": "is called the\ndiagonalization method.", "start": 159.24, "duration": 2.175}, {"text": "I mean, ultimately, we're going\nto show the reducibility method", "start": 161.415, "duration": 2.625}, {"text": "as well.", "start": 164.04, "duration": 0.85}, {"text": "But it really depends\non having already", "start": 164.89, "duration": 2.18}, {"text": "shown some other\nproblem undecidable", "start": 167.07, "duration": 2.22}, {"text": "via the diagonalization method.", "start": 169.29, "duration": 3.76}, {"text": "So ultimately, everything hinges\non the diagonalization method,", "start": 173.05, "duration": 2.9}, {"text": "which is really what we have.", "start": 175.95, "duration": 2.89}, {"text": "And so to introduce the\ndiagonalization method,", "start": 178.84, "duration": 6.98}, {"text": "I'm going to make a\nbit of a digression", "start": 185.82, "duration": 3.72}, {"text": "into a branch of mathematics\ncalled set theory", "start": 189.54, "duration": 3.84}, {"text": "or it's a part of\nmathematical logic, where", "start": 193.38, "duration": 3.48}, {"text": "the method of\ndiagonalization was first", "start": 196.86, "duration": 3.18}, {"text": "conceived of back in the late\n19th century by a mathematician", "start": 200.04, "duration": 5.28}, {"text": "called Georg Cantor.", "start": 205.32, "duration": 2.37}, {"text": "And Cantor was considering\nthe problem of,", "start": 207.69, "duration": 3.45}, {"text": "how do you compare the relative\nsizes of infinite sets?", "start": 211.14, "duration": 5.39}, {"text": "For finite sets, the problem\nof comparing-- somebody", "start": 216.53, "duration": 6.96}, {"text": "said that Cantor went crazy.", "start": 223.49, "duration": 1.89}, {"text": "That is true.", "start": 225.38, "duration": 2.07}, {"text": "And maybe I don't know\nwhy he went crazy.", "start": 227.45, "duration": 3.18}, {"text": "But he did go--", "start": 230.63, "duration": 3.33}, {"text": "he had some mental\nproblems, unfortunately.", "start": 233.96, "duration": 2.82}, {"text": "And so how do we compare the\nsizes of sets in general?", "start": 240.56, "duration": 2.91}, {"text": "If they're finite sets,\nwe can just count them up.", "start": 243.47, "duration": 2.78}, {"text": "We can say, whoa, this\nset has 11 elements,", "start": 246.25, "duration": 3.49}, {"text": "and the other set\nhas 10 elements.", "start": 249.74, "duration": 2.88}, {"text": "So the one with 11 is bigger.", "start": 252.62, "duration": 2.01}, {"text": "Or if they both have 11,\nthey're the same size.", "start": 254.63, "duration": 1.99}, {"text": "Well, that's not going\nto work for infinite sets", "start": 256.62, "duration": 2.0}, {"text": "because you can't count them up.", "start": 258.62, "duration": 2.49}, {"text": "And so he had--", "start": 261.11, "duration": 1.74}, {"text": "Cantor had the following\nidea for comparing", "start": 262.85, "duration": 2.07}, {"text": "the sizes of infinite sets.", "start": 264.92, "duration": 2.52}, {"text": "And that was, basically,\nto see whether you", "start": 267.44, "duration": 4.5}, {"text": "can have a function\nthat would map", "start": 271.94, "duration": 3.37}, {"text": "from one set to the other\nset with certain properties.", "start": 275.31, "duration": 4.26}, {"text": "And those properties are\ncalled, traditionally, well,", "start": 279.57, "duration": 4.34}, {"text": "I mean, in the past, have\nbeen called the one-to-one", "start": 283.91, "duration": 3.4}, {"text": "and onto properties\nfor the function.", "start": 287.31, "duration": 2.1}, {"text": "I'll tell you what that means.", "start": 289.41, "duration": 1.89}, {"text": "But the concept is very simple.", "start": 291.3, "duration": 2.88}, {"text": "So a one-to-one\nfunction is a function", "start": 294.18, "duration": 2.64}, {"text": "that's mapping from A to B.\nThose are the two sets whose", "start": 296.82, "duration": 3.42}, {"text": "sizes we're trying to compare.", "start": 300.24, "duration": 3.03}, {"text": "And the function\nbeing one-to-one just", "start": 303.27, "duration": 3.54}, {"text": "means that there\nare no collisions.", "start": 306.81, "duration": 1.62}, {"text": "If you have two\ndifferent elements of A,", "start": 308.43, "duration": 2.31}, {"text": "they're never going to map\nonto the same element of B.", "start": 310.74, "duration": 2.79}, {"text": "So two different\nelements of A always", "start": 313.53, "duration": 1.68}, {"text": "map onto two different\nelements of B.", "start": 315.21, "duration": 2.07}, {"text": "So that's the\none-to-one property.", "start": 317.28, "duration": 2.44}, {"text": "It's also called injective.", "start": 319.72, "duration": 2.69}, {"text": "And the other property is\ncalled onto or surjective,", "start": 322.41, "duration": 6.55}, {"text": "which is that the range\nof f has to be all of B.", "start": 328.96, "duration": 2.398}, {"text": "So you're not allowed\nto miss any elements,", "start": 331.358, "duration": 1.792}, {"text": "but you have to hit everything.", "start": 333.15, "duration": 2.52}, {"text": "And when you have both\nof those properties,", "start": 335.67, "duration": 2.85}, {"text": "the function is called a\none-to-one correspondence", "start": 338.52, "duration": 3.06}, {"text": "or a bijection also, OK?", "start": 341.58, "duration": 4.26}, {"text": "Now another way\nof looking at it--", "start": 345.84, "duration": 2.67}, {"text": "I don't want to make this\nmore complicated than it", "start": 348.51, "duration": 2.22}, {"text": "needs to be.", "start": 350.73, "duration": 0.75}, {"text": "It just simply\nmeans that two sets", "start": 351.48, "duration": 2.37}, {"text": "are considered to be the\nsame size if we can match up", "start": 353.85, "duration": 3.39}, {"text": "the elements with one set with\nelements of the other set.", "start": 357.24, "duration": 2.82}, {"text": "You just pair them up.", "start": 360.06, "duration": 2.01}, {"text": "For example, well, if you\nhave finite sets, that idea,", "start": 362.07, "duration": 6.69}, {"text": "that informal idea just works\nexactly as you would expect.", "start": 368.76, "duration": 3.15}, {"text": "For example, if we have two\nsets-- here's a set of puppies.", "start": 371.91, "duration": 3.72}, {"text": "Here's a set of kittens.", "start": 375.63, "duration": 2.85}, {"text": "Now we want to show that those\ntwo sets have the same size.", "start": 378.48, "duration": 4.82}, {"text": "We could count them up, as I\nmentioned, and see that there", "start": 383.3, "duration": 2.82}, {"text": "are six elements in both.", "start": 386.12, "duration": 1.92}, {"text": "But counting up does not\nwork for infinite sets.", "start": 388.04, "duration": 3.7}, {"text": "So we can just match up\nthe elements of the puppies", "start": 391.74, "duration": 5.42}, {"text": "with the kittens,\nand then we know", "start": 397.16, "duration": 2.63}, {"text": "we have the same number\nof puppies as kittens.", "start": 399.79, "duration": 2.82}, {"text": "OK, now that has the\nadvantage of it making sense", "start": 402.61, "duration": 4.53}, {"text": "when you have infinite sets.", "start": 407.14, "duration": 1.57}, {"text": "So we're just going\nto extend that idea", "start": 408.71, "duration": 1.813}, {"text": "and apply it to\ninfinite sets too.", "start": 410.523, "duration": 1.417}, {"text": "And then we'll have\na notion of what", "start": 411.94, "duration": 1.56}, {"text": "it means for two infinite\nsets to have the same size.", "start": 413.5, "duration": 3.78}, {"text": "And you might wonder,\nwhat do you get?", "start": 417.28, "duration": 1.74}, {"text": "Are all infinite\nsets of the same size", "start": 419.02, "duration": 2.61}, {"text": "when you use this notion or not?", "start": 421.63, "duration": 2.6}, {"text": "What happens?", "start": 424.23, "duration": 1.63}, {"text": "Well, some strange\nthings do happen.", "start": 425.86, "duration": 2.31}, {"text": "But there actually are quite\nsome interesting structure", "start": 428.17, "duration": 4.62}, {"text": "there that emerges.", "start": 432.79, "duration": 1.87}, {"text": "So anyway, I don't\nwant to rush on.", "start": 434.66, "duration": 5.0}, {"text": "Questions on any of this?", "start": 439.66, "duration": 1.14}, {"text": "If you want to--", "start": 440.8, "duration": 1.62}, {"text": "quick question.", "start": 442.42, "duration": 0.66}, {"text": "This, hopefully, was not too\nhard, but I want to make sure", "start": 443.08, "duration": 2.417}, {"text": "everybody's together with\nme on it, so we can pop in--", "start": 445.497, "duration": 3.373}, {"text": "I'll give a few seconds for a\nchat if you have any questions.", "start": 448.87, "duration": 3.66}, {"text": "The range of the set\nis all of the elements", "start": 458.58, "duration": 2.64}, {"text": "that you hit as you look at\nthe different possible elements", "start": 461.22, "duration": 3.39}, {"text": "of A. So all of the things that\nf hits, the standard notion", "start": 464.61, "duration": 2.64}, {"text": "of a range of a function.", "start": 467.25, "duration": 1.17}, {"text": "So the range of f\nhas to be equal to B.", "start": 471.22, "duration": 2.15}, {"text": "You have to hit everything.", "start": 473.37, "duration": 1.478}, {"text": "All right, can you think of\na one-to-one correspondence", "start": 478.06, "duration": 4.12}, {"text": "as a relabeling?", "start": 482.18, "duration": 0.923}, {"text": "Yeah, I'm not sure\nthat's helpful.", "start": 483.103, "duration": 1.417}, {"text": "But, yes, you can think of as\na relabeling of the elements", "start": 484.52, "duration": 3.06}, {"text": "in a sense, but yeah.", "start": 487.58, "duration": 2.2}, {"text": "I just think of it\nas a matching up.", "start": 489.78, "duration": 3.1}, {"text": "All right, so let's move on.", "start": 492.88, "duration": 2.44}, {"text": "So coming out of\nthis notion of sets", "start": 495.32, "duration": 4.8}, {"text": "being of the same\nsize with this notion", "start": 500.12, "duration": 2.55}, {"text": "of countable sets, as\nwe'll see in a minute,", "start": 502.67, "duration": 2.89}, {"text": "so let's do an example.", "start": 505.56, "duration": 2.84}, {"text": "Let's take the set\nof natural numbers--", "start": 508.4, "duration": 2.6}, {"text": "1, 2, 3, 4, dot, dot,\ndot, dot, and the set", "start": 511.0, "duration": 2.429}, {"text": "of integers, which includes\nthe natural numbers", "start": 513.429, "duration": 3.48}, {"text": "but also the negative\nnumbers and zero.", "start": 516.909, "duration": 2.461}, {"text": "So the natural\nnumbers are typically", "start": 522.159, "duration": 2.611}, {"text": "referred to as N. The integers\nare typically referred to as Z.", "start": 524.77, "duration": 4.74}, {"text": "And what do we think of the\nrelative sizes of N and Z?", "start": 529.51, "duration": 8.7}, {"text": "Well, N is a subset of Z.\nIt's a proper subset of Z.", "start": 538.21, "duration": 3.39}, {"text": "So you might, at first glance,\nthink that Z is larger,", "start": 541.6, "duration": 2.88}, {"text": "and they're not really going\nto be of the same size.", "start": 544.48, "duration": 2.76}, {"text": "But actually, it turns out that\nthere's a very simple way--", "start": 547.24, "duration": 3.72}, {"text": "the arithmetic and the\nproperties of infinite sets", "start": 550.96, "duration": 3.63}, {"text": "can be a little bit surprising.", "start": 554.59, "duration": 1.41}, {"text": "And there is a way of matching\nup all of the elements of Z", "start": 556.0, "duration": 3.66}, {"text": "with their own elements\nof N. And so you", "start": 559.66, "duration": 3.93}, {"text": "can show that following that\ndefinition, that these two sets", "start": 563.59, "duration": 4.89}, {"text": "are, in fact, of the same size.", "start": 568.48, "duration": 1.54}, {"text": "So let's just quickly go\nmake sure you see that.", "start": 570.02, "duration": 4.86}, {"text": "So here is N. Here\nis Z I'm going", "start": 574.88, "duration": 2.57}, {"text": "to write down a table which\nshows how they match up.", "start": 577.45, "duration": 3.06}, {"text": "Here is the function f of n\nthat I'm going to be describing.", "start": 580.51, "duration": 2.91}, {"text": "That's the one-to-one\ncorrespondence.", "start": 583.42, "duration": 3.63}, {"text": "And so here are the\nnatural numbers--", "start": 587.05, "duration": 1.83}, {"text": "dot, dot, dot, 1 through\n7, dot, dot, dot.", "start": 588.88, "duration": 3.15}, {"text": "And here are the\nelements of Z that I'm", "start": 592.03, "duration": 1.64}, {"text": "going to be matching up.", "start": 593.67, "duration": 1.0}, {"text": "This is how the\nfunction is working.", "start": 594.67, "duration": 1.63}, {"text": "So one, I'm going to--", "start": 596.3, "duration": 1.55}, {"text": "f of 1 is going to be 0.", "start": 597.85, "duration": 1.71}, {"text": "F of 2 is going to be minus 1.", "start": 599.56, "duration": 2.28}, {"text": "F of 3 is going to be 1.", "start": 601.84, "duration": 3.33}, {"text": "And I'm just giving\nyou a way to match up", "start": 605.17, "duration": 3.57}, {"text": "each of the natural\nnumbers with integers", "start": 608.74, "duration": 3.97}, {"text": "so that every single integer has\nits own natural number and vice", "start": 612.71, "duration": 4.2}, {"text": "versa.", "start": 616.91, "duration": 0.5}, {"text": "So 4 goes to minus 2.", "start": 619.99, "duration": 1.505}, {"text": "5 goes to 2.", "start": 621.495, "duration": 1.23}, {"text": "6 goes to minus 3.", "start": 622.725, "duration": 1.575}, {"text": "7 goes to 3, and then minus\n4, and then 4, and minus 5,", "start": 624.3, "duration": 3.44}, {"text": "and then 5, and so on.", "start": 627.74, "duration": 1.96}, {"text": "You're clearly\ngoing to cover all", "start": 629.7, "duration": 1.44}, {"text": "of the integers in this way.", "start": 631.14, "duration": 1.5}, {"text": "And each of the\nnatural numbers is", "start": 632.64, "duration": 1.47}, {"text": "going to have its own integer.", "start": 634.11, "duration": 1.275}, {"text": "And there's never going\nto be any collisions.", "start": 635.385, "duration": 1.875}, {"text": "There's never going to be\ntwo natural numbers assigned", "start": 637.26, "duration": 1.83}, {"text": "to the same integer.", "start": 639.09, "duration": 1.18}, {"text": "So this meets the conditions\nof a one-to-one correspondence", "start": 640.27, "duration": 4.91}, {"text": "and shows that the natural\nnumbers and the integers", "start": 645.18, "duration": 4.23}, {"text": "have the same size\nfollowing this definition.", "start": 649.41, "duration": 5.99}, {"text": "OK, let's do one that's slightly\nmore complicated and, perhaps,", "start": 655.4, "duration": 6.63}, {"text": "slightly more surprising,\nwhich is that if you consider", "start": 662.03, "duration": 4.2}, {"text": "all of the rational numbers,\nand then they have the same--", "start": 666.23, "duration": 5.485}, {"text": "that is a collection.", "start": 671.715, "duration": 0.875}, {"text": "Even though the\nrational numbers seem", "start": 672.59, "duration": 2.4}, {"text": "to be much richer and more\nnumerous than the integers,", "start": 674.99, "duration": 4.2}, {"text": "from this perspective,\nthey have the same size.", "start": 679.19, "duration": 2.31}, {"text": "And for the simplicity\nof my presentation,", "start": 681.5, "duration": 3.61}, {"text": "I'm going to consider only\nthe positive rational numbers,", "start": 685.11, "duration": 2.61}, {"text": "which I'm going to write as Q+.", "start": 687.72, "duration": 2.36}, {"text": "So those are all of the\npositive rational numbers", "start": 690.08, "duration": 2.85}, {"text": "that can be expressed as a\nratio of two natural numbers.", "start": 