WEBVTT 00:00:00.000 --> 00:00:01.988 [SQUEAKING] 00:00:01.988 --> 00:00:04.473 [RUSTLING] 00:00:04.473 --> 00:00:07.455 [CLICKING] 00:00:24.910 --> 00:00:26.390 MICHAEL SIPSER: Welcome, everyone. 00:00:26.390 --> 00:00:29.800 Welcome back to theory of computation. 00:00:29.800 --> 00:00:38.130 And just to recap where we are, we 00:00:38.130 --> 00:00:40.470 have been looking at time complexity and space 00:00:40.470 --> 00:00:41.790 complexity. 00:00:41.790 --> 00:00:48.570 And we just finished proving what 00:00:48.570 --> 00:00:53.790 are called the hierarchy theorems, which, in a nutshell, 00:00:53.790 --> 00:00:58.360 basically say that, if you allow the computational model 00:00:58.360 --> 00:01:00.360 to have a little bit more resource, a little bit 00:01:00.360 --> 00:01:03.780 more time, a little bit more space, then 00:01:03.780 --> 00:01:09.500 you can do more things with certain conditions. 00:01:09.500 --> 00:01:11.890 So we proved that last time. 00:01:11.890 --> 00:01:14.600 It was a proof, basically, by a diagonalization. 00:01:14.600 --> 00:01:17.620 I don't know if you recognized the diagonalization there, 00:01:17.620 --> 00:01:21.730 but when you're encoding a machine by an input 00:01:21.730 --> 00:01:24.550 and then basically running all possible different machines, 00:01:24.550 --> 00:01:28.910 that's essentially a diagonalization. 00:01:28.910 --> 00:01:33.050 So today, we're going to build on that work 00:01:33.050 --> 00:01:35.810 to give an example of what we call 00:01:35.810 --> 00:01:39.260 a natural intractable problem. 00:01:39.260 --> 00:01:41.755 We'll say a bit more about what that means. 00:01:41.755 --> 00:01:43.880 And then, we're going to talk about something which 00:01:43.880 --> 00:01:47.030 is a different topic, but nevertheless related, 00:01:47.030 --> 00:01:52.220 having to do with oracles and methods which may or may not 00:01:52.220 --> 00:01:59.750 work to solve the P versus NP problem, which, of course, is 00:01:59.750 --> 00:02:02.390 a big open problem in the field. 00:02:02.390 --> 00:02:03.470 OK. 00:02:03.470 --> 00:02:07.100 So the time and space hierarchy theorems-- 00:02:07.100 --> 00:02:09.560 because we're going to be using those today-- 00:02:09.560 --> 00:02:13.670 they say that if you give a little bit more space here-- so 00:02:13.670 --> 00:02:16.190 for space constructible functions, functions 00:02:16.190 --> 00:02:19.400 that you can actually compute within the amount of space 00:02:19.400 --> 00:02:24.230 that they specify, you can show that the things that you 00:02:24.230 --> 00:02:29.060 can do in that much space is probably larger than what 00:02:29.060 --> 00:02:30.500 you can do in less space. 00:02:30.500 --> 00:02:33.390 And you can prove a similar slightly weaker fact 00:02:33.390 --> 00:02:36.270 about the time complexity classes. 00:02:36.270 --> 00:02:43.050 So what that means is that these classes form a hierarchy. 00:02:43.050 --> 00:02:46.220 So as you add more time, or let's 00:02:46.220 --> 00:02:49.010 say, in this case, space, from n squared, to n cubed, 00:02:49.010 --> 00:02:53.810 to n to the 4th, you get larger and larger classes, 00:02:53.810 --> 00:02:57.660 which I'm kind illustrating here by putting a dot there, 00:02:57.660 --> 00:02:59.360 which shows that there's something 00:02:59.360 --> 00:03:02.180 that we know that's new in those classes 00:03:02.180 --> 00:03:07.830 as you go up these different bounds. 00:03:07.830 --> 00:03:09.890 And this is going to be true for space complexity 00:03:09.890 --> 00:03:12.500 and it's also going to be true for time complexity. 00:03:17.900 --> 00:03:21.470 And one of the corollaries that we pointed out last time 00:03:21.470 --> 00:03:30.320 is that, PSPACE is a-- properly includes non-deterministic log 00:03:30.320 --> 00:03:31.340 space, NL. 00:03:31.340 --> 00:03:33.920 So NL is a proper subset of PSPACE. 00:03:33.920 --> 00:03:38.750 So there's stuff in PSPACE that is not in NL. 00:03:38.750 --> 00:03:41.420 And remember this notation here, this means proper subset. 00:03:44.470 --> 00:03:46.600 One of the things that-- 00:03:46.600 --> 00:03:49.520 a follow-on corollary that we didn't mention last time, 00:03:49.520 --> 00:03:52.450 but that's something that you should know, 00:03:52.450 --> 00:03:57.970 is that the TQBF problem, our PSPACE based complete problem, 00:03:57.970 --> 00:04:02.140 is an example of a problem that's in PSPACE, obviously, 00:04:02.140 --> 00:04:05.620 but we know it's also not in NL. 00:04:05.620 --> 00:04:07.540 And in order to get that conclusion, 00:04:07.540 --> 00:04:14.020 you have to look, again, at the proof that TQBF is PSPACE 00:04:14.020 --> 00:04:19.300 complete, and observe that the reductions that we gave 00:04:19.300 --> 00:04:23.410 in that proof can be carried out not only in polynomial time, 00:04:23.410 --> 00:04:26.110 but they can be carried out in log space. 00:04:26.110 --> 00:04:32.200 And therefore, if TQBF turned out to go down to NL, 00:04:32.200 --> 00:04:34.660 then because everything in PSPACE 00:04:34.660 --> 00:04:38.650 is log space reducible to TQBF, that would bring all of PSPACE 00:04:38.650 --> 00:04:40.480 down to NL. 00:04:40.480 --> 00:04:42.950 But that we just proved is not the case. 00:04:42.950 --> 00:04:46.530 So therefore, TQBF could not be in NL. 00:04:46.530 --> 00:04:50.340 OK, and we're going to be using that kind of reasoning 00:04:50.340 --> 00:04:54.070 again in this lecture. 00:04:54.070 --> 00:04:56.740 So just a quick check-in. 00:04:56.740 --> 00:05:00.030 These are a few, more or less easy, 00:05:00.030 --> 00:05:04.370 maybe more or less tricky, follow-ons 00:05:04.370 --> 00:05:09.357 that you can conclude from the time and space hierarchy 00:05:09.357 --> 00:05:10.940 theorems plus some of the other things 00:05:10.940 --> 00:05:12.450 we've proven along the way. 00:05:12.450 --> 00:05:16.957 And so just as a check of your understanding, maybe 00:05:16.957 --> 00:05:18.540 these a little bit on the tricky side, 00:05:18.540 --> 00:05:21.020 so you have to read them carefully. 00:05:21.020 --> 00:05:25.520 Which of these are known to be true based on the material 00:05:25.520 --> 00:05:26.510 that we've presented? 00:05:26.510 --> 00:05:29.210 And this is also just material that's 00:05:29.210 --> 00:05:35.300 the facts that we know to be true in complexity theory. 