transcripts/4dFPVJrNLDs.json

[{"text": "[SQUEAKING]", "start": 0.0, "duration": 1.98}, {"text": "[RUSTLING]", "start": 1.98, "duration": 1.98}, {"text": "[CLICKING]", "start": 3.96, "duration": 2.97}, {"text": "MICHAEL SIPSER: Let's get\nback to space complexity.", "start": 24.76, "duration": 5.8}, {"text": "So just to review\nwhat we've been doing", "start": 30.56, "duration": 4.07}, {"text": "from last lecture, which\nfeels like a long time ago", "start": 34.63, "duration": 3.21}, {"text": "but it was only two\ndays ago, we were", "start": 37.84, "duration": 5.19}, {"text": "looking at those three\ntheorems, which all", "start": 43.03, "duration": 5.7}, {"text": "had basically the same proof.", "start": 48.73, "duration": 3.03}, {"text": "One was about the\nladder problem,", "start": 51.76, "duration": 3.45}, {"text": "where you're trying to see if\nyou can get from one string", "start": 55.21, "duration": 3.78}, {"text": "to another string changing\none symbol at a time.", "start": 58.99, "duration": 4.02}, {"text": "And all of the strings\nwere in the language", "start": 63.01, "duration": 3.06}, {"text": "of some finite automaton, or\nyou had some reasonable rule", "start": 66.07, "duration": 2.94}, {"text": "for saying which are the\nvalid strings, which are not.", "start": 69.01, "duration": 4.08}, {"text": "And then we built on that idea.", "start": 73.09, "duration": 3.57}, {"text": "We showed Savitch's\ntheorem, that", "start": 76.66, "duration": 1.77}, {"text": "going from any\nnondeterministic machine", "start": 78.43, "duration": 2.61}, {"text": "to a deterministic machine only\nsquares the amount of space.", "start": 81.04, "duration": 3.12}, {"text": "And then we finally\nproved that TQBF,", "start": 84.16, "duration": 7.05}, {"text": "this problem about\nquantified Boolean formulas,", "start": 91.21, "duration": 3.18}, {"text": "testing whether they are\ntrue, that that problem is", "start": 94.39, "duration": 3.0}, {"text": "complete for PSPACE.", "start": 97.39, "duration": 2.98}, {"text": "So we're going to build\noff of that last theorem", "start": 100.37, "duration": 2.72}, {"text": "today, and talk about one\nother complete problem,", "start": 103.09, "duration": 7.02}, {"text": "and also show a\nconnection between PSPACE", "start": 110.11, "duration": 4.62}, {"text": "and testing which side can\nwin in certain kinds of games.", "start": 114.73, "duration": 5.745}, {"text": "And then at the end of the\nlecture, the second half", "start": 124.3, "duration": 2.7}, {"text": "of the lecture, we'll\ntalk about diving deeper", "start": 127.0, "duration": 3.419}, {"text": "into space complexity.", "start": 130.419, "duration": 1.231}, {"text": "We're going to talk about a\ndifferent part of the parameter", "start": 131.65, "duration": 3.48}, {"text": "space where instead of\nlooking at polynomial space,", "start": 135.13, "duration": 2.627}, {"text": "we're going to talk about\nwhat happens if you have", "start": 137.757, "duration": 2.083}, {"text": "logarithmic space, which is\nanother natural point to look", "start": 139.84, "duration": 3.96}, {"text": "at for space complexity.", "start": 143.8, "duration": 2.76}, {"text": "But we'll get to\nthat after the break.", "start": 146.56, "duration": 2.73}, {"text": "Talk about game\ncomplexity and games--", "start": 153.07, "duration": 3.84}, {"text": "so when I was little, we used\nto play, me and my sisters,", "start": 156.91, "duration": 5.19}, {"text": "would play a game\ncalled Geography.", "start": 162.1, "duration": 1.92}, {"text": "I don't know how many\nof you have heard it", "start": 164.02, "duration": 1.98}, {"text": "or maybe it goes under\na different name.", "start": 166.0, "duration": 3.33}, {"text": "But it was a game basically\nabout words and places.", "start": 169.33, "duration": 4.92}, {"text": "And it has a very\nsimple set of rules.", "start": 174.25, "duration": 3.9}, {"text": "Let's say there were two people.", "start": 178.15, "duration": 1.41}, {"text": "You take turns picking\nnames of places.", "start": 179.56, "duration": 3.3}, {"text": "And then each person has\nto pick a place which", "start": 185.91, "duration": 4.53}, {"text": "starts with the letter that the\nprevious person's place ended", "start": 190.44, "duration": 5.01}, {"text": "with.", "start": 195.45, "duration": 2.53}, {"text": "So for example, you\nhave two people playing.", "start": 197.98, "duration": 4.41}, {"text": "Perhaps you agree on\nsome starting place", "start": 202.39, "duration": 3.06}, {"text": "like a major city\nnearby, like Boston.", "start": 205.45, "duration": 3.45}, {"text": "So one player, the first\nplayer, picks Boston.", "start": 208.9, "duration": 3.48}, {"text": "And then after that,\nthe next player", "start": 212.38, "duration": 1.71}, {"text": "has to pick a place that starts\nwith N, because Boston ends", "start": 214.09, "duration": 4.74}, {"text": "with N. And so maybe\nNebraska would be", "start": 218.83, "duration": 4.97}, {"text": "a possible response to Boston.", "start": 223.8, "duration": 2.47}, {"text": "And then the first\nplayer would have", "start": 226.27, "duration": 1.55}, {"text": "to respond to that with a\nplace that starts with an A,", "start": 227.82, "duration": 2.52}, {"text": "because Nebraska\nends with an A, maybe", "start": 230.34, "duration": 2.22}, {"text": "Alaska, which also starts\nwith A, ends with A.", "start": 232.56, "duration": 4.78}, {"text": "So then Arkansas,\nmaybe, would be", "start": 237.34, "duration": 3.14}, {"text": "a reasonable response to that.", "start": 240.48, "duration": 1.99}, {"text": "And then that ends with\nS, so San Francisco,", "start": 242.47, "duration": 4.52}, {"text": "and then so on, and so on.", "start": 246.99, "duration": 2.22}, {"text": "And of course, you\nwant to forbid people", "start": 249.21, "duration": 5.07}, {"text": "from reusing names because then\nyou'll just get into a loop.", "start": 254.28, "duration": 4.739}, {"text": "Saying, \"Alaska,\"\n\"Alaska,\" \"Alaska,\"", "start": 259.019, "duration": 2.25}, {"text": "won't make the game\nvery interesting,", "start": 261.269, "duration": 1.801}, {"text": "so you have to forbid\nthat possibility.", "start": 263.07, "duration": 3.66}, {"text": "Names cannot be reused.", "start": 266.73, "duration": 1.51}, {"text": "They can be used at most once.", "start": 268.24, "duration": 1.88}, {"text": "And then, eventually,\none side or the other", "start": 270.12, "duration": 4.11}, {"text": "is going to run out of\nnames and not have--", "start": 274.23, "duration": 3.15}, {"text": "either because of the lack of\ntheir geographical knowledge", "start": 277.38, "duration": 3.42}, {"text": "or maybe you've just exhausted\nall the possibilities--", "start": 280.8, "duration": 2.52}, {"text": "and will not have\na good response.", "start": 283.32, "duration": 2.61}, {"text": "And that person will be stuck.", "start": 285.93, "duration": 1.47}, {"text": "And that person is the loser.", "start": 287.4, "duration": 1.92}, {"text": "The other person\nhas won the game.", "start": 289.32, "duration": 1.87}, {"text": "So the objective is to try\nto avoid getting stuck.", "start": 291.19, "duration": 3.03}, {"text": "So let's just take a look.", "start": 294.22, "duration": 1.97}, {"text": "I'm going to think about\nthat game in a slightly more", "start": 296.19, "duration": 4.14}, {"text": "abstracted, formalized way.", "start": 300.33, "duration": 2.46}, {"text": "So you write down all of\nthe legitimate places.", "start": 302.79, "duration": 2.91}, {"text": "They become nodes of a graph\nsuch as I've shown here.", "start": 305.7, "duration": 3.45}, {"text": "And then you draw an edge\nfrom one node to another", "start": 309.15, "duration": 4.71}, {"text": "if that corresponds to a legal\nmove in the game, a legal step", "start": 313.86, "duration": 7.09}, {"text": "of the game.", "start": 320.95, "duration": 0.58}, {"text": "So Boston connects to\nplaces that start with N,", "start": 321.53, "duration": 3.5}, {"text": "like New York and Nebraska.", "start": 325.03, "duration": 2.43}, {"text": "So the first person\nmight pick Boston.", "start": 327.46, "duration": 1.77}, {"text": "The second person might\npick either of these two,", "start": 329.23, "duration": 2.573}, {"text": "and then back to\nthe first person.", "start": 331.803, "duration": 1.417}, {"text": "If they picked Nebraska,\nthey could pick Arkansas.", "start": 333.22, "duration": 2.083}, {"text": "They can pick Alaska, and\nso on, just as I described.", "start": 335.303, "duration": 4.327}, {"text": "Believe it or not, we\nactually played this game.", "start": 339.63, "duration": 2.97}, {"text": "I can remember playing\nthis game when I was a kid.", "start": 342.6, "duration": 2.16}, {"text": "That was, of course, before\nwe had \"League of Legends\"", "start": 344.76, "duration": 5.58}, {"text": "and other fun stuff, but\nthis is what kids used to do.", "start": 350.34, "duration": 4.02}, {"text": "So anyway, just to get it down\non the slide, the rules are--", "start": 354.36, "duration": 9.85}, {"text": "assuming we have\ntwo players, we'll", "start": 364.21, "duration": 2.73}, {"text": "call them Player\nI and Player II.", "start": 366.94, "duration": 1.98}, {"text": "Player I goes first.", "start": 368.92, "duration": 2.34}, {"text": "And they take turns\npicking places", "start": 371.26, "duration": 1.71}, {"text": "that start with the letter\nwhich ended the previous place,", "start": 372.97, "duration": 3.36}, {"text": "no repeats allowed.", "start": 376.33, "duration": 1.32}, {"text": "And the first player stuck\nloses, the person who", "start": 377.65, "duration": 5.67}, {"text": "doesn't have a move to make.", "start": 383.32, "duration": 3.61}, {"text": "So what we're going\nto look at today", "start": 386.93, "duration": 2.57}, {"text": "is a generalized version\nof that, where we're just", "start": 389.5, "duration": 3.99}, {"text": "going to take away\nthe names, but just", "start": 393.49, "duration": 2.25}, {"text": "leave the underlying graph.", "start": 395.74, "duration": 2.62}, {"text": "And we call it going to call\nthat \"generalized geography.\"", "start": 398.36, "duration": 3.69}, {"text": "So here, that can be played\non any directed graph.", "start": 402.05, "duration": 3.51}, {"text": "You take away the\nnames of the nodes.", "start": 405.56, "duration": 3.6}, {"text": "Now you just have\nsome arbitrary graph.", "start": 409.16, "duration": 3.99}, {"text": "You're going to designate\na particular node", "start": 413.15, "duration": 1.86}, {"text": "as the starting node.", "start": 415.01, "duration": 1.23}, {"text": "And then the players take\nturns following those edges.", "start": 416.24, "duration": 3.435}, {"text": "And because you're going to\nforbid having any loops reusing", "start": 422.18, "duration": 7.62}, {"text": "any nodes, what you end up\ndoing is, working together,", "start": 429.8, "duration": 5.71}, {"text": "you're going to be constructing\na simple path in that graph.", "start": 435.51, "duration": 4.61}, {"text": "Simple path just\nfollows those edges", "start": 440.12, "duration": 2.67}, {"text": "and never intersects itself.", "start": 442.79, "duration": 3.99}, {"text": "And the first player to\nbe stuck is the loser.", "start": 446.78, "duration": 3.09}, {"text": "So the other one is the winner.", "start": 449.87, "duration": 1.86}, {"text": "So we're going to call that\n\"generalized geography.\"", "start": 451.73, "duration": 3.9}, {"text": "And we're going to look at\nthe computational complexity", "start": 455.63, "duration": 2.67}, {"text": "of deciding, for a\ngiven graph and starting", "start": 458.3, "duration": 2.7}, {"text": "point, which side would win if\nboth players played optimally,", "start": 461.0, "duration": 8.73}, {"text": "the best possible.", "start": 469.73, "duration": 1.96}, {"text": "And we'll name that\nproblem GG, or look", "start": 471.69, "duration": 3.47}, {"text": "at its associated language.", "start": 475.16, "duration": 1.6}, {"text": "So here is a graph\nand the starting node.", "start": 476.76, "duration": 3.41}, {"text": "And you want to say that\npair is in the language GG", "start": 480.17, "duration": 6.66}, {"text": "if the first player, Player I,\nwho must play at that node A", "start": 486.83, "duration": 6.39}, {"text": "to start off with\nhas a forced win", "start": 493.22, "duration": 6.04}, {"text": "in that generalized geography\ngame on that graph starting", "start": 499.26, "duration": 3.66}, {"text": "at A. Again, what I\nmean by a forced win--", "start": 502.92, "duration": 3.187}, {"text": "I'm not going to\ndefine this formally,", "start": 506.107, "duration": 1.583}, {"text": "though we could do that--", "start": 507.69, "duration": 1.05}, {"text": "it's not hard to do.", "start": 508.74, "duration": 1.08}, {"text": "But a forced win is also\ncalled a \"winning strategy.\"", "start": 509.82, "duration": 4.05}, {"text": "That just means that if\nboth sides play optimally,", "start": 513.87, "duration": 3.419}, {"text": "they play the best possible--", "start": 517.289, "duration": 2.431}, {"text": "not of their capability,\nI mean absolutely", "start": 519.72, "duration": 2.43}, {"text": "the best possible play--", "start": 522.15, "duration": 2.25}, {"text": "then that Player\nI would still win.", "start": 524.4, "duration": 5.49}, {"text": "There's nothing\nthat Player II can", "start": 529.89, "duration": 1.74}, {"text": "do to prevent Player\nI from winning", "start": 531.63, "duration": 3.75}, {"text": "if Player I has a winning\nstrategy or a forced win.", "start": 535.38, "duration": 3.75}, {"text": "You may have seen, for example--", "start": 539.13, "duration": 3.03}, {"text": "there are examples\nof this that--", "start": 542.16, "duration": 1.623}, {"text": "I mean, I don't even\nknow if they still exist,", "start": 543.783, "duration": 1.917}, {"text": "but it used to be\nin the newspaper.", "start": 545.7, "duration": 1.83}, {"text": "There used to be\nexamples of chess boards", "start": 547.53, "duration": 2.37}, {"text": "and they would start\nfrom a certain position.", "start": 549.9, "duration": 3.15}, {"text": "And they would say,\n\"White to win in three.