692.93, "duration": 3.6}, {"text": "And I'm going to show that there\nis a one-to-one correspondence", "start": 696.53, "duration": 4.47}, {"text": "between these positive\nrational numbers", "start": 701.0, "duration": 3.18}, {"text": "and the natural numbers.", "start": 704.18, "duration": 2.64}, {"text": "OK, so here, I'm\ngoing to, again,", "start": 706.82, "duration": 3.21}, {"text": "make a table, just\nas I did up above--", "start": 710.03, "duration": 3.48}, {"text": "so comparing the\nnatural numbers and", "start": 713.51, "duration": 2.01}, {"text": "the positive rational numbers.", "start": 715.52, "duration": 2.01}, {"text": "And to do that, I'm going\nto write down over here", "start": 717.53, "duration": 3.48}, {"text": "on the side, just\nto help you see", "start": 721.01, "duration": 1.98}, {"text": "how I'm getting this table,\na separate table that", "start": 722.99, "duration": 5.19}, {"text": "has all of the positive\nrational numbers appearing", "start": 728.18, "duration": 3.63}, {"text": "in a nicely organized way.", "start": 731.81, "duration": 3.36}, {"text": "Here are all of the\nrational numbers", "start": 735.17, "duration": 2.28}, {"text": "here that have 1 as a numerator,\nthat have 2 as a numerator,", "start": 737.45, "duration": 3.51}, {"text": "and so on, and going\nthrough across the columns,", "start": 740.96, "duration": 3.18}, {"text": "the different denominators.", "start": 744.14, "duration": 2.17}, {"text": "So the numbers\ninside here represent", "start": 746.31, "duration": 3.057}, {"text": "all of the different\nrational numbers.", "start": 749.367, "duration": 1.583}, {"text": "And so whatever rational\nnumber you have,", "start": 750.95, "duration": 1.95}, {"text": "m over n, that's\ngoing to appear down", "start": 752.9, "duration": 3.87}, {"text": "the nth row in the nth column.", "start": 756.77, "duration": 2.248}, {"text": "That number is going to appear.", "start": 759.018, "duration": 1.292}, {"text": "So they're all going to show up.", "start": 760.31, "duration": 1.89}, {"text": "And I'm going to use this table\nhere to fill out this function.", "start": 762.2, "duration": 6.87}, {"text": "So here are all the\nnatural numbers.", "start": 769.07, "duration": 3.29}, {"text": "And the way I'm going to assign\nthe rational numbers to appear", "start": 772.36, "duration": 4.59}, {"text": "over here is I'm just going to\nwork my way in from the corner.", "start": 776.95, "duration": 5.73}, {"text": "And I'm going to do\nthat by doing layers.", "start": 782.68, "duration": 3.52}, {"text": "So first, I'll take\nthe 1 in this number", "start": 786.2, "duration": 3.17}, {"text": "here in this corner.", "start": 789.37, "duration": 0.96}, {"text": "Then, I'll do these\nthree that surrounded it,", "start": 790.33, "duration": 2.82}, {"text": "and then these guys\nthat surround that,", "start": 793.15, "duration": 2.4}, {"text": "and these guys sort of in shells\ngoing around the previous ones", "start": 795.55, "duration": 3.36}, {"text": "that I've already covered.", "start": 798.91, "duration": 1.92}, {"text": "OK, so let me illustrate that.", "start": 800.83, "duration": 2.86}, {"text": "So here's 1 over 1, my\nfirst rational number", "start": 803.69, "duration": 2.84}, {"text": "that I'm going to\nenter into the table,", "start": 806.53, "duration": 2.04}, {"text": "appearing right over there.", "start": 808.57, "duration": 1.35}, {"text": "So next, I'm going\nto go 2 over 1.", "start": 809.92, "duration": 2.64}, {"text": "That's going to appear\nin the table over there.", "start": 812.56, "duration": 2.71}, {"text": "Now, we have 2 over 2.", "start": 815.27, "duration": 1.41}, {"text": "Now that's actually a little\nbit problematic because 2 over 2", "start": 816.68, "duration": 4.53}, {"text": "has already been done.", "start": 821.21, "duration": 2.32}, {"text": "And if we put that in--", "start": 823.53, "duration": 1.88}, {"text": "because 2 over 2\nequals 1 over 1.", "start": 825.41, "duration": 1.65}, {"text": "They're both the equal\nto the rational number 1.", "start": 827.06, "duration": 3.24}, {"text": "So if we put 2 over 2\nin this table over here,", "start": 830.3, "duration": 2.61}, {"text": "then we would no longer\nhave the one-to-one property", "start": 832.91, "duration": 2.67}, {"text": "because both 1 and 3 would\nbe mapping to the same point.", "start": 835.58, "duration": 4.27}, {"text": "So we're just going to\nsimply skip over 2 over 2.", "start": 839.85, "duration": 5.0}, {"text": "I'll show that as kind\nof graying it out.", "start": 844.85, "duration": 2.08}, {"text": "So we're not going to add\nthat one on to the table", "start": 846.93, "duration": 3.08}, {"text": "onto my function.", "start": 850.01, "duration": 2.64}, {"text": "But so we'll skip over that.", "start": 852.65, "duration": 1.62}, {"text": "We'll go to 1 over 2, which\nhas not been seen before.", "start": 854.27, "duration": 4.262}, {"text": "And then we're just going to\ncontinue doing the same thing,", "start": 858.532, "duration": 2.458}, {"text": "now going to this\nnext shell out here.", "start": 860.99, "duration": 2.61}, {"text": "3 over 1, 3 over 2, 3 over 3--", "start": 863.6, "duration": 5.16}, {"text": "same thing, we're going\nto skip over that one--", "start": 868.76, "duration": 3.08}, {"text": "2 over 3, 1 over 3.", "start": 871.84, "duration": 2.76}, {"text": "I hope you're getting the idea.", "start": 874.6, "duration": 2.08}, {"text": "So I'm not going\nto fill this table.", "start": 876.68, "duration": 1.5}, {"text": "I ran out of room to fill\nout this table some more.", "start": 878.18, "duration": 3.23}, {"text": "But just to look at how the\nprocess is going to continue", "start": 881.41, "duration": 2.73}, {"text": "here, we get 4 over 1.", "start": 884.14, "duration": 1.44}, {"text": "Now 4 over 2, again, is a\nnumber we previously seen", "start": 885.58, "duration": 4.05}, {"text": "because that's the\nrational number 2.", "start": 889.63, "duration": 2.67}, {"text": "We saw that up here\nwhen we had 2 over 1,", "start": 892.3, "duration": 2.1}, {"text": "so we're going to\nhave to skip that one.", "start": 894.4, "duration": 1.77}, {"text": "4 over 3, 4 over 4--", "start": 896.17, "duration": 2.31}, {"text": "that was going to skip.", "start": 898.48, "duration": 1.74}, {"text": "3 over 4, and so on.", "start": 900.22, "duration": 2.54}, {"text": "OK?", "start": 902.76, "duration": 0.5}, {"text": "So by following\nthis procedure, I'm", "start": 903.26, "duration": 3.78}, {"text": "going to be able to\ndefine this function.", "start": 907.04, "duration": 1.938}, {"text": "Now this function\nis not particularly", "start": 908.978, "duration": 1.542}, {"text": "nice in terms of having\nan elegant, closed form.", "start": 910.52, "duration": 3.6}, {"text": "But it is a totally\nlegitimate function", "start": 914.12, "duration": 2.7}, {"text": "in the sense of being a\nmapping from natural numbers", "start": 916.82, "duration": 3.18}, {"text": "to the positive\nrational numbers.", "start": 920.0, "duration": 3.63}, {"text": "And it has the one-to-one\ncorrespondence property.", "start": 923.63, "duration": 3.9}, {"text": "So it pairs up each\nof the natural numbers", "start": 927.53, "duration": 3.81}, {"text": "with each of the positive\nrational numbers.", "start": 931.34, "duration": 5.4}, {"text": "They each get their\nown mate, in a sense.", "start": 936.74, "duration": 3.477}, {"text": "And so that shows that\nthe rational numbers,", "start": 940.217, "duration": 1.833}, {"text": "despite the fact\nthat they're dense,", "start": 942.05, "duration": 2.1}, {"text": "and they have all sorts of\nvery more much bigger seeming,", "start": 944.15, "duration": 3.87}, {"text": "they really, from this\nperspective of the sizes", "start": 948.02, "duration": 3.09}, {"text": "of the sets, they have the same\nsize as the natural numbers do.", "start": 951.11, "duration": 5.1}, {"text": "And so with that, it suggests\nthat the following definition", "start": 956.21, "duration": 3.81}, {"text": "that a set is\ncountable if it has", "start": 960.02, "duration": 2.58}, {"text": "the same size as the natural\nnumbers or it's finite.", "start": 962.6, "duration": 3.983}, {"text": "From this perspective,\nwe're going", "start": 966.583, "duration": 1.417}, {"text": "to be focusing on infinite sets.", "start": 968.0, "duration": 2.22}, {"text": "But yeah, countable or countably\ninfinite, sometimes people say,", "start": 970.22, "duration": 3.72}, {"text": "if it has the same size\nas the natural numbers.", "start": 973.94, "duration": 2.008}, {"text": "Or otherwise, you have to\ninclude the finite sets", "start": 975.948, "duration": 2.042}, {"text": "as well.", "start": 977.99, "duration": 0.66}, {"text": "And countable means you can\njust go through the elements", "start": 978.65, "duration": 4.8}, {"text": "of the set as a list.", "start": 983.45, "duration": 3.93}, {"text": "So you can count them-- the\nfirst one, the second one,", "start": 987.38, "duration": 2.25}, {"text": "the third one, the fourth\none, dot, dot, dot.", "start": 989.63, "duration": 2.576}, {"text": "And once you can do\nthat, and that counting", "start": 992.206, "duration": 2.784}, {"text": "is going to hit everything,\nthen you know can match them--", "start": 994.99, "duration": 3.61}, {"text": "you can pair them up\nwith the natural numbers.", "start": 998.6, "duration": 1.97}, {"text": "And so therefore, you\nhave a countable set.", "start": 1000.57, "duration": 2.19}, {"text": "OK.", "start": 1002.76, "duration": 2.43}, {"text": "So as we've shown,\nboth Z, the integers,", "start": 1005.19, "duration": 4.56}, {"text": "and the positive rational\nnumbers are countable sets.", "start": 1009.75, "duration": 3.94}, {"text": "OK?", "start": 1013.69, "duration": 0.5}, {"text": "Now you might think\nthat, well, since we're", "start": 1014.19, "duration": 3.65}, {"text": "allowing ourselves\nto do something", "start": 1017.84, "duration": 2.88}, {"text": "as arbitrary, in a way,\nas this kind of a function", "start": 1020.72, "duration": 5.82}, {"text": "to match up the natural numbers\nwith whatever set you're trying", "start": 1026.54, "duration": 4.2}, {"text": "to measure, that every set\nis going to be countable,", "start": 1030.74, "duration": 4.832}, {"text": "if you think about it that way\nbecause it seems like you're", "start": 1035.572, "duration": 2.458}, {"text": "allowing too many possibilities.", "start": 1038.03, "duration": 2.422}, {"text": "And then all the\ninfinite sets are", "start": 1040.452, "duration": 1.417}, {"text": "going to end up being the same\nsize as the natural numbers.", "start": 1041.869, "duration": 2.73}, {"text": "Well, that is, in\nfact, is not true.", "start": 1044.599, "duration": 3.401}, {"text": "And I don't know.", "start": 1048.0, "duration": 1.97}, {"text": "Cantor, is the one\nwho discovered that.", "start": 1049.97, "duration": 2.49}, {"text": "I don't know if that\nwas surprising or not.", "start": 1052.46, "duration": 3.04}, {"text": "But it is interesting,\nI think, that there are", "start": 1055.5, "duration": 3.95}, {"text": "different sizes of infinities.", "start": 1059.45, "duration": 2.49}, {"text": "And so we're going to now\ntake a look and see that.", "start": 1061.94, "duration": 4.93}, {"text": "But I want to, again,\nI want to stop and make", "start": 1066.87, "duration": 4.79}, {"text": "sure we're all together.", "start": 1071.66, "duration": 1.32}, {"text": "Can we always find\na closed formula", "start": 1072.98, "duration": 1.71}, {"text": "for f, somebody's asking me.", "start": 1074.69, "duration": 2.16}, {"text": "I don't know.", "start": 1076.85, "duration": 1.35}, {"text": "For this particular\nf, you probably could,", "start": 1078.2, "duration": 5.26}, {"text": "but it would probably\nbe very messy.", "start": 1083.46, "duration": 4.14}, {"text": "But in general, that's\nnot a requirement,", "start": 1087.6, "duration": 2.34}, {"text": "having a closed form for\nthe mapping function.", "start": 1089.94, "duration": 3.87}, {"text": "Any function is allowed as long\nas a one-to-one correspondence.", "start": 1093.81, "duration": 3.51}, {"text": "Are we all together on this?", "start": 1100.53, "duration": 1.59}, {"text": "Are we comfortable\nwith the notion", "start": 1102.12, "duration": 1.8}, {"text": "of some infinite sets\nhaving the same size", "start": 1103.92, "duration": 4.007}, {"text": "as the natural numbers,\nand therefore, we're going", "start": 1107.927, "duration": 2.083}, {"text": "to call them countable sets?", "start": 1110.01, "duration": 1.41}, {"text": "And we're going to show\nsome other sets are", "start": 1111.42, "duration": 2.34}, {"text": "going to have more elements.", "start": 1113.76, "duration": 2.76}, {"text": "They're going to be uncountable.", "start": 1116.52, "duration": 1.68}, {"text": "They're going to be beyond--", "start": 1118.2, "duration": 1.35}, {"text": "strictly speaking, strictly\nlarger sets in the sense", "start": 1119.55, "duration": 3.75}, {"text": "that we won't be\nable to put them", "start": 1123.3, "duration": 1.44}, {"text": "into a one-to-one correspondence\nwith the integers.", "start": 1124.74, "duration": 3.49}, {"text": "So is N the smallest infinity?", "start": 1128.23, "duration": 1.31}, {"text": "Yes, N is the smallest one.", "start": 1129.54, "duration": 3.27}, {"text": "So any infinity\nis going to have--", "start": 1132.81, "duration": 4.11}, {"text": "you're going to always--", "start": 1136.92, "duration": 1.187}, {"text": "I don't think you even need it--", "start": 1138.107, "duration": 1.333}, {"text": "you need a special--", "start": 1139.44, "duration": 3.18}, {"text": "whenever you have an\ninfinite collection,", "start": 1142.62, "duration": 1.98}, {"text": "you can always\nfind a subset which", "start": 1144.6, "duration": 2.28}, {"text": "is going to be a\ncountable subset", "start": 1146.88, "duration": 5.567}, {"text": "and is going to be the\nsmallest of the infinities,", "start": 1152.447, "duration": 2.083}, {"text": "but there are bigger ones.", "start": 1154.53, "duration": 2.117}, {"text": "All right, why don't we move on?", "start": 1156.647, "duration": 1.333}, {"text": "OK, so the example of a set that\nwe will show is not countable--", "start": 1160.68, "duration": 6.6}, {"text": "an uncountable set,\nas we call it--", "start": 1167.28, "duration": 4.02}, {"text": "is a set of real numbers, which\nwe all know what these are,", "start": 1171.3, "duration": 6.61}, {"text": "I hope.", "start": 1177.91, "duration": 0.77}, {"text": "These are all of\nthe numbers that you", "start": 1178.68, "duration": 2.31}, {"text": "can express by possibly\ninfinite decimal representations", "start": 1180.99, "duration": 3.63}, {"text": "like pi, or E squared of 2, or\nany of the other familiar ones.", "start": 1184.62, "duration": 6.27}, {"text": "Rational numbers are also\nmembers of or also count", "start": 1190.89, "duration": 5.7}, {"text": "as real numbers\nand integers too.", "start": 1196.59, "duration": 2.82}, {"text": "And so but there are\nthese additional numbers", "start": 1199.41, "duration": 4.98}, {"text": "that you can get by the\ndecimal expansions which", "start": 1204.39, "duration": 2.85}, {"text": "may not be rational.", "start": 1207.24, "duration": 2.28}, {"text": "And that collection, even\nthough, in some ways,", "start": 1209.52, "duration": 2.58}, {"text": "it looks somewhat similar\nto the rational numbers,", "start": 1212.1, "duration": 3.96}, {"text": "the real numbers--", "start": 1216.06, "duration": 1.268}, {"text": "I hope I said that right.", "start": 1217.328, "duration": 1.042}, {"text": "The real numbers,\nwhich is the set", "start": 