00:05:35.300 --> 00:05:36.640 So let me launch that poll. 00:05:36.640 --> 00:05:43.290 And just check off the ones that we can prove. 00:05:43.290 --> 00:05:44.970 Hmm. 00:05:44.970 --> 00:05:46.710 OK. 00:05:46.710 --> 00:05:48.220 I'm going to close it down. 00:05:48.220 --> 00:05:53.010 So please answer quickly if you're going to. 00:05:53.010 --> 00:05:54.360 OK, 1, 2, 3, end. 00:05:57.120 --> 00:05:59.220 OK. 00:05:59.220 --> 00:06:04.620 Well, the two leading candidates are correct. 00:06:04.620 --> 00:06:06.750 And the two that are the laggards here 00:06:06.750 --> 00:06:09.960 are, in fact, the ones that are not true. 00:06:09.960 --> 00:06:14.910 So A and D are not true, based on what we know. 00:06:14.910 --> 00:06:18.070 And B and C are true. 00:06:18.070 --> 00:06:20.250 So let's understand, first of all, A, we 00:06:20.250 --> 00:06:25.620 know it's false because 2 to the n plus 1 is just 2 times 2 00:06:25.620 --> 00:06:27.140 to the n. 00:06:27.140 --> 00:06:32.250 And so these two bounds differ only by a constant factor. 00:06:32.250 --> 00:06:35.570 And so in fact, they're the same complexity class. 00:06:35.570 --> 00:06:37.940 And so you don't get proper containment for A. 00:06:37.940 --> 00:06:39.755 So that one we absolutely know is false. 00:06:42.820 --> 00:06:49.440 D, well, if we could prove that, then we 00:06:49.440 --> 00:06:51.120 would have solved the famous problem, 00:06:51.120 --> 00:06:56.530 because we don't know whether even P equals PSPACE. 00:06:56.530 --> 00:06:59.280 So if P equals PSPACE, then certainly PSPACE 00:06:59.280 --> 00:07:02.200 would equal NP, which is in between the two. 00:07:02.200 --> 00:07:04.890 And so we don't know how to prove 00:07:04.890 --> 00:07:08.130 PSPACE is different from NP, that's 00:07:08.130 --> 00:07:11.777 based on the current state of knowledge of the field. 00:07:11.777 --> 00:07:13.360 So this would not be something that we 00:07:13.360 --> 00:07:17.530 know to be true based on what things that we've said. 00:07:17.530 --> 00:07:23.980 Now, B follows directly from the time hierarchy theorem, 00:07:23.980 --> 00:07:28.210 because 2 to the 2n is the square of 2 to the n. 00:07:28.210 --> 00:07:33.650 And that is, asymptotically, a significantly larger bound. 00:07:33.650 --> 00:07:41.660 And so you can prove that time 2 the n is properly 00:07:41.660 --> 00:07:44.560 contains time 2 the n. 00:07:44.560 --> 00:07:49.970 C is a little trickier because you need 00:07:49.970 --> 00:07:52.630 to remember Savitch's theorem. 00:07:52.630 --> 00:07:54.970 Savitch's theorem applies to space. 00:07:54.970 --> 00:07:56.650 But you also need to remember that what 00:07:56.650 --> 00:07:59.440 you can do in time, in non-deterministic time 00:07:59.440 --> 00:08:02.230 n squared, you can also do in non-deterministic space 00:08:02.230 --> 00:08:04.720 n squared, which, then, in turn, you 00:08:04.720 --> 00:08:09.550 can do in deterministic space n to the 4th, which is properly 00:08:09.550 --> 00:08:12.380 contained within space n to the 5th. 00:08:12.380 --> 00:08:15.800 So you can prove that PSPACE properly 00:08:15.800 --> 00:08:19.940 contains non-deterministic time n squared. 00:08:19.940 --> 00:08:22.850 OK, just a bunch of containments there. 00:08:22.850 --> 00:08:27.470 A and C are perhaps, in a sense, it may be the most tricky 00:08:27.470 --> 00:08:29.380 of this group. 00:08:29.380 --> 00:08:32.049 OK. 00:08:32.049 --> 00:08:34.460 So let's move on. 00:08:34.460 --> 00:08:38.350 So we're going to introduce, today, two new classes. 00:08:38.350 --> 00:08:41.069 And actually, I want to go back to here. 00:08:43.750 --> 00:08:46.930 What are we going to be trying to accomplish 00:08:46.930 --> 00:08:48.620 in today's lecture? 00:08:48.620 --> 00:08:53.170 So we're going to be looking at provable intractability. 00:08:53.170 --> 00:08:58.780 So a problem being intractable for us means it's outside of P. 00:08:58.780 --> 00:09:02.560 So we can't solve it in polynomial time. 00:09:02.560 --> 00:09:06.160 For our perspective, we're going to call that 00:09:06.160 --> 00:09:08.710 an intractable problem. 00:09:08.710 --> 00:09:13.870 Now, this problem over here, that's sitting in time 2 the n, 00:09:13.870 --> 00:09:17.740 but not in smaller classes, so this is an intractable problem. 00:09:17.740 --> 00:09:25.420 That's outside of P. But this example 00:09:25.420 --> 00:09:30.280 of a language, if you remember how the time hierarchy 00:09:30.280 --> 00:09:32.560 theorem or the space hierarchy theorem 00:09:32.560 --> 00:09:36.430 was proved, basically, this language itself is not 00:09:36.430 --> 00:09:39.490 an interesting language for other than the purpose 00:09:39.490 --> 00:09:43.270 that it serves, to be in that class and not in a lower class. 00:09:43.270 --> 00:09:45.670 But it's not a language that anyone would care about. 00:09:45.670 --> 00:09:48.470 And it's not even a language that is easy to describe. 00:09:48.470 --> 00:09:53.110 It's just the language that some Turing machine decides, 00:09:53.110 --> 00:09:54.790 where that Turing machine is especially 00:09:54.790 --> 00:09:58.210 designed to have the property that its language is 00:09:58.210 --> 00:10:01.610 at a particular complexity level. 00:10:01.610 --> 00:10:04.388 But otherwise, there's no nice description of that language. 00:10:04.388 --> 00:10:05.930 It's not like a to the n, b to the n, 00:10:05.930 --> 00:10:13.320 or some equivalence of 2 dfa's or something like that. 00:10:13.320 --> 00:10:16.190 So I would say that that language is, in a sense, 00:10:16.190 --> 00:10:18.830 it serves its purpose, but it's not a natural language that you 00:10:18.830 --> 00:10:19.950 really care about. 00:10:19.950 --> 00:10:23.150 So one one of the goals of today's lecture 00:10:23.150 --> 00:10:26.090 is to give an example of a natural language, 00:10:26.090 --> 00:10:30.260 a naturally-occurring language, in a sense, that's 00:10:30.260 --> 00:10:33.830 easy to describe, where you can prove that that language is 00:10:33.830 --> 00:10:39.170 intractable, is actually outside of P. 00:10:39.170 --> 00:10:42.840 So that's a bit of motivation where we're going. 00:10:42.840 --> 00:10:45.230 So along the way, we're going to introduce 00:10:45.230 --> 00:10:48.810 these exponential complexity classes, exponential time 00:10:48.810 --> 00:10:52.490 and exponential space, which are exponentially 00:10:52.490 --> 00:10:57.170 bigger than polynomial time and polynomial space classes. 