\"", "start": 553.05, "duration": 2.85}, {"text": "And so that means\nwhite has a forced win.", "start": 555.9, "duration": 2.28}, {"text": "No matter what black does, white\nis going to end up winning.", "start": 558.18, "duration": 2.68}, {"text": "But you have to think\nof what the strategy is.", "start": 560.86, "duration": 1.917}, {"text": "It may not be so\nobvious to think", "start": 562.777, "duration": 1.463}, {"text": "through what are the right\nmoves that white makes.", "start": 564.24, "duration": 2.85}, {"text": "But the point is that no\nmatter what black would do,", "start": 567.09, "duration": 3.06}, {"text": "white would end up winning.", "start": 570.15, "duration": 2.52}, {"text": "And what you can show--", "start": 572.67, "duration": 1.26}, {"text": "we're not going to do that--\nbut for a class of games that", "start": 573.93, "duration": 3.84}, {"text": "includes this\ngeneralized geography,", "start": 577.77, "duration": 3.12}, {"text": "either one side or\nthe other is going", "start": 580.89, "duration": 2.49}, {"text": "to be guaranteed to\nhave a forced win.", "start": 583.38, "duration": 2.67}, {"text": "That's not necessarily true for\nall possible games, obviously,", "start": 586.05, "duration": 3.51}, {"text": "but for a large class of\ngames, one side or the other", "start": 589.56, "duration": 4.2}, {"text": "is guaranteed to have a winning\nstrategy or a forced win.", "start": 593.76, "duration": 5.73}, {"text": "So let's just review\nthat, because this", "start": 599.49, "duration": 1.68}, {"text": "is going to be essential for\nthe first half of the lecture--", "start": 601.17, "duration": 4.2}, {"text": "to understand what\nwe mean by a game", "start": 605.37, "duration": 2.35}, {"text": "and what we mean by one side or\nthe other to have a forced win.", "start": 607.72, "duration": 3.92}, {"text": "And what we're going to show\nis that GG is PSPACE complete,", "start": 614.37, "duration": 3.35}, {"text": "that the problem of\ngiven one of these graphs", "start": 617.72, "duration": 2.16}, {"text": "and a starting\npoint, figuring out", "start": 619.88, "duration": 2.25}, {"text": "does Player I have a forced\nwin or is it Player II that", "start": 622.13, "duration": 2.34}, {"text": "has a forced win?", "start": 624.47, "duration": 2.94}, {"text": "Which side is guaranteed\nto win under optimal play?", "start": 627.41, "duration": 3.09}, {"text": "That's a PSPACE\ncomplete problem.", "start": 630.5, "duration": 3.19}, {"text": "And so let's do a\nlittle checking on that.", "start": 633.69, "duration": 2.85}, {"text": "And so I'm going to give\nyou a very small example", "start": 636.54, "duration": 2.96}, {"text": "of a generalized geography game.", "start": 639.5, "duration": 1.77}, {"text": "You have to figure out whether\nit's Player I or Player II that", "start": 644.13, "duration": 3.93}, {"text": "has the forced win, the\nwinning strategy, or maybe", "start": 648.06, "duration": 3.39}, {"text": "neither, or both.", "start": 651.45, "duration": 1.35}, {"text": "Those are the four options.", "start": 652.8, "duration": 1.65}, {"text": "And you understand Player\nI has to play here,", "start": 654.45, "duration": 2.58}, {"text": "because that's\nthe starting place", "start": 657.03, "duration": 1.542}, {"text": "of this generalized\ngeography game.", "start": 658.572, "duration": 1.458}, {"text": "So then it's up to\nPlayer II to continue on.", "start": 660.03, "duration": 3.13}, {"text": "All right, so let's\nlaunch that poll.", "start": 663.16, "duration": 4.07}, {"text": "You'll have to think\nabout it a little bit,", "start": 667.23, "duration": 3.55}, {"text": "and let me know what you think.", "start": 670.78, "duration": 4.62}, {"text": "Which side has a forced win?", "start": 675.4, "duration": 3.92}, {"text": "Which player makes\nthe first move?", "start": 679.32, "duration": 1.53}, {"text": "It's Player I makes the first\nmove, and Player I plays,", "start": 680.85, "duration": 3.21}, {"text": "as I showed you here, this\nis Player I playing at A.", "start": 684.06, "duration": 3.96}, {"text": "So Player I picks A,\nand then Player II", "start": 688.02, "duration": 2.812}, {"text": "has to pick one of these two.", "start": 690.832, "duration": 1.208}, {"text": "All right, I think we're\nthe end of the poll.", "start": 696.43, "duration": 2.17}, {"text": "Everybody's in?", "start": 698.6, "duration": 1.67}, {"text": "Pick something.", "start": 700.27, "duration": 0.84}, {"text": "You guys did not do\nwell on this poll.", "start": 710.09, "duration": 4.8}, {"text": "At least not too\nmany of you picked", "start": 714.89, "duration": 2.01}, {"text": "D. That's a little\nreassuring because you", "start": 716.9, "duration": 4.2}, {"text": "can't have both players\nhaving a forced win.", "start": 721.1, "duration": 2.31}, {"text": "I did say that in games of this\nkind, one side or the other", "start": 723.41, "duration": 3.21}, {"text": "is going to have a\nforced win, so C is not", "start": 726.62, "duration": 2.04}, {"text": "a very good choice, either.", "start": 728.66, "duration": 2.34}, {"text": "So maybe you should spend\nless time reading your email", "start": 731.0, "duration": 3.745}, {"text": "and more time listening\nto the lecture.", "start": 734.745, "duration": 1.625}, {"text": "The actual correct answer is B.", "start": 740.73, "duration": 2.46}, {"text": "Player II, which obviously\nis a minority position here--", "start": 743.19, "duration": 4.89}, {"text": "Player II has the forced win.", "start": 748.08, "duration": 1.56}, {"text": "Let's understand why.", "start": 749.64, "duration": 2.25}, {"text": "So Player I plays here.", "start": 751.89, "duration": 3.63}, {"text": "As I said, that's\nthe first move.", "start": 755.52, "duration": 2.22}, {"text": "Player II can either take the\nupper node or the lower node.", "start": 757.74, "duration": 3.6}, {"text": "Player II takes the\nupper node, then", "start": 764.05, "duration": 2.55}, {"text": "Player I has no choice but\nto take the right-most node,", "start": 766.6, "duration": 4.26}, {"text": "and now there's no more\nmove for Player II.", "start": 770.86, "duration": 4.52}, {"text": "So Player II will lose if\nPlayer II takes the upper node,", "start": 775.38, "duration": 5.22}, {"text": "the upper choice there.", "start": 780.6, "duration": 1.74}, {"text": "Because Player II will go\nhere, Player I will go there,", "start": 782.34, "duration": 3.84}, {"text": "and it'll be game over.", "start": 786.18, "duration": 2.04}, {"text": "And Player I will have won\nbecause Player II was stuck.", "start": 788.22, "duration": 3.45}, {"text": "However, Player II\nhad another choice.", "start": 791.67, "duration": 2.64}, {"text": "Player II could have\nalso gone down here.", "start": 801.8, "duration": 3.91}, {"text": "So Player II, if it\nplayed down here,", "start": 805.71, "duration": 4.503}, {"text": "things are looking a lot\nbetter for Player II.", "start": 810.213, "duration": 1.917}, {"text": "If Player II goes here, Player\nI's move is forced, goes here.", "start": 812.13, "duration": 3.66}, {"text": "But now, there's another move\nleft that Player II could make.", "start": 815.79, "duration": 3.54}, {"text": "So Player II goes here, I goes\nthere, Player II goes up there.", "start": 819.33, "duration": 6.05}, {"text": "And now Player I is stuck.", "start": 825.38, "duration": 1.68}, {"text": "So it's Player II's\nchoice, Player II", "start": 827.06, "duration": 4.42}, {"text": "can make a move\nwhich will end up", "start": 831.48, "duration": 2.91}, {"text": "causing Player I to get\nstuck, because that's", "start": 834.39, "duration": 4.83}, {"text": "the nature of what we mean\nby having a winning strategy.", "start": 839.22, "duration": 2.73}, {"text": "There is some way to guarantee\nif you're playing optimally,", "start": 841.95, "duration": 3.175}, {"text": "and you're certainly\nplaying optimally,", "start": 845.125, "duration": 1.625}, {"text": "means you'll take\nthat lower move.", "start": 846.75, "duration": 3.45}, {"text": "Then you will take that.", "start": 850.2, "duration": 3.1}, {"text": "So under optimal play,\nPlayer II will win this game,", "start": 853.3, "duration": 6.69}, {"text": "and so Player II has\na winning strategy.", "start": 859.99, "duration": 2.1}, {"text": "It has a forced win.", "start": 862.09, "duration": 2.94}, {"text": "So I see.", "start": 865.03, "duration": 1.02}, {"text": "So I'm not going to define\n\"playing optimally.\"", "start": 866.05, "duration": 4.49}, {"text": "There's some questions\nhere about that.", "start": 870.54, "duration": 3.66}, {"text": "I'm going to-- I'm going to\nleave it somewhat informal,", "start": 879.18, "duration": 7.35}, {"text": "but we could make\nthat all precise.", "start": 886.53, "duration": 2.31}, {"text": "I don't think it\nwould be clarifying", "start": 888.84, "duration": 2.13}, {"text": "to go through a precise\ndefinition of that.", "start": 890.97, "duration": 2.7}, {"text": "We don't even have\nto think about it", "start": 898.92, "duration": 1.56}, {"text": "in terms of optimal play.", "start": 900.48, "duration": 1.38}, {"text": "You can say that a\nplayer has a forced", "start": 901.86, "duration": 13.04}, {"text": "win if they have moves\nthat they could make", "start": 914.9, "duration": 5.07}, {"text": "in response to every possible\nmove the opponent could make", "start": 919.97, "duration": 4.4}, {"text": "that will guarantee that\nthey will end up winning.", "start": 924.37, "duration": 3.2}, {"text": "So you don't even have to\ntalk about optimality here.", "start": 927.57, "duration": 2.76}, {"text": "Just talk about every\npossible opponent.", "start": 930.33, "duration": 3.18}, {"text": "Every possible opponent\nis going to lose", "start": 933.51, "duration": 2.4}, {"text": "to somebody who has\na winning strategy", "start": 935.91, "duration": 7.96}, {"text": "and follows that strategy.", "start": 943.87, "duration": 1.47}, {"text": "I can see there was\nsome confusion about who", "start": 949.37, "duration": 2.13}, {"text": "was going first and so on.", "start": 951.5, "duration": 1.14}, {"text": "I'm glad.", "start": 952.64, "duration": 0.51}, {"text": "That was part of the reason\nI gave this check, just", "start": 953.15, "duration": 2.125}, {"text": "to hopefully to clear\nup some of those points.", "start": 955.275, "duration": 2.805}, {"text": "So Player I plays here\nthen Player II goes.", "start": 958.08, "duration": 3.83}, {"text": "And I got some comments\nthat some people", "start": 961.91, "duration": 2.01}, {"text": "were confused about\nwhat does it mean where", "start": 963.92, "duration": 5.22}, {"text": "does the first person move.", "start": 969.14, "duration": 1.95}, {"text": "Hopefully, that's clearer.", "start": 971.09, "duration": 3.84}, {"text": "All right, so let's\nmove on because we're", "start": 974.93, "duration": 2.67}, {"text": "going to be spending a\nwhile proving something", "start": 977.6, "duration": 4.77}, {"text": "about generalized\ngeography in these graphs.", "start": 982.37, "duration": 2.64}, {"text": "So let's continue.", "start": 988.42, "duration": 2.76}, {"text": "So in order to make the\nconnection between games", "start": 991.18, "duration": 6.22}, {"text": "and complexity,\nwe're going to need", "start": 997.4, "duration": 2.82}, {"text": "to talk about a problem,\none related to one", "start": 1000.22, "duration": 5.73}, {"text": "that we've already seen,\nwhere we can reformulate", "start": 1005.95, "duration": 2.58}, {"text": "that problem as a game.", "start": 1008.53, "duration": 1.87}, {"text": "And that's going to be a problem\non quantified Boolean formulas", "start": 1010.4, "duration": 3.51}, {"text": "called the \"Formula Game.\"", "start": 1013.91, "duration": 2.94}, {"text": "So this is a different game\nnow, not one that you're", "start": 1016.85, "duration": 5.0}, {"text": "likely to end up playing.", "start": 1021.85, "duration": 1.29}, {"text": "But in a sense in\nwhich you'll see,", "start": 1023.14, "duration": 3.179}, {"text": "this is still a reasonable\ngame where two players", "start": 1026.319, "duration": 2.881}, {"text": "can play against one another.", "start": 1029.2, "duration": 1.419}, {"text": "And one of them is going to\nend up winning and one of them", "start": 1030.619, "duration": 2.45}, {"text": "is going to end up having--", "start": 1033.069, "duration": 2.461}, {"text": "one or the other will\nhave a winning strategy.", "start": 1035.53, "duration": 2.92}, {"text": "So let's understand\nwhat the game is.", "start": 1038.45, "duration": 3.03}, {"text": "The game is played on a formula.", "start": 1041.48, "duration": 7.83}, {"text": "So you write down a\nquantified Boolean formula.", "start": 1049.31, "duration": 2.81}, {"text": "Remember, we talked about\nthat for several lectures now.", "start": 1052.12, "duration": 4.41}, {"text": "It's a bunch of quanti--\nvariables with quantifiers.", "start": 1056.53, "duration": 3.96}, {"text": "And then there's a part that\nhas no quantifiers, which", "start": 1060.49, "duration": 6.36}, {"text": "you can always put into\nconjunctive normal form", "start": 1066.85, "duration": 3.36}, {"text": "if you want.", "start": 1070.21, "duration": 0.55}, {"text": "But for the purposes of\nthis discussion so far,", "start": 1070.76, "duration": 2.93}, {"text": "it doesn't matter.", "start": 1073.69, "duration": 0.75}, {"text": "Later on, we'll actually\nwant it to be in CNF.", "start": 1074.44, "duration": 4.56}, {"text": "But for now, we just\nhave some formula.", "start": 1079.0, "duration": 4.85}, {"text": "And associated to\nthat formula there", "start": 1083.85, "duration": 3.27}, {"text": "is a game, in a\nvery natural sense.", "start": 1087.12, "duration": 2.08}, {"text": "So first of all, there are\ngoing to be two players.", "start": 1089.2, "duration": 3.33}, {"text": "But now the names\nof those players", "start": 1092.53, "duration": 1.52}, {"text": "are going to be Player\nExist and Player For All.", "start": 1094.05, "duration": 2.31}, {"text": "And this is how the game goes--", "start": 1100.8, "duration": 2.07}, {"text": "both players are going to be\npicking values for variables.", "start": 1108.63, "duration": 3.72}, {"text": "They're going to say variable x1\nis true, variable x2 is false,", "start": 1112.35, "duration": 5.49}, {"text": "variable x3 is false,\nvariable x4 is true.", "start": 1117.84, "duration": 3.45}, {"text": "And that's how the\nmoves of the game go.", "start": 1121.29, "duration": 4.59}, {"text": "They're just assigning values\nto variables so that in the end,", "start": 1125.88, "duration": 3.87}, {"text": "all of the variables have\nbeen assigned a Boolean value.", "start": 1129.75, "duration": 3.42}, {"text": "And so we end up\nwith an assignment.", "start": 1133.17, "duration": 1.95}, {"text": "But before we get to\nthat stage, which players", "start": 1135.12, "duration": 5.41}, {"text": "get to pick which variables?", "start": 1140.53, "duration": 2.58}, {"text": "And the way it works\nis the Exist player", "start": 1143.11, "duration": 2.22}, {"text": "is going to assign values\nto the Exist variables.", "start": 1145.33, "duration": 3.52}, {"text": "So Exist will pick x1 and\nx3 and assign them values.", "start": 1148.85, "duration": 5.57}, {"text": "The For All player is\ngoing to assign values", "start": 1154.42, "duration": 3.96}, {"text": "to the variables that have a\nFor All quantifier attached.", "start": 1158.38, "duration": 3.04}, {"text": "So in this case,\nthe For All player", "start": 1161.42, "duration": 6.56}, {"text": "will pick the value\nfor the x2 variable.", "start": 1167.98, "duration": 4.5}, {"text": "Fading out here.", "start": 1172.48, "duration": 0.84}, {"text": "And the order of\nplay is according", "start": 1179.36, "duration": 2.58}, {"text": "to the order of the\nappearance of the qualifiers", "start": 1181.94, "duration": 3.63}, {"text": "in the formula.", "start": 1185.57, "duration": 2.04}, {"text": "So the way I have it written\nout in this particular formula,", "start": 1187.61, "duration": 2.96}, {"text": "it's Exist x1--", "start": 1190.57, "duration": 1.26}, {"text": "so Exist will pick\nthe value for x1.", "start": 1191.83, "duration": 2.79}, {"text": "Then For All's turn.", "start": 1194.62, "duration": 1.5}, {"text": "For All will pick\nthe value of x2.", "start": 1196.12, "duration": 2.13}, {"text": "Then it's going to be Exist\npicking the value for x3,", "start": 1198.25, "duration": 5.56}, {"text": "and so on, until they get to the\nend, and all of the