1218.37, "duration": 4.53}, {"text": "I'm considering here, the\nones with the infinite decimal", "start": 1222.9, "duration": 2.73}, {"text": "expansions, they\nactually are much larger.", "start": 1225.63, "duration": 4.57}, {"text": "So the theorem is-- and this\nwas discovered by Cantor--", "start": 1230.2, "duration": 6.45}, {"text": "that R is an uncountable set.", "start": 1236.65, "duration": 3.07}, {"text": "And the reason why\nI'm introducing", "start": 1239.72, "duration": 2.39}, {"text": "this is, besides that I think\nit's interesting and has", "start": 1242.11, "duration": 5.67}, {"text": "some relation to the kinds\nof things we're doing,", "start": 1247.78, "duration": 3.58}, {"text": "but it's really for\nthe purposes right now,", "start": 1251.36, "duration": 2.27}, {"text": "is to introduce this method\ncalled diagonalization.", "start": 1253.63, "duration": 2.42}, {"text": "OK, that's what we're going\nto use later on again.", "start": 1259.13, "duration": 2.97}, {"text": "But this is the first\ntime it appeared.", "start": 1262.1, "duration": 2.43}, {"text": "So we're going to use a proof\nby contradiction in order", "start": 1264.53, "duration": 3.66}, {"text": "to show that R, the\ncollection of real numbers,", "start": 1268.19, "duration": 3.87}, {"text": "is uncountable.", "start": 1272.06, "duration": 1.68}, {"text": "And we'll assume,\nfor contradiction,", "start": 1273.74, "duration": 2.1}, {"text": "that R countable.", "start": 1275.84, "duration": 2.07}, {"text": "OK?", "start": 1277.91, "duration": 2.34}, {"text": "So if we assume\nthat R is countable,", "start": 1280.25, "duration": 3.09}, {"text": "it means we have, by definition,\na one-to-one correspondence", "start": 1283.34, "duration": 3.3}, {"text": "from the natural numbers\nto the real numbers.", "start": 1286.64, "duration": 2.89}, {"text": "OK, so we can match up\nall of the natural numbers", "start": 1289.53, "duration": 3.03}, {"text": "with the real numbers in a\none-to-one and onto fashion.", "start": 1292.56, "duration": 4.96}, {"text": "So we can pair up the natural\nnumbers with the real numbers.", "start": 1297.52, "duration": 3.12}, {"text": "And we will cover all of the\nreal numbers by doing that.", "start": 1300.64, "duration": 3.6}, {"text": "And we can make a table,\njust like I did before.", "start": 1304.24, "duration": 3.6}, {"text": "Here it is.", "start": 1307.84, "duration": 2.93}, {"text": "And you can fill this out\nwith all of the real numbers.", "start": 1310.77, "duration": 3.93}, {"text": "And that was what\nthe assumption means.", "start": 1314.7, "duration": 3.24}, {"text": "And I'm going to show you\nthat that's impossible.", "start": 1317.94, "duration": 2.19}, {"text": "Something is going to go\nterribly wrong if you do that.", "start": 1320.13, "duration": 4.57}, {"text": "Now you might disagree.", "start": 1324.7, "duration": 2.71}, {"text": "You might be surprised.", "start": 1327.41, "duration": 1.17}, {"text": "You might think,\nwell, Professor Sipser", "start": 1328.58, "duration": 4.62}, {"text": "is wrong, that I'm going\nto work on this overnight", "start": 1333.2, "duration": 4.44}, {"text": "and forget the\nrest of my classes,", "start": 1337.64, "duration": 1.44}, {"text": "and I'm going to come up with\na list of all the real numbers.", "start": 1339.08, "duration": 4.96}, {"text": "I'm going to fill\nthem out here and show", "start": 1344.04, "duration": 2.24}, {"text": "that that's impossible.", "start": 1346.28, "duration": 1.74}, {"text": "So let's say, hypothetically,\njust for my example's sake,", "start": 1348.02, "duration": 3.72}, {"text": "here is the list that\nyou came up with.", "start": 1351.74, "duration": 3.54}, {"text": "So you have to match up\nsomething with the 1.", "start": 1355.28, "duration": 1.92}, {"text": "Let's say, E. That\nwas a special case,", "start": 1357.2, "duration": 2.28}, {"text": "so you're going to\nstick that with 1.", "start": 1359.48, "duration": 2.07}, {"text": "And then pi came\nout to be the number", "start": 1361.55, "duration": 1.77}, {"text": "that you connected up with 2.", "start": 1363.32, "duration": 2.662}, {"text": "You understand what\nI'm doing here?", "start": 1365.982, "duration": 1.458}, {"text": "I'm making a list.", "start": 1367.44, "duration": 0.89}, {"text": "I'm trying to make my\ncorrespondence-- my matching up", "start": 1368.33, "duration": 3.33}, {"text": "between the natural\nnumbers and my real numbers", "start": 1371.66, "duration": 3.21}, {"text": "that I'm hypothesizing to\nexist for a contradiction.", "start": 1374.87, "duration": 4.57}, {"text": "So 3, I don't know, 3\nis gets connected to 0.", "start": 1379.44, "duration": 3.56}, {"text": "I'm making this up, obviously--", "start": 1383.0, "duration": 2.82}, {"text": "not on the spot.", "start": 1385.82, "duration": 0.84}, {"text": "I wrote the slide,\nyou know, yesterday.", "start": 1386.66, "duration": 2.55}, {"text": "But here is the square root of 2\nshowing up for whatever reason.", "start": 1389.21, "duration": 4.11}, {"text": "Here is 1/7.", "start": 1393.32, "duration": 2.64}, {"text": "Here is some other number,\nwhich, I'll be interested", "start": 1395.96, "duration": 2.7}, {"text": "if you recognize that one.", "start": 1398.66, "duration": 1.747}, {"text": "That's a subtle one.", "start": 1400.407, "duration": 0.833}, {"text": "This is 1.234-- yeah, some\nfamiliar looking sequence here.", "start": 1403.99, "duration": 5.94}, {"text": "And whatever it is--", "start": 1409.93, "duration": 1.68}, {"text": "whatever it is, this is\nwhat you came up with.", "start": 1411.61, "duration": 2.16}, {"text": "You claim that this\nis going to work as--", "start": 1413.77, "duration": 3.39}, {"text": "very good.", "start": 1417.16, "duration": 0.78}, {"text": "Somebody got it.", "start": 1417.94, "duration": 1.17}, {"text": "It's i to the ith power.", "start": 1419.11, "duration": 1.53}, {"text": "But let's not get hung\nup on that, please.", "start": 1420.64, "duration": 4.0}, {"text": "I to the ith power,\noddly enough,", "start": 1424.64, "duration": 1.46}, {"text": "can be seen as a\nreal number under--", "start": 1426.1, "duration": 2.13}, {"text": "if you define the imaginary\nexponentiation, which is weird.", "start": 1428.23, "duration": 7.33}, {"text": "But anyway, let's not\nget too distracted.", "start": 1435.56, "duration": 4.34}, {"text": "So here's my list of\nnumbers, my real numbers", "start": 1439.9, "duration": 3.33}, {"text": "that I've matched up in my table\nhere with the natural numbers.", "start": 1443.23, "duration": 4.72}, {"text": "Now I claim\nsomething goes wrong.", "start": 1447.95, "duration": 1.76}, {"text": "What goes wrong?", "start": 1449.71, "duration": 0.9}, {"text": "I'm going to show you\nthat you actually did not", "start": 1450.61, "duration": 5.22}, {"text": "do as you claimed,\nthat there's actually", "start": 1455.83, "duration": 3.57}, {"text": "at least one number that's\nmissing from this list.", "start": 1459.4, "duration": 3.622}, {"text": "And I'll exhibit that number.", "start": 1463.022, "duration": 1.208}, {"text": "I'm going to show you\nwhat that number is.", "start": 1464.23, "duration": 1.75}, {"text": "I'm going to explicitly\ncome up with a number", "start": 1465.98, "duration": 1.96}, {"text": "and show you that\nthere's that number it's", "start": 1467.94, "duration": 1.75}, {"text": "missing from the list.", "start": 1469.69, "duration": 1.81}, {"text": "So here, it's going\nto be a number x.", "start": 1471.5, "duration": 3.41}, {"text": "X is the one that's missing.", "start": 1474.91, "duration": 1.93}, {"text": "So how am I going to get x?", "start": 1476.84, "duration": 1.58}, {"text": "So x, well, I'm going to\nstart it off with 0 point,", "start": 1478.42, "duration": 2.88}, {"text": "and then I'm going to fill\nout the rest of the places.", "start": 1481.3, "duration": 2.25}, {"text": "And how am I going\nto get those places?", "start": 1483.55, "duration": 1.44}, {"text": "I'm going to look\nat the table here.", "start": 1484.99, "duration": 1.5}, {"text": "So I want to look at--", "start": 1488.94, "duration": 1.22}, {"text": "to get my first digit after\nthe decimal point of x,", "start": 1490.16, "duration": 3.45}, {"text": "I'm going to look\nat the first digit", "start": 1493.61, "duration": 1.5}, {"text": "after the decimal point\nof the very first number.", "start": 1495.11, "duration": 4.28}, {"text": "So I take the first number,\nlook at the first digit", "start": 1499.39, "duration": 2.15}, {"text": "after the decimal point,\nwhich happens to be a 7.", "start": 1501.54, "duration": 3.87}, {"text": "And the number I'm\ngoing to put in for x", "start": 1505.41, "duration": 2.1}, {"text": "It's not going to be 7.", "start": 1507.51, "duration": 1.44}, {"text": "It's going to be\nanything except 7.", "start": 1508.95, "duration": 2.94}, {"text": "Let say, 8.", "start": 1511.89, "duration": 0.57}, {"text": "For my next digit of x,\nI'm going to look at--", "start": 1515.41, "duration": 3.31}, {"text": "so my second digit\nof x, I'm going to--", "start": 1518.72, "duration": 2.05}, {"text": "everything's going to be\nup to the decimal point", "start": 1520.77, "duration": 2.0}, {"text": "now, so I'm not going\nto keep saying that.", "start": 1522.77, "duration": 2.09}, {"text": "I'm going to look\nat the second digit", "start": 1524.86, "duration": 1.65}, {"text": "of the second number, which is\na 4 after the decimal point.", "start": 1526.51, "duration": 5.37}, {"text": "There it is.", "start": 1534.43, "duration": 1.83}, {"text": "And for the second\ndigit of x, I'm", "start": 1536.26, "duration": 2.04}, {"text": "going to use something which\nis anything besides 4--", "start": 1538.3, "duration": 3.99}, {"text": "5.", "start": 1542.29, "duration": 1.86}, {"text": "I have some flexibility here.", "start": 1544.15, "duration": 2.342}, {"text": "So I could have used 6.", "start": 1546.492, "duration": 0.958}, {"text": "I could have used 7.", "start": 1547.45, "duration": 2.37}, {"text": "For those of you who\nhave seen this before,", "start": 1549.82, "duration": 2.04}, {"text": "who are going to\nnitpick on me, I'm", "start": 1551.86, "duration": 2.58}, {"text": "going to stay away from 9\nand 0 just because there", "start": 1554.44, "duration": 3.6}, {"text": "is some little edge cases that\narise there, which I don't want", "start": 1558.04, "duration": 3.127}, {"text": "to get into because\nI don't think it's", "start": 1561.167, "duration": 1.583}, {"text": "interesting for this argument.", "start": 1562.75, "duration": 2.16}, {"text": "But since I have\nsome flexibility,", "start": 1564.91, "duration": 1.65}, {"text": "I'm going to avoid\nzeros and nines--", "start": 1566.56, "duration": 1.71}, {"text": "probably just nines\nis all I really need.", "start": 1568.27, "duration": 1.87}, {"text": "And then the argument is\njust going to work fine, OK?", "start": 1570.14, "duration": 5.66}, {"text": "So the next digit is a 0.", "start": 1575.8, "duration": 4.49}, {"text": "The third digit, or the\nthird number is a 0.", "start": 1580.29, "duration": 2.61}, {"text": "The third digit of x is going\nto be anything except for 0.", "start": 1582.9, "duration": 2.685}, {"text": "It could be a 1.", "start": 1585.585, "duration": 1.785}, {"text": "Then I have a 2 here.", "start": 1587.37, "duration": 3.39}, {"text": "Anything except a 2, 6.", "start": 1590.76, "duration": 5.35}, {"text": "Then a 5-- anything\nexcept a 5, a 1.", "start": 1596.11, "duration": 3.0}, {"text": "A 9-- anything except\na 9, an 8, and so on.", "start": 1599.11, "duration": 4.56}, {"text": "There's an 8.", "start": 1603.67, "duration": 1.8}, {"text": "There's a 2, and dot, dot, dot.", "start": 1605.47, "duration": 1.8}, {"text": "I'm just going to\nkeep doing that.", "start": 1607.27, "duration": 2.0}, {"text": "And I'm going to say you missed\nthat number, whatever it is.", "start": 1609.27, "duration": 5.06}, {"text": "But that number,\nthat's a number.", "start": 1614.33, "duration": 1.64}, {"text": "After I'm all done,\nit's going to be", "start": 1615.97, "duration": 1.8}, {"text": "the decimal representation\nof something.", "start": 1617.77, "duration": 2.43}, {"text": "And that number, I claim,\nis absent from the table.", "start": 1620.2, "duration": 6.28}, {"text": "And you might say,\nwell, it's really there.", "start": 1626.48, "duration": 6.09}, {"text": "It's just on the 23rd row.", "start": 1632.57, "duration": 4.68}, {"text": "You just didn't get\nto it and your slide.", "start": 1637.25, "duration": 2.79}, {"text": "But it's there.", "start": 1640.04, "duration": 1.14}, {"text": "Well, I'm going to say I\nknow it's not in the 23rd row", "start": 1641.18, "duration": 3.18}, {"text": "because it differs from\nthe number in the 23rd row", "start": 1644.36, "duration": 2.91}, {"text": "at the 23rd digit because\nI constructed it that way.", "start": 1647.27, "duration": 4.65}, {"text": "This number is\ndesigned explicitly", "start": 1651.92, "duration": 2.82}, {"text": "to be different from each of\nthe numbers that's on the list.", "start": 1654.74, "duration": 2.94}, {"text": "So it can't be on\nthe list because it's", "start": 1661.255, "duration": 1.625}, {"text": "been constructed to be\ndifferent from each of them", "start": 1662.88, "duration": 1.8}, {"text": "in at least one place.", "start": 1664.68, "duration": 0.917}, {"text": "So I think this is a\nbeautiful argument,", "start": 1668.29, "duration": 2.12}, {"text": "and it shows that\nno matter how you", "start": 1670.41, "duration": 3.09}, {"text": "try to come up with a\none-to-one correspondence,", "start": 1673.5, "duration": 4.08}, {"text": "you're going to fail.", "start": 1677.58, "duration": 1.41}, {"text": "You might say, oh,\nlike, just one number?", "start": 1678.99, "duration": 4.79}, {"text": "I did so much hard work.", "start": 1683.78, "duration": 1.003}, {"text": "I'm missing just one number.", "start": 1684.783, "duration": 1.167}, {"text": "Can't I get partial credit\nand put this one on the list", "start": 1685.95, "duration": 3.695}, {"text": "now I fixed it?", "start": 1689.645, "duration": 0.625}, {"text": "No.", "start": 1690.27, "duration": 0.36}, {"text": "Obviously, there are many,\nmany numbers that are missing.", "start": 1690.63, "duration": 2.1}, {"text": "If you put this\none on the list, I", "start": 1692.73, "duration": 1.425}, {"text": "was going to construct another\nnumber that was missing.", "start": 1694.155, "duration": 2.415}, {"text": "So there's just not no\npossibility of fixing this.", "start": 1696.57, "duration": 7.74}, {"text": "And in fact, the real\nnumbers are uncountable--", "start": 1704.31, "duration": 5.013}, {"text": "cannot be matched up\nwith the natural numbers.", "start": 1709.323, "duration": 1.917}, {"text": "And there's nothing.", "start": 1711.24, "duration": 1.77}, {"text": "It doesn't even come close.", "start": 1713.01, "duration": 1.125}, {"text": "So here, a summary\nof what I just said.", "start": 1716.65, "duration": 4.4}, {"text": "F is not a one-to-one\ncorrespondence", "start": 1721.05, "duration": 1.582}, {"text": "no matter what you try to do.", "start": 1722.632, "duration": 1.208}, {"text": "You can't come up with a\none-to-one correspondence.", "start": 1723.84, "duration": 2.125}, {"text": "That's the contradiction.", "start": 1725.965, "duration": 2.935}, {"text": "So that proves that\nR is uncountable.", "start": 1728.9, "duration": 4.41}, {"text": "And that's why, by the way,\nwe call this a diagonalization", "start": 1733.31, "duration": 3.45}, {"text": "because we're going\ndown this diagonal here", "start": 1736.76, "duration": 2.19}, {"text": "that's where the