00:10:57.170 --> 00:11:00.200 So it's 2 to the n to the k in both cases. 00:11:00.200 --> 00:11:03.930 2 to a polynomial. 00:11:03.930 --> 00:11:08.790 And the first five of these classes, L 00:11:08.790 --> 00:11:10.430 through PSPACE we've already seen, 00:11:10.430 --> 00:11:12.890 and exponential time and exponential space 00:11:12.890 --> 00:11:17.880 extend the containments that we've already seen. 00:11:17.880 --> 00:11:24.200 So you have to double check that you understand why PSPACE is 00:11:24.200 --> 00:11:27.090 a subset of exponential time. 00:11:27.090 --> 00:11:30.170 But that's because that, as we showed, 00:11:30.170 --> 00:11:33.110 going from space to time, you can do that 00:11:33.110 --> 00:11:35.190 with an exponential increase. 00:11:35.190 --> 00:11:37.130 That's the cost of the simulation. 00:11:37.130 --> 00:11:39.050 And going from time to space, you 00:11:39.050 --> 00:11:40.370 don't need any increase at all. 00:11:40.370 --> 00:11:42.578 Anything that you can do in a certain amount of time, 00:11:42.578 --> 00:11:44.293 you can do in that much space. 00:11:44.293 --> 00:11:46.460 So anything you can do in a certain amount of space, 00:11:46.460 --> 00:11:50.290 you can also do in exponentially more amount of time. 00:11:50.290 --> 00:11:53.090 OK, so those were simple theorems 00:11:53.090 --> 00:11:57.590 that we proved right at the very beginning. 00:11:57.590 --> 00:11:59.560 Now, the hierarchy theorems allow 00:11:59.560 --> 00:12:02.890 us to conclude some separations among these classes. 00:12:02.890 --> 00:12:06.880 So we already looked at this one, NL versus PSPACE. 00:12:06.880 --> 00:12:11.800 And we saw that because NL is, by Savitch's theorem, 00:12:11.800 --> 00:12:15.550 in deterministic log squared space, which is properly 00:12:15.550 --> 00:12:19.270 contained in polynomial space, you 00:12:19.270 --> 00:12:24.260 get a separation between those two classes, provably. 00:12:24.260 --> 00:12:26.780 And for similar reasons, polynomial space 00:12:26.780 --> 00:12:28.490 to exponential space, you're going 00:12:28.490 --> 00:12:32.000 to get a separation from the space hierarchy theorem. 00:12:32.000 --> 00:12:34.280 And polynomial time to exponential time, 00:12:34.280 --> 00:12:37.480 you get a provable separation by virtue 00:12:37.480 --> 00:12:38.980 of the hierarchy theorem. 00:12:41.930 --> 00:12:45.710 Now we're going to define complete problems 00:12:45.710 --> 00:12:48.333 for these two classes, exponential time 00:12:48.333 --> 00:12:49.250 and exponential space. 00:12:49.250 --> 00:12:51.500 So we have exponential time complete. 00:12:51.500 --> 00:12:56.880 It's going to be analogous to what we showed before, 00:12:56.880 --> 00:13:01.860 which is that it's a member of exponential time. 00:13:01.860 --> 00:13:06.173 And every problem in exponential time is reducible to it, 00:13:06.173 --> 00:13:08.090 let's say, in polynomial time, though it's not 00:13:08.090 --> 00:13:09.680 going to really turn out to be matter. 00:13:09.680 --> 00:13:12.320 It could be in log space. 00:13:12.320 --> 00:13:14.135 Some simple method of doing the reduction 00:13:14.135 --> 00:13:15.260 is going to be good enough. 00:13:15.260 --> 00:13:18.800 Let's say polynomial time is the typical definition. 00:13:18.800 --> 00:13:21.350 And the same thing for exponential space complete. 00:13:21.350 --> 00:13:23.630 We'll say it's exponential space complete, 00:13:23.630 --> 00:13:25.040 if it's an exponential space. 00:13:25.040 --> 00:13:27.230 And anything else in exponential space 00:13:27.230 --> 00:13:31.050 is polynomial time reducible to it. 00:13:31.050 --> 00:13:31.890 OK. 00:13:31.890 --> 00:13:37.320 But the important thing is that if something 00:13:37.320 --> 00:13:40.156 is exponential time complete, you 00:13:40.156 --> 00:13:45.770 know it's outside of P, for the same reasons we've now 00:13:45.770 --> 00:13:46.745 seen several times. 00:13:49.860 --> 00:13:55.110 Namely, that if an exponential time complete problem ended up 00:13:55.110 --> 00:14:00.220 being in P, then because everything 00:14:00.220 --> 00:14:03.190 else in exponential time is reducible to the complete 00:14:03.190 --> 00:14:05.560 problem, they would also be in P. 00:14:05.560 --> 00:14:08.650 And so exponential time and P would be equal. 00:14:08.650 --> 00:14:12.820 But we just said they're not equal because of the hierarchy 00:14:12.820 --> 00:14:15.140 theorem. 00:14:15.140 --> 00:14:19.550 So the logic is the hierarchy theorem separates the class, 00:14:19.550 --> 00:14:24.020 and then the complete problem inherits the difficulty 00:14:24.020 --> 00:14:26.010 of the larger class. 00:14:26.010 --> 00:14:30.202 So the complete problem cannot be any lower than the other 00:14:30.202 --> 00:14:32.660 problems in the class, because they're all reducible to it. 00:14:36.150 --> 00:14:39.390 So the same thing is going to be true for an exponential space 00:14:39.390 --> 00:14:40.170 complete problem. 00:14:40.170 --> 00:14:43.650 Can't be even in PSPACE because exponential space and PSPACE 00:14:43.650 --> 00:14:45.120 are different. 00:14:45.120 --> 00:14:47.940 And if it's not in PSPACE, it's not going to be in P. 00:14:47.940 --> 00:14:51.630 And so in both cases, if you have a problem that's 00:14:51.630 --> 00:14:54.780 complete for exponential space or exponential time, 00:14:54.780 --> 00:14:59.110 we know that those problems are intractable. 00:14:59.110 --> 00:15:02.080 And our strategy, then, for giving 00:15:02.080 --> 00:15:07.420 a natural intractable problem is to show 00:15:07.420 --> 00:15:09.760 it's complete for one of these classes. 00:15:09.760 --> 00:15:11.290 And it's actually going to turn out 00:15:11.290 --> 00:15:14.800 to be an exponential space complete problem that we're 00:15:14.800 --> 00:15:17.880 going to give as our example. 00:15:17.880 --> 00:15:19.890 OK, so that is the plan. 00:15:22.500 --> 00:15:24.390 I think it's a good time to-- 00:15:24.390 --> 00:15:27.120 let's just take a few questions here 00:15:27.120 --> 00:15:33.300 to make sure we're all on the same page as what we're doing. 