variables", "start": 1203.81, "duration": 4.05}, {"text": "have been assigned to value\nby one side or the other.", "start": 1207.86, "duration": 3.81}, {"text": "And now the game is over.", "start": 1211.67, "duration": 3.74}, {"text": "So what's left is to understand\nwho has won at that point.", "start": 1215.41, "duration": 8.505}, {"text": "And the way we say who has won\nis Exist wins if in the end,", "start": 1226.98, "duration": 10.25}, {"text": "this part psi, which is the\npart without quantifiers,", "start": 1237.23, "duration": 4.23}, {"text": "ended up being satisfied\nby that assignment", "start": 1241.46, "duration": 5.31}, {"text": "that the two players\ntogether ended up picking.", "start": 1246.77, "duration": 3.39}, {"text": "And then For All will win\nif that part of the formula", "start": 1250.16, "duration": 6.81}, {"text": "is not satisfied.", "start": 1256.97, "duration": 1.8}, {"text": "So let me just write\nthat down here.", "start": 1258.77, "duration": 1.5}, {"text": "After the variables have\nbeen assigned values,", "start": 1260.27, "duration": 2.19}, {"text": "we determine the winner.", "start": 1262.46, "duration": 1.44}, {"text": "Exist wins if the assignment\nthat built cooperatively", "start": 1263.9, "duration": 5.07}, {"text": "together satisfied psi,\nand otherwise For All,", "start": 1268.97, "duration": 4.77}, {"text": "if it doesn't satisfy psi.", "start": 1273.74, "duration": 2.87}, {"text": "So think about it this way--", "start": 1276.61, "duration": 3.18}, {"text": "Exist is picking the\nvalues of those variables.", "start": 1279.79, "duration": 2.58}, {"text": "He's trying to make this part\nof the formula satisfied,", "start": 1282.37, "duration": 5.91}, {"text": "trying to satisfy this\npart of the formula.", "start": 1288.28, "duration": 1.89}, {"text": "For All is picking\nvalues, but she's", "start": 1290.17, "duration": 3.51}, {"text": "trying to make this part\nof the formula unsatisfied.", "start": 1293.68, "duration": 3.21}, {"text": "So they're in opposition\nto one another.", "start": 1296.89, "duration": 5.2}, {"text": "And in the end, one of them\nis going to have succeeded", "start": 1302.09, "duration": 4.64}, {"text": "and the other one\nwill have failed.", "start": 1306.73, "duration": 2.7}, {"text": "And now, thinking\nabout it as a game,", "start": 1309.43, "duration": 3.0}, {"text": "depending upon the formula,\none side or the other", "start": 1312.43, "duration": 3.0}, {"text": "is going to have the ability\nto dominate the other one", "start": 1315.43, "duration": 2.76}, {"text": "and always win.", "start": 1318.19, "duration": 0.9}, {"text": "And so the question\nis, which side", "start": 1321.93, "duration": 4.43}, {"text": "has that ability to always win?", "start": 1326.36, "duration": 3.09}, {"text": "Which side has the\nwinning strategy?", "start": 1329.45, "duration": 2.522}, {"text": "Of course, it'll depend\nupon the formula.", "start": 1331.972, "duration": 1.708}, {"text": "Some formulas, it'll\nbe the Exist player", "start": 1333.68, "duration": 1.762}, {"text": "who has the winning strategy.", "start": 1335.442, "duration": 1.208}, {"text": "And other formulas, it\nwill be the For All player", "start": 1336.65, "duration": 2.042}, {"text": "who has the winning strategy.", "start": 1338.692, "duration": 1.318}, {"text": "And computationally, we'd like\nto make that determination.", "start": 1340.01, "duration": 3.25}, {"text": "Which side is going to\nwin for a given formula?", "start": 1343.26, "duration": 5.93}, {"text": "And there's a very, very\nnice, and actually very easy", "start": 1349.19, "duration": 5.19}, {"text": "and simple way to understand\nthat because we've already", "start": 1354.38, "duration": 3.58}, {"text": "run into that problem before.", "start": 1357.96, "duration": 2.7}, {"text": "The problem of determining\nwhich side has a forced win", "start": 1360.66, "duration": 4.02}, {"text": "is exactly the same\nthing as determining", "start": 1364.68, "duration": 1.98}, {"text": "whether this quantified\nBoolean formula is true", "start": 1366.66, "duration": 4.1}, {"text": "because the Exist player has a\nforced win when that formula--", "start": 1370.76, "duration": 5.1}, {"text": "let me say it again--", "start": 1381.46, "duration": 0.93}, {"text": "the Exist player has a forced\nwin when that formula is true.", "start": 1382.39, "duration": 2.94}, {"text": "So if this quantified Boolean\nformula were a true quantified", "start": 1388.57, "duration": 4.16}, {"text": "Boolean formula, then\nExist is guaranteed", "start": 1392.73, "duration": 2.76}, {"text": "to be able to win that game\nif it plays its hand right.", "start": 1395.49, "duration": 6.05}, {"text": "If this is a false\nformula, the For All player", "start": 1401.54, "duration": 3.75}, {"text": "will always be able to win\nif it plays its hand right.", "start": 1405.29, "duration": 5.47}, {"text": "So why is that?", "start": 1410.76, "duration": 2.22}, {"text": "It really follows kind of almost\nwithout doing any work at all.", "start": 1416.07, "duration": 5.89}, {"text": "This proof, even though\nit looks superficially", "start": 1421.96, "duration": 3.75}, {"text": "like it might be not\nso easy to prove,", "start": 1425.71, "duration": 2.79}, {"text": "it follows for a\nvery simple reason", "start": 1428.5, "duration": 2.19}, {"text": "because the meaning of what\nit means for Exist to have", "start": 1430.69, "duration": 5.52}, {"text": "a winning strategy is exactly\ncaptured by the semantics", "start": 1436.21, "duration": 4.32}, {"text": "of the quantifiers.", "start": 1440.53, "duration": 1.8}, {"text": "Let's see what that means.", "start": 1442.33, "duration": 1.36}, {"text": "What does it mean that\nExist has a forced", "start": 1443.69, "duration": 2.45}, {"text": "win, let's say up here?", "start": 1446.14, "duration": 4.28}, {"text": "That means that Exist has\na move that it could make,", "start": 1450.42, "duration": 3.91}, {"text": "that Exist can pick some value\nfor x1-- so there was some way", "start": 1454.33, "duration": 7.59}, {"text": "to assign x1 so that no\nmatter what the opponent does,", "start": 1461.92, "duration": 4.77}, {"text": "no matter what For\nAll does, there's", "start": 1466.69, "duration": 2.77}, {"text": "going to be a response\nthat Exist can make,", "start": 1469.46, "duration": 3.52}, {"text": "some assignment to x3, so that\nno matter what For All does,", "start": 1472.98, "duration": 7.345}, {"text": "there's going to\nbe an assignment", "start": 1480.325, "duration": 1.375}, {"text": "to the next variable,\nand so on and so on,", "start": 1481.7, "duration": 4.84}, {"text": "so that this thing\nends up being true.", "start": 1486.54, "duration": 2.137}, {"text": "So let me just say\nthat again somehow.", "start": 1488.677, "duration": 1.583}, {"text": "I didn't feel it\ncame out very clear.", "start": 1490.26, "duration": 2.3}, {"text": "So Exist has a winning\nstrategy exactly when", "start": 1492.56, "duration": 4.02}, {"text": "there is some assignment\nto x1 that Exist", "start": 1496.58, "duration": 2.49}, {"text": "can make such that no\nmatter what assignment", "start": 1499.07, "duration": 2.55}, {"text": "that For All makes to x2,\nthere is some assignment", "start": 1501.62, "duration": 3.57}, {"text": "x3 such that no matter what\nassignment that For All", "start": 1505.19, "duration": 4.59}, {"text": "makes to x4 and so on,\nmakes this thing true.", "start": 1509.78, "duration": 5.08}, {"text": "That's what it means for Exist\nto have a winning strategy.", "start": 1514.86, "duration": 4.17}, {"text": "That's exactly what\nthe quantifiers", "start": 1519.03, "duration": 2.04}, {"text": "are saying in the first place.", "start": 1521.07, "duration": 1.69}, {"text": "So even really without\ndoing anything,", "start": 1522.76, "duration": 2.51}, {"text": "you can see that \"Exist\nhas a winning strategy\"", "start": 1525.27, "duration": 3.51}, {"text": "means exactly the same thing\nas the quantified Boolean", "start": 1528.78, "duration": 3.63}, {"text": "formula being true.", "start": 1532.41, "duration": 1.6}, {"text": "So making the test of which\nside has a winning strategy", "start": 1534.01, "duration": 5.0}, {"text": "is the same as testing\nwhether the formula is true.", "start": 1539.01, "duration": 2.34}, {"text": "So let me try to\nturn to the chat", "start": 1545.55, "duration": 2.97}, {"text": "before we move on\nto anything else.", "start": 1548.52, "duration": 1.92}, {"text": "Here we go.", "start": 1557.84, "duration": 0.63}, {"text": "So I'm getting some questions\nabout the order of play", "start": 1570.26, "duration": 2.7}, {"text": "and about the alternation\nof the quantifiers here.", "start": 1572.96, "duration": 2.74}, {"text": "So the way I'm showing it is\nthat the quantifiers always", "start": 1575.7, "duration": 3.2}, {"text": "alternate--", "start": 1578.9, "duration": 1.02}, {"text": "Exist, For All, Exist, For All.", "start": 1579.92, "duration": 1.98}, {"text": "That doesn't really matter.", "start": 1581.9, "duration": 3.24}, {"text": "You can have several\nExists in a row,", "start": 1585.14, "duration": 1.98}, {"text": "and that would\ncorrespond to Exist", "start": 1587.12, "duration": 1.5}, {"text": "making several moves\nin a row before turning", "start": 1588.62, "duration": 2.91}, {"text": "it over to For All.", "start": 1591.53, "duration": 1.83}, {"text": "But alternatively, we could\njust add extra variables which", "start": 1593.36, "duration": 6.29}, {"text": "don't play any role in psi.", "start": 1599.65, "duration": 3.31}, {"text": "And they're just kind\nof dummy variables", "start": 1602.96, "duration": 2.76}, {"text": "that serve as spacers so if\nyou have two Exists in a row", "start": 1605.72, "duration": 3.21}, {"text": "and you don't like\nthat, you can just add a", "start": 1608.93, "duration": 2.19}, {"text": "For All of some variables that\nyou never see again in between.", "start": 1611.12, "duration": 4.35}, {"text": "So you can always get it to\nbe alternating if you want.", "start": 1615.47, "duration": 2.31}, {"text": "OK, also question-- the\ndifference between phi and psi.", "start": 1625.94, "duration": 6.66}, {"text": "So psi is the part here that\ndoes not have any quantifiers.", "start": 1632.6, "duration": 3.91}, {"text": "So the way we always\nwrite down our QBFs", "start": 1636.51, "duration": 2.9}, {"text": "is they're leading quantifiers\nwith the variables,", "start": 1639.41, "duration": 4.2}, {"text": "and all variables have to\nbe within the scope of one", "start": 1643.61, "duration": 2.7}, {"text": "of the quantified variables.", "start": 1646.31, "duration": 2.83}, {"text": "So there are no\nfree variables that", "start": 1649.14, "duration": 2.45}, {"text": "are unquantified in our QBFs.", "start": 1651.59, "duration": 2.245}, {"text": "And so all of these\nQBFs have a run", "start": 1656.57, "duration": 3.54}, {"text": "of quantifiers and then a\npart without quantifiers, so", "start": 1660.11, "duration": 2.85}, {"text": "the Boolean logic part.", "start": 1662.96, "duration": 1.86}, {"text": "Psi here is the\nBoolean logic part.", "start": 1664.82, "duration": 3.19}, {"text": "And phi is the whole thing\ntogether with the quantifiers.", "start": 1668.01, "duration": 2.895}, {"text": "So I'm not sure I\nunderstand some of these--", "start": 1673.58, "duration": 7.47}, {"text": "why not have the For All\nplayer have a forced win?", "start": 1681.05, "duration": 2.2}, {"text": "The For All player\nmight have a forced win.", "start": 1683.25, "duration": 3.8}, {"text": "One side or the\nother is guaranteed", "start": 1687.05, "duration": 1.92}, {"text": "to have a forced win.", "start": 1688.97, "duration": 1.27}, {"text": "So if Exist does not\nhave a forced win,", "start": 1690.24, "duration": 1.94}, {"text": "then For All will\nhave a forced win.", "start": 1692.18, "duration": 2.58}, {"text": "I'm not going to prove that.", "start": 1694.76, "duration": 1.95}, {"text": "That's a fairly simple\nproof by induction, which", "start": 1696.71, "duration": 2.553}, {"text": "I'm not going to go through.", "start": 1699.263, "duration": 1.167}, {"text": "Just take it as a fact.", "start": 1700.43, "duration": 2.86}, {"text": "Somebody is asking, why does\nFor All want to not satisfy", "start": 1703.29, "duration": 4.65}, {"text": "this expression, the psi part?", "start": 1707.94, "duration": 4.53}, {"text": "Well, that's the way\nwe set up the game.", "start": 1712.47, "duration": 2.67}, {"text": "In order to make this\ncorrespond to TQBF,", "start": 1715.14, "duration": 4.92}, {"text": "just the truth of\nthis expression,", "start": 1720.06, "duration": 4.21}, {"text": "we want to make Exist\ntry to satisfy this part", "start": 1724.27, "duration": 4.56}, {"text": "and For All try to make it\nnot satisfied because that's", "start": 1728.83, "duration": 7.7}, {"text": "what works.", "start": 1736.53, "duration": 0.69}, {"text": "I'm not sure what else, what\nother why I can answer there.", "start": 1737.22, "duration": 5.13}, {"text": "I'm going to answer some of\nthe most basic questions here.", "start": 1742.35, "duration": 2.95}, {"text": "Somebody is asking, how do we\nknow how many variables to use?", "start": 1745.3, "duration": 3.69}, {"text": "So the variables are the\nvariables of the formula.", "start": 1748.99, "duration": 4.2}, {"text": "So somebody is going\nto hand you a formula.", "start": 1753.19, "duration": 2.3}, {"text": "That's going to have x1 to\nx10, whatever that formula has,", "start": 1755.49, "duration": 3.54}, {"text": "some number of variables.", "start": 1759.03, "duration": 1.32}, {"text": "And so that game is\ngoing to have 10 moves.", "start": 1760.35, "duration": 3.63}, {"text": "If the quantifiers alternate,\nthen the moves will alternate,", "start": 1763.98, "duration": 4.14}, {"text": "but whatever the pattern\nof the quantifiers", "start": 1768.12, "duration": 2.43}, {"text": "is going to be the\npattern of the moves.", "start": 1770.55, "duration": 1.71}, {"text": "So Exist is going to follow\nthe places where the Exist", "start": 1772.26, "duration": 5.73}, {"text": "quantifier occurs,\nand For All is", "start": 1777.99, "duration": 2.245}, {"text": "going to follow the places where\nthe For All quantifier occurs.", "start": 1780.235, "duration": 2.625}, {"text": "So if you have Exist, For\nAll, Exist, For All, Exist", "start": 1782.86, "duration": 3.123}, {"text": "will move first, then\nFor All will move,", "start": 1785.983, "duration": 1.667}, {"text": "then Exist will move,\nthen For All will move.", "start": 1787.65, "duration": 1.875}, {"text": "Together, they're picking values\nto their respective variables", "start": 1789.525, "duration": 3.115}, {"text": "with their opposing goals.", "start": 1792.64, "duration": 2.09}, {"text": "Exist is trying to make\nthe psi part satisfied.", "start": 1794.73, "duration": 4.98}, {"text": "For All is trying to\navoid making it satisfied,", "start": 1799.71, "duration": 3.26}, {"text": "to make the assignment\nnot satisfy that part.", "start": 1802.97, "duration": 4.4}, {"text": "And if you think about\nwhat that means, just", "start": 1807.37, "duration": 3.32}, {"text": "in terms of the\nmeaning of that, it's", "start": 1810.69, "duration": 2.85}, {"text": "going to mean exactly\nthe same thing,", "start": 1813.54, "duration": 1.74}, {"text": "that