term-- that's\nwhere the name is come from.", "start": 1738.95, "duration": 4.04}, {"text": "OK, so I'm happy to--", "start": 1742.99, "duration": 5.73}, {"text": "[CHUCKLES] OK, that's a--", "start": 1748.72, "duration": 2.52}, {"text": "somebody's asking me--\n[CHUCKLES] I'm actually--", "start": 1754.26, "duration": 3.0}, {"text": "I have a-- is it here?", "start": 1757.26, "duration": 3.31}, {"text": "I can't remember if it's here.", "start": 1760.57, "duration": 1.25}, {"text": "I have a check-in\ncoming about this.", "start": 1761.82, "duration": 3.27}, {"text": "And somebody is\nanticipating that", "start": 1765.09, "duration": 2.55}, {"text": "by asking the actually rather\nfamous question about there", "start": 1767.64, "duration": 3.81}, {"text": "being a possible infinity in\nbetween the natural numbers", "start": 1771.45, "duration": 5.1}, {"text": "and the real numbers.", "start": 1776.55, "duration": 1.435}, {"text": "We know the real\nnumbers now are bigger", "start": 1777.985, "duration": 1.625}, {"text": "than the natural numbers.", "start": 1779.61, "duration": 1.047}, {"text": "But is there something\nthat's an in between?", "start": 1780.657, "duration": 1.833}, {"text": "I mean, this is a very,\nvery famous problem,", "start": 1782.49, "duration": 3.51}, {"text": "which I'll talk about when\nwe get to our check-in, which", "start": 1786.0, "duration": 3.09}, {"text": "is coming up.", "start": 1789.09, "duration": 0.72}, {"text": "Somebody is objecting\njust the way", "start": 1795.02, "duration": 1.92}, {"text": "I've defined x using\nthis procedure that", "start": 1796.94, "duration": 2.76}, {"text": "really can't, in a sense, state,\nyou know-- but x is a number.", "start": 1799.7, "duration": 4.0}, {"text": "X is the result of\nwhat this procedure is.", "start": 1803.7, "duration": 4.28}, {"text": "Following this procedure,\nwe're converging", "start": 1807.98, "duration": 1.75}, {"text": "on some particular value.", "start": 1809.73, "duration": 1.53}, {"text": "And so that is a value.", "start": 1811.26, "duration": 2.43}, {"text": "If you want, we can make a more\nprecise way of determining.", "start": 1813.69, "duration": 4.682}, {"text": "We don't have the\nflexibility of choosing", "start": 1818.372, "duration": 1.708}, {"text": "the way I did in my example.", "start": 1820.08, "duration": 1.54}, {"text": "We can make a precise\nprocedure for coming up", "start": 1821.62, "duration": 3.23}, {"text": "with these digits.", "start": 1824.85, "duration": 0.97}, {"text": "But I don't think there's\nanybody thinks there's anything", "start": 1825.82, "duration": 3.255}, {"text": "that's--", "start": 1829.075, "duration": 0.5}, {"text": "there's any shortcoming\nin this argument in terms", "start": 1832.46, "duration": 3.24}, {"text": "of the way we're defining x.", "start": 1835.7, "duration": 1.77}, {"text": "I think it's worth understanding\nthis because it's really going", "start": 1837.47, "duration": 3.39}, {"text": "to set the groundwork for our\nproof that A,TM is undecidable,", "start": 1840.86, "duration": 6.57}, {"text": "which is a little,\nI think, perhaps,", "start": 1847.43, "duration": 1.59}, {"text": "slightly more abstract in a\nway in the way it sort of comes", "start": 1849.02, "duration": 3.33}, {"text": "across.", "start": 1852.35, "duration": 0.87}, {"text": "I'm going to try\nto connect the two.", "start": 1853.22, "duration": 3.03}, {"text": "But I think it's\nhelpful to understand", "start": 1856.25, "duration": 2.07}, {"text": "at least this argument\nbecause this argument is", "start": 1858.32, "duration": 2.22}, {"text": "diagonalization in\nits most raw form.", "start": 1860.54, "duration": 3.93}, {"text": "All right, I think we're good.", "start": 1869.6, "duration": 1.69}, {"text": "So why don't we continue, then?", "start": 1871.29, "duration": 2.93}, {"text": "So there are a\nnumber of corollaries", "start": 1874.22, "duration": 2.34}, {"text": "that follow from the statement\nthat the real numbers is", "start": 1876.56, "duration": 3.12}, {"text": "uncountable.", "start": 1879.68, "duration": 0.81}, {"text": "First of all, if\nyou let script L", "start": 1883.91, "duration": 3.27}, {"text": "be the collection\nof all languages,", "start": 1887.18, "duration": 2.305}, {"text": "if you want to consider it\nover some particular alphabet,", "start": 1889.485, "duration": 2.375}, {"text": "that's fine.", "start": 1891.86, "duration": 0.51}, {"text": "But that's not going to be\nreally important for this point", "start": 1892.37, "duration": 6.64}, {"text": "that I'm going to make.", "start": 1899.01, "duration": 1.59}, {"text": "So script L is the\ncollection of all languages.", "start": 1900.6, "duration": 5.94}, {"text": "So every subset of sigma star--", "start": 1906.54, "duration": 3.74}, {"text": "every subset of sigma\nstar is a language.", "start": 1910.28, "duration": 2.97}, {"text": "So look at all those\npossible subsets.", "start": 1913.25, "duration": 3.51}, {"text": "So that includes 0\nto the k, 1 to the k,", "start": 1916.76, "duration": 2.28}, {"text": "plus every other language you\ncan ever think of and more--", "start": 1919.04, "duration": 2.82}, {"text": "all possible languages.", "start": 1921.86, "duration": 2.87}, {"text": "So the collection of all\nlanguages is uncountable.", "start": 1924.73, "duration": 4.38}, {"text": "There's uncountably many\ndifferent languages out there.", "start": 1929.11, "duration": 3.0}, {"text": "I don't want to\nbelabor this point.", "start": 1932.11, "duration": 1.832}, {"text": "You can just take this\nif you don't quite", "start": 1933.942, "duration": 1.708}, {"text": "follow the quick argument\nI'm going to make here.", "start": 1935.65, "duration": 2.88}, {"text": "But you can make a\none-to-one correspondence", "start": 1938.53, "duration": 4.89}, {"text": "from all languages to the\nreals so that each language", "start": 1943.42, "duration": 4.23}, {"text": "gets its own real number.", "start": 1947.65, "duration": 2.678}, {"text": "And the way I'm going to\nthink about that-- let's", "start": 1950.328, "duration": 2.042}, {"text": "put the real numbers\ninto binary form.", "start": 1952.37, "duration": 2.85}, {"text": "And if you imagine\nhere being sigma star,", "start": 1955.22, "duration": 2.37}, {"text": "all of the possible\nstrings of sigma", "start": 1957.59, "duration": 1.68}, {"text": "star written out in\ntheir standard order.", "start": 1959.27, "duration": 4.29}, {"text": "And now if you have\na language here, A,", "start": 1963.56, "duration": 1.94}, {"text": "it's just some\narbitrary language.", "start": 1965.5, "duration": 2.35}, {"text": "So that's going to have some\nof the strings of sigma star", "start": 1967.85, "duration": 4.36}, {"text": "appearing.", "start": 1972.21, "duration": 0.5}, {"text": "Like 0 is appearing,\nbut 1 is not appearing.", "start": 1972.71, "duration": 2.64}, {"text": "00 is appearing and 01,\nbut not these three.", "start": 1975.35, "duration": 3.24}, {"text": "And I'm going to associate\nto A, my language,", "start": 1981.27, "duration": 3.33}, {"text": "some real number in binary by\nputting in a 0 in the decimal", "start": 1984.6, "duration": 6.84}, {"text": "representation-- well,\nthe binary representation,", "start": 1991.44, "duration": 2.46}, {"text": "I should say--", "start": 1993.9, "duration": 1.71}, {"text": "for that string if it's not\nthere and a 1 if it is there.", "start": 1995.61, "duration": 5.16}, {"text": "And so a real number--\nbecause there's infinitely", "start": 2004.45, "duration": 3.41}, {"text": "many yes/no choices in\nthe binary representation", "start": 2007.86, "duration": 4.02}, {"text": "can represent a\nlanguage because of each", "start": 2011.88, "duration": 3.21}, {"text": "of the yes/no choices of a\nstring being present or not.", "start": 2015.09, "duration": 3.15}, {"text": "I'm going to put a 1 for\nwhen the string is present,", "start": 2018.24, "duration": 2.21}, {"text": "a 0 when it's not present.", "start": 2020.45, "duration": 1.3}, {"text": "So each language has\nits own real number,", "start": 2021.75, "duration": 3.072}, {"text": "and each real number\nis going to be", "start": 2024.822, "duration": 1.458}, {"text": "associated to its own language.", "start": 2026.28, "duration": 2.22}, {"text": "Here, I'm restricting myself to\nreal numbers between 0 and 1.", "start": 2028.5, "duration": 3.92}, {"text": "That's not going to\nbe an essential point.", "start": 2032.42, "duration": 1.75}, {"text": "So let's not get\nhung up on that.", "start": 2034.17, "duration": 1.71}, {"text": "So the fact that the\nlanguages can be putting it", "start": 2040.05, "duration": 10.629}, {"text": "into a one-to-one correspondence\nwith the real numbers", "start": 2050.679, "duration": 3.75}, {"text": "shows that the collection of all\nlanguages is also uncountable.", "start": 2054.429, "duration": 4.801}, {"text": "Now another observation here--", "start": 2059.23, "duration": 2.67}, {"text": "another point worth noting is\nthat if you look at sigma star", "start": 2061.9, "duration": 3.15}, {"text": "itself, the strings,\nall possible strings,", "start": 2065.05, "duration": 5.41}, {"text": "that's a countable set.", "start": 2070.46, "duration": 2.42}, {"text": "The collection of all possible\nlanguages is uncountable.", "start": 2072.88, "duration": 2.46}, {"text": "But the collection of all\npossible finite strings", "start": 2075.34, "duration": 3.0}, {"text": "is countable because\nI can just list them.", "start": 2078.34, "duration": 2.93}, {"text": "Here's my list of all\npossible strings, which", "start": 2081.27, "duration": 2.1}, {"text": "you can put into a\ntable if you like", "start": 2083.37, "duration": 1.5}, {"text": "to think of it matching up\nwith the natural numbers", "start": 2084.87, "duration": 2.125}, {"text": "in that way.", "start": 2086.995, "duration": 2.605}, {"text": "Now I'm trying to\nmake a point here,", "start": 2089.6, "duration": 2.41}, {"text": "which is that if you take M,\nwhich is all possible Turing", "start": 2092.01, "duration": 3.62}, {"text": "machines-- script M\nis all possible Turing", "start": 2095.63, "duration": 2.4}, {"text": "machines-- the collection of\nall possible Turing machines", "start": 2098.03, "duration": 2.82}, {"text": "is countable.", "start": 2100.85, "duration": 1.52}, {"text": "There are only countably many\ndifferent Turing machines.", "start": 2102.37, "duration": 4.14}, {"text": "And you can argue that in all\nsorts of messy different ways.", "start": 2106.51, "duration": 2.707}, {"text": "But I think the most\nsimple way to see", "start": 2109.217, "duration": 1.583}, {"text": "that is to think about\neach Turing machine", "start": 2110.8, "duration": 3.3}, {"text": "as having a description,\nwhich is a string.", "start": 2114.1, "duration": 5.13}, {"text": "So the collection of all\ndescriptions of Turing machines", "start": 2119.23, "duration": 3.3}, {"text": "is just a subset of sigma\nstar, which we already", "start": 2122.53, "duration": 2.91}, {"text": "know is countable.", "start": 2125.44, "duration": 1.29}, {"text": "And the subset of any countable\nset is going to be countable.", "start": 2126.73, "duration": 5.48}, {"text": "So anyway, I think\nit's worth remembering", "start": 2132.21, "duration": 9.04}, {"text": "that the collection of old\nTuring machines is countable.", "start": 2141.25, "duration": 5.64}, {"text": "Whereas the collection of\nall languages is uncountable.", "start": 2146.89, "duration": 2.4}, {"text": "And that tells you right there\nthat some language is not", "start": 2149.29, "duration": 3.6}, {"text": "decided because there are more\nlanguages than Turing machines.", "start": 2152.89, "duration": 6.94}, {"text": "We've unaccountably\nmany language, only", "start": 2159.83, "duration": 2.27}, {"text": "countably many Turing machines,\nso that's fewer Turing machines", "start": 2162.1, "duration": 3.37}, {"text": "than languages.", "start": 2165.47, "duration": 1.02}, {"text": "There's no way to map all the\nlanguages onto Turing machines.", "start": 2166.49, "duration": 2.908}, {"text": "So there's going to always\nbe some languages that", "start": 2169.398, "duration": 2.042}, {"text": "got unmapped.", "start": 2171.44, "duration": 1.89}, {"text": "And so, in particular,\nthere are going", "start": 2173.33, "duration": 1.988}, {"text": "to be some languages\nwhich are undecidable.", "start": 2175.318, "duration": 1.792}, {"text": "There are going to be\nsome languages which", "start": 2177.11, "duration": 1.75}, {"text": "are not Turing-recognizable.", "start": 2178.86, "duration": 1.22}, {"text": "And anything based on some\nautomata kind of a definition", "start": 2180.08, "duration": 5.37}, {"text": "process is going to be some\nlanguages that they're not", "start": 2185.45, "duration": 3.277}, {"text": "going to be defined.", "start": 2188.727, "duration": 0.833}, {"text": "OK, now what this\ncorollary shows you", "start": 2192.16, "duration": 2.58}, {"text": "that there is some language out\nthere, which is not decidable.", "start": 2194.74, "duration": 3.21}, {"text": "What we're going to\nshow next is that there", "start": 2197.95, "duration": 2.25}, {"text": "is a specific language--", "start": 2200.2, "duration": 1.41}, {"text": "A,TM-- which is not decidable.", "start": 2201.61, "duration": 3.08}, {"text": "And but first, I think we\nhave a check-in coming up.", "start": 2204.69, "duration": 3.24}, {"text": "And let me give you a\nlittle bit of background", "start": 2207.93, "duration": 2.643}, {"text": "here because this is\nrelevant to this question", "start": 2210.573, "duration": 1.917}, {"text": "that I got about\nintermediate size sets.", "start": 2212.49, "duration": 3.06}, {"text": "So the question of\nwhether there is", "start": 2215.55, "duration": 2.55}, {"text": "a set of size between\nthe natural numbers", "start": 2218.1, "duration": 3.27}, {"text": "and the real numbers\nstrictly in between--", "start": 2221.37, "duration": 3.21}, {"text": "so bigger than the natural\nnumbers, not the same size", "start": 2224.58, "duration": 2.52}, {"text": "as the natural numbers,\nbut not the same size", "start": 2227.1, "duration": 2.22}, {"text": "as the real numbers either,\nbut in between in size.", "start": 2229.32, "duration": 5.67}, {"text": "That was Hilbert's question\nnumber 1 out of his list of 23", "start": 2234.99, "duration": 3.51}, {"text": "that I talked about\na few lectures back.", "start": 2238.5, "duration": 3.277}, {"text": "It's interesting that he put\nit as number 1 in that very", "start": 2241.777, "duration": 2.333}, {"text": "privileged special place because\nI know Hilbert was very--", "start": 2244.11, "duration": 4.86}, {"text": "he felt that the\nunderstanding infinity", "start": 2248.97, "duration": 1.95}, {"text": "was a really central\nissue in mathematics.", "start": 2250.92, "duration": 2.248}, {"text": "And that if we can't answer\na question like this,", "start": 2253.168, "duration": 2.042}, {"text": "we don't really\nunderstand infinity.", "start": 2255.21, "duration": 3.198}, {"text": "You want to understand what\nkind of sizes of infinities", "start": 2258.408, "duration": 2.292}, {"text": "are there.", "start": 2260.7, "duration": 0.752}, {"text": "We know there's the\nreal number is bigger", "start": 2261.452, "duration": 1.708}, {"text": "than the natural numbers.", "start": 2263.16, "duration": 1.8}, {"text": "Is there something in between?", "start": 2264.96, "duration": 1.38}, {"text": "So fundamental, really.", "start": 2266.34, "duration": 2.46}, {"text": "But it was shown during the\ncourse of the 20th century,", "start": 2268.8, "duration": 3.45}, {"text": "really, in two separate