00:15:33.300 --> 00:15:34.140 So let me just read. 00:15:34.140 --> 00:15:36.060 I got a couple of questions already in here. 00:15:45.360 --> 00:15:46.830 So this is a little bit of a side 00:15:46.830 --> 00:15:49.230 comment that somebody-- that's an interesting question. 00:15:49.230 --> 00:15:53.910 Basically, is it possible that we may not 00:15:53.910 --> 00:15:57.270 be able to prove, solve the P versus NP problem, 00:15:57.270 --> 00:16:01.050 that it's not a problem that one can answer 00:16:01.050 --> 00:16:02.830 from the basic axioms of mathematics, 00:16:02.830 --> 00:16:06.740 if I'm interpreting the question correctly. 00:16:06.740 --> 00:16:10.280 There are certain problems in mathematics-- 00:16:10.280 --> 00:16:11.810 and I think I, perhaps, I mentioned 00:16:11.810 --> 00:16:14.150 earlier in the term, the problem of whether there 00:16:14.150 --> 00:16:21.080 is a set whose size is in between the integers 00:16:21.080 --> 00:16:22.670 and the real numbers. 00:16:22.670 --> 00:16:24.890 We know the real numbers are larger in size 00:16:24.890 --> 00:16:26.540 than the integers. 00:16:26.540 --> 00:16:28.790 That was our first example of a diagonalization. 00:16:28.790 --> 00:16:32.660 And is there a problem of size strictly in between the two? 00:16:32.660 --> 00:16:35.850 Bigger than the integers, smaller than the real numbers. 00:16:35.850 --> 00:16:39.680 So that's a problem that was posed a long time ago. 00:16:39.680 --> 00:16:41.240 It was one of Hilbert's problems. 00:16:41.240 --> 00:16:46.300 And was eventually shown to be unanswerable 00:16:46.300 --> 00:16:49.310 using the basic axioms of mathematics. 00:16:49.310 --> 00:16:51.175 So the question is, maybe P versus NP 00:16:51.175 --> 00:16:52.510 is in the same category. 00:16:55.370 --> 00:16:55.870 Could be. 00:16:55.870 --> 00:16:58.350 That could be true of any unsolved problems 00:16:58.350 --> 00:17:00.000 in mathematics. 00:17:00.000 --> 00:17:02.130 But at least our experience has shown 00:17:02.130 --> 00:17:04.980 that the kinds of problems that, at least, have been shown 00:17:04.980 --> 00:17:09.420 to be unsolvable from mathematical axioms 00:17:09.420 --> 00:17:12.270 tend to involve infinities and very large 00:17:12.270 --> 00:17:14.940 things, things that are very far from our intuitions. 00:17:14.940 --> 00:17:18.810 And something as down to earth as P versus NP, at least, 00:17:18.810 --> 00:17:20.700 it would be very surprising to me 00:17:20.700 --> 00:17:23.430 if that turned out to be unanswerable 00:17:23.430 --> 00:17:25.530 using our mathematical axioms. 00:17:25.530 --> 00:17:26.692 But, who knows? 00:17:26.692 --> 00:17:28.109 Oh, this is another good question. 00:17:28.109 --> 00:17:31.260 Do the time and space hierarchy theorems 00:17:31.260 --> 00:17:33.130 have non-deterministic variants? 00:17:33.130 --> 00:17:34.380 Yes, they do. 00:17:34.380 --> 00:17:36.000 They're much harder to prove, however, 00:17:36.000 --> 00:17:37.417 and we're not going to cover that. 00:17:37.417 --> 00:17:42.900 But you can also prove that non-deterministic time, n cubed 00:17:42.900 --> 00:17:45.180 properly includes non-deterministic time 00:17:45.180 --> 00:17:45.750 n squared. 00:17:45.750 --> 00:17:47.583 You're not going to be responsible for that. 00:17:47.583 --> 00:17:48.990 Don't worry. 00:17:48.990 --> 00:17:51.480 If you try to actually prove that, 00:17:51.480 --> 00:17:58.520 you'll see the diagonalization doesn't directly work. 00:17:58.520 --> 00:18:01.730 And so you have to do something fancier. 00:18:04.560 --> 00:18:07.320 People are asking about which reduction method to use. 00:18:07.320 --> 00:18:11.550 Again, the kinds of reductions that we encounter 00:18:11.550 --> 00:18:13.210 are always very simple. 00:18:13.210 --> 00:18:16.200 So we're just going to be working with very weak notions 00:18:16.200 --> 00:18:17.460 of reductions. 00:18:17.460 --> 00:18:20.130 Not interesting yet, generally, to consider powerful kinds 00:18:20.130 --> 00:18:25.037 of reductions like polynomial exponential time reductions 00:18:25.037 --> 00:18:25.870 or things like that. 00:18:25.870 --> 00:18:30.210 So it's just not something that people really think about much. 00:18:30.210 --> 00:18:33.360 I mean, I can talk about it at length offline. 00:18:33.360 --> 00:18:36.870 But let's just assume that our reduction strength 00:18:36.870 --> 00:18:38.370 is something very low. 00:18:38.370 --> 00:18:39.930 Log space is going to be good enough 00:18:39.930 --> 00:18:41.745 to do all of the reductions in this class. 00:18:45.330 --> 00:18:47.490 OK, so let's move on, then. 00:18:47.490 --> 00:18:51.090 So here is the problem that we're 00:18:51.090 --> 00:18:53.250 going to spend the next 20 minutes 00:18:53.250 --> 00:18:59.040 or so proving to be exponential space complete. 00:18:59.040 --> 00:19:01.080 I have got to do a little introduction first. 00:19:01.080 --> 00:19:07.130 So this is not the problem, but this is related to the problem. 00:19:07.130 --> 00:19:10.060 So the problem of testing if two regular expressions 00:19:10.060 --> 00:19:12.083 are equivalent. 00:19:12.083 --> 00:19:13.500 Write down to regular expressions, 00:19:13.500 --> 00:19:15.830 do they generate the same language? 00:19:15.830 --> 00:19:18.915 So that problem actually turns out to be in PSPACE. 00:19:18.915 --> 00:19:21.040 So it's not going to be exponential space complete. 00:19:21.040 --> 00:19:22.732 It's actually in PSPACE. 00:19:22.732 --> 00:19:24.190 I don't think we're going to have-- 00:19:24.190 --> 00:19:26.260 I thought about presenting it in the lecture. 00:19:26.260 --> 00:19:27.820 It's not that hard to show. 00:19:27.820 --> 00:19:30.190 But it just took too much time and doesn't really 00:19:30.190 --> 00:19:31.780 introduce new methods. 00:19:31.780 --> 00:19:35.890 It's a good exercise, actually, using Savitch's theorem. 00:19:35.890 --> 00:19:38.740 But maybe we'll do it in recitation, 00:19:38.740 --> 00:19:43.150 or if the lecture miraculously ends earlier, 00:19:43.150 --> 00:19:44.073 I'll do it at the end. 00:19:44.073 --> 00:19:45.490 But I don't think we'll have time. 00:19:50.570 --> 00:19:55.520 But that's a setup for the intractable problem 00:19:55.520 --> 00:19:58.850 that we're going to talk about, which is very related. 