the formula is true.", "start": 1815.28, "duration": 1.08}, {"text": "So anyway, if you don't--", "start": 1820.6, "duration": 4.24}, {"text": "let's see.", "start": 1824.84, "duration": 0.87}, {"text": "I don't know if there's\nany more questions", "start": 1825.71, "duration": 1.75}, {"text": "that I can try to answer here.", "start": 1827.46, "duration": 1.31}, {"text": "It's still about the\nnumbers of variables.", "start": 1837.89, "duration": 2.055}, {"text": "Each formula is going to\ndefine a different game.", "start": 1843.26, "duration": 3.48}, {"text": "You see the formula.", "start": 1846.74, "duration": 2.76}, {"text": "So the formula, however big--\nit may have 50 variables,", "start": 1849.5, "duration": 2.81}, {"text": "it may have two variables.", "start": 1852.31, "duration": 1.347}, {"text": "Well, we'll do an example.", "start": 1853.657, "duration": 1.083}, {"text": "Maybe that'll help.", "start": 1854.74, "duration": 2.94}, {"text": "The next check in, which\nis coming up pretty soon--", "start": 1857.68, "duration": 2.29}, {"text": "might actually be now--", "start": 1859.97, "duration": 1.52}, {"text": "we'll give you an example.", "start": 1861.49, "duration": 2.05}, {"text": "So we'll see who is\nunderstanding and who is not.", "start": 1863.54, "duration": 4.64}, {"text": "So let's continue here.", "start": 1868.18, "duration": 1.08}, {"text": "So therefore, the\nproblem of determining", "start": 1872.04, "duration": 3.67}, {"text": "does Exist have a forced win\nis exactly the TQBF problem,", "start": 1875.71, "duration": 3.75}, {"text": "because Exist has a forced win\nexactly when the formula is", "start": 1879.46, "duration": 2.85}, {"text": "true.", "start": 1882.31, "duration": 2.51}, {"text": "And what we're going to show is\nthat TQBF is a polynomial time", "start": 1884.82, "duration": 5.58}, {"text": "reducible to\ngeneralized geography,", "start": 1890.4, "duration": 2.7}, {"text": "but conceptually, we're going to\nthink about TQBF now as a game", "start": 1893.1, "duration": 4.11}, {"text": "because that's how generalized\ngeography is set up.", "start": 1897.21, "duration": 3.39}, {"text": "So given a formula\nlike this, we're", "start": 1900.6, "duration": 3.06}, {"text": "going to construct an instance\nof generalized geography.", "start": 1903.66, "duration": 2.7}, {"text": "We're going to construct a\ngraph where play on that graph", "start": 1906.36, "duration": 4.37}, {"text": "is going to mimic play\nin the formula game.", "start": 1910.73, "duration": 4.42}, {"text": "So making the\nmoves in that graph", "start": 1915.15, "duration": 1.65}, {"text": "are going to correspond\nto setting variables", "start": 1916.8, "duration": 2.67}, {"text": "true and false\nbecause the graph is", "start": 1919.47, "duration": 1.83}, {"text": "going to be specially designed\nto force that behavior.", "start": 1921.3, "duration": 3.015}, {"text": "So I think we have a\ncheck-in coming in.", "start": 1926.85, "duration": 1.97}, {"text": "Yeah, so let's just see how--", "start": 1928.82, "duration": 2.43}, {"text": "I suggest you pay\nattention to this check-in,", "start": 1931.25, "duration": 2.36}, {"text": "really try to think\nabout it and solve it,", "start": 1933.61, "duration": 1.75}, {"text": "because I think\nthat'll help you make", "start": 1935.36, "duration": 1.542}, {"text": "the connection between\nthe truth of the formula", "start": 1936.902, "duration": 4.488}, {"text": "and Exist having a\nwinning strategy.", "start": 1941.39, "duration": 2.895}, {"text": "So if you take\nthis formula here--", "start": 1948.87, "duration": 2.52}, {"text": "this is some particular\nformula, Exist x for all y,", "start": 1951.39, "duration": 2.88}, {"text": "blah, blah, blah.", "start": 1954.27, "duration": 2.25}, {"text": "There's going to be associated\nto that formula, a formula", "start": 1956.52, "duration": 2.49}, {"text": "game where first\nExist moves, assigns", "start": 1959.01, "duration": 3.18}, {"text": "a value for x, and\ndepending upon that value,", "start": 1962.19, "duration": 2.67}, {"text": "For All is going to move and is\ngoing to assign a value for y.", "start": 1964.86, "duration": 3.18}, {"text": "And after that's\ndone, this part here", "start": 1968.04, "duration": 2.28}, {"text": "is either going to\nbe true or false,", "start": 1970.32, "duration": 1.5}, {"text": "is either going to be\nsatisfied or unsatisfied.", "start": 1971.82, "duration": 3.09}, {"text": "And I want to know,\ncan Exist always find", "start": 1974.91, "duration": 2.64}, {"text": "a way to guarantee that this\npart is going to be satisfied,", "start": 1977.55, "duration": 4.38}, {"text": "or can For All always find a way\nto guarantee that this part is", "start": 1981.93, "duration": 3.57}, {"text": "not satisfied?", "start": 1985.5, "duration": 2.17}, {"text": "So you have to look\nat this formula", "start": 1987.67, "duration": 5.22}, {"text": "to understand which side is\ngoing to end up succeeding.", "start": 1992.89, "duration": 5.78}, {"text": "So if you're not clear on\nthe rules, look back here.", "start": 1998.67, "duration": 3.2}, {"text": "Exist is trying to make\nthis part satisfied.", "start": 2001.87, "duration": 2.61}, {"text": "For All is trying to make\nthis part unsatisfied.", "start": 2004.48, "duration": 2.49}, {"text": "But they neither of them\nhas the totally upper hand,", "start": 2006.97, "duration": 2.82}, {"text": "because Exist is picking\none of the variables,", "start": 2009.79, "duration": 2.67}, {"text": "For All is picking\nthe other variable.", "start": 2012.46, "duration": 1.89}, {"text": "But they're doing it\nin a certain order.", "start": 2014.35, "duration": 4.07}, {"text": "First Exist is going to pick,\nthen For All is going to pick.", "start": 2018.42, "duration": 2.82}, {"text": "So that's the way\nthe game works--", "start": 2024.15, "duration": 2.03}, {"text": "Exist picks a value, then\nFor All picks a value,", "start": 2026.18, "duration": 2.53}, {"text": "and then you see who wins.", "start": 2028.71, "duration": 2.82}, {"text": "So who wins?", "start": 2031.53, "duration": 1.46}, {"text": "Who's guaranteed to win?", "start": 2032.99, "duration": 2.58}, {"text": "Can we make sure\nthat Exists always", "start": 2035.57, "duration": 2.4}, {"text": "wins or can we make sure\nthat For All always wins", "start": 2037.97, "duration": 2.64}, {"text": "for this particular formula?", "start": 2040.61, "duration": 1.63}, {"text": "So there's just\ntwo variables here.", "start": 2042.24, "duration": 2.885}, {"text": "You can think about this in\neither of two ways-- strictly", "start": 2045.125, "duration": 2.375}, {"text": "speaking, purely as a\ngame, or you can look at,", "start": 2047.5, "duration": 3.42}, {"text": "understand whether that\nformula is true or not.", "start": 2050.92, "duration": 2.489}, {"text": "They're equivalent ways\nof looking at the problem.", "start": 2060.11, "duration": 3.79}, {"text": "In fact, in a certain\nsense, it's the same.", "start": 2063.9, "duration": 2.279}, {"text": "That's what I'm\ntrying to get across.", "start": 2066.179, "duration": 2.642}, {"text": "Thinking about this as a\ngame or thinking about it", "start": 2068.821, "duration": 2.083}, {"text": "in terms of quantifiers-- it\ndoesn't make any difference.", "start": 2070.904, "duration": 2.821}, {"text": "Almost done here.", "start": 2077.79, "duration": 1.72}, {"text": "So all right, last call.", "start": 2079.51, "duration": 5.0}, {"text": "Closing the poll.", "start": 2084.51, "duration": 0.93}, {"text": "I think this one you got.", "start": 2089.01, "duration": 1.979}, {"text": "You did pretty well on.", "start": 2090.989, "duration": 1.621}, {"text": "So the correct answer\nis the For All player", "start": 2092.61, "duration": 2.97}, {"text": "has a winning strategy,\nhas a forced win.", "start": 2095.58, "duration": 5.4}, {"text": "And also, similarly, the\nexpression, phi is false.", "start": 2100.98, "duration": 8.1}, {"text": "Because if you try to\nfind some x such that", "start": 2109.08, "duration": 3.9}, {"text": "no matter what y you pick,\nthis is going to be true,", "start": 2112.98, "duration": 4.307}, {"text": "this is going to be\nsatisfied, you're out of luck.", "start": 2117.287, "duration": 2.083}, {"text": "Because if you make\nx true, well, then--", "start": 2119.37, "duration": 4.215}, {"text": "now I'm confused.", "start": 2132.332, "duration": 0.708}, {"text": "What is going on.", "start": 2137.972, "duration": 0.708}, {"text": "If you make x true,\nthen x bar is false,", "start": 2138.68, "duration": 10.26}, {"text": "so y bar has to be true.", "start": 2148.94, "duration": 1.355}, {"text": "Now I'm completely confused.", "start": 2155.85, "duration": 1.26}, {"text": "Oh, no.", "start": 2157.11, "duration": 2.6}, {"text": "If you make x true, you\nhave to work For All y.", "start": 2159.71, "duration": 2.23}, {"text": "So this thing is clearly false.", "start": 2161.94, "duration": 1.32}, {"text": "If you make x true, then this\nx bar is going to be false,", "start": 2163.26, "duration": 7.05}, {"text": "and it's not the case that\nfor every y, setting for y--", "start": 2170.31, "duration": 4.77}, {"text": "because y is forced here. y\nis going to have to be true.", "start": 2175.08, "duration": 2.703}, {"text": "y bar is going to\nhave to be true,", "start": 2177.783, "duration": 1.417}, {"text": "so y is going to\nhave to be false.", "start": 2179.2, "duration": 1.417}, {"text": "So if you try again\nto make x false--", "start": 2180.617, "duration": 8.773}, {"text": "maybe that's one to think about\nfirst-- if you make x false,", "start": 2189.39, "duration": 4.887}, {"text": "you're going to be stuck\non the first clause", "start": 2194.277, "duration": 1.833}, {"text": "because it has to work\nfor both settings of y.", "start": 2196.11, "duration": 3.48}, {"text": "But maybe thinking about\nit as a game is better.", "start": 2199.59, "duration": 4.68}, {"text": "No matter what x\ndoes, y is going", "start": 2204.27, "duration": 2.94}, {"text": "to have a way of making one of\nthose two clauses unsatisfied.", "start": 2207.21, "duration": 5.04}, {"text": "So if x is true, we can set\ny to make the second clause", "start": 2212.25, "duration": 10.1}, {"text": "unsatisfied, and\nif x is false, y", "start": 2222.35, "duration": 3.3}, {"text": "can be assigned to make the\nfirst clause unsatisfied.", "start": 2225.65, "duration": 3.9}, {"text": "So anyway, the right answer\nis the For All player", "start": 2229.55, "duration": 5.56}, {"text": "has the winning strategy here.", "start": 2235.11, "duration": 1.41}, {"text": "And at the end of the\nday, it's not critical", "start": 2236.52, "duration": 2.4}, {"text": "that you understand\nthis correspondence,", "start": 2238.92, "duration": 1.68}, {"text": "but you have to then,\nif you don't quite", "start": 2240.6, "duration": 2.898}, {"text": "understand that,\nyou're going to have", "start": 2243.498, "duration": 1.542}, {"text": "to take on faith that the\nformula game is the same as", "start": 2245.04, "duration": 3.96}, {"text": "TQBF because that's\nwhat we're going to use.", "start": 2249.0, "duration": 2.55}, {"text": "And by the way, I would\nlike to also mention here", "start": 2251.55, "duration": 2.28}, {"text": "that this correspondence\nbetween games and quantifiers", "start": 2253.83, "duration": 6.6}, {"text": "is something that\nmathematicians use all the time.", "start": 2260.43, "duration": 4.0}, {"text": "Because if you have\nsome expression which", "start": 2264.43, "duration": 2.96}, {"text": "comes up often in\nmathematics, where there", "start": 2267.39, "duration": 2.25}, {"text": "are a whole run of quantifiers\nin front of some statement,", "start": 2269.64, "duration": 7.53}, {"text": "it's really kind\nof hard for anybody", "start": 2277.17, "duration": 3.24}, {"text": "to really get a feeling\nfor what that means.", "start": 2280.41, "duration": 4.14}, {"text": "If you have six alternating\nquantifiers, which", "start": 2284.55, "duration": 3.21}, {"text": "happens often in\nmathematics-- you're", "start": 2287.76, "duration": 1.717}, {"text": "going to have statements that\nhave a lot of alternations", "start": 2289.477, "duration": 2.333}, {"text": "with quantifiers--\nvery hard to get", "start": 2291.81, "duration": 1.5}, {"text": "a feeling for what that means.", "start": 2293.31, "duration": 1.39}, {"text": "But if you think about it as a\ngame, it's much more intuitive.", "start": 2294.7, "duration": 3.29}, {"text": "And it's completely equivalent\nto think about quantifiers.", "start": 2297.99, "duration": 3.24}, {"text": "Go back and forth between\nquantifiers and games.", "start": 2301.23, "duration": 3.84}, {"text": "Anyway, let's look\nat the reduction,", "start": 2305.07, "duration": 4.92}, {"text": "the construction that\nshows generalized geography", "start": 2309.99, "duration": 2.79}, {"text": "is complete.", "start": 2312.78, "duration": 1.56}, {"text": "A little longer than I thought--", "start": 2314.34, "duration": 1.96}, {"text": "I just want to make sure\neverybody is with me on this.", "start": 2316.3, "duration": 2.28}, {"text": "So why don't we call the break\nnow, and then we'll come back", "start": 2318.58, "duration": 4.76}, {"text": "and we'll look at the\nconstruction after that.", "start": 2323.34, "duration": 2.633}, {"text": "Because the\nconstruction itself is", "start": 2325.973, "duration": 1.417}, {"text": "going to take about 10 minutes\nto work through to, figure out", "start": 2327.39, "duration": 3.3}, {"text": "how to reduce TQBF\nand build a graph that", "start": 2330.69, "duration": 5.45}, {"text": "simulates the formula.", "start": 2336.14, "duration": 2.34}, {"text": "So I'm going to move\nright on to the--", "start": 2338.48, "duration": 3.795}, {"text": "and we'll come back.", "start": 2345.58, "duration": 2.07}, {"text": "So feel free to ask some\nquestions and get ready for us", "start": 2347.65, "duration": 6.78}, {"text": "diving in to a construction\nfor the reduction of TQBF", "start": 2354.43, "duration": 6.93}, {"text": "to generalized geography.", "start": 2361.36, "duration": 2.23}, {"text": "So it is an interesting\nquestion about what's", "start": 2363.59, "duration": 4.67}, {"text": "the relationship\nbetween a formula", "start": 2368.26, "duration": 3.52}, {"text": "and when you swap the\norder of the quantifiers.", "start": 2371.78, "duration": 2.69}, {"text": "The questioner is asking, does\nthat somehow relate to negating", "start": 2377.66, "duration": 4.89}, {"text": "the quantifier-free part, the\npart without the quantifiers?", "start": 2382.55, "duration": 5.46}, {"text": "And I don't think that that's\nthe right correspondence.", "start": 2388.01, "duration": 3.06}, {"text": "There actually is\nsome relationship.", "start": 2391.07, "duration": 3.09}, {"text": "When you say \"there\nexists for all,\"", "start": 2394.16, "duration": 5.1}, {"text": "it's actually a\nstronger statement", "start": 2399.26, "duration": 1.65}, {"text": "than saying that \"for\nall there exists.