steps--\none in the 1930s by Godel,", "start": 2272.25, "duration": 2.73}, {"text": "one in the early\n1970s by Cohen--", "start": 2274.98, "duration": 2.76}, {"text": "that we can't\nanswer this question", "start": 2277.74, "duration": 2.1}, {"text": "by using the standard\naxioms of mathematics.", "start": 2279.84, "duration": 4.6}, {"text": "The answer can go either way.", "start": 2284.44, "duration": 3.31}, {"text": "And both of them are consistent\nwith the axes mathematics.", "start": 2287.75, "duration": 3.57}, {"text": "So you're never going\nto be able to prove", "start": 2291.32, "duration": 1.94}, {"text": "that there is a set\nwhose size is in between", "start": 2293.26, "duration": 5.4}, {"text": "or that there is no such set.", "start": 2298.66, "duration": 1.68}, {"text": "It's just impossible to\nprove either way using", "start": 2300.34, "duration": 3.545}, {"text": "the standard axiom of\nmathematics, which, actually,", "start": 2303.885, "duration": 2.125}, {"text": "is kind of remarkable.", "start": 2306.01, "duration": 3.137}, {"text": "And so my question for you\nis-- and this is really", "start": 2309.147, "duration": 2.083}, {"text": "a philosophical question,\nnot one that is directly you", "start": 2311.23, "duration": 4.07}, {"text": "need to know about\nmaterial in the course.", "start": 2315.3, "duration": 1.75}, {"text": "I think it's just a matter\nof your own interest.", "start": 2317.05, "duration": 2.488}, {"text": "I hope you find it interesting.", "start": 2319.538, "duration": 1.292}, {"text": "If you don't, you can\njust answer it randomly.", "start": 2320.83, "duration": 3.96}, {"text": "But what's going on here\nthat we can't answer", "start": 2324.79, "duration": 5.47}, {"text": "that question\nabout whether there", "start": 2330.26, "duration": 1.89}, {"text": "is a set of intermediate size?", "start": 2332.15, "duration": 1.89}, {"text": "Is it because our axioms for\nmathematics are inadequate?", "start": 2334.04, "duration": 3.93}, {"text": "Or maybe there is no such thing\nas a mathematical reality.", "start": 2337.97, "duration": 4.5}, {"text": "You can talk about\nwhat's real here?", "start": 2342.47, "duration": 2.95}, {"text": "What's the reality?", "start": 2345.42, "duration": 1.7}, {"text": "Either there is a set\nin between or not.", "start": 2347.12, "duration": 2.785}, {"text": "If you can imagine,\nall of these things", "start": 2349.905, "duration": 1.625}, {"text": "have their own reality\nto them, well, then,", "start": 2351.53, "duration": 2.67}, {"text": "there's going to be an answer.", "start": 2354.2, "duration": 1.267}, {"text": "And then you would\nexpect, well, maybe", "start": 2355.467, "duration": 1.583}, {"text": "we can find better axioms,\nwhich will actually", "start": 2357.05, "duration": 1.917}, {"text": "give us that answer.", "start": 2358.967, "duration": 0.873}, {"text": "Or you can say, well,\nthere is no reality.", "start": 2359.84, "duration": 2.7}, {"text": "And infinite sets are\nkind of human constructs", "start": 2362.54, "duration": 4.68}, {"text": "anyway, so we can make them kind\nof play out any way we like.", "start": 2367.22, "duration": 2.91}, {"text": "Mathematicians argue\nabout that to this day.", "start": 2370.13, "duration": 4.44}, {"text": "And it is, as I say, really,\nit's a philosophical question.", "start": 2374.57, "duration": 3.72}, {"text": "But just out of curiosity,\nlet's see how you guys end up", "start": 2378.29, "duration": 3.87}, {"text": "deciding on that one.", "start": 2382.16, "duration": 2.11}, {"text": "So here is the poll.", "start": 2384.27, "duration": 2.04}, {"text": "5 seconds to go.", "start": 2386.31, "duration": 1.14}, {"text": "Please vote.", "start": 2387.45, "duration": 1.48}, {"text": "And we're going to end\nthe polling, 1, 2, 3, now.", "start": 2388.93, "duration": 3.44}, {"text": "All right, here we go.", "start": 2395.2, "duration": 1.87}, {"text": "So there's no right answer.", "start": 2397.07, "duration": 3.15}, {"text": "I think if most mathematicians\nwere to, I think,", "start": 2400.22, "duration": 5.0}, {"text": "the instinct of most\nlogicians, especially, I'm", "start": 2405.22, "duration": 2.61}, {"text": "not sure if general\nmathematicians really even care", "start": 2407.83, "duration": 2.4}, {"text": "about this question, but\nlogicians would probably", "start": 2410.23, "duration": 3.27}, {"text": "have an instinct that, probably,\nthere are sets in between.", "start": 2413.5, "duration": 3.55}, {"text": "There's no reason that\nthere shouldn't be.", "start": 2417.05, "duration": 3.02}, {"text": "It seems kind of\nstrange that there", "start": 2420.07, "duration": 1.77}, {"text": "should be this sort of jump\nfrom the natural numbers", "start": 2421.84, "duration": 4.44}, {"text": "to the real numbers and\nwhy nothing in between?", "start": 2426.28, "duration": 4.81}, {"text": "But I don't think that\nquestion is really settled.", "start": 2431.09, "duration": 5.505}, {"text": "All right, let's continue on.", "start": 2439.962, "duration": 1.208}, {"text": "I think we are--", "start": 2441.17, "duration": 2.46}, {"text": "OK, so we're going to--", "start": 2443.63, "duration": 1.08}, {"text": "our coffee break is coming,\nin case you're wondering.", "start": 2447.872, "duration": 2.208}, {"text": "So this is my last\nslide before then.", "start": 2450.08, "duration": 1.71}, {"text": "But this is an important\nslide, so please hold out.", "start": 2451.79, "duration": 6.09}, {"text": "So here is our promised\ntheorem of the day.", "start": 2457.88, "duration": 4.29}, {"text": "I'm going to show that\nA,TM is not decidable,", "start": 2462.17, "duration": 2.34}, {"text": "the acceptance problem\nfor Turing machines.", "start": 2464.51, "duration": 2.77}, {"text": "And it's all going to be\ncontained on this one slide.", "start": 2467.28, "duration": 2.877}, {"text": "We're going to give a\nproof by contradiction", "start": 2470.157, "duration": 1.833}, {"text": "using diagonalization.", "start": 2471.99, "duration": 3.0}, {"text": "And we're going to assume\nsome Turing machine, H,", "start": 2474.99, "duration": 3.63}, {"text": "decides A,TM.", "start": 2478.62, "duration": 2.74}, {"text": "And we're going to\nget a contradiction.", "start": 2481.36, "duration": 2.94}, {"text": "So let's, first of\nall, make sure we", "start": 2484.3, "duration": 1.5}, {"text": "understand what H is doing.", "start": 2485.8, "duration": 3.1}, {"text": "So H gets an input--", "start": 2488.9, "duration": 1.44}, {"text": "a Turing machine and an input.", "start": 2490.34, "duration": 3.57}, {"text": "And H is going to be a\ndecider, so it always", "start": 2493.91, "duration": 3.27}, {"text": "halts with an\naccept or a reject.", "start": 2497.18, "duration": 3.65}, {"text": "It's supposed to accept if\nM accepts w and reject if it", "start": 2500.83, "duration": 5.0}, {"text": "doesn't.", "start": 2505.83, "duration": 0.91}, {"text": "So in other words,\nit's going to reject", "start": 2506.74, "duration": 1.88}, {"text": "if M rejects w either by\nhalting or by looping.", "start": 2508.62, "duration": 4.71}, {"text": "That's the job of H. And\nwe're assuming we can do that.", "start": 2516.34, "duration": 6.19}, {"text": "But we will see a\ncontradiction occur.", "start": 2522.53, "duration": 1.61}, {"text": "So the way we're going to do\nthat is really kind of just one", "start": 2526.9, "duration": 4.55}, {"text": "step here in a way.", "start": 2531.45, "duration": 1.77}, {"text": "And we're going to use H\nto construct another Turing", "start": 2533.22, "duration": 3.69}, {"text": "machine D. H is going to be a\nsubroutine to D. We've already", "start": 2536.91, "duration": 3.93}, {"text": "seen us doing that in the past.", "start": 2540.84, "duration": 1.65}, {"text": "D is going to do something\na little strange,", "start": 2545.17, "duration": 2.66}, {"text": "just to warn you.", "start": 2547.83, "duration": 3.22}, {"text": "D's input is just\na Turing machine--", "start": 2551.05, "duration": 2.3}, {"text": "no w.", "start": 2553.35, "duration": 2.82}, {"text": "And what D is going to\ndo using its subroutine", "start": 2556.17, "duration": 5.46}, {"text": "H is going to simulate H on\ninput M, comma, the description", "start": 2561.63, "duration": 6.59}, {"text": "of M. Now what is that?", "start": 2568.22, "duration": 3.64}, {"text": "Well, the description of\nM is just some string.", "start": 2571.86, "duration": 3.45}, {"text": "So what H is trying--", "start": 2575.31, "duration": 1.61}, {"text": "what it's asking H to\ntell, to answer is,", "start": 2576.92, "duration": 4.08}, {"text": "does M accept the\nstring representing", "start": 2581.0, "duration": 6.01}, {"text": "M's own description?", "start": 2587.01, "duration": 3.55}, {"text": "It's as if we're\nfeeding M into itself,", "start": 2590.56, "duration": 3.58}, {"text": "which seems like a totally\ntwisted thing to do,", "start": 2594.14, "duration": 6.67}, {"text": "you might say.", "start": 2600.81, "duration": 0.71}, {"text": "Why would you ever feed\na program into itself?", "start": 2605.12, "duration": 2.64}, {"text": "Somebody has written\ncannibalism here--", "start": 2610.37, "duration": 2.34}, {"text": "yeah, kind of.", "start": 2612.71, "duration": 1.77}, {"text": "I'd say it's worse\n[LAUGHS] because it's not", "start": 2614.48, "duration": 3.57}, {"text": "eating somebody else.", "start": 2618.05, "duration": 0.96}, {"text": "It's eating yourself.", "start": 2619.01, "duration": 2.89}, {"text": "OK, but I claim that there are\nactual cases in practice where", "start": 2621.9, "duration": 7.86}, {"text": "we do this.", "start": 2629.76, "duration": 2.18}, {"text": "We feed programs\ninto themselves.", "start": 2631.94, "duration": 4.56}, {"text": "And the example that I\nknow of where this is done", "start": 2636.5, "duration": 4.2}, {"text": "is when you're\nmaking a compiler.", "start": 2640.7, "duration": 3.33}, {"text": "You might want to\nmake a compiler", "start": 2644.03, "duration": 1.98}, {"text": "and then written in the same\nlanguage that it's compiling.", "start": 2646.01, "duration": 3.27}, {"text": "And then you feed the\ncompiler into itself.", "start": 2649.28, "duration": 3.66}, {"text": "You may say, why even\nbother because it's already,", "start": 2652.94, "duration": 2.43}, {"text": "obviously, if it,\nonce it's running,", "start": 2655.37, "duration": 1.63}, {"text": "you don't need to\ncompile it again.", "start": 2657.0, "duration": 4.84}, {"text": "But actually, an example\nwhere this was really used", "start": 2661.84, "duration": 3.42}, {"text": "was when there was an\noptimizing compiler, I think,", "start": 2665.26, "duration": 2.7}, {"text": "for C written on\nearly Unix machines.", "start": 2667.96, "duration": 3.98}, {"text": "And the optimizing compiler\nfor C was written in C.", "start": 2671.94, "duration": 5.08}, {"text": "So you would feed the\noptimizing compiler", "start": 2677.02, "duration": 4.26}, {"text": "into the regular\ncompiler, first of all.", "start": 2681.28, "duration": 2.67}, {"text": "Now, you had the\ncompiler running,", "start": 2683.95, "duration": 5.02}, {"text": "but it was an optimized.", "start": 2688.97, "duration": 2.37}, {"text": "So but now that it's the\noptimizing compiler is running,", "start": 2691.34, "duration": 3.12}, {"text": "you can feed the\noptimizing compiler", "start": 2694.46, "duration": 1.83}, {"text": "into that, which is itself.", "start": 2696.29, "duration": 1.75}, {"text": "Now you have an optimized\noptimizing compiler.", "start": 2698.04, "duration": 3.25}, {"text": "So it really makes some--", "start": 2701.29, "duration": 2.07}, {"text": "there is at least one case\nwhere this has actually", "start": 2703.36, "duration": 2.97}, {"text": "been done in practice.", "start": 2706.33, "duration": 3.29}, {"text": "Not that we really care.", "start": 2709.62, "duration": 1.11}, {"text": "This is a theory class-- but\njust for fun to observe that.", "start": 2710.73, "duration": 4.17}, {"text": "So here, H is trying to\nsay, well, does in a M end", "start": 2714.9, "duration": 3.98}, {"text": "up accepting when it's fed\nthe description of itself?", "start": 2718.88, "duration": 4.287}, {"text": "You know, at least,\nmathematically speaking,", "start": 2723.167, "duration": 1.833}, {"text": "that's a reasonable\nthing to ask.", "start": 2725.0, "duration": 2.61}, {"text": "And then what D is going to do\nwhen it gets the answer back", "start": 2727.61, "duration": 3.72}, {"text": "from H--", "start": 2731.33, "duration": 1.11}, {"text": "H is a decider, don't forget--", "start": 2732.44, "duration": 1.8}, {"text": "is D's going to do the\nopposite of whatever H does.", "start": 2734.24, "duration": 2.74}, {"text": "It's going to\naccept if H rejects", "start": 2736.98, "duration": 1.49}, {"text": "and reject if H accepts.", "start": 2738.47, "duration": 3.56}, {"text": "So let's, in summary,\nlet's pull this together,", "start": 2742.03, "duration": 3.02}, {"text": "so it's easy to understand,\nin the end, what is D doing?", "start": 2745.05, "duration": 6.18}, {"text": "D is going to accept.", "start": 2751.23, "duration": 2.01}, {"text": "D is also going to be\ndecider, by the way.", "start": 2753.24, "duration": 1.77}, {"text": "So D is always going to\neither accept or reject--", "start": 2755.01, "duration": 2.74}, {"text": "Just.", "start": 2757.75, "duration": 0.5}, {"text": "The opposite of what\nH tells it to do.", "start": 2758.25, "duration": 1.9}, {"text": "So D is going to accept M\nexactly when M doesn't accept M", "start": 2760.15, "duration": 6.21}, {"text": "because when M doesn't accept\nM, H is going to reject,", "start": 2766.36, "duration": 6.07}, {"text": "and so then D is\ngoing to accept.", "start": 2772.43, "duration": 1.92}, {"text": "So D accepts M if and\nonly if M doesn't accept", "start": 2774.35, "duration": 3.14}, {"text": "M. That's exactly the\ncondition in which D accepts", "start": 2777.49, "duration": 2.64}, {"text": "M. I think it's important to\njust step back and make sure", "start": 2780.13, "duration": 4.12}, {"text": "you understand this line\nbecause we have only one line", "start": 2784.25, "duration": 2.43}, {"text": "left to get our contradiction.", "start": 2786.68, "duration": 4.98}, {"text": "Right?", "start": 2791.66, "duration": 0.5}, {"text": "Are we together?", "start": 2792.16, "duration": 0.69}, {"text": "D accepts M if and\nonly if M doesn't", "start": 2792.85, "duration": 2.22}, {"text": "accept M. That's just by the\nway we've defined setup D.", "start": 2795.07, "duration": 3.7}, {"text": "Now what we're going to do is\nfeed in, instead of M to D,", "start": 2798.77, "duration": 5.02}, {"text": "and not some arbitrary feed,\nwe're going to use feed in D", "start": 2803.79, "duration": 3.11}, {"text": "into D. And that is going\nto be our contradiction", "start": 2806.9, "duration": 3.86}, {"text": "because D is now\ngoing to accept D if", "start": 2810.76, "duration": 2.49}, {"text": "and only if D doesn't\naccept D, and that's", "start": 2813.25, "duration": 2.01}, {"text": "certainly impossible.", "start": 2815.26, "duration": 1.415}, {"text": "That's our contradiction which\nproves that H cannot exist.", "start": 2819.52, "duration": 3.93}, {"text": "And therefore, A,TM\nis undecidable.", "start": 2823.45, "duration": 3.82}, {"text": "OK, so we're done,\nexcept for the one", "start": 2827.27, "duration": 2.79}, {"text": "point, which is that why\nis this a diagonalization?", "start": 2830.06, "duration": 6.03}, {"text": "And I think you can get that\nfrom the following picture.", "start": 2836.09, "duration": 4.71}, {"text": "If you imagine here writing\ndown all possible