00:19:58.850 --> 00:20:01.310 Now, OK, before we get to that, so 00:20:01.310 --> 00:20:04.370 if I have a regular expression, I'm 00:20:04.370 --> 00:20:11.450 going to enhance our regular expression in one simple way, 00:20:11.450 --> 00:20:15.490 by allowing exponents or exponentiation. 00:20:15.490 --> 00:20:21.060 And that means if I have a regular expression R, 00:20:21.060 --> 00:20:25.170 I can write R to the k to mean R concatenated with itself k 00:20:25.170 --> 00:20:26.190 times. 00:20:26.190 --> 00:20:28.770 We've been sort of informally using that all the way along 00:20:28.770 --> 00:20:31.350 anyway, like when we talk about 0 to the k, 1 to the k. 00:20:34.260 --> 00:20:36.330 So if we're going to formally allow that 00:20:36.330 --> 00:20:40.080 when we write down regular expressions, in some cases, 00:20:40.080 --> 00:20:42.090 that might allow the regular expression 00:20:42.090 --> 00:20:46.030 to be much smaller, especially if we're 00:20:46.030 --> 00:20:48.800 writing down k in binary. 00:20:48.800 --> 00:20:52.220 Because I can write R to the million with just a few 00:20:52.220 --> 00:20:55.490 symbols if I have exponentiation. 00:20:55.490 --> 00:20:57.500 But if I don't have exponentiation, 00:20:57.500 --> 00:20:59.900 then I have to write R concatenated 00:20:59.900 --> 00:21:03.750 with R out a million times, and I get a much, 00:21:03.750 --> 00:21:07.340 much longer, an exponentially longer expression 00:21:07.340 --> 00:21:11.300 if I don't have that exponent as a way of describing 00:21:11.300 --> 00:21:13.010 regular expressions. 00:21:13.010 --> 00:21:15.870 And that's going to make a big difference. 00:21:15.870 --> 00:21:21.410 So now, the equivalence problem for regular expressions with 00:21:21.410 --> 00:21:25.890 exponentiation-- that's what that little up arrow means, 00:21:25.890 --> 00:21:27.840 what it signifies-- 00:21:27.840 --> 00:21:30.390 now I'm giving you two regular expressions. 00:21:30.390 --> 00:21:33.780 But they're allowed to have the exponentiation 00:21:33.780 --> 00:21:41.860 operation in addition to the standard regular operations. 00:21:41.860 --> 00:21:46.560 So now, testing whether two of these regular expressions that 00:21:46.560 --> 00:21:49.140 have exponentiation, that problem 00:21:49.140 --> 00:21:51.975 turns out to be exponential space complete. 00:21:56.495 --> 00:21:58.870 So here's the equivalence problem for regular expressions 00:21:58.870 --> 00:22:00.250 with exponentiation. 00:22:00.250 --> 00:22:02.840 That's an exponential space complete problem. 00:22:02.840 --> 00:22:05.500 And as we pointed out, that means this problem 00:22:05.500 --> 00:22:08.360 is provably intractable. 00:22:08.360 --> 00:22:11.930 So there's just no way, in general, 00:22:11.930 --> 00:22:14.180 to solve that problem in polynomial time. 00:22:14.180 --> 00:22:15.740 That's proven, that's known. 00:22:19.120 --> 00:22:23.170 So we're going to go through the reduction. 00:22:23.170 --> 00:22:25.690 I think it's going to be our last reduction of the term, 00:22:25.690 --> 00:22:28.660 of proving problems complete for some class. 00:22:28.660 --> 00:22:34.660 But each one of those has their own kind of thing 00:22:34.660 --> 00:22:37.120 that makes it special. 00:22:37.120 --> 00:22:40.870 So first of all, we have to show that it's in exponential space. 00:22:40.870 --> 00:22:42.820 That's really going to rely on this other fact 00:22:42.820 --> 00:22:43.870 that we didn't prove. 00:22:43.870 --> 00:22:47.360 So I'm going to go over that very quickly. 00:22:47.360 --> 00:22:49.400 But the interesting part is doing the reduction. 00:22:49.400 --> 00:22:51.970 So if I have something in exponential space 00:22:51.970 --> 00:22:54.820 that I can show that I can reduce it 00:22:54.820 --> 00:22:58.630 to the equivalence problem for regular expressions 00:22:58.630 --> 00:23:00.870 with exponentiation. 00:23:00.870 --> 00:23:04.530 OK, so quickly arguing part one that we're 00:23:04.530 --> 00:23:07.950 in exponential space, basically, what you do 00:23:07.950 --> 00:23:09.690 is you take your two regular expressions 00:23:09.690 --> 00:23:12.120 that you want to test to see if they're equivalent, 00:23:12.120 --> 00:23:14.010 but now they have exponentiation. 00:23:14.010 --> 00:23:17.400 And as a first step, you get rid of the exponentiation. 00:23:17.400 --> 00:23:22.620 You just expand things out by repeating the parts that 00:23:22.620 --> 00:23:25.050 have the exponents. 00:23:25.050 --> 00:23:27.900 And of course, as I said, that's going 00:23:27.900 --> 00:23:31.050 to make the expression themselves exponentially 00:23:31.050 --> 00:23:33.040 bigger. 00:23:33.040 --> 00:23:36.610 But now, you run the PSPACE algorithm 00:23:36.610 --> 00:23:40.220 on those two exponentially larger expressions. 00:23:40.220 --> 00:23:42.970 So the input that the PSPACE algorithm is now 00:23:42.970 --> 00:23:47.620 exponential in the original input size, 00:23:47.620 --> 00:23:50.330 but it's PSPACE in that enlarged input. 00:23:50.330 --> 00:23:52.690 So that's going to give you an exponential space 00:23:52.690 --> 00:23:57.070 algorithm in the original input size, because you expanded 00:23:57.070 --> 00:23:58.570 it to become exponentially bigger, 00:23:58.570 --> 00:24:04.655 and then you run the PSPACE algorithm on that expanded 00:24:04.655 --> 00:24:05.155 problem. 00:24:08.620 --> 00:24:10.750 So that gives you an exponential space algorithm 00:24:10.750 --> 00:24:15.380 for this problem. 00:24:15.380 --> 00:24:17.170 But now, what we're going to do-- 00:24:17.170 --> 00:24:20.480 the interesting part is the reduction. 00:24:20.480 --> 00:24:24.280 So given some language and exponential space, say, 00:24:24.280 --> 00:24:27.880 decided by some Turing machine in that amount of space, 00:24:27.880 --> 00:24:33.730 2 to the n to the k, we're going to give a reduction that maps a 00:24:33.730 --> 00:24:38.710 to this equivalence problem. 00:24:38.710 --> 00:24:40.150 Got it? 00:24:40.150 --> 00:24:41.530 That is the plan. 00:24:44.640 --> 00:24:47.780 So let's make sure we're all together on the plan 00:24:47.780 --> 00:24:51.320 before we go ahead and carry out that plan. 00:24:56.160 --> 00:24:57.680 We just sort of set things up here, 00:24:57.680 --> 00:25:01.130 in a sense, for what we're going to be doing. 00:25:01.130 --> 00:25:08.240 So feel free to ask a question on just the plan. 00:25:08.240 --> 00:25:09.890 It's going to get technical. 