\"", "start": 2400.91, "duration": 2.31}, {"text": "And in general, that\nactually implies--", "start": 2403.22, "duration": 3.57}, {"text": "exist x for a y implies\nfor a y there exists an x,", "start": 2406.79, "duration": 7.65}, {"text": "because what you're saying\nis that if there exists an x", "start": 2414.44, "duration": 7.62}, {"text": "for a y, there is one x that\nworks for every one of the y's.", "start": 2422.06, "duration": 3.48}, {"text": "If you're saying for\na y there exists an x,", "start": 2425.54, "duration": 2.37}, {"text": "the choice of that x\ncan depend on the y.", "start": 2427.91, "duration": 4.03}, {"text": "So there is a\nconnection, but you", "start": 2431.94, "duration": 3.51}, {"text": "have to think through\nwhat it means in order", "start": 2435.45, "duration": 1.95}, {"text": "to understand that connection.", "start": 2437.4, "duration": 3.3}, {"text": "You won't need to\nknow that in order", "start": 2440.7, "duration": 2.37}, {"text": "to process what we're\ngoing to be doing,", "start": 2443.07, "duration": 1.87}, {"text": "but maybe it just helps\nyou think about it.", "start": 2444.94, "duration": 1.792}, {"text": "Other questions here.", "start": 2449.67, "duration": 1.08}, {"text": "I think we're out of time here.", "start": 2459.98, "duration": 3.63}, {"text": "So let's return to our lecture.", "start": 2466.39, "duration": 5.055}, {"text": "So I'm going to go back.", "start": 2475.02, "duration": 2.7}, {"text": "Hopefully that won't\ncrash everything.", "start": 2477.72, "duration": 3.0}, {"text": "There we go.", "start": 2484.25, "duration": 2.43}, {"text": "So now we're going\nto be reducing TQBF", "start": 2486.68, "duration": 3.18}, {"text": "to generalized geography.", "start": 2489.86, "duration": 1.4}, {"text": "Are we all together now?", "start": 2491.26, "duration": 1.0}, {"text": "OK.", "start": 2496.4, "duration": 1.14}, {"text": "So I'm going to illustrate this\nconstruction, which is a very", "start": 2497.54, "duration": 3.81}, {"text": "nice construction, by the way.", "start": 2501.35, "duration": 1.47}, {"text": "I'm going to illustrate\nthis construction just", "start": 2502.82, "duration": 1.917}, {"text": "by doing an example,\nor a partial example,", "start": 2504.737, "duration": 3.433}, {"text": "but I think it'll\ngive you the idea.", "start": 2508.17, "duration": 2.45}, {"text": "So you understand what\nI'm trying to do here,", "start": 2510.62, "duration": 2.15}, {"text": "is I'm starting off with a\nquantified Boolean formula.", "start": 2512.77, "duration": 5.97}, {"text": "So here it is.", "start": 2518.74, "duration": 2.12}, {"text": "I'm going to assume it's in\nconjunctive normal form, which", "start": 2520.86, "duration": 4.59}, {"text": "I can always convert\nit into that form,", "start": 2525.45, "duration": 2.97}, {"text": "maybe by adding some\nadditional variables,", "start": 2528.42, "duration": 2.43}, {"text": "but without doing anything\ntoo drastic to it.", "start": 2530.85, "duration": 7.36}, {"text": "And so now, starting\nfrom that formula,", "start": 2538.21, "duration": 5.37}, {"text": "I'm going to now build a graph,\nwhere playing the geography", "start": 2543.58, "duration": 3.99}, {"text": "game on that graph-- which\nis, remember, taking turns,", "start": 2547.57, "duration": 2.94}, {"text": "picking nodes\nwhich form a path--", "start": 2550.51, "duration": 3.3}, {"text": "is going to correspond to\nplaying the formula game, which", "start": 2553.81, "duration": 4.77}, {"text": "is picking the variables\nof that formula.", "start": 2558.58, "duration": 2.67}, {"text": "And then you want--", "start": 2561.25, "duration": 3.396}, {"text": "if the Exist player\nwins in the formula,", "start": 2564.646, "duration": 2.514}, {"text": "then Player I is supposed to\nwin in the geography game.", "start": 2567.16, "duration": 4.2}, {"text": "So here is how the\ngraph is going to look.", "start": 2574.24, "duration": 2.31}, {"text": "So try to follow this.", "start": 2579.2, "duration": 2.31}, {"text": "So there's going to\nbe kind of two parts,", "start": 2581.51, "duration": 4.0}, {"text": "one that's going to\ncorrespond to the variables,", "start": 2585.51, "duration": 2.9}, {"text": "and the other part that's going\nto correspond to the clauses.", "start": 2588.41, "duration": 4.35}, {"text": "And so for each\nvariable, I'm going", "start": 2592.76, "duration": 2.22}, {"text": "to have a diamond\nstructure here.", "start": 2594.98, "duration": 1.883}, {"text": "So there's a little\nstarting thing", "start": 2596.863, "duration": 1.417}, {"text": "that's going to be unique\nto the first variable.", "start": 2598.28, "duration": 2.8}, {"text": "But then every variable is\ngoing to have a little diamond", "start": 2601.08, "duration": 2.72}, {"text": "structure here.", "start": 2603.8, "duration": 2.16}, {"text": "And they're going to be\nattached one to the next.", "start": 2605.96, "duration": 3.56}, {"text": "And we'll understand what this\nmeans when we play geography", "start": 2609.52, "duration": 7.35}, {"text": "on this piece of the\ngraph, but let's just", "start": 2616.87, "duration": 3.383}, {"text": "understand the structure first.", "start": 2620.253, "duration": 1.292}, {"text": "So this is the start node.", "start": 2624.34, "duration": 2.4}, {"text": "And now, here Player I is going\nto play the role of Exist.", "start": 2626.74, "duration": 4.8}, {"text": "Player II is going to\nplay the role of For All.", "start": 2631.54, "duration": 2.25}, {"text": "And in fact, I'm going\nto identify that.", "start": 2633.79, "duration": 2.5}, {"text": "So I'm going to\ncall-- because it's", "start": 2636.29, "duration": 1.82}, {"text": "going to be helpful just\nto think about Player I", "start": 2638.11, "duration": 5.28}, {"text": "as being the Exist player.", "start": 2643.39, "duration": 1.47}, {"text": "So the Exist player is going\nto be playing on this graph.", "start": 2644.86, "duration": 3.665}, {"text": "The Exist and the\nFor All players", "start": 2648.525, "duration": 1.375}, {"text": "are going to be\nplaying on the graph.", "start": 2649.9, "duration": 1.542}, {"text": "It's really just\nPlayers I and II.", "start": 2651.442, "duration": 1.458}, {"text": "So the Exist player,\nPlayer I, by the rules,", "start": 2652.9, "duration": 6.93}, {"text": "has to pick the start node.", "start": 2659.83, "duration": 2.81}, {"text": "And now it's Player II's move,\nthe For All player's move.", "start": 2662.64, "duration": 3.92}, {"text": "Now, if you're with\nme on this graph,", "start": 2666.56, "duration": 6.82}, {"text": "the For All player's move\nis now not very interesting", "start": 2673.38, "duration": 3.39}, {"text": "because it's forced.", "start": 2676.77, "duration": 1.71}, {"text": "It has to go to here,\nbecause again, they're", "start": 2678.48, "duration": 2.61}, {"text": "just picking the nodes of a\nsimple path in this graph.", "start": 2681.09, "duration": 7.67}, {"text": "So the For All, Player\nII, the For All player,", "start": 2688.76, "duration": 3.95}, {"text": "has to go here if Exist\nstarted over there.", "start": 2692.71, "duration": 3.6}, {"text": "Now it's the Exist\nplayer's turn.", "start": 2696.31, "duration": 4.41}, {"text": "And now something\ninteresting happens.", "start": 2700.72, "duration": 2.91}, {"text": "The Exist player has a choice--", "start": 2703.63, "duration": 1.92}, {"text": "it can either go left or right.", "start": 2705.55, "duration": 3.02}, {"text": "There are two possibilities.", "start": 2708.57, "duration": 2.97}, {"text": "And then after that,\nthe For All player's", "start": 2711.54, "duration": 3.47}, {"text": "turn, who is,\nagain, just forced.", "start": 2715.01, "duration": 3.24}, {"text": "So far, For All has not\nhad much interesting stuff", "start": 2718.25, "duration": 2.43}, {"text": "to do in this, has\nnot had to think hard.", "start": 2720.68, "duration": 5.77}, {"text": "So no matter whether\nExist went left or right,", "start": 2726.45, "duration": 4.1}, {"text": "the For All player's move is\nforced and ends up over here.", "start": 2730.55, "duration": 4.27}, {"text": "And now it's this player's turn.", "start": 2734.82, "duration": 1.62}, {"text": "And again, this is\nkind of an easy one.", "start": 2736.44, "duration": 2.21}, {"text": "Oh, before I do that, let's just\nlook at how that play could go.", "start": 2738.65, "duration": 6.03}, {"text": "So the first two moves, as\nkind of I was suggesting,", "start": 2744.68, "duration": 2.9}, {"text": "Exist goes here and\nthen For All goes there.", "start": 2747.58, "duration": 4.57}, {"text": "I'm going to illustrate possible\nplays through this graph", "start": 2752.15, "duration": 4.53}, {"text": "by tracing them out in green.", "start": 2756.68, "duration": 1.92}, {"text": "So now Exist goes here,\nFor All goes there.", "start": 2761.57, "duration": 2.39}, {"text": "Those were forced.", "start": 2763.96, "duration": 1.23}, {"text": "But now, the Exist player\ngoes, could either go this way", "start": 2765.19, "duration": 3.45}, {"text": "or could go that way.", "start": 2768.64, "duration": 2.16}, {"text": "So those are two\npossible different ways", "start": 2775.12, "duration": 3.63}, {"text": "of playing the game\nthat we're building.", "start": 2778.75, "duration": 4.59}, {"text": "And now, whose turn is it here?", "start": 2783.34, "duration": 6.62}, {"text": "This was the For All\nplayer picked this one,", "start": 2792.547, "duration": 1.833}, {"text": "so the next turn is the Exist\nplayer, and that's forced.", "start": 2794.38, "duration": 6.03}, {"text": "But now notice-- now\nfor the first time,", "start": 2800.41, "duration": 3.33}, {"text": "the For All player has to think,\nbecause the For All player", "start": 2803.74, "duration": 3.3}, {"text": "can either go left or right,\nso could either go left,", "start": 2807.04, "duration": 9.445}, {"text": "or it could go right.", "start": 2816.485, "duration": 0.875}, {"text": "And so then there's going to\nbe a sequence of these diamond", "start": 2820.98, "duration": 4.94}, {"text": "structures, where they're\nconstructed so that alternately", "start": 2825.92, "duration": 4.92}, {"text": "either the Exist\nhas a choice or the", "start": 2830.84, "duration": 2.19}, {"text": "For All player has a choice,\nuntil you get all the way down", "start": 2833.03, "duration": 3.407}, {"text": "to the bottom.", "start": 2836.437, "duration": 0.583}, {"text": "That is the graph\nwe've built so far.", "start": 2841.38, "duration": 2.28}, {"text": "Now, if we stopped here, then\nwhoever ended up at this point,", "start": 2843.66, "duration": 8.31}, {"text": "whether it was Exist or For\nAll, would be the winner", "start": 2851.97, "duration": 2.25}, {"text": "because there's nowhere\nfor the opponent to go.", "start": 2854.22, "duration": 4.17}, {"text": "Of course, that wouldn't\nbe very interesting.", "start": 2858.39, "duration": 2.62}, {"text": "So there's going to be more\nstuff we're going to add on,", "start": 2861.01, "duration": 2.93}, {"text": "which are going to\ncorrespond to the clauses.", "start": 2863.94, "duration": 3.31}, {"text": "Now, how to think about\nwhat's happened so far?", "start": 2867.25, "duration": 5.09}, {"text": "Maybe some of you\nmight be guessing", "start": 2872.34, "duration": 3.47}, {"text": "that when the Exist player\nover here has a choice,", "start": 2875.81, "duration": 3.467}, {"text": "could have either gone\nleft or right, that's going", "start": 2879.277, "duration": 2.083}, {"text": "to correspond-- because this\nis supposed to be mimicking", "start": 2881.36, "duration": 3.99}, {"text": "the formula game-- that's going\nto correspond to the Exist", "start": 2885.35, "duration": 4.62}, {"text": "player in the formula\npicking the variable", "start": 2889.97, "duration": 3.48}, {"text": "either true or false.", "start": 2893.45, "duration": 1.93}, {"text": "So to the left or right is going\nto correspond to true or false.", "start": 2895.38, "duration": 3.99}, {"text": "And so let's just\narbitrarily say", "start": 2899.37, "duration": 2.82}, {"text": "left is going to correspond\nto picking it true and right", "start": 2902.19, "duration": 3.358}, {"text": "is going to correspond\nto picking false.", "start": 2905.548, "duration": 1.667}, {"text": "So the way I've set it\nso far is Exist ended up", "start": 2910.3, "duration": 5.67}, {"text": "picking the first\nvariable false,", "start": 2915.97, "duration": 2.37}, {"text": "then For All picked the\nsecond variable false.", "start": 2918.34, "duration": 3.57}, {"text": "And the nth variable\nalso got set false.", "start": 2921.91, "duration": 2.25}, {"text": "Everything got set false so far.", "start": 2924.16, "duration": 1.383}, {"text": "Who knows what happened\nover here, of course--", "start": 2925.543, "duration": 1.917}, {"text": "so just to understand\nwhat we've done so far.", "start": 2931.5, "duration": 3.19}, {"text": "Now I'm going to show you how\nto build the rest of the graph.", "start": 2934.69, "duration": 3.51}, {"text": "So let's take away\nthe green part.", "start": 2938.2, "duration": 4.08}, {"text": "And now we're just back\nto the construction.", "start": 2942.28, "duration": 3.06}, {"text": "The green part is actually\nhow you use that construction.", "start": 2945.34, "duration": 2.38}, {"text": "So back to the\nconstruction here-- now,", "start": 2947.72, "duration": 1.85}, {"text": "what do we want to achieve?", "start": 2949.57, "duration": 2.64}, {"text": "By the time the players on the\ngraph have got down to here,", "start": 2952.21, "duration": 5.97}, {"text": "they've effectively made an\nassignment to the variables", "start": 2958.18, "duration": 3.21}, {"text": "by going either left or right\nat each one of these diamonds.", "start": 2961.39, "duration": 3.18}, {"text": "So the assignment is done.", "start": 2964.57, "duration": 2.43}, {"text": "From the perspective of the\nformula game, the game is over,", "start": 2967.0, "duration": 4.14}, {"text": "and now one side or\nthe other has won.", "start": 2971.14, "duration": 3.68}, {"text": "Here, we want to build\nsome extra structure here", "start": 2974.82, "duration": 3.78}, {"text": "as a kind of an endgame, to make\nsure that the For All player", "start": 2978.6, "duration": 5.69}, {"text": "gets stuck if the\nExist player has", "start": 2984.29, "duration": 3.15}, {"text": "made an assignment\nwhich satisfies", "start": 2987.44, "duration": 3.13}, {"text": "this part of the formula.", "start": 2990.57, "duration": 1.37}, {"text": "And the Exist player should get\nstuck if the For All player,", "start": 2991.94, "duration": 4.69}, {"text": "if the assignment that they made\ndoes not satisfy the formula.", "start": 2996.63, "duration": 5.44}, {"text": "So let's see how we're\ngoing to achieve that", "start": 3002.07, "duration": 2.01}, {"text": "with some additional structure.", "start": 3004.08, "duration": 2.23}, {"text": "So there's going to be\nsome extra node over here.", "start": 3006.31, "duration": 3.153}, {"text": "Let me just tell you what the\nstructure is, and then we'll", "start": 3009.463, "duration": 2.417}, {"text": "argue why it works.", "start": 3011.88, "duration": 1.41}, {"text": "So going from this\nbottom node, we're", "start": 3013.29, "duration": 3.18}, {"text": "going to start the\nsecond part of the graph.", "start": 3016.47, "duration": 4.83}, {"text": "There's going to be\na node which fans out", "start": 3021.3, "duration": 3.09}, {"text": "to a node for each\none of the clauses.", "start": 3024.39, "duration": 1.86}, {"text": "And each one of those\nclauses, in turn,", "start": 3029.31, "duration": 1.68}, {"text": "fans out