times--", "start": 2840.8, "duration": 3.48}, {"text": "I'm going to make a table here.", "start": 2844.28, "duration": 1.33}, {"text": "Here is the list of\nall Turing machines,", "start": 2845.61, "duration": 4.88}, {"text": "including D, which\nis a machine which", "start": 2850.49, "duration": 2.37}, {"text": "I built under the\nassumption that H exists.", "start": 2852.86, "duration": 5.46}, {"text": "So D appears here somewhere.", "start": 2858.32, "duration": 1.47}, {"text": "But here are all the\nother Turing machines.", "start": 2859.79, "duration": 1.8}, {"text": "And here are all of these\ndescriptions of the Turing", "start": 2861.59, "duration": 2.167}, {"text": "machines along the labeling\nall of the columns.", "start": 2863.757, "duration": 4.103}, {"text": "OK, so these are\nthe rows labeled.", "start": 2867.86, "duration": 1.53}, {"text": "These are the column labels.", "start": 2869.39, "duration": 1.32}, {"text": "And inside, I'm\ngoing to tell you", "start": 2870.71, "duration": 3.39}, {"text": "the answer for whether a given\nmachine accepts a given input.", "start": 2874.1, "duration": 7.02}, {"text": "So for example, M1 accepts\nits own description", "start": 2881.12, "duration": 3.51}, {"text": "but rejects the\ndescription of M2,", "start": 2884.63, "duration": 2.37}, {"text": "but accepts the\ndescription of M3.", "start": 2887.0, "duration": 1.92}, {"text": "I don't you know.", "start": 2888.92, "duration": 0.84}, {"text": "I'm obviously I'm\nmaking this up.", "start": 2889.76, "duration": 1.2}, {"text": "I don't know what M1 is.", "start": 2890.96, "duration": 1.0}, {"text": "But just hypothetically, that's\nwhat the machine M1 does.", "start": 2891.96, "duration": 4.152}, {"text": "So I'm just filling\nout this table.", "start": 2896.112, "duration": 1.458}, {"text": "H could get the answer to any\nof the elements in this table", "start": 2901.632, "duration": 2.868}, {"text": "because it can test whether M4\naccepts the description of M3.", "start": 2904.5, "duration": 5.94}, {"text": "So H could fill out this table.", "start": 2910.44, "duration": 2.58}, {"text": "So maybe M2 is a machine\nthat always rejects.", "start": 2913.02, "duration": 2.7}, {"text": "It's a very unfriendly,\nrejecting machine.", "start": 2915.72, "duration": 2.85}, {"text": "M3 is a very friendly machine.", "start": 2918.57, "duration": 1.48}, {"text": "It accepts all inputs.", "start": 2920.05, "duration": 1.85}, {"text": "M4 rejects some and accepts\nothers, dot, dot, dot.", "start": 2921.9, "duration": 4.495}, {"text": "Now, I want to look and\nsee, what does D do?", "start": 2926.395, "duration": 4.715}, {"text": "Now based on the information\nthat I've already given you,", "start": 2931.11, "duration": 2.73}, {"text": "we can understand what D does.", "start": 2933.84, "duration": 3.4}, {"text": "So for example, what\ndoes D do when I", "start": 2937.24, "duration": 2.19}, {"text": "feed it the description of M1?", "start": 2939.43, "duration": 4.51}, {"text": "What does D do?", "start": 2943.94, "duration": 1.66}, {"text": "Well, we can look here.", "start": 2945.6, "duration": 1.01}, {"text": "D accepts M if and only\nif M doesn't accept M. So", "start": 2946.61, "duration": 3.6}, {"text": "D is going to accept M1 if and\nonly if M1 does not accept M1.", "start": 2950.21, "duration": 7.88}, {"text": "Well, M1 does accept M1.", "start": 2958.09, "duration": 2.4}, {"text": "So D does the opposite.", "start": 2960.49, "duration": 2.49}, {"text": "D rejects.", "start": 2962.98, "duration": 1.26}, {"text": "So OK, I'm highlighting here.", "start": 2964.24, "duration": 2.01}, {"text": "D rejects because\nD is going to do", "start": 2966.25, "duration": 3.0}, {"text": "the opposite of what the machine\ndoes on its own it on input.", "start": 2969.25, "duration": 3.54}, {"text": "So D on M2, you have to\nlook what M2 does on M2.", "start": 2975.37, "duration": 3.78}, {"text": "It rejects, so D does\nthe opposite of that.", "start": 2979.15, "duration": 2.41}, {"text": "It accepts.", "start": 2981.56, "duration": 1.66}, {"text": "And similarly, each\none of these things--", "start": 2983.22, "duration": 5.88}, {"text": "D's answer is going to\nbe the opposite of what", "start": 2989.1, "duration": 2.96}, {"text": "the machine does on\nits own description,", "start": 2992.06, "duration": 2.76}, {"text": "just by virtue of the\ndefinition of D. OK, and so on--", "start": 2994.82, "duration": 5.23}, {"text": "so far, so good.", "start": 3000.05, "duration": 1.49}, {"text": "But the problem is, what\nhappens when D is fed itself?", "start": 3001.54, "duration": 4.86}, {"text": "Because, as you can see,\nwe're heading for trouble", "start": 3006.4, "duration": 5.19}, {"text": "because this is a\ndiagonal down here.", "start": 3011.59, "duration": 2.4}, {"text": "D is just one of the rows.", "start": 3013.99, "duration": 1.35}, {"text": "That diagonal is going\nto intersect that row", "start": 3015.34, "duration": 2.55}, {"text": "at this point.", "start": 3017.89, "duration": 1.41}, {"text": "And D is defined to be going to\nbe doing the opposite of what", "start": 3019.3, "duration": 3.472}, {"text": "that element is, but it can't\nbe the opposite of itself.", "start": 3022.772, "duration": 2.333}, {"text": "And so that's the contradiction.", "start": 3027.98, "duration": 3.32}, {"text": "So I think we're--", "start": 3031.3, "duration": 3.44}, {"text": "I'm going to call us, give\nus a little break here.", "start": 3034.74, "duration": 4.45}, {"text": "And then you can also\ntext me in the meantime.", "start": 3039.19, "duration": 2.01}, {"text": "I'll be happy to answer\nsome questions during that.", "start": 3041.2, "duration": 2.66}, {"text": "A little over 4 minutes\nto go-- so let's see.", "start": 3043.86, "duration": 2.31}, {"text": "Let me see if I can\nanswer your questions.", "start": 3046.17, "duration": 3.6}, {"text": "OK, what's so special about\nA,TM that enables us to do this?", "start": 3049.77, "duration": 3.04}, {"text": "Why can't we do this\non ADFA, for example?", "start": 3052.81, "duration": 4.21}, {"text": "That's a good question.", "start": 3057.02, "duration": 2.47}, {"text": "And the answer is that, in a\nsense, we can do this on DFA.", "start": 3059.49, "duration": 15.4}, {"text": "I mean, I think this is,\nperhaps, a bit of a stretch.", "start": 3074.89, "duration": 3.03}, {"text": "But DFAs could not answer ADFA.", "start": 3077.92, "duration": 4.44}, {"text": "I mean, we could prove that in\nother ways as well by just--", "start": 3082.36, "duration": 4.338}, {"text": "we could use things\nlike the pumping lemma,", "start": 3086.698, "duration": 1.792}, {"text": "and it's not clear, even\nhow you'd formulate ADFA.", "start": 3088.49, "duration": 2.72}, {"text": "But what's important\nhere is that it's really", "start": 3091.21, "duration": 5.73}, {"text": "the model talking about\nitself that really", "start": 3096.94, "duration": 3.33}, {"text": "is where the problem comes up.", "start": 3100.27, "duration": 2.44}, {"text": "So if you try to push\nthis argument through", "start": 3102.71, "duration": 2.78}, {"text": "to show that ADFA is not\ndecidable by Turing machines,", "start": 3105.49, "duration": 3.9}, {"text": "you're going to fail\nbecause we're starting off", "start": 3109.39, "duration": 5.3}, {"text": "with a Turing machine.", "start": 3114.69, "duration": 1.44}, {"text": "And I think I'm going\nto confuse myself", "start": 3116.13, "duration": 3.93}, {"text": "if I try to just repeat it.", "start": 3120.06, "duration": 1.51}, {"text": "But you won't get\na contradiction", "start": 3121.57, "duration": 2.78}, {"text": "because the Turing machine\nis not a finite automaton.", "start": 3124.35, "duration": 2.758}, {"text": "OK.", "start": 3130.245, "duration": 0.5}, {"text": "Will this argument\nget into self-loops?", "start": 3133.87, "duration": 2.01}, {"text": "I don't see why it would--\nthere is some self-reference,", "start": 3135.88, "duration": 3.18}, {"text": "perhaps.", "start": 3139.06, "duration": 0.633}, {"text": "We're going to talk about\nthat a little later.", "start": 3139.693, "duration": 1.917}, {"text": "So we're going to come back and\nrevisit this argument in a week", "start": 3141.61, "duration": 5.19}, {"text": "or so when we talk about\nthe recursion theorem which", "start": 3146.8, "duration": 2.37}, {"text": "talks about machines that\ncan refer to themselves.", "start": 3149.17, "duration": 3.57}, {"text": "But this is a little\nbit of a head--", "start": 3152.74, "duration": 2.635}, {"text": "getting ahead of ourselves.", "start": 3155.375, "duration": 1.125}, {"text": "So somebody's commenting\non this reminding them", "start": 3156.5, "duration": 4.01}, {"text": "of the barber paradox,\nif you remember that,", "start": 3160.51, "duration": 2.64}, {"text": "which has some similarity.", "start": 3163.15, "duration": 4.972}, {"text": "There is a town\nin which there was", "start": 3168.122, "duration": 2.758}, {"text": "a barber which shaves every man\nwho doesn't shave themselves.", "start": 3170.88, "duration": 3.7}, {"text": "It seems he's a\nvery good barber.", "start": 3174.58, "duration": 1.79}, {"text": "So there are some people\nwho shave themselves.", "start": 3176.37, "duration": 2.34}, {"text": "And all the rest,\nthe barber shaves.", "start": 3178.71, "duration": 3.24}, {"text": "The question is, does\nthe barber shave himself?", "start": 3181.95, "duration": 2.61}, {"text": "Because he shaves only those\nmen who don't shave themselves.", "start": 3184.56, "duration": 3.09}, {"text": "So you've got a same\nkind of a contradiction.", "start": 3187.65, "duration": 2.24}, {"text": "There is a relationship there.", "start": 3189.89, "duration": 1.69}, {"text": "So someone wants to know, where\ndid we use the decidability?", "start": 3194.56, "duration": 2.84}, {"text": "So we used the decidability\nto come up with H.", "start": 3197.4, "duration": 5.81}, {"text": "Once we know that\nA,TM is decidable,", "start": 3203.21, "duration": 2.01}, {"text": "then we have that H function,\nand then we can build D.", "start": 3205.22, "duration": 4.38}, {"text": "So that's the chain of reason.", "start": 3209.6, "duration": 5.95}, {"text": "So you assume A,TM is decidable.", "start": 3215.55, "duration": 1.82}, {"text": "Then you have the\ndecider called H,", "start": 3217.37, "duration": 3.21}, {"text": "and then you can build\nD. Somebody wants", "start": 3220.58, "duration": 3.67}, {"text": "to see the previous slide.", "start": 3224.25, "duration": 3.1}, {"text": "What part of the\nslide do you want?", "start": 3227.35, "duration": 1.8}, {"text": "So I'll leave that up there.", "start": 3232.88, "duration": 2.348}, {"text": "Why can we apply the proof\nthat all Turing machines are", "start": 3235.228, "duration": 2.292}, {"text": "accountable to all languages?", "start": 3237.52, "duration": 1.5}, {"text": "Well, because Turing\nmachines have descriptions.", "start": 3239.02, "duration": 3.18}, {"text": "General languages don't\nhave descriptions.", "start": 3242.2, "duration": 2.64}, {"text": "And so that's why.", "start": 3244.84, "duration": 1.8}, {"text": "OK, the candle has\nburnt to the bottom,", "start": 3246.64, "duration": 3.21}, {"text": "and it's time to move on.", "start": 3249.85, "duration": 1.62}, {"text": "So now, let's look at\nthe acceptance problem", "start": 3254.2, "duration": 3.17}, {"text": "for queue automata.", "start": 3257.37, "duration": 3.01}, {"text": "I'll give you a queue automaton\non input, and I want to know,", "start": 3260.38, "duration": 3.66}, {"text": "does it accept the input?", "start": 3264.04, "duration": 1.41}, {"text": "Is that going to be decidable?", "start": 3265.45, "duration": 2.52}, {"text": "And you have your choices.", "start": 3267.97, "duration": 1.125}, {"text": "It's either yes, it is\ndecidable because these", "start": 3269.095, "duration": 3.735}, {"text": "are similar to pushdown\nautomata and APDA is decidable,", "start": 3272.83, "duration": 2.37}, {"text": "or no because yes\ncontradicts results", "start": 3275.2, "duration": 5.53}, {"text": "that we know at\nthis point, or we", "start": 3280.73, "duration": 1.83}, {"text": "don't have enough information\nto answer the question.", "start": 3282.56, "duration": 2.76}, {"text": "OK, let's put that up.", "start": 3285.32, "duration": 1.2}, {"text": "One second-- [LAUGHS]\nall right, that's it.", "start": 3289.1, "duration": 6.02}, {"text": "Ready, going, gone.", "start": 3295.12, "duration": 0.99}, {"text": "So yes, the answer is, well, no.", "start": 3300.18, "duration": 4.08}, {"text": "[LAUGHS] The answer\nis, indeed, the answer", "start": 3304.26, "duration": 2.64}, {"text": "is B. True, that\nqueue automata are", "start": 3306.9, "duration": 5.663}, {"text": "similar to pushdown automata,\nbut all these automata", "start": 3312.563, "duration": 2.167}, {"text": "are similar to each\nother, and that's not", "start": 3314.73, "duration": 2.01}, {"text": "going to be good enough.", "start": 3316.74, "duration": 1.0}, {"text": "What the homework\nhas asked you to do", "start": 3320.67, "duration": 1.88}, {"text": "is to show that you\ncan simulate Turing", "start": 3322.55, "duration": 1.71}, {"text": "machines with queue automata.", "start": 3324.26, "duration": 2.07}, {"text": "So if you can\nanswer the question", "start": 3326.33, "duration": 1.47}, {"text": "about whether the queue automata\nwill accept their input,", "start": 3327.8, "duration": 3.51}, {"text": "that would allow you to be\nable to answer questions", "start": 3331.31, "duration": 2.16}, {"text": "about whether Turing\nmachines accept their input.", "start": 3333.47, "duration": 2.34}, {"text": "And we just proved\nthat's not possible.", "start": 3335.81, "duration": 2.91}, {"text": "So it would be a contradiction\nif we could answer--", "start": 3338.72, "duration": 8.4}, {"text": "if we could decide A, queue, A.", "start": 3347.12, "duration": 3.36}, {"text": "Now, we have an example of\nan undecidable language.", "start": 3350.48, "duration": 2.34}, {"text": "Let's look at an example of\nan unrecognizable language.", "start": 3352.82, "duration": 3.39}, {"text": "Now A,TM is not going to serve\nthat purpose because A,TM is", "start": 3356.21, "duration": 12.35}, {"text": "Turing-recognizable, as we\npointed out by the universal", "start": 3368.56, "duration": 2.34}, {"text": "Turing machine.", "start": 3370.9, "duration": 1.57}, {"text": "So A,TM is undecidable, however.", "start": 3372.47, "duration": 2.87}, {"text": "How about an\nunrecognizable language?", "start": 3375.34, "duration": 3.18}, {"text": "For that, we will see that the\ncomplement of A,TM will serve.", "start": 3378.52, "duration": 5.5}, {"text": "So the complement of A,TM is\nneither decidable nor even", "start": 3384.02, "duration": 4.61}, {"text": "recognizable.", "start": 3388.63, "duration": 2.07}, {"text": "Now it's not\nTuring-recognizable.", "start": 3390.7, "duration": 2.13}, {"text": "And that's going to follow from\na pretty basic theorem that", "start": 3392.83, "duration": 3.0}, {"text": "connects recognizability and\ndecidability that I've put up", "start": 3395.83, "duration": 3.3}, {"text": "here on the screen,\nwhich is that if you have", "start": 3399.13, "duration": 2.49}, {"text": "a language where it\nand its complement", "start": 3401.62, "duration": 2.55}, {"text": "are both recognizable, then\nthe language turnout turns out", "start": 3404.17, "duration": 3.3}, {"text": "to be decidable.", "start": 3407.47, "duration": 0.75}, {"text": "In fact, a language and its\ncomplement are decidable.", "start": 3408.22, "duration": 2.24}, {"text": "But being decidable is\nclosed under complement,", "start": 