00:25:09.890 --> 00:25:12.740 Because, as doing these reductions always is, 00:25:12.740 --> 00:25:15.080 there's a simulation involved, and you 00:25:15.080 --> 00:25:19.290 have to kind of describe that simulation in its own way. 00:25:19.290 --> 00:25:22.100 So now, we're going to be simulating, 00:25:22.100 --> 00:25:28.290 in a certain sense, M on w, the decider 00:25:28.290 --> 00:25:32.970 for this exponential space, problem A, 00:25:32.970 --> 00:25:34.470 we're going to take M on w and we're 00:25:34.470 --> 00:25:39.130 going to somehow have to express the fact that M accepts w using 00:25:39.130 --> 00:25:41.130 this equivalence problem for regular expressions 00:25:41.130 --> 00:25:42.367 with exponentiation. 00:25:47.260 --> 00:25:48.365 So no questions? 00:25:48.365 --> 00:25:49.240 Why don't we move on? 00:25:52.120 --> 00:25:56.320 I have three slides on this, but they're kind of dense, 00:25:56.320 --> 00:25:57.340 I'm sorry to say. 00:26:00.810 --> 00:26:04.920 So here is the plan as usual. 00:26:04.920 --> 00:26:09.420 We're going to map A with a polynomial time reduction 00:26:09.420 --> 00:26:11.910 to the equivalence problem for regular expressions 00:26:11.910 --> 00:26:13.590 with exponentiation. 00:26:13.590 --> 00:26:16.680 So that means we're going to have to take an input, which 00:26:16.680 --> 00:26:21.480 may or may not be in A, and produce two regular expressions 00:26:21.480 --> 00:26:28.160 with exponentiation, which are going to be equivalent when 00:26:28.160 --> 00:26:33.220 w is in A. Or when M accepts w. 00:26:40.220 --> 00:26:45.570 So it's going to be, as these things always are, 00:26:45.570 --> 00:26:47.660 these are going to be in terms of the computation 00:26:47.660 --> 00:26:49.760 history for M under w. 00:26:49.760 --> 00:26:51.590 But in this case, it's going to turn out 00:26:51.590 --> 00:26:57.230 to be convenient to work with the rejecting computation 00:26:57.230 --> 00:26:59.270 history for M on w. 00:26:59.270 --> 00:27:04.240 So remember, now we have a Turing machine M. 00:27:04.240 --> 00:27:08.140 It's a decider, so that means it always holds-- 00:27:08.140 --> 00:27:11.140 for the strings in the language, it ends up at a Q accept state, 00:27:11.140 --> 00:27:15.340 for things not in the language, it ends up at a Q reject state. 00:27:15.340 --> 00:27:17.710 So a rejecting computation history 00:27:17.710 --> 00:27:19.330 is the sequence of configurations 00:27:19.330 --> 00:27:22.870 the machine goes through from the start configuration 00:27:22.870 --> 00:27:25.240 until it ends up at a configuration 00:27:25.240 --> 00:27:29.890 with a reject state, a rejecting configuration. 00:27:29.890 --> 00:27:32.830 And we're going to make a regular expression that 00:27:32.830 --> 00:27:38.640 describes all strings except for that one. 00:27:38.640 --> 00:27:43.050 It's going to avoid describing a rejecting computation 00:27:43.050 --> 00:27:44.670 history for M on w. 00:27:44.670 --> 00:27:47.445 Otherwise, it's going to describe all possible strings. 00:27:50.480 --> 00:27:54.530 Now, if M does not reject w, so there 00:27:54.530 --> 00:27:57.170 is no rejecting computation history-- 00:27:57.170 --> 00:27:59.000 namely, M accepts w, by the way. 00:27:59.000 --> 00:28:01.790 So if M accepts w, does not reject w, 00:28:01.790 --> 00:28:05.270 it does not have a rejecting computation history, 00:28:05.270 --> 00:28:09.470 what is R1 describing? 00:28:09.470 --> 00:28:12.610 Well, it's describing, in that case, everything, 00:28:12.610 --> 00:28:16.450 because there is no rejecting computation history. 00:28:16.450 --> 00:28:19.150 So it's describing every other string besides. 00:28:19.150 --> 00:28:23.170 So that means it's describing all strings, if there 00:28:23.170 --> 00:28:25.390 is no rejecting computation history in the case 00:28:25.390 --> 00:28:27.370 that M accepts w. 00:28:27.370 --> 00:28:30.890 So what does that suggest we should use for R2? 00:28:30.890 --> 00:28:33.740 R2 is going to be the regular expression that 00:28:33.740 --> 00:28:36.480 just generates all strings. 00:28:36.480 --> 00:28:40.920 So we'll be testing whether R1 generates all strings or not, 00:28:40.920 --> 00:28:49.410 which is the same as saying does M accept w or not. 00:28:49.410 --> 00:28:52.470 So R2 is going to be-- 00:28:52.470 --> 00:28:55.710 I would like to say sigma star, but sigma is really 00:28:55.710 --> 00:29:00.622 the input to M, and gamma is the tape alphabet for M. 00:29:00.622 --> 00:29:02.580 So we have a lot of Greek letters to play with, 00:29:02.580 --> 00:29:07.190 so we're going to use delta for the alphabet 00:29:07.190 --> 00:29:10.910 that we write the computation histories in. 00:29:10.910 --> 00:29:17.760 If you want to get reminded what that delta is, a computation 00:29:17.760 --> 00:29:20.760 history can have a tape alphabet symbol for M, 00:29:20.760 --> 00:29:23.970 it can have a state symbol for M, 00:29:23.970 --> 00:29:25.590 or it can have a delimiter pound-- 00:29:25.590 --> 00:29:26.880 hashtag. 00:29:26.880 --> 00:29:32.530 So it's either a capital delta alphabet 00:29:32.530 --> 00:29:37.330 is a tape alphabet symbol, or state, something representing 00:29:37.330 --> 00:29:40.740 a state symbol, or a hashtag. 00:29:40.740 --> 00:29:41.490 That's just delta. 00:29:41.490 --> 00:29:44.190 So don't get-- I always feel bad if somebody 00:29:44.190 --> 00:29:45.840 gets confused by something that's 00:29:45.840 --> 00:29:47.140 supposed to be very simple. 00:29:47.140 --> 00:29:49.140 Don't get confused by delta star. 00:29:49.140 --> 00:29:51.150 This is just all possible strings 00:29:51.150 --> 00:29:52.275 over the alphabet delta. 00:29:56.450 --> 00:29:58.820 OK, so what does R1-- 00:29:58.820 --> 00:30:01.130 so my job is to do R1. 00:30:01.130 --> 00:30:04.160 R2, I already told you. 00:30:04.160 --> 00:30:07.380 R1 now has to describe all those strings 00:30:07.380 --> 00:30:10.830 except for the rejecting computation history. 00:30:10.830 --> 00:30:16.260 So everything that fails to be a rejecting computation history-- 00:30:16.260 --> 00:30:19.380 so it fails either because it started wrong, 00:30:19.380 --> 00:30:22.320 or it ended wrong, or it's wrong somewhere in the middle. 00:30:22.320 --> 00:30:27.510 And by wrong I mean, it fails to correctly describe 00:30:27.510 --> 00:30:32.235 the way the machine operates if it's ending up rejecting w. 00:30:35.550 --> 00:30:36.050 All right. 