to a node for\neach one of its literals.", "start": 3030.99, "duration": 5.01}, {"text": "So you see we have\nclause c1, and it", "start": 3036.0, "duration": 3.03}, {"text": "has the literals x1, x2\nbar, x3, x1, x2 bar, x3.", "start": 3039.03, "duration": 4.92}, {"text": "So there's a node for each\nof the literals in clause c1,", "start": 3043.95, "duration": 3.33}, {"text": "same for clause c2, and so on.", "start": 3047.28, "duration": 2.37}, {"text": "Now we're almost done.", "start": 3052.34, "duration": 1.43}, {"text": "Now I'm going to tell you how to\nconnect up these literal nodes.", "start": 3053.77, "duration": 4.38}, {"text": "So x1, each node is\ngoing to correspond back", "start": 3058.15, "duration": 4.98}, {"text": "to its own diamond.", "start": 3063.13, "duration": 2.37}, {"text": "So now we're going\nto tie it back", "start": 3065.5, "duration": 1.47}, {"text": "to the first part, where\nwe made the assignment", "start": 3066.97, "duration": 3.15}, {"text": "part of the graph.", "start": 3070.12, "duration": 1.23}, {"text": "x1 is going to connect\nback to the true part", "start": 3071.35, "duration": 3.69}, {"text": "of the x1 diamond.", "start": 3075.04, "duration": 3.23}, {"text": "And x2 because, it's\nnegated, is going", "start": 3078.27, "duration": 2.19}, {"text": "to tie back to the false\npart of the x2 diamond", "start": 3080.46, "duration": 4.46}, {"text": "because it's an x2 variable.", "start": 3084.92, "duration": 3.24}, {"text": "Similarly, x1 bar--\nbecause x1 bar--", "start": 3088.16, "duration": 7.88}, {"text": "here's x1 bar in clause 2--", "start": 3096.04, "duration": 3.31}, {"text": "x1 bar is going\nto connect up now", "start": 3099.35, "duration": 3.66}, {"text": "to the false side\nof the x1 diamond.", "start": 3103.01, "duration": 5.09}, {"text": "And x2 bar is\ngoing to connect up", "start": 3108.1, "duration": 2.25}, {"text": "to the false side of the\nx2 diamond, and so on.", "start": 3110.35, "duration": 4.79}, {"text": "I don't have the\nother diamonds here,", "start": 3115.14, "duration": 1.83}, {"text": "but the x4 would connect\nup to the true side", "start": 3116.97, "duration": 4.08}, {"text": "of the x4 diamond, for example.", "start": 3121.05, "duration": 4.5}, {"text": "That's the whole construction.", "start": 3125.55, "duration": 1.26}, {"text": "Let's understand why it works.", "start": 3126.81, "duration": 2.81}, {"text": "So the endgame here\nis we want Exist", "start": 3129.62, "duration": 2.25}, {"text": "to win if the assignment\nsatisfied all the clauses.", "start": 3131.87, "duration": 6.47}, {"text": "And For All should win if there\nwas some unsatisfied clauses.", "start": 3138.34, "duration": 4.06}, {"text": "So why does this happen?", "start": 3142.4, "duration": 1.51}, {"text": "So let's put back an\nassignment here that was made.", "start": 3143.91, "duration": 2.46}, {"text": "So the one I'm putting\nback, the assignment", "start": 3146.37, "duration": 5.66}, {"text": "that they cooperatively\nbuilt had x1 being true,", "start": 3152.03, "duration": 4.41}, {"text": "x2 being false, and\nsome other stuff.", "start": 3156.44, "duration": 4.26}, {"text": "Now, why does this work?", "start": 3160.7, "duration": 1.15}, {"text": "So what we want to\nhave happen now--", "start": 3161.85, "duration": 2.81}, {"text": "so now, the move\nproceeds over to here,", "start": 3164.66, "duration": 2.91}, {"text": "and you need to arrange\nit so that Exist", "start": 3167.57, "duration": 5.25}, {"text": "is the one who picks that node.", "start": 3172.82, "duration": 1.83}, {"text": "If this would have been For\nAll, just add an extra node", "start": 3180.73, "duration": 2.64}, {"text": "to switch whose turn it is.", "start": 3183.37, "duration": 3.49}, {"text": "But you want Exist\nto be up here.", "start": 3186.86, "duration": 1.56}, {"text": "And what you want is for For All\nto be picking the clause node.", "start": 3188.42, "duration": 5.34}, {"text": "Now, here is the part to\nunderstand what's going on--", "start": 3193.76, "duration": 4.32}, {"text": "we want For All to win\nif this is not satisfied.", "start": 3202.24, "duration": 5.92}, {"text": "So For All is going\nto make a claim.", "start": 3208.16, "duration": 3.91}, {"text": "If it is not satisfied,\nthere is some clause", "start": 3212.07, "duration": 1.98}, {"text": "which is not satisfied.", "start": 3214.05, "duration": 1.14}, {"text": "There's some clause\nwhich ended up", "start": 3215.19, "duration": 1.68}, {"text": "being false in the assignment.", "start": 3216.87, "duration": 1.54}, {"text": "So the For All player going\nto say, \"I think I won.", "start": 3218.41, "duration": 4.12}, {"text": "And I won because clause\nnumber 1 is unsatisfied.\"", "start": 3222.53, "duration": 4.97}, {"text": "So it's going to pick\nclause number one.", "start": 3227.5, "duration": 3.147}, {"text": "So here's the For All\nplayer's [INAUDIBLE]..", "start": 3230.647, "duration": 1.833}, {"text": "Here's Exist player.", "start": 3232.48, "duration": 1.44}, {"text": "And so the For All player picks\nthe clause claimed unsatisfied.", "start": 3233.92, "duration": 5.19}, {"text": "So over here, For All player\nsays, clause c1 is unsatisfied.", "start": 3239.11, "duration": 4.02}, {"text": "I'm going to move here.", "start": 3243.13, "duration": 3.04}, {"text": "Exist player says, \"Uh-uh,\nI don't agree with you.", "start": 3246.17, "duration": 3.88}, {"text": "The assignment, your line--", "start": 3250.05, "duration": 2.79}, {"text": "the assignment actually\nmakes one of the literals", "start": 3255.66, "duration": 4.44}, {"text": "true in clause c1.", "start": 3260.1, "duration": 3.5}, {"text": "Let's say you made x1 was\ntrue, as in fact, it is here--", "start": 3263.6, "duration": 4.59}, {"text": "x1, in fact, is true.", "start": 3268.19, "duration": 1.39}, {"text": "So Exist is now going to pick\nthe literal that was true", "start": 3269.58, "duration": 7.76}, {"text": "in the assignment in that\nclause, that the For All player", "start": 3277.34, "duration": 3.48}, {"text": "is claiming is unsatisfied.", "start": 3280.82, "duration": 2.27}, {"text": "One of them is\nlying at this point.", "start": 3283.09, "duration": 3.66}, {"text": "This is either now true in here,\nwhich means the For All player", "start": 3286.75, "duration": 2.88}, {"text": "did not pick an\nunsatisfied clause,", "start": 3289.63, "duration": 1.89}, {"text": "or the Exist player\npicked the false literal,", "start": 3291.52, "duration": 2.89}, {"text": "which is the Exist player picked\nthe false literal so the Exist", "start": 3294.41, "duration": 4.37}, {"text": "player is lying.", "start": 3298.78, "duration": 1.29}, {"text": "Now, the moment of truth\nbecause let's see--", "start": 3300.07, "duration": 6.44}, {"text": "if Exist picked here, now it's\nthe For All player's turn.", "start": 3306.51, "duration": 6.08}, {"text": "It's connected back to the\ntrue side of the diamond.", "start": 3312.59, "duration": 2.415}, {"text": "So that node was already used.", "start": 3318.53, "duration": 2.04}, {"text": "That's the only place that\nFor All could possibly go.", "start": 3320.57, "duration": 2.82}, {"text": "For All is stuck,\nand Exist has won", "start": 3323.39, "duration": 2.64}, {"text": "the game, which is what you\nwant because For All's claims", "start": 3326.03, "duration": 5.16}, {"text": "are false and the Exist\nwas correct in saying", "start": 3331.19, "duration": 4.92}, {"text": "that x1 satisfied clause c1.", "start": 3336.11, "duration": 3.3}, {"text": "So x1 is true, so there's\nnowhere for For All to go.", "start": 3339.41, "duration": 4.08}, {"text": "But compare that\nwith the situation", "start": 3343.49, "duration": 2.34}, {"text": "if the assignment, on\nthis part of the graph,", "start": 3345.83, "duration": 3.3}, {"text": "had gone through the\nfalse, had assigned x1", "start": 3349.13, "duration": 3.42}, {"text": "to be false, so the path\nhad gone through the", "start": 3352.55, "duration": 3.16}, {"text": "For All side of the diamond.", "start": 3355.71, "duration": 2.18}, {"text": "Now this node would\nstill be unoccupied,", "start": 3357.89, "duration": 2.76}, {"text": "would still be available.", "start": 3360.65, "duration": 4.21}, {"text": "So now the For All\nplay-- so if Exist", "start": 3364.86, "duration": 3.35}, {"text": "was claiming that x1 was\ntrue and it really was false,", "start": 3368.21, "duration": 5.71}, {"text": "the For All player would be\nable to move onto that node.", "start": 3373.92, "duration": 2.688}, {"text": "And now it's the\nExist player's turn,", "start": 3376.608, "duration": 1.542}, {"text": "and the Exist would be stuck\nbecause this node down here is", "start": 3378.15, "duration": 3.84}, {"text": "guaranteed to have been used.", "start": 3381.99, "duration": 1.23}, {"text": "So that is the idea.", "start": 3385.78, "duration": 3.68}, {"text": "It's really very beautiful, and\nactually not that complicated.", "start": 3389.46, "duration": 4.71}, {"text": "You have to stare at it a bit.", "start": 3394.17, "duration": 1.84}, {"text": "So let's just see what\nhappens, for example,", "start": 3396.01, "duration": 2.37}, {"text": "if I had one other case-- maybe\nit's not necessary at this", "start": 3398.38, "duration": 3.8}, {"text": "point, but if the For All player\nclaimed it's the second clause", "start": 3402.18, "duration": 6.9}, {"text": "which is unsatisfied in the\nmutually selected assignment,", "start": 3409.08, "duration": 4.63}, {"text": "then--", "start": 3413.71, "duration": 0.5}, {"text": "so this is sort of the case\nwhere the For All player", "start": 3417.93, "duration": 2.83}, {"text": "is correct, in that if\nthe Exist player now says,", "start": 3420.76, "duration": 4.74}, {"text": "well, I think x1 bar\nsatisfied that second clause,", "start": 3425.5, "duration": 6.02}, {"text": "but this assignment made x1\ntrue, so x1 bar is false.", "start": 3431.52, "duration": 5.76}, {"text": "And so the Exist\nplayer's lying this time.", "start": 3437.28, "duration": 2.89}, {"text": "And so now the For All player\ncan take this last edge", "start": 3440.17, "duration": 3.08}, {"text": "and go here, and has\none last move to make.", "start": 3443.25, "duration": 4.54}, {"text": "And then the Exist\nplayer's stuck.", "start": 3447.79, "duration": 1.58}, {"text": "So the For All will\nwin in that case.", "start": 3449.37, "duration": 1.96}, {"text": "So let's turn now,\nshifting gears entirely", "start": 3451.33, "duration": 3.92}, {"text": "to a different part\nof space complexity.", "start": 3455.25, "duration": 7.23}, {"text": "Instead of looking at polynomial\nspace, which is very powerful,", "start": 3462.48, "duration": 3.73}, {"text": "we're going to look at log\nspace, which is comparatively", "start": 3466.21, "duration": 2.78}, {"text": "speaking much, much weaker.", "start": 3468.99, "duration": 1.65}, {"text": "So log space are the things\nthat you can do when you only", "start": 3470.64, "duration": 3.54}, {"text": "have enough space to\nwrite down, essentially,", "start": 3474.18, "duration": 3.42}, {"text": "a pointer or some fixed number\nof pointers into the input.", "start": 3477.6, "duration": 3.21}, {"text": "That's what you get when\nyou have logarithmic space,", "start": 3480.81, "duration": 7.2}, {"text": "because log space is\nenough to write down", "start": 3488.01, "duration": 2.42}, {"text": "an index of something.", "start": 3490.43, "duration": 1.56}, {"text": "So this is what you can do\nwith a bunch of pointers.", "start": 3491.99, "duration": 3.84}, {"text": "And in order to\nmake sense of this,", "start": 3495.83, "duration": 4.38}, {"text": "we have to change the\nmodel of computation.", "start": 3500.21, "duration": 2.16}, {"text": "Because if we use the ordinary\none-tape Turing machine, just", "start": 3502.37, "duration": 3.75}, {"text": "scanning across the input,\njust reading the input the way", "start": 3506.12, "duration": 5.85}, {"text": "we've defined it,\nwould cost space n.", "start": 3511.97, "duration": 2.61}, {"text": "And so you can't\neven read the input", "start": 3514.58, "duration": 2.07}, {"text": "if you have less than n space\navailable, like log space.", "start": 3516.65, "duration": 3.46}, {"text": "So we're going to introduce\na different model just", "start": 3520.11, "duration": 3.25}, {"text": "to allow us to define this.", "start": 3523.36, "duration": 1.5}, {"text": "And that's a model\nwhere it's going", "start": 3524.86, "duration": 2.01}, {"text": "to have two tapes, where the\npart that contains the input", "start": 3526.87, "duration": 4.84}, {"text": "doesn't count.", "start": 3531.71, "duration": 1.235}, {"text": "And in order to make sure\nit's not being cheated,", "start": 3532.945, "duration": 4.275}, {"text": "the input tape is\ngoing to be read only,", "start": 3537.22, "duration": 3.848}, {"text": "but it doesn't count\ntoward the space used.", "start": 3541.068, "duration": 1.792}, {"text": "Only the work tape, which\nnow can be written or read,", "start": 3542.86, "duration": 7.232}, {"text": "is going to count\ntoward the space bound.", "start": 3550.092, "duration": 1.708}, {"text": "So now what we're\ngoing to define is,", "start": 3554.72, "duration": 3.06}, {"text": "using this definition\nfor our space complexity,", "start": 3557.78, "duration": 6.84}, {"text": "we'll define SPACE\nlog n, the things", "start": 3564.62, "duration": 3.93}, {"text": "that you can do if you have\nonly a log amount of space", "start": 3568.55, "duration": 2.25}, {"text": "here, on the work tape.", "start": 3570.8, "duration": 3.9}, {"text": "So the length of the input is n.", "start": 3574.7, "duration": 2.16}, {"text": "The length of the work\ntape is order log n.", "start": 3576.86, "duration": 7.39}, {"text": "So we have the deterministic\nand nondeterministic associated", "start": 3584.25, "duration": 4.86}, {"text": "classes--", "start": 3589.11, "duration": 1.5}, {"text": "SPACE log n and NSPACE log n.", "start": 3590.61, "duration": 2.31}, {"text": "We're calling it L and NL--", "start": 3592.92, "duration": 2.34}, {"text": "log space and\nnondeterministic log space.", "start": 3595.26, "duration": 4.17}, {"text": "And as I said, that's\njust enough space", "start": 3599.43, "duration": 1.71}, {"text": "to write down some\npointers into the input.", "start": 3601.14, "duration": 2.1}, {"text": "And let's do some\nquick examples.", "start": 3603.24, "duration": 1.455}, {"text": "So if you take the language of\npalindromes, essentially, or ww", "start": 3608.09, "duration": 3.95}, {"text": "reverse, that's in log space.", "start": 3612.04, "duration": 5.89}, {"text": "So here is a string.", "start": 3617.93, "duration": 1.88}, {"text": "Here's a string\nthat's in ww reverse.", "start": 3619.81, "duration": 3.84}, {"text": "And the ordinary\nway you might use", "start": 3623.65, "duration": 4.95}, {"text": "to test whether a string is\nof this form in ww reverse", "start": 3628.6, "duration": 3.99}, {"text": "might be to cross off\ncorresponding locations here.", "start": 3632.59, "duration": 3.073}, {"text": "But you can't do that.", "start": 3635.663, "duration": 0.917}, {"text": "It's a read-only input tape.", "start": 3636.58, "duration": 1.81}, {"text": "So you have to somehow avoid\nwriting on the input tape,", "start": 3638.39, "duration": 3.11}, {"text": "but still testing