3410.46, "duration": 2.56}, {"text": "so that's something\nyou should be aware.", "start": 3413.02, "duration": 3.2}, {"text": "But being-- OK.", "start": 3416.22, "duration": 3.5}, {"text": "We'll get to that in a minute.", "start": 3419.72, "duration": 1.25}, {"text": "But if-- so anyway, so\nif you have a language", "start": 3420.97, "duration": 6.63}, {"text": "and it's complement\nboth recognizable,", "start": 3427.6, "duration": 1.95}, {"text": "how do we know the\nlanguage is decidable?", "start": 3429.55, "duration": 2.91}, {"text": "So first of all, let's take\nthe two Turing machines,", "start": 3432.46, "duration": 2.43}, {"text": "M1 and M2 that recognizes\nA and A-complement.", "start": 3434.89, "duration": 5.49}, {"text": "And we're going to\nput those together", "start": 3440.38, "duration": 1.89}, {"text": "to get a decider for A. And\nthat's going to work like this.", "start": 3442.27, "duration": 6.31}, {"text": "It's going to be\ncalled T. So T says,", "start": 3448.58, "duration": 3.16}, {"text": "on input w, what\nit's going to do,", "start": 3451.74, "duration": 2.28}, {"text": "it's going to feed w\ninto M1 and M2 both.", "start": 3454.02, "duration": 3.36}, {"text": "A is the language,\nby the way, yes.", "start": 3460.582, "duration": 1.458}, {"text": "A is-- when I say it's\nTuring-recognizable,", "start": 3462.04, "duration": 3.24}, {"text": "you know, Turing-recognizable\nonly applies to languages.", "start": 3465.28, "duration": 2.91}, {"text": "So yes, A is often this symbol\nI'm going to use for languages,", "start": 3468.19, "duration": 4.11}, {"text": "sometimes for an automaton.", "start": 3472.3, "duration": 1.35}, {"text": "But A is typically\ngoing to be a language.", "start": 3473.65, "duration": 3.88}, {"text": "So now, I'm trying to\nmake T be a decider for A", "start": 3477.53, "duration": 9.448}, {"text": "from the recognizers\nfor A and A-complement.", "start": 3486.978, "duration": 3.252}, {"text": "So I'm going to\ntake an input to T", "start": 3490.23, "duration": 3.69}, {"text": "and feed it into both\nrecognizers, M1 and M2.", "start": 3493.92, "duration": 3.245}, {"text": "OK, I'm going to run\nthem in parallel.", "start": 3500.96, "duration": 3.27}, {"text": "What's nice is that\nbecause M2 recognizes", "start": 3504.23, "duration": 7.71}, {"text": "the complement of\nwhat M1 recognizes,", "start": 3511.94, "duration": 2.76}, {"text": "every string is going to be\naccepted either by M1 or by M2", "start": 3514.7, "duration": 5.37}, {"text": "because every string is either\nan A or an A-complement.", "start": 3520.07, "duration": 3.81}, {"text": "So if I run M1 and M2 on w until\none of the halts or one of them", "start": 3523.88, "duration": 4.91}, {"text": "accepts, I know I'm not\ngoing to run forever", "start": 3528.79, "duration": 2.61}, {"text": "because, eventually, one or\nthe other one have to accept.", "start": 3531.4, "duration": 4.46}, {"text": "So and then I got my answer\nbecause if M1 accepts,", "start": 3535.86, "duration": 2.747}, {"text": "then I know I'm in the language.", "start": 3538.607, "duration": 1.333}, {"text": "But if M2 accepts, I know I'm in\nthe complement of the language,", "start": 3539.94, "duration": 3.36}, {"text": "so I'm out of the language.", "start": 3543.3, "duration": 1.42}, {"text": "So if M1 accepts,\nthen T should accept.", "start": 3544.72, "duration": 2.42}, {"text": "But if M2 accepts,\nthen T should reject. ,", "start": 3547.14, "duration": 7.5}, {"text": "So that proves that nice little\ntheorem written at the top", "start": 3554.64, "duration": 5.7}, {"text": "in blue.", "start": 3560.34, "duration": 2.24}, {"text": "So I got my decider\nfor A built out", "start": 3562.58, "duration": 3.05}, {"text": "of the recognizers for\nA and A-complement.", "start": 3565.63, "duration": 3.27}, {"text": "Now immediately, it follows that\nthe complement of A,TM is not", "start": 3568.9, "duration": 5.31}, {"text": "Turing-recognizable because\nwe know that A,TM itself is", "start": 3574.21, "duration": 7.44}, {"text": "recognizable, but\nit's undecidable.", "start": 3581.65, "duration": 2.81}, {"text": "If the complement was\nalso recognizable,", "start": 3587.41, "duration": 2.82}, {"text": "then A,TM would be\ndecidable, but it isn't.", "start": 3590.23, "duration": 4.02}, {"text": "So when something is decidable,\neither it or its complement", "start": 3594.25, "duration": 3.99}, {"text": "have to be unrecognizable.", "start": 3598.24, "duration": 1.443}, {"text": "And in the case for A,TM, it has\nto be the complement because we", "start": 3599.683, "duration": 2.667}, {"text": "already showed that it is\nitself is recognizable.", "start": 3602.35, "duration": 3.18}, {"text": "So that's the proof of that.", "start": 3609.13, "duration": 1.48}, {"text": "So here is a little\npicture of the way", "start": 3610.61, "duration": 1.73}, {"text": "the world looks right now if\nyou have here, in the middle are", "start": 3612.34, "duration": 2.55}, {"text": "the decidable languages--\nso these are all", "start": 3614.89, "duration": 2.34}, {"text": "languages, this Venn\ndiagram of languages.", "start": 3617.23, "duration": 2.58}, {"text": "We showed earlier, the regular,\nthe context-free, decidable,", "start": 3619.81, "duration": 3.93}, {"text": "recognizable.", "start": 3623.74, "duration": 2.34}, {"text": "Here, I've got the recognizable\nand what I'm calling", "start": 3626.08, "duration": 3.09}, {"text": "the co-Turing recognizable.", "start": 3629.17, "duration": 4.12}, {"text": "This is the collection\nof all complements", "start": 3633.29, "duration": 3.06}, {"text": "of recognizable languages.", "start": 3636.35, "duration": 1.59}, {"text": "So A,TM bar, A,TM\ncomplement is the complement", "start": 3637.94, "duration": 3.6}, {"text": "of a recognizable language.", "start": 3641.54, "duration": 1.59}, {"text": "This region here are\nall the complements", "start": 3643.13, "duration": 5.16}, {"text": "of the recognizable languages\nor the so-called co-Turing", "start": 3648.29, "duration": 4.74}, {"text": "recognizable languages--\ncomplement of.", "start": 3653.03, "duration": 4.08}, {"text": "So A,TM is on this side.", "start": 3657.11, "duration": 1.77}, {"text": "A,TM-complement is on that side.", "start": 3658.88, "duration": 2.07}, {"text": "But if something's in both, by\nvirtue of this theorem here,", "start": 3660.95, "duration": 3.66}, {"text": "then it's decidable.", "start": 3664.61, "duration": 1.315}, {"text": "OK, last check-in for the day.", "start": 3668.6, "duration": 3.45}, {"text": "From what we learned so far,\nwhich closure properties", "start": 3672.05, "duration": 3.9}, {"text": "can we prove for the\nclass of deterring", "start": 3675.95, "duration": 2.25}, {"text": "recognizable languages?", "start": 3678.2, "duration": 2.52}, {"text": "Choose all that apply.", "start": 3680.72, "duration": 0.99}, {"text": "Well, as I say, you don't\nhave to get it right.", "start": 3684.73, "duration": 3.35}, {"text": "Let's not spend too\nmuch more time on this", "start": 3688.08, "duration": 2.17}, {"text": "because we'll talk a\nlittle bit about it.", "start": 3690.25, "duration": 4.49}, {"text": "Almost all-- its closed\nunder almost all of them,", "start": 3694.74, "duration": 2.257}, {"text": "but not all of them.", "start": 3696.997, "duration": 0.833}, {"text": "Because-- are we done here?", "start": 3705.38, "duration": 3.81}, {"text": "I think we're done--", "start": 3709.19, "duration": 2.24}, {"text": "5 seconds.", "start": 3711.43, "duration": 0.525}, {"text": "OK, here we go, ending polling.", "start": 3714.7, "duration": 1.425}, {"text": "I'm not sure what the meaning\nof [LAUGHS] deleting your answer", "start": 3720.16, "duration": 3.78}, {"text": "is here.", "start": 3723.94, "duration": 1.38}, {"text": "Everybody likes union, I guess.", "start": 3725.32, "duration": 1.92}, {"text": "They're closed under\nall of these operations", "start": 3727.24, "duration": 1.89}, {"text": "except complement.", "start": 3729.13, "duration": 1.36}, {"text": "But we just proved it's not\nclosed under complement,", "start": 3733.753, "duration": 2.167}, {"text": "so I'm a little puzzled by\nwhy we have so many votes", "start": 3735.92, "duration": 4.84}, {"text": "for closure under complement.", "start": 3740.76, "duration": 2.73}, {"text": "We have here, A,TM is\nTuring-recognizable,", "start": 3743.49, "duration": 3.6}, {"text": "but A,TM-complement is\nnot Turing-recognizable.", "start": 3747.09, "duration": 1.965}, {"text": "it's right here on the slide.", "start": 3749.055, "duration": 3.215}, {"text": "I'm not trying to\nmake you feel bad,", "start": 3752.27, "duration": 1.5}, {"text": "but I'm trying to just point\nout that you think, please.", "start": 3753.77, "duration": 8.05}, {"text": "So now closure under union\nand intersection, I mean,", "start": 3761.82, "duration": 6.2}, {"text": "you could kind of\nget those answers", "start": 3768.02, "duration": 1.56}, {"text": "just by running things\nin parallel the way", "start": 3769.58, "duration": 2.13}, {"text": "we did the proof here.", "start": 3771.71, "duration": 1.17}, {"text": "You just run both machines.", "start": 3772.88, "duration": 3.36}, {"text": "And if they both give--", "start": 3776.24, "duration": 5.803}, {"text": "I mean, it's a little\ntricky, I suppose.", "start": 3782.043, "duration": 1.667}, {"text": "If either one of them\naccept, then you can accept.", "start": 3783.71, "duration": 5.06}, {"text": "Or if they both\naccept, you just wait", "start": 3788.77, "duration": 1.59}, {"text": "until they both have accepted.", "start": 3790.36, "duration": 1.25}, {"text": "Otherwise, you\njust keep running.", "start": 3791.61, "duration": 1.85}, {"text": "So the first two are\npretty straightforward.", "start": 3793.46, "duration": 2.503}, {"text": "Closure under\nconcatenation-- this", "start": 3795.963, "duration": 1.417}, {"text": "is also going to be similar.", "start": 3797.38, "duration": 1.21}, {"text": "You just try every possible\nway of cutting the string up", "start": 3798.59, "duration": 3.23}, {"text": "into two pieces and\nrun in parallel on.", "start": 3801.82, "duration": 2.62}, {"text": "And if you ever find a\nway of cutting it up,", "start": 3804.44, "duration": 2.48}, {"text": "and you run those two, and put\nthose two sides in parallel,", "start": 3806.92, "duration": 2.62}, {"text": "and if they both accept,\nthen you can accept.", "start": 3809.54, "duration": 4.97}, {"text": "And star is, again,\nvery similar.", "start": 3814.51, "duration": 2.26}, {"text": "So these are not too bad.", "start": 3816.77, "duration": 2.57}, {"text": "But I admit, you know, it's\nnot a whole lot of time", "start": 3819.34, "duration": 2.46}, {"text": "to have to contend with\nsomething that you're just", "start": 3821.8, "duration": 2.94}, {"text": "getting used to.", "start": 3824.74, "duration": 2.25}, {"text": "So let's talk about the very\nlast topic of the day, which", "start": 3826.99, "duration": 6.28}, {"text": "is really going to be setting\nourselves up for Tuesday's", "start": 3833.27, "duration": 3.78}, {"text": "lecture next week.", "start": 3837.05, "duration": 1.86}, {"text": "And that's how we are going to\nbe showing other languages are", "start": 3838.91, "duration": 4.32}, {"text": "undecidable, which is something\nthat I'm going to be expecting", "start": 3843.23, "duration": 4.26}, {"text": "you guys to be able to do.", "start": 3847.49, "duration": 3.39}, {"text": "This is the standard\nprocedure for showing", "start": 3850.88, "duration": 5.07}, {"text": "languages are\nundecidable using what's", "start": 3855.95, "duration": 2.67}, {"text": "called the reducibility method.", "start": 3858.62, "duration": 2.77}, {"text": "And what that does is it\ntakes, as a starting point,", "start": 3861.39, "duration": 3.86}, {"text": "a language that we already\nknow is undecidable--", "start": 3865.25, "duration": 3.84}, {"text": "typically, A,TM-- or it\ncould be another one that", "start": 3869.09, "duration": 3.54}, {"text": "you've previously shown\nto be undecidable--", "start": 3872.63, "duration": 2.13}, {"text": "and leverages that information\nto show other languages are", "start": 3874.76, "duration": 3.3}, {"text": "undecidable.", "start": 3878.06, "duration": 1.98}, {"text": "And it's using what's\ncalled reducibility.", "start": 3880.04, "duration": 4.69}, {"text": "We're going to go into this\nmore carefully next time.", "start": 3884.73, "duration": 3.84}, {"text": "But basically,\nreducibility is a way", "start": 3888.57, "duration": 2.25}, {"text": "of using one problem to\nsolve another problem.", "start": 3890.82, "duration": 3.93}, {"text": "And so we are going to\nshow, for example, let's", "start": 3894.75, "duration": 3.38}, {"text": "take a look at\nthe problem called", "start": 3898.13, "duration": 2.25}, {"text": "the halting problem, which\nis like the famous problem", "start": 3900.38, "duration": 2.22}, {"text": "for Turing machines.", "start": 3902.6, "duration": 2.97}, {"text": "You just want to know whether\nit halts, not necessarily", "start": 3905.57, "duration": 2.67}, {"text": "whether it accepts.", "start": 3908.24, "duration": 1.5}, {"text": "So it's very similar,\nbut not exactly the same.", "start": 3909.74, "duration": 2.1}, {"text": "And we're going to show\nthat this halting problem is", "start": 3914.64, "duration": 4.04}, {"text": "similarly undecidable.", "start": 3918.68, "duration": 1.41}, {"text": "Now we could go back and do\nthe whole diagonalization,", "start": 3920.09, "duration": 2.43}, {"text": "but that seems like--", "start": 3922.52, "duration": 1.44}, {"text": "well, that's more work than\nnecessary now that we already", "start": 3923.96, "duration": 2.58}, {"text": "know A,TM is undecidable because\nwe're going to show that we can", "start": 3926.54, "duration": 4.29}, {"text": "reduce the A,TM problem\nto the halting problem.", "start": 3930.83, "duration": 6.0}, {"text": "And we'll explain what\nthat means again later.", "start": 3936.83, "duration": 2.68}, {"text": "But the idea is-- and as\nwe'll show in an illustration", "start": 3939.51, "duration": 4.55}, {"text": "shortly--", "start": 3944.06, "duration": 1.44}, {"text": "that by proving by contradiction\nif the if HALT,TM were", "start": 3945.5, "duration": 8.2}, {"text": "decidable, then A,TM\nwould be decided.", "start": 3953.7, "duration": 3.79}, {"text": "And we know A,TM\nis not decidable.", "start": 3957.49, "duration": 1.96}, {"text": "And so that's our contradiction.", "start": 3959.45, "duration": 5.16}, {"text": "Now the way we're going to show\nthat if HALT,TM is decidable,", "start": 3964.61, "duration": 7.98}, {"text": "then A,TM is decidable is use\na decider for HALT,TM to decide", "start": 3972.59, "duration": 5.7}, {"text": "A,TM with a suitable\nmodification.", "start": 3978.29, "duration": 10.63}, {"text": "So basically, we want to turn\na HALT,TM decider into an A,TM", "start": 3988.92, "duration": 6.7}, {"text": "decider.", "start": 3995.62, "duration": 1.02}, {"text": "And that's how we're going to\nreduce the problem of solving", "start": 3996.64, "duration": 4.38}, {"text": "A,TM to the problem\nof solving HALT,TM.", "start": 4001.02, "duration": 3.12}, {"text": "Let's just do an example.", "start": 4004.14, "duration": 1.832}, {"text": "If you've seen it\nbefore, obviously, this", "start": 4005.972, "duration": 1.708}, {"text": "is not going to be hard.", "start": 4007.68, "duration": 1.36}, {"text": "But for the many of you who\nhave not seen it before,", "start": 4009.04, "duration": 3.91}, {"text": "I'm partly doing it this time\njust so we can do it again", "start": 4012.95, "duration": 2.52}, {"text": "next