00:30:36.050 --> 00:30:40.970 So I'm going to describe all those possible strings 00:30:40.970 --> 00:30:44.780 by breaking it down into those three categories. 00:30:44.780 --> 00:30:47.870 Starts wrong, ends wrong, or somewhere computes 00:30:47.870 --> 00:30:51.180 wrong along the way. 00:30:51.180 --> 00:30:51.680 OK. 00:30:51.680 --> 00:30:56.780 So rejecting computation history looks something like this. 00:30:56.780 --> 00:31:04.400 Here's the start configuration as we usually envision it. 00:31:04.400 --> 00:31:07.160 It's a start state looking at the first symbol of the input, 00:31:07.160 --> 00:31:09.650 and there's the rest of the input. 00:31:09.650 --> 00:31:10.955 So let me just write this out. 00:31:13.615 --> 00:31:15.760 This is a rejecting computation history now. 00:31:15.760 --> 00:31:19.310 So the first configuration, the second one, 00:31:19.310 --> 00:31:21.790 and so on and so on, until we end up at a rejecting 00:31:21.790 --> 00:31:26.350 computation-- rejecting configuration. 00:31:26.350 --> 00:31:32.530 Now, for convenience, I'm going to insist 00:31:32.530 --> 00:31:38.960 that all of these configurations are the same length. 00:31:38.960 --> 00:31:44.620 It's going to make my life easier in doing the proof. 00:31:44.620 --> 00:31:47.630 But why can I do that? 00:31:47.630 --> 00:31:49.193 Well, I'm just going to take them-- 00:31:49.193 --> 00:31:51.610 you know, because usually you think of the configurations, 00:31:51.610 --> 00:31:54.340 they start small because they're just basically of length n, 00:31:54.340 --> 00:31:56.380 but this is using exponential space, 00:31:56.380 --> 00:31:57.820 they're getting longer and longer. 00:31:57.820 --> 00:32:00.280 Let's just pair them all out with blanks 00:32:00.280 --> 00:32:02.780 so that they're all the same size. 00:32:02.780 --> 00:32:04.900 So as I've indicated over here, we're 00:32:04.900 --> 00:32:06.340 adding in a bunch of blanks. 00:32:06.340 --> 00:32:09.436 It's going to be a lot of blanks here, 00:32:09.436 --> 00:32:12.280 to make sure they all have length 2 to the n 00:32:12.280 --> 00:32:15.260 to the k, which is the maximum size of a configuration 00:32:15.260 --> 00:32:16.510 when you have that much space. 00:32:24.440 --> 00:32:26.720 I'm going to construct-- so basically, that's my job. 00:32:26.720 --> 00:32:29.000 I'm going to construct R1 so that it 00:32:29.000 --> 00:32:30.980 generates all those strings. 00:32:30.980 --> 00:32:37.500 I wrote a little box around that thing I'm trying to-- 00:32:41.703 --> 00:32:44.030 that's my to do. 00:32:44.030 --> 00:32:46.400 It's going to help me in the coming slides 00:32:46.400 --> 00:32:49.370 because they're a little bit dense. 00:32:49.370 --> 00:32:52.550 When I'm going to draw this sort of reddish, pinkish box 00:32:52.550 --> 00:32:55.760 around something, that means that I'm 00:32:55.760 --> 00:32:58.970 going to try to describe all strings except for that one. 00:33:10.770 --> 00:33:12.600 I want to avoid describing that one, 00:33:12.600 --> 00:33:14.730 because that's the rejecting computation history, 00:33:14.730 --> 00:33:16.410 but I want to describe everything else. 00:33:16.410 --> 00:33:18.585 That's my wish. 00:33:21.870 --> 00:33:25.180 So here's a check in before we move forward. 00:33:25.180 --> 00:33:26.790 But we can also-- maybe we should just 00:33:26.790 --> 00:33:30.463 take some questions, even before we launch the check in. 00:33:30.463 --> 00:33:31.380 How are we doing here? 00:33:36.000 --> 00:33:40.380 So, is our one describing-- 00:33:40.380 --> 00:33:43.520 well, R1 is a regular expression. 00:33:43.520 --> 00:33:45.620 Over here, we're talking about a-- 00:33:45.620 --> 00:33:47.630 this is just an ordinary computation history, 00:33:47.630 --> 00:33:48.720 but it ends with a reject. 00:33:48.720 --> 00:33:49.220 That's all. 00:33:49.220 --> 00:33:52.040 A rejecting computation history is just one that's 00:33:52.040 --> 00:33:53.540 a little different at the end. 00:33:53.540 --> 00:33:56.180 The machine just ended up rejecting instead of accepting. 00:33:56.180 --> 00:34:03.620 Otherwise everything has to be spelled out in accordance 00:34:03.620 --> 00:34:06.254 with the rules of the machine and the start configuration. 00:34:09.710 --> 00:34:13.597 Yeah, we were assuming one rejecting state. 00:34:13.597 --> 00:34:15.889 Yeah, that's the way we actually define Turing machines 00:34:15.889 --> 00:34:16.681 in the first place. 00:34:16.681 --> 00:34:19.159 But, who's arguing. 00:34:19.159 --> 00:34:21.460 Yeah, there's one reject state. 00:34:21.460 --> 00:34:24.460 We're all deterministic, correct. 00:34:24.460 --> 00:34:25.960 Why do we need the padding? 00:34:25.960 --> 00:34:29.560 Because I want to make these all the same size, all of these 00:34:29.560 --> 00:34:30.550 configurations. 00:34:30.550 --> 00:34:32.739 That's going to help me later in terms 00:34:32.739 --> 00:34:37.810 of describing the invalid configurations, the ones that 00:34:37.810 --> 00:34:42.980 are not legal configurations, legal rejecting configurations. 00:34:42.980 --> 00:34:45.409 So just simply a matter of convenience, 00:34:45.409 --> 00:34:47.449 but just accept it for now. 00:34:47.449 --> 00:34:49.580 I just want all of those configurations 00:34:49.580 --> 00:34:55.540 to be the same length in my rejecting computation history. 00:34:55.540 --> 00:34:57.370 Otherwise I'm not going to-- 00:34:57.370 --> 00:34:59.980 I'm just coding that rejecting computation history 00:34:59.980 --> 00:35:01.210 in this particular way. 00:35:06.610 --> 00:35:09.340 So people are asking about the details of bad start. 00:35:09.340 --> 00:35:10.540 That's yet to come. 00:35:10.540 --> 00:35:13.010 I have two more slides on this. 00:35:13.010 --> 00:35:16.950 So I'll tell you about how we're going to do those. 00:35:16.950 --> 00:35:21.480 So R bad-start-- that's a good question-- is R bad-start all-- 00:35:21.480 --> 00:35:25.770 these are all the strings that don't start this way. 00:35:25.770 --> 00:35:27.060 We'll see it in a second. 00:35:27.060 --> 00:35:30.480 But R bad-start are all the things that don't start with 00:35:30.480 --> 00:35:31.890 the-- 00:35:31.890 --> 00:35:33.000 they start bad. 00:35:36.720 --> 00:35:39.577 They're not starting with the start configuration. 