whether\nyou're in this language.", "start": 3641.5, "duration": 2.61}, {"text": "It's not hard to do.", "start": 3644.11, "duration": 1.35}, {"text": "You can use the work tape\njust to keep track of where", "start": 3645.46, "duration": 3.75}, {"text": "your pointers would be.", "start": 3649.21, "duration": 2.49}, {"text": "And in so doing,\nyou can just make", "start": 3651.7, "duration": 4.56}, {"text": "sure you're matching off\ncorresponding locations", "start": 3656.26, "duration": 4.38}, {"text": "with each other in the input.", "start": 3660.64, "duration": 2.8}, {"text": "So log space, because\nof its ability", "start": 3663.44, "duration": 2.15}, {"text": "to store some fixed\nnumber of pointers,", "start": 3665.59, "duration": 2.79}, {"text": "gives you enough to test\nmembership in this language.", "start": 3668.38, "duration": 3.86}, {"text": "So this is in\nsolvable in log space.", "start": 3672.24, "duration": 4.67}, {"text": "The path problem, which\nwe've seen before--", "start": 3676.91, "duration": 2.64}, {"text": "you're given a graph, and a\nstart node, and an ending node,", "start": 3679.55, "duration": 3.66}, {"text": "or a start and a target.", "start": 3683.21, "duration": 1.5}, {"text": "You're given a graph\nof an s and t--", "start": 3684.71, "duration": 2.87}, {"text": "it's a directed a graph--", "start": 3687.58, "duration": 1.2}, {"text": "and can I know, can\nI get from s to t?", "start": 3688.78, "duration": 3.42}, {"text": "So that problem is solvable\nin nondeterministic log space", "start": 3692.2, "duration": 5.58}, {"text": "because the way we would do\nthat, not writing this down", "start": 3697.78, "duration": 6.42}, {"text": "in any detail,\nbut what you would", "start": 3704.2, "duration": 3.27}, {"text": "do in your nondeterministic\nlog space machine--", "start": 3707.47, "duration": 4.8}, {"text": "you have your input graph\nwritten here in the input,", "start": 3712.27, "duration": 4.02}, {"text": "and you're just going to guess\nthe sequence of nodes one", "start": 3716.29, "duration": 6.36}, {"text": "by one, which takes you from the\nstart node to the target node.", "start": 3722.65, "duration": 6.69}, {"text": "You're not going to write down\nthat whole sequence in advance", "start": 3729.34, "duration": 2.7}, {"text": "because that cost you way\ntoo much space to write down.", "start": 3732.04, "duration": 3.93}, {"text": "The only space that\nyou're going to use", "start": 3735.97, "duration": 1.68}, {"text": "is to remember the current\nlocation where you're sitting.", "start": 3737.65, "duration": 2.94}, {"text": "So you start out, you\nwrite down on the work tape", "start": 3740.59, "duration": 2.73}, {"text": "the start node.", "start": 3743.32, "duration": 1.26}, {"text": "Then you're going to\nnondeterministically pick", "start": 3744.58, "duration": 2.34}, {"text": "one of the outgoing\nedges from the start node", "start": 3746.92, "duration": 2.73}, {"text": "and look at its associated node,\nand replace that on your work", "start": 3749.65, "duration": 4.02}, {"text": "tape, and keep repeating that.", "start": 3753.67, "duration": 3.12}, {"text": "If you ever get to the node\nt, then you can accept.", "start": 3756.79, "duration": 5.4}, {"text": "And you also have to\nbe a little careful--", "start": 3762.19, "duration": 1.75}, {"text": "I don't have this on the slide.", "start": 3763.94, "duration": 2.3}, {"text": "You also have to make sure that\nif the graph has a loop in it,", "start": 3766.24, "duration": 7.03}, {"text": "because that's not allowed\nin space complexity,", "start": 3773.27, "duration": 3.1}, {"text": "so you're going to need\nalso a counter to make sure", "start": 3776.37, "duration": 2.36}, {"text": "that if you count up to--", "start": 3778.73, "duration": 4.82}, {"text": "you've gone through\na number of nodes", "start": 3783.55, "duration": 1.96}, {"text": "which exceeds the total\nnumber of nodes in the graph,", "start": 3785.51, "duration": 2.97}, {"text": "then you can shut down that\nbranch of the nondeterministic,", "start": 3788.48, "duration": 3.03}, {"text": "because it's just\ngoing in a loop.", "start": 3791.51, "duration": 8.05}, {"text": "If there's any path\nthat connects s to t,", "start": 3799.56, "duration": 2.46}, {"text": "there's certainly\ngoing to be a path that", "start": 3802.02, "duration": 1.728}, {"text": "has the most of the number\nof nodes of the graph in it.", "start": 3803.748, "duration": 2.292}, {"text": "So path is in NL.", "start": 3808.73, "duration": 1.65}, {"text": "This language here\nis in L. What's", "start": 3810.38, "duration": 3.87}, {"text": "the relationship\nbetween L and NL?", "start": 3814.25, "duration": 2.22}, {"text": "Well, certainly the\ndeterministic class", "start": 3816.47, "duration": 2.31}, {"text": "is contained in the\nnondeterministic class.", "start": 3818.78, "duration": 2.37}, {"text": "That's all that's known.", "start": 3821.15, "duration": 2.17}, {"text": "Whether these two collapse down\nto be the same is unsolved.", "start": 3823.32, "duration": 4.08}, {"text": "And so we're going to spend\na little time next lecture", "start": 3827.4, "duration": 2.4}, {"text": "exploring this.", "start": 3829.8, "duration": 2.55}, {"text": "But let's first look at some\nof the basic facts about log", "start": 3832.35, "duration": 6.33}, {"text": "space, sort of setting\nourselves up for next lecture.", "start": 3838.68, "duration": 3.915}, {"text": "So first of all,\nanything that you", "start": 3846.71, "duration": 2.21}, {"text": "can do in log space you\ncan do in polynomial time--", "start": 3848.92, "duration": 3.21}, {"text": "in a sense kind of\ntrivially, and in a sense", "start": 3852.13, "duration": 3.24}, {"text": "we almost kind of really\nproved this already.", "start": 3855.37, "duration": 2.76}, {"text": "Because if you remember,\nwe said that anything", "start": 3858.13, "duration": 3.48}, {"text": "that you can do in a\ncertain amount of space", "start": 3861.61, "duration": 2.19}, {"text": "you can do in time that's\nexponential in the amount", "start": 3863.8, "duration": 10.17}, {"text": "of space, in the\ncorresponding amount of space.", "start": 3873.97, "duration": 2.46}, {"text": "So going from space to time\nblows you up exponentially.", "start": 3876.43, "duration": 4.24}, {"text": "And the exponential of\norder log n is polynomial.", "start": 3880.67, "duration": 5.24}, {"text": "So the way we're\ngoing to prove this--", "start": 3885.91, "duration": 2.55}, {"text": "and again, we kind of\nproved this already", "start": 3888.46, "duration": 2.19}, {"text": "but let's just go through it\nagain, specific to log space.", "start": 3890.65, "duration": 5.19}, {"text": "We'll say-- and this is going\nto be a useful definition for us", "start": 3895.84, "duration": 2.76}, {"text": "anyway--", "start": 3898.6, "duration": 1.53}, {"text": "a configuration for M on w.", "start": 3900.13, "duration": 2.73}, {"text": "So we also already talked\nabout the configuration", "start": 3902.86, "duration": 2.97}, {"text": "of the Turing machine\nM, which is just", "start": 3905.83, "duration": 2.13}, {"text": "a snapshot, which is the\ntape, the state, and the head", "start": 3907.96, "duration": 4.71}, {"text": "position.", "start": 3912.67, "duration": 1.74}, {"text": "When we have an\ninput, which is sort", "start": 3914.41, "duration": 3.57}, {"text": "of a read-only\ninput, which is not", "start": 3917.98, "duration": 2.67}, {"text": "being counted\ntoward the space, we", "start": 3920.65, "duration": 1.86}, {"text": "don't include that\nin the configuration.", "start": 3922.51, "duration": 3.66}, {"text": "We just say it's a configuration\nof M on that particular input,", "start": 3926.17, "duration": 4.08}, {"text": "but we're going to be\ncounting configurations.", "start": 3930.25, "duration": 2.37}, {"text": "And I don't want to count all\nthe different inputs as well.", "start": 3932.62, "duration": 2.7}, {"text": "I'm going to fix\nthe input and count", "start": 3935.32, "duration": 1.65}, {"text": "all of the configurations\nrelative to that input.", "start": 3936.97, "duration": 2.415}, {"text": "And so the number of\nsuch configurations", "start": 3942.72, "duration": 3.06}, {"text": "is just simply going to\nbe the number of states", "start": 3945.78, "duration": 3.75}, {"text": "times the number of head\npositions for the two heads", "start": 3949.53, "duration": 3.33}, {"text": "now, times the number of\npossible tape contents, which", "start": 3952.86, "duration": 3.72}, {"text": "is, as I mentioned,\nhere is a number", "start": 3956.58, "duration": 2.04}, {"text": "of different possible\ntape contents.", "start": 3958.62, "duration": 2.17}, {"text": "It's d, which is the tape\nalphabet, to the order log n.", "start": 3960.79, "duration": 4.4}, {"text": "And that's just going to\nbe n to the k for some k.", "start": 3965.19, "duration": 2.34}, {"text": "So it's going to be polynomial.", "start": 3967.53, "duration": 1.59}, {"text": "So that tells us that\nbecause the total number", "start": 3969.12, "duration": 3.15}, {"text": "of configurations for\nM on w is polynomial,", "start": 3972.27, "duration": 5.49}, {"text": "this machine can only be\nrunning for a polynomial number", "start": 3977.76, "duration": 3.75}, {"text": "of steps.", "start": 3981.51, "duration": 0.773}, {"text": "Otherwise, it'll be looping.", "start": 3982.283, "duration": 1.167}, {"text": "It'll be repeating\na configuration.", "start": 3983.45, "duration": 1.93}, {"text": "And so therefore, that\nmachine has to run itself", "start": 3985.38, "duration": 3.06}, {"text": "in polynomial time.", "start": 3988.44, "duration": 1.21}, {"text": "So you don't really, in a\nsense, have to do any work.", "start": 3989.65, "duration": 2.54}, {"text": "If a machine is\ndeterministically", "start": 3992.19, "duration": 2.49}, {"text": "deciding a language\nin log space,", "start": 3994.68, "duration": 2.34}, {"text": "it's also deterministically\ndeciding the language", "start": 3997.02, "duration": 3.065}, {"text": "in polynomial time\nbecause that's", "start": 4000.085, "duration": 1.375}, {"text": "all the time you\ncan use when you", "start": 4001.46, "duration": 1.62}, {"text": "have log space, unless you're\nlooping, which is not allowed.", "start": 4003.08, "duration": 3.81}, {"text": "So therefore, the\nsame machine shows", "start": 4006.89, "duration": 2.64}, {"text": "that you're also in P. I'll\nget to that in a second.", "start": 4009.53, "duration": 4.175}, {"text": "One thing I forgot to\nmention on the previous slide", "start": 4013.705, "duration": 2.125}, {"text": "when I'm talking about\nthe model is by the way,", "start": 4015.83, "duration": 1.958}, {"text": "this model is not\nso unreasonable,", "start": 4017.788, "duration": 3.232}, {"text": "where you have kind\nof-- imagine having", "start": 4021.02, "duration": 2.25}, {"text": "a very large, read-only\ninput and your local storage", "start": 4023.27, "duration": 5.44}, {"text": "is much smaller.", "start": 4028.71, "duration": 0.85}, {"text": "It's much too small to\npull in the entire input.", "start": 4029.56, "duration": 4.2}, {"text": "The way I used to\nexplain this years ago", "start": 4033.76, "duration": 2.79}, {"text": "was your input as a CD-ROM,\nwhich you guys probably barely", "start": 4036.55, "duration": 4.24}, {"text": "even know what it\nis anymore, but you", "start": 4040.79, "duration": 1.88}, {"text": "used to distribute\nsoftware that way.", "start": 4042.67, "duration": 1.86}, {"text": "So the ROM was on a\nDVD or a CD, which", "start": 4044.53, "duration": 4.44}, {"text": "contained whatever you're trying\nto-- like your software you're", "start": 4048.97, "duration": 4.05}, {"text": "trying to distribute.", "start": 4053.02, "duration": 1.44}, {"text": "It was some large\nthing, relatively,", "start": 4054.46, "duration": 3.09}, {"text": "and so you imagine having\na smaller amount of memory", "start": 4057.55, "duration": 4.8}, {"text": "relative to that.", "start": 4062.35, "duration": 0.708}, {"text": "So you didn't want\nto necessarily copy", "start": 4063.058, "duration": 1.584}, {"text": "that whole thing\ninto your storage.", "start": 4064.642, "duration": 1.608}, {"text": "Maybe even a better example\nnow is you think of the input", "start": 4066.25, "duration": 2.88}, {"text": "as being the entire internet.", "start": 4069.13, "duration": 2.88}, {"text": "Obviously, unless\nyou're Google, you", "start": 4072.01, "duration": 2.07}, {"text": "can't download\nthe whole internet", "start": 4074.08, "duration": 3.27}, {"text": "into your own local\nmemory, but you", "start": 4077.35, "duration": 3.26}, {"text": "can have references,\npointers into the input", "start": 4080.61, "duration": 2.61}, {"text": "into different places.", "start": 4083.22, "duration": 1.2}, {"text": "That's perhaps more analogous\nto this sort of read-only input", "start": 4084.42, "duration": 3.24}, {"text": "Turing machine model\nthat I'm describing.", "start": 4087.66, "duration": 2.58}, {"text": "And there's another fact\nthat I want to mention,", "start": 4096.12, "duration": 2.04}, {"text": "is that anything that you can\ndo in nondeterministic log", "start": 4098.16, "duration": 6.39}, {"text": "space you can do in\ndeterministic space,", "start": 4104.55, "duration": 3.66}, {"text": "but now with the squaring.", "start": 4108.21, "duration": 2.13}, {"text": "And that's using the same proof.", "start": 4110.34, "duration": 2.04}, {"text": "That's using Savitch's theorem,\nwhich you have to check also", "start": 4112.38, "duration": 3.24}, {"text": "works down to log space--", "start": 4115.62, "duration": 2.19}, {"text": "same proof.", "start": 4117.81, "duration": 0.629}, {"text": "So anything that you\ndo nondeterministically", "start": 4121.89, "duration": 2.19}, {"text": "in log n space you can use\ndeterministically in log", "start": 4124.08, "duration": 3.09}, {"text": "squared n space.", "start": 4127.17, "duration": 1.984}, {"text": "So let me just see if\nthere's a couple of questions", "start": 4129.154, "duration": 2.125}, {"text": "I want to answer here.", "start": 4131.279, "duration": 0.917}, {"text": "The relationship between L and\nNL is not known to be strict.", "start": 4135.359, "duration": 6.681}, {"text": "Nobody knows of an example.", "start": 4142.04, "duration": 1.47}, {"text": "No one knows whether\nthey're equal.", "start": 4143.51, "duration": 3.28}, {"text": "And have people looked at\nsublinear time classes?", "start": 4146.79, "duration": 9.71}, {"text": "Yes, generally when you\nhave nondeterministic", "start": 4156.5, "duration": 3.42}, {"text": "or probabilistic-- which\nwe haven't defined yet,", "start": 4159.92, "duration": 2.859}, {"text": "but we will--", "start": 4162.779, "duration": 1.071}, {"text": "people have looked at those\nsublinear time classes as well.", "start": 4163.85, "duration": 3.969}, {"text": "Deterministically, it\ndoesn't make so much sense", "start": 4167.819, "duration": 2.0}, {"text": "because then you can't even\ntalk about the whole input.", "start": 4169.819, "duration": 3.451}, {"text": "OK, let me move on.", "start": 4179.029, "duration": 1.861}, {"text": "This is my last slide.", "start": 4183.95, "duration": 1.119}, {"text": "Let me see if we can do\nthis before we break.", "start": 4185.069, "duration": 3.29}, {"text": "Not only is L\ncontained within P,", "start": 4188.359, "duration": 2.521}, {"text": "but the much\nstronger statement is", "start": 4190.88, "duration": 1.95}, {"text": "that NL is contained\nwithin P. And for that", "start": 4192.83, "duration": 4.41}, {"text": "you'll have to do some\nwork because converting", "start": 4197.24, "duration": 2.34}, {"text": "your nondeterministic\nlog space machine", "start": 4199.58, "duration": 2.1}, {"text": "to a deterministic machine--", "start": 4201.68, "duration": 3.99}, {"text": "obviously, you're going to\nhave to change the machine.", "start": 4205.67, "duration": 4.08}, {"text": "And so we'll introduce\na new method here.", "start": 4209.75, "duration": 1.95}, {"text": "Maybe we'll quickly go\nover this at the beginning", "start": 4211.7, "duration": 2.97}, {"text": "of the next lecture.", "start": 4214.67, "duration": 2.325}, {"text": "I don't like rushing\nthrough things at the end.", "start": 4216.995, "duration": 2.955}, {"text": "But for this, if I'm given a\nnondeterministic machine that", "start": 4219.95, "duration": 8.0}, {"text": "runs in log space, I want\nto make a new machine that", "start": 4227.95, "duration": 3.39}, {"text": "runs in polynomial\ntime deterministically", "start": 4231.34, "duration": 2.1}, {"text": "for the same language.", "start": 4233.44, "duration": 1.38}, {"text": "And I'm going to\ndefine something", "start": 4234.82, "duration": 1.5}, {"text": "called the \"configuration\ngraph\" for M on an input w.", "start": 4236.32, "duration": 5.38}, {"text": "And that's just you\ntake M and w, its input,", "start": 4241.7, "duration": 7.08}, {"text": "and you look at all of the\nconfigurations for M on w--", "start": 4248.78, "duration": 6.34}, {"text": "actually, configuration--", "start": 4255.12, "duration": 1.71}, {"text": "actually, that should be\ncalled the \"computation graph.