time.", "start": 4015.47, "duration": 0.85}, {"text": "And maybe it'll sink in\nby virtue of repetition.", "start": 4016.32, "duration": 2.66}, {"text": "So again, so as I just\nsaid, we're going to assume", "start": 4023.52, "duration": 2.52}, {"text": "the HALT,TM problem is decidable\nand use that to show that A,TM", "start": 4026.04, "duration": 3.15}, {"text": "is decidable, which\nwe know is false.", "start": 4029.19, "duration": 1.98}, {"text": "We showed it just\nearlier that it's not.", "start": 4031.17, "duration": 3.29}, {"text": "So assume we have a\ndecider for HALT,TM.", "start": 4034.46, "duration": 2.02}, {"text": "We'll call it R. And we're going\nto construct from R a decider", "start": 4036.48, "duration": 7.52}, {"text": "for A,TM we'll call S. OK?", "start": 4044.0, "duration": 4.43}, {"text": "So we're, again, typical\nproof by contradiction.", "start": 4048.43, "duration": 2.22}, {"text": "We're assuming the opposite\nof what we're trying to prove.", "start": 4050.65, "duration": 3.45}, {"text": "And then we're going\nto get something crazy.", "start": 4054.1, "duration": 3.54}, {"text": "OK, so here, my job now,\nis I'm assuming I have R,", "start": 4057.64, "duration": 4.29}, {"text": "which is a HALT,TM decider.", "start": 4061.93, "duration": 1.42}, {"text": "So now, I'm assuming\nI know how to decide", "start": 4063.35, "duration": 6.32}, {"text": "if a Turing machine and an\ninput eventually halts--", "start": 4069.67, "duration": 4.33}, {"text": "Not.", "start": 4074.0, "duration": 0.5}, {"text": "Necessarily would it accept,\njust whether it halts.", "start": 4074.5, "duration": 3.89}, {"text": "It's conceivable.", "start": 4078.39, "duration": 1.07}, {"text": "You have to bear with me here.", "start": 4079.46, "duration": 1.86}, {"text": "It's conceivable that\nyou could find a way", "start": 4081.32, "duration": 2.34}, {"text": "to test whether Turing\nmachines halt on their input,", "start": 4083.66, "duration": 3.78}, {"text": "even though we now know that\ntesting whether they accept", "start": 4087.44, "duration": 2.61}, {"text": "their input is not decidable.", "start": 4090.05, "duration": 3.034}, {"text": "So you have to be open-minded\nto the possibility", "start": 4095.87, "duration": 2.55}, {"text": "that the HALT\nproblem is decidable,", "start": 4098.42, "duration": 3.42}, {"text": "and we're going to show\nthat that's can't be.", "start": 4101.84, "duration": 1.979}, {"text": "So we're going to show that\nif we could decide the halting", "start": 4103.819, "duration": 3.301}, {"text": "problem, then we can use that to\ndecide the acceptance problem.", "start": 4107.12, "duration": 5.359}, {"text": "OK, so how are we\ngoing to do that?", "start": 4112.479, "duration": 2.111}, {"text": "So imagine how we can\nsolve the halting problem.", "start": 4114.59, "duration": 2.49}, {"text": "So to solve the A,TM, which\nis what my job is to do,", "start": 4117.08, "duration": 6.26}, {"text": "so S is supposed to solve A,TM.", "start": 4123.34, "duration": 2.9}, {"text": "I'm constructing a Turing\nmachine as decide A,TM.", "start": 4126.24, "duration": 3.6}, {"text": "I'm going to use first--", "start": 4129.84, "duration": 1.229}, {"text": "I'm giving it M and w.", "start": 4131.069, "duration": 1.321}, {"text": "I'm going to feed it into R\nsince that's really all I got.", "start": 4132.39, "duration": 3.44}, {"text": "See if R tells me what happens.", "start": 4135.83, "duration": 4.98}, {"text": "Does M on w at least halt?", "start": 4140.81, "duration": 3.87}, {"text": "Well, if R says,\nno, it doesn't halt,", "start": 4144.68, "duration": 3.719}, {"text": "well then I'm actually done\nbecause if M doesn't even", "start": 4148.399, "duration": 3.721}, {"text": "halt on w, then it\ncouldn't be accepting w.", "start": 4152.12, "duration": 2.25}, {"text": "So at that point, I know\nthat M doesn't accept w,", "start": 4154.37, "duration": 3.029}, {"text": "and I can reject right off.", "start": 4157.399, "duration": 3.441}, {"text": "So R, you can see how it\ncould potentially be helpful.", "start": 4160.84, "duration": 3.059}, {"text": "But it's going to be\nhelpful in either way", "start": 4163.899, "duration": 1.83}, {"text": "because if R says M\ndoes hold, well, then,", "start": 4165.729, "duration": 5.071}, {"text": "I'm also good because I\ndon't know the answer yet,", "start": 4170.8, "duration": 3.22}, {"text": "but what I do know is I\ncan now simulate M on w", "start": 4174.02, "duration": 4.669}, {"text": "until it halts because\nR has told me it halts.", "start": 4178.689, "duration": 3.451}, {"text": "So I don't have to worry\nabout getting into a loop.", "start": 4182.14, "duration": 5.02}, {"text": "So S can be confident in being\na decider for whatever it's", "start": 4187.16, "duration": 3.47}, {"text": "doing because I'm running\nnow M on w with a guarantee", "start": 4190.63, "duration": 6.64}, {"text": "that it halts.", "start": 4197.27, "duration": 3.11}, {"text": "And now, that's going to\ntell me-- now eventually,", "start": 4200.38, "duration": 2.58}, {"text": "the simulation of M on w is\ngoing to end up at an accept", "start": 4202.96, "duration": 2.76}, {"text": "or reject.", "start": 4205.72, "duration": 0.5}, {"text": "And that's going to\nbe the answer I need.", "start": 4206.22, "duration": 1.708}, {"text": "So if M is accepted,\nthen accept.", "start": 4207.928, "duration": 1.392}, {"text": "And if M is rejected,\nthen reject.", "start": 4209.32, "duration": 3.78}, {"text": "And that's how S solves A,TM\nusing R, which solves HALT,TM.", "start": 4213.1, "duration": 6.89}, {"text": "But S can't exist.", "start": 4219.99, "duration": 2.55}, {"text": "And so therefore, R can't exist.", "start": 4222.54, "duration": 1.59}, {"text": "And therefore, HALT,TM\ncan't be decidable.", "start": 4224.13, "duration": 4.33}, {"text": "OK?", "start": 4228.46, "duration": 0.71}, {"text": "So that quickly--", "start": 4229.17, "duration": 1.125}, {"text": "OK, I'm not sure which\ndiagram you wanted me to show.", "start": 4235.63, "duration": 4.24}, {"text": "But anyway, maybe\nwe can do that.", "start": 4239.87, "duration": 1.62}, {"text": "We're basically at the end\nof the hour or end of the 90", "start": 4241.49, "duration": 2.82}, {"text": "minutes.", "start": 4244.31, "duration": 0.61}, {"text": "So let's do a quick review.", "start": 4244.92, "duration": 1.7}, {"text": "And if you stick around, I'm\nhappy to go back and look", "start": 4246.62, "duration": 2.295}, {"text": "at any of the other slides that\nyou might have missed something", "start": 4248.915, "duration": 2.625}, {"text": "on.", "start": 4251.54, "duration": 0.95}, {"text": "OK, so just to recap, we\nshowed that the natural numbers", "start": 4252.49, "duration": 5.46}, {"text": "and the real numbers are not the\nsame size using that definition", "start": 4257.95, "duration": 3.0}, {"text": "of one-to-one correspondence to\nintroduce the diagonalization", "start": 4260.95, "duration": 2.79}, {"text": "method.", "start": 4263.74, "duration": 0.75}, {"text": "We used the diagonalization\nmethod to show that A,TM is", "start": 4264.49, "duration": 2.46}, {"text": "undecidable.", "start": 4266.95, "duration": 1.29}, {"text": "We also showed\nthat little theorem", "start": 4268.24, "duration": 2.065}, {"text": "that if the language\nand its complement", "start": 4270.305, "duration": 1.625}, {"text": "are recognizable, then\nthe language is decidable.", "start": 4271.93, "duration": 3.36}, {"text": "And from that, we concluded\nthat A,TM complement is not", "start": 4275.29, "duration": 3.66}, {"text": "recognizable.", "start": 4278.95, "duration": 1.26}, {"text": "And then we showed, at least\nby virtue of an introduction", "start": 4280.21, "duration": 3.18}, {"text": "to the method, the reducibility\nmethod to show that HALT,TM is", "start": 4283.39, "duration": 3.0}, {"text": "undecidable.", "start": 4286.39, "duration": 1.38}, {"text": "And that was today's lecture.", "start": 4287.77, "duration": 1.89}, {"text": "And we're and we're at\nthe end of the hour.", "start": 4289.66, "duration": 3.22}, {"text": "So why don't I--", "start": 4292.88, "duration": 3.02}, {"text": "we are finished.", "start": 4295.9, "duration": 0.99}, {"text": "You can log out.", "start": 4296.89, "duration": 2.37}, {"text": "And if you want, I\nwill stick around.", "start": 4299.26, "duration": 4.04}, {"text": "OK, OK, this is kind of\na good question here.", "start": 4303.3, "duration": 3.21}, {"text": "So I'm getting a question about\nthe A,TM-complement, which is--", "start": 4310.13, "duration": 10.66}, {"text": "since we have a\nrecognizer for A,TM,", "start": 4320.79, "duration": 6.17}, {"text": "if I'm doing justice\nto this question,", "start": 4326.96, "duration": 5.19}, {"text": "we have a recogniser for A,TM,\nso why can't we just invert", "start": 4332.15, "duration": 2.79}, {"text": "the answer?", "start": 4334.94, "duration": 3.01}, {"text": "Flip the answer around, and\nnow, we have a recognizer", "start": 4337.95, "duration": 5.36}, {"text": "for the complement of\nA,TM, A,TM-complement.", "start": 4343.31, "duration": 4.59}, {"text": "So why doesn't that work?", "start": 4347.9, "duration": 2.63}, {"text": "Well, the reason that doesn't\nwork is because the recognizer", "start": 4350.53, "duration": 5.64}, {"text": "for A,TM might be rejecting\nsome things by looping.", "start": 4356.17, "duration": 5.91}, {"text": "And now, if you just flip\nthe accepting and rejecting,", "start": 4362.08, "duration": 6.39}, {"text": "when it hits one of\nthose halting states,", "start": 4368.47, "duration": 2.94}, {"text": "it's going to give\nthe reverse answer.", "start": 4371.41, "duration": 2.25}, {"text": "But when it rejects\nby looping, it'll", "start": 4373.66, "duration": 4.47}, {"text": "continue to reject by looping.", "start": 4378.13, "duration": 1.41}, {"text": "So you won't get the\ncomplementary language", "start": 4379.54, "duration": 2.49}, {"text": "coming out.", "start": 4382.03, "duration": 0.57}, {"text": "So if it would be helpful, I\ncan go back to that slide here,", "start": 4386.56, "duration": 9.22}, {"text": "which proves that\nA,TM-complement is", "start": 4395.78, "duration": 2.45}, {"text": "unrecognizable because maybe we\nshould start with the bottom.", "start": 4398.23, "duration": 7.9}, {"text": "We know that A,TM is\nrecognizable and undecidable,", "start": 4406.13, "duration": 7.04}, {"text": "right?", "start": 4413.17, "duration": 0.5}, {"text": "We already proved\nthose two facts.", "start": 4413.67, "duration": 1.56}, {"text": "A,TM is recognizable from\nthe universal Turing machine,", "start": 4415.23, "duration": 5.79}, {"text": "and it's undecidable by the\ndiagonalization argument.", "start": 4421.02, "duration": 4.34}, {"text": "Those two things\ntogether tell us", "start": 4425.36, "duration": 2.7}, {"text": "that the complement has\nto be unrecognizable", "start": 4428.06, "duration": 4.14}, {"text": "because if a language\nand its complement", "start": 4432.2, "duration": 3.15}, {"text": "are both recognizable-- and\nwe already know the language", "start": 4435.35, "duration": 2.4}, {"text": "itself is recognizable.", "start": 4437.75, "duration": 1.74}, {"text": "So now, if the complement\nis also recognizable,", "start": 4439.49, "duration": 2.207}, {"text": "the language is going to be\ndecidable by the upper theorem.", "start": 4441.697, "duration": 2.458}, {"text": "So it must be the case had\neither the language itself", "start": 4450.67, "duration": 2.31}, {"text": "is unrecognizable, or its\ncomplement is unrecognizable.", "start": 4452.98, "duration": 3.968}, {"text": "We know the language\nis recognizable.", "start": 4456.948, "duration": 1.542}, {"text": "That's what the universal\nTuring machine told us.", "start": 4458.49, "duration": 2.07}, {"text": "So the only thing left\nis for the complement", "start": 4460.56, "duration": 1.98}, {"text": "to be unrecognizable.", "start": 4462.54, "duration": 2.76}, {"text": "You should review that\nif you didn't get it", "start": 4465.3, "duration": 3.39}, {"text": "because this is the\nkind of reasoning", "start": 4468.69, "duration": 3.87}, {"text": "we're going to be building\non things like that.", "start": 4472.56, "duration": 3.37}, {"text": "So I think it's good to\nmake sure you understand.", "start": 4475.93, "duration": 4.58}, {"text": "OK.", "start": 4480.51, "duration": 0.5}, {"text": "OK, the diagram on\nthe right-- so this", "start": 4484.147, "duration": 1.583}, {"text": "is just a Venn diagram here.", "start": 4485.73, "duration": 1.62}, {"text": "I threw this in at\nthe last minute here.", "start": 4487.35, "duration": 2.07}, {"text": "I was worried about\nit being confusing.", "start": 4489.42, "duration": 4.22}, {"text": "That part is--", "start": 4493.64, "duration": 1.39}, {"text": "I'm trying to show that the\nthree classes that we've", "start": 4500.57, "duration": 2.67}, {"text": "already talked about--", "start": 4503.24, "duration": 1.35}, {"text": "the languages which\nare decidable,", "start": 4504.59, "duration": 2.19}, {"text": "the languages which are\nTuring-recognizable,", "start": 4506.78, "duration": 2.43}, {"text": "and the languages\nwhose complements", "start": 4509.21, "duration": 1.95}, {"text": "are Turing-recognizable.", "start": 4511.16, "duration": 1.62}, {"text": "Those are three separate\nclasses of languages.", "start": 4512.78, "duration": 4.51}, {"text": "And those come up here\nin those three regions.", "start": 4517.29, "duration": 5.34}, {"text": "These are the decidable ones.", "start": 4522.63, "duration": 1.82}, {"text": "Here are the recognizable ones.", "start": 4524.45, "duration": 2.79}, {"text": "And here are the ones whose\ncomplements are recognizable.", "start": 4527.24, "duration": 4.69}, {"text": "Now if a language is in\nboth the recognizable,", "start": 4531.93, "duration": 4.85}, {"text": "and its complement\nis recognizable--", "start": 4536.78, "duration": 1.53}, {"text": "so it's in both of these\nbigger regions here--", "start": 4538.31, "duration": 3.15}, {"text": "then this theorem tells\nyou it's decidable.", "start": 4541.46, "duration": 4.52}, {"text": "So that's why the intersection\nof these two regions", "start": 4545.98, "duration": 3.62}, {"text": "is marked as being\ndecidable because that", "start": 4549.6, "duration": 2.34}, {"text": "means you're in both.", "start": 4551.94, "duration": 0.93}, {"text": "OK?", "start": 4557.51, "duration": 2.07}, {"text": "But we know that A,TM is\nsitting out here as recognizable", "start": 4559.58, "duration": 3.51}, {"text": "but not decidable.", "start": 4563.09, "duration": 1.08}, {"text": "So A,TM is in the\nrecognizable side,", "start": 4564.17, "duration": 2.55}, {"text": "but it's not on the complement\nof recognizable, A,TM itself.", "start": 4566.72, "duration": 5.73}, {"text": "The complement of A,TM is the\ncomplement of a recognizable", "start": 4572.45, "duration": 3.39}, {"text": "but itself is not recognizable\nand not decidable.", "start": 4575.84, "duration": 6.28}, {"text": "So you got this side\nof nice, you know--", "start": 4582.12, "duration": 4.06}, {"text": "I hope you think\nit's nice, but it's", "start": 4586.18, "duration": 2.16}, {"text": "sort of a try to summarize\nthings in this little Venn", "start": 4588.34, "duration": 3.37}, {"text": "diagram.", "start": 4591.71, "duration": 0.5}, {"text": "So I think I'm going\nto then sign off.", "start": 4595.2, "duration": 5.41}, {"text": "And I'll see you all on Tuesday.", "start": 4600.61, "duration": 5.08}, {"text": "And have a good weekend.", "start": 4605.69, "duration": 1.41}, {"text": "Bye bye.", "start": 4607.1, "duration": 1.67}]