\"", "start": 4256.83, "duration": 3.27}, {"text": "That's what it's called in\nthe book, but it's a typo.", "start": 4260.1, "duration": 2.7}, {"text": "Yeah, I'll fix that next time.", "start": 4262.8, "duration": 2.16}, {"text": "It doesn't matter.", "start": 4264.96, "duration": 1.08}, {"text": "Computation graph, configuration\ngraph, all the same.", "start": 4266.04, "duration": 4.212}, {"text": "Basically, you're going to take\nall of the configurations of M", "start": 4270.252, "duration": 2.583}, {"text": "on w, of which we\nalready observed", "start": 4272.835, "duration": 4.155}, {"text": "are only polynomial in number.", "start": 4276.99, "duration": 2.58}, {"text": "And they become the\nnodes of a graph.", "start": 4279.57, "duration": 3.75}, {"text": "And the edges connect\ntwo configurations", "start": 4283.32, "duration": 3.15}, {"text": "if one follows another according\nto the rules of the machine.", "start": 4286.47, "duration": 4.49}, {"text": "So here's a picture.", "start": 4290.96, "duration": 1.26}, {"text": "This is some graph.", "start": 4292.22, "duration": 2.49}, {"text": "The nodes of this graph are\nthe configurations of M on w.", "start": 4294.71, "duration": 5.86}, {"text": "So each node here\ncorresponds to a snapshot", "start": 4300.57, "duration": 4.29}, {"text": "of the machine, a tape contents,\nhead position, and state.", "start": 4304.86, "duration": 7.87}, {"text": "So writing down all those\ndifferent configurations,", "start": 4312.73, "duration": 2.34}, {"text": "I connect one to\nthe other, if I can", "start": 4315.07, "duration": 1.8}, {"text": "get to C j from C i in one\nstep legally on the machine.", "start": 4316.87, "duration": 5.74}, {"text": "And then, the\nnondeterministic machine M", "start": 4322.61, "duration": 4.67}, {"text": "accepts w exactly when\nthere's a path from the start", "start": 4327.28, "duration": 4.48}, {"text": "configuration to an\naccept configuration.", "start": 4331.76, "duration": 3.42}, {"text": "Let's assume we have just a\nsingle accept configuration,", "start": 4335.18, "duration": 3.09}, {"text": "as we argued we can do\none or two lectures back,", "start": 4338.27, "duration": 5.14}, {"text": "if we clean up the tapes.", "start": 4343.41, "duration": 1.44}, {"text": "So testing whether you can get\nfrom the start to the accept", "start": 4344.85, "duration": 7.05}, {"text": "is the same as testing whether\nthe machine accepts its input.", "start": 4351.9, "duration": 4.84}, {"text": "And so because we can\ntest whether there's", "start": 4356.74, "duration": 4.17}, {"text": "a path in a graph connecting\ntwo nodes in polynomial time,", "start": 4360.91, "duration": 4.17}, {"text": "we can solve this problem on\nthis computation/configuration", "start": 4365.08, "duration": 3.15}, {"text": "graph in polynomial time.", "start": 4368.23, "duration": 1.81}, {"text": "And so we can figure out whether\nthe nondeterministic machine", "start": 4370.04, "duration": 2.69}, {"text": "accepts its input.", "start": 4372.73, "duration": 1.415}, {"text": "Sorry, that came a little\nfaster than I like to do,", "start": 4374.145, "duration": 2.125}, {"text": "so we'll see it again.", "start": 4376.27, "duration": 2.8}, {"text": "So here's the polynomial\ntime algorithm.", "start": 4379.07, "duration": 2.15}, {"text": "You construct the\ngraph, you accept", "start": 4381.22, "duration": 2.602}, {"text": "if there's a path from\nthe start to the accept,", "start": 4383.822, "duration": 1.958}, {"text": "and you reject if there is not.", "start": 4385.78, "duration": 1.292}, {"text": "And so that tells us that not\nonly is L contained within NL,", "start": 4389.8, "duration": 5.03}, {"text": "but NL itself is also\ncontained within P.", "start": 4394.83, "duration": 2.88}, {"text": "So here's a kind of a nice\nhierarchy of languages.", "start": 4397.71, "duration": 5.43}, {"text": "Not only do we not know\nwhether L equals NL,", "start": 4403.14, "duration": 2.73}, {"text": "we don't know\nwhether L equals P.", "start": 4405.87, "duration": 2.82}, {"text": "It's possible that anything\nyou can do in polynomial time,", "start": 4408.69, "duration": 2.79}, {"text": "you can do deterministically\nin log space,", "start": 4411.48, "duration": 2.31}, {"text": "shocking as that\nmight be, because this", "start": 4413.79, "duration": 1.74}, {"text": "is a pretty weak class.", "start": 4415.53, "duration": 2.55}, {"text": "But we don't know how\nto prove that there's", "start": 4418.08, "duration": 2.1}, {"text": "anything different, anything\nin P that's not in here.", "start": 4420.18, "duration": 4.8}, {"text": "Last check in.", "start": 4424.98, "duration": 2.17}, {"text": "So we showed that PATH is in NL.", "start": 4427.15, "duration": 1.97}, {"text": "What's the best thing we can do\nabout the deterministic space", "start": 4429.12, "duration": 2.73}, {"text": "complexity of PATH?", "start": 4431.85, "duration": 1.14}, {"text": "So deterministic.", "start": 4432.99, "duration": 1.12}, {"text": "So this is\nnondeterministic log space.", "start": 4434.11, "duration": 1.94}, {"text": "What can we say\ndeterministically about PATH?", "start": 4436.05, "duration": 4.59}, {"text": "Hint-- this should not be\nhard if you think back to what", "start": 4440.64, "duration": 5.01}, {"text": "we've shown very recently.", "start": 4445.65, "duration": 2.55}, {"text": "Get your check in points.", "start": 4453.36, "duration": 1.53}, {"text": "Closing up.", "start": 4454.89, "duration": 1.11}, {"text": "Closing shop here.", "start": 4456.0, "duration": 1.68}, {"text": "All set, 1, 2, 3.", "start": 4457.68, "duration": 1.095}, {"text": "Yeah, so the correct answer\nis log squared space,", "start": 4461.55, "duration": 2.79}, {"text": "because this is just\nSavitch's theorem.", "start": 4464.34, "duration": 2.84}, {"text": "We can do it in log space\nnondeterministically,", "start": 4467.18, "duration": 2.29}, {"text": "so you can do it in log squared\nspace deterministically.", "start": 4469.47, "duration": 3.23}, {"text": "So this is what we did today.", "start": 4472.7, "duration": 2.73}, {"text": "And as I mentioned,\nI will do this again", "start": 4478.9, "duration": 6.68}, {"text": "on Tuesday's lecture,\njust to recap that.", "start": 4485.58, "duration": 3.21}, {"text": "All right, so I'll stick around\na little bit for questions.", "start": 4488.79, "duration": 2.97}, {"text": "And someone's asking me\nabout the nomenclature.", "start": 4491.76, "duration": 4.71}, {"text": "Why is it L and not L space?", "start": 4496.47, "duration": 1.85}, {"text": "Because people don't\nusually talk about L time.", "start": 4503.16, "duration": 3.12}, {"text": "So L is sort of--", "start": 4506.28, "duration": 1.61}, {"text": "everybody knows the only\nreasonable option is space,", "start": 4507.89, "duration": 3.12}, {"text": "so people just say L and NL.", "start": 4511.01, "duration": 3.54}, {"text": "I mean, some of these names have\na little bit evolved over time.", "start": 4514.55, "duration": 9.18}, {"text": "And even now, some\npeople talk about--", "start": 4523.73, "duration": 2.385}, {"text": "I call \"time classes,\" some\npeople call \"detime classes.\"", "start": 4529.21, "duration": 3.42}, {"text": "You can make different\nchoices there.", "start": 4538.16, "duration": 3.73}, {"text": "Let's see.", "start": 4541.89, "duration": 0.5}, {"text": "Good.", "start": 4544.775, "duration": 0.5}, {"text": "Why does Savitch's\ntheorem work for log n?", "start": 4547.78, "duration": 3.12}, {"text": "You have to look back\nand see what you needed.", "start": 4550.9, "duration": 3.36}, {"text": "And all you needed\nto be able to do", "start": 4554.26, "duration": 2.1}, {"text": "was to write down\nthe configurations.", "start": 4556.36, "duration": 7.36}, {"text": "And if you look back at\nhow Savitch's film works,", "start": 4563.72, "duration": 3.333}, {"text": "you're just needing to write\ndown the configuration.", "start": 4567.053, "duration": 2.167}, {"text": "So the deterministic\nmachine can write down", "start": 4569.22, "duration": 2.323}, {"text": "the configurations for the\nnondeterministic machine.", "start": 4571.543, "duration": 2.167}, {"text": "They take exactly the same size.", "start": 4573.71, "duration": 2.74}, {"text": "And then you look\nat the recursion.", "start": 4576.45, "duration": 1.91}, {"text": "And the depth of\nthe recursion is", "start": 4578.36, "duration": 2.55}, {"text": "going to be exponential in the\nsize of the configurations.", "start": 4580.91, "duration": 4.773}, {"text": "And so you're going to end\nup with a squaring again.", "start": 4585.683, "duration": 2.167}, {"text": "You have to go back and\njust rethink that proof.", "start": 4587.85, "duration": 3.29}, {"text": "And you'll see nowhere did it\nneed a linear amount of space.", "start": 4591.14, "duration": 3.27}, {"text": "It works down to--", "start": 4594.41, "duration": 1.32}, {"text": "it actually does not\nwork for less than log n.", "start": 4595.73, "duration": 2.37}, {"text": "Log n is sort of the\nlower threshold there.", "start": 4598.1, "duration": 3.58}, {"text": "And the reason for that\nis because you also", "start": 4601.68, "duration": 3.05}, {"text": "need to store the\ninput location,", "start": 4604.73, "duration": 3.78}, {"text": "and that already\ntakes log space.", "start": 4608.51, "duration": 1.53}, {"text": "The tape heads-- the\ntape heads already kind", "start": 4613.92, "duration": 2.54}, {"text": "have kind of a log\nspace aspect to them.", "start": 4616.46, "duration": 4.93}, {"text": "And so if you're going to\nuse less than log space,", "start": 4621.39, "duration": 3.35}, {"text": "then funky things happen\nwith storing the tape heads.", "start": 4624.74, "duration": 3.81}, {"text": "And so less than\nlog space usually", "start": 4628.55, "duration": 2.67}, {"text": "turns out not to be\ninteresting, very", "start": 4631.22, "duration": 3.33}, {"text": "specific to Turing machines\nand not general models.", "start": 4634.55, "duration": 3.39}, {"text": "Yeah, so somebody's asking,\nsuppose in the reduction", "start": 4641.17, "duration": 5.37}, {"text": "to generalized\ngeography from TQBF,", "start": 4646.54, "duration": 2.88}, {"text": "if the formula had\ntwo Exists in a row,", "start": 4649.42, "duration": 3.325}, {"text": "then you would do kind of the\nnatural thing in the graph.", "start": 4652.745, "duration": 2.375}, {"text": "Instead of having that spacer\nedge between the two diamonds,", "start": 4655.12, "duration": 3.03}, {"text": "you could just have one\ndiamond connecting directly", "start": 4658.15, "duration": 2.49}, {"text": "to the other diamond\nwithout a spacer edge.", "start": 4660.64, "duration": 2.61}, {"text": "And that would give you\nthe effect of not switching", "start": 4663.25, "duration": 5.31}, {"text": "whose turn it is.", "start": 4668.56, "duration": 2.26}, {"text": "Somebody is asking me\njust a general question,", "start": 4670.82, "duration": 3.54}, {"text": "are people thinking about\nthese open problems?", "start": 4674.36, "duration": 3.06}, {"text": "I don't know.", "start": 4682.63, "duration": 2.9}, {"text": "People don't say.", "start": 4685.53, "duration": 4.23}, {"text": "There was a lot of work\non problems related,", "start": 4689.76, "duration": 3.36}, {"text": "that seemed to be related to\nthose many open questions,", "start": 4693.12, "duration": 4.41}, {"text": "like P versus NP, L versus\nNL or L versus P and so on,", "start": 4697.53, "duration": 4.68}, {"text": "P versus PSPACE.", "start": 4702.21, "duration": 1.99}, {"text": "We'll talk a little bit\nmore about some of that.", "start": 4704.2, "duration": 2.0}, {"text": "There's some very\ninteresting things", "start": 4706.2, "duration": 1.23}, {"text": "that have been developed.", "start": 4707.43, "duration": 1.12}, {"text": "I think there's a sense\nwithin the community", "start": 4708.55, "duration": 2.21}, {"text": "that people are\nstuck, and you're", "start": 4710.76, "duration": 3.39}, {"text": "going to need some\nsort of major new idea", "start": 4714.15, "duration": 3.6}, {"text": "in order to push\nthe thing forward.", "start": 4717.75, "duration": 1.78}, {"text": "So I don't know how many people\nare still thinking about them.", "start": 4719.53, "duration": 4.665}, {"text": "I hope people are\nbecause I would", "start": 4724.195, "duration": 1.375}, {"text": "like to see the answer at\nsome point, or get the answer.", "start": 4725.57, "duration": 4.56}, {"text": "I think about them\nsometimes myself,", "start": 4730.13, "duration": 1.68}, {"text": "but one has to acknowledge\nchances of success", "start": 4731.81, "duration": 5.55}, {"text": "are not high.", "start": 4737.36, "duration": 0.63}, {"text": "So we're a little after past\nthe end of the hour here.", "start": 4743.79, "duration": 4.53}, {"text": "Unless there's any\nother questions,", "start": 4748.32, "duration": 1.5}, {"text": "if I didn't answer your\nquestion, I may have missed it,", "start": 4749.82, "duration": 2.292}, {"text": "so you can ask again.", "start": 4752.112, "duration": 2.938}, {"text": "But otherwise, I'm going to\nclose this, close the session", "start": 4755.05, "duration": 3.95}, {"text": "here.", "start": 4759.0, "duration": 1.08}, {"text": "OK, bye-bye all.", "start": 4760.08, "duration": 1.86}, {"text": "Have a good weekend.", "start": 4761.94, "duration": 1.0}, {"text": "See you next week.", "start": 4762.94, "duration": 1.62}, {"text": "Take care.", "start": 4764.56, "duration": 1.6}]