00:35:39.577 --> 00:35:41.160 They're starting with some other junk. 00:35:46.130 --> 00:35:48.850 Do we need only one rejecting computation history? 00:35:48.850 --> 00:35:51.410 What about the other ones? 00:35:51.410 --> 00:35:54.500 This is a deterministic machine, so there's only going to be-- 00:35:54.500 --> 00:35:57.060 if I prescribe the lengths as I've done, 00:35:57.060 --> 00:35:59.607 there's going to be only one rejecting computation history. 00:35:59.607 --> 00:36:01.190 Because it's deterministic, everything 00:36:01.190 --> 00:36:06.380 is going to be forced from the beginning. 00:36:06.380 --> 00:36:09.020 Should R1 be the not of those three? 00:36:09.020 --> 00:36:09.880 No. 00:36:09.880 --> 00:36:11.870 R1 is describing all of the strings 00:36:11.870 --> 00:36:18.660 except, except this one string. 00:36:18.660 --> 00:36:21.800 So I'm capturing all the different possible ways 00:36:21.800 --> 00:36:24.140 a string could fail to be the string. 00:36:24.140 --> 00:36:26.180 It could start wrong. 00:36:26.180 --> 00:36:28.760 Could be wrong along the middle somewhere. 00:36:28.760 --> 00:36:31.870 So I have to union them together. 00:36:31.870 --> 00:36:35.290 Because I'm describing-- as I always believe, 00:36:35.290 --> 00:36:38.420 negations are the most confusing thing to everybody, 00:36:38.420 --> 00:36:41.510 including me. 00:36:41.510 --> 00:36:43.460 So we're describing all the things 00:36:43.460 --> 00:36:47.143 that are not this string. 00:36:47.143 --> 00:36:48.810 We're trying to stay away from that one. 00:36:48.810 --> 00:36:50.580 We want to describe everything else. 00:36:55.308 --> 00:36:57.100 All right, I think I'd better move on here. 00:36:57.100 --> 00:36:58.740 We've got a lot of questions. 00:36:58.740 --> 00:37:01.470 Talk to the TAs. 00:37:01.470 --> 00:37:02.700 All right, check in. 00:37:09.190 --> 00:37:15.110 How big is this rejecting computation history anyway? 00:37:15.110 --> 00:37:16.430 Interesting. 00:37:16.430 --> 00:37:18.620 There's a lesson here. 00:37:18.620 --> 00:37:21.680 I got a big burst of answers right at the very beginning. 00:37:21.680 --> 00:37:24.550 All wrong. 00:37:24.550 --> 00:37:26.020 But then the bright-- 00:37:26.020 --> 00:37:29.770 the people who took a little bit more time to think 00:37:29.770 --> 00:37:35.620 started getting the right answer, which is-- 00:37:35.620 --> 00:37:36.130 let's look. 00:37:36.130 --> 00:37:39.622 We've got a close election here folks, so now I have to report. 00:37:39.622 --> 00:37:41.080 Hope we don't have to do a recount. 00:37:45.680 --> 00:37:47.450 OK, come on guys. 00:37:47.450 --> 00:37:48.500 Answer up. 00:37:48.500 --> 00:37:50.152 10 seconds. 00:37:50.152 --> 00:37:51.110 This is not super hard. 00:37:53.690 --> 00:37:54.395 Stop the count. 00:37:57.500 --> 00:38:00.905 Yeah, I think we'd better stop at this, we're on the edge. 00:38:03.880 --> 00:38:05.845 OK, 3 seconds. 00:38:10.030 --> 00:38:12.780 End polling. 00:38:12.780 --> 00:38:13.470 Share results. 00:38:20.670 --> 00:38:23.100 The correct answer is, in fact, c. 00:38:23.100 --> 00:38:23.730 Why is that? 00:38:23.730 --> 00:38:27.900 Because each configuration is 2 to the n to the k. 00:38:27.900 --> 00:38:31.770 So that's how much space the machine has, exponential space. 00:38:31.770 --> 00:38:35.328 But the amount of time, which is each one-- 00:38:35.328 --> 00:38:36.870 the number of configurations is going 00:38:36.870 --> 00:38:39.600 to be the amount of time that's used. 00:38:39.600 --> 00:38:42.300 It's going to be exponentially more even than that. 00:38:42.300 --> 00:38:45.450 So it's going to be 2 to the 2 to the n of the k, 00:38:45.450 --> 00:38:47.550 is how many steps the machine can run. 00:38:47.550 --> 00:38:51.420 And that's going to be how long the computation history could 00:38:51.420 --> 00:38:52.380 be. 00:38:52.380 --> 00:38:55.650 So it's a very long thing. 00:38:55.650 --> 00:39:02.070 And when you think about it, the regular expression 00:39:02.070 --> 00:39:03.860 we are generating, how big is that? 00:39:06.880 --> 00:39:09.490 The regular expression-- again, a lot 00:39:09.490 --> 00:39:11.830 of people playing off my comments here. 00:39:15.660 --> 00:39:17.100 Were the votes legal or not? 00:39:17.100 --> 00:39:18.810 OK. 00:39:18.810 --> 00:39:20.050 Let's focus here. 00:39:22.990 --> 00:39:26.260 So this is doubly exponentially large. 00:39:26.260 --> 00:39:28.360 How big is the regular expression 00:39:28.360 --> 00:39:29.680 that we're generating? 00:39:29.680 --> 00:39:32.780 Well that has to be produced in polynomial time, 00:39:32.780 --> 00:39:34.330 so it's only polynomially big. 00:39:34.330 --> 00:39:38.260 So we have this little teensy weensy, relatively speaking, 00:39:38.260 --> 00:39:42.100 regular expression, which is only n to the k. 00:39:42.100 --> 00:39:44.830 It's having to describe all strings 00:39:44.830 --> 00:39:49.300 except for this particular string, which is 2 to the 2 00:39:49.300 --> 00:39:51.050 to the n to the k. 00:39:51.050 --> 00:39:54.410 So in a sense, this string that is 00:39:54.410 --> 00:39:56.270 related to that regular expression 00:39:56.270 --> 00:39:58.520 is doubly exponentially larger than that. 00:39:58.520 --> 00:40:01.070 And that kind of presents some of the challenge 00:40:01.070 --> 00:40:05.000 in doing the reduction, in constructing 00:40:05.000 --> 00:40:07.640 that regular expression. 00:40:07.640 --> 00:40:09.740 So let's move on and start doing-- 00:40:09.740 --> 00:40:12.660 this is the hard stuff. 00:40:12.660 --> 00:40:17.430 Here is the bad start, which is challenging enough. 00:40:17.430 --> 00:40:20.160 Even this little piece is going to be a little bit challenging 00:40:20.160 --> 00:40:22.680 to describe. 00:40:22.680 --> 00:40:26.230 Just rewriting from the previous slide. 00:40:26.230 --> 00:40:28.050 So we're trying to make R1, which 00:40:28.050 --> 00:40:30.540 is generating all the strings except the rejecting 00:40:30.540 --> 00:40:34.230 computation history for M on w. 00:40:34.230 --> 00:40:35.940 It's in those three parts. 00:40:35.940 --> 00:40:39.150 Right now I'm describing the bad start piece. 00:40:39.150 --> 00:40:41.820 So that's going to describe all strings that 00:40:41.820 --> 00:40:46.630 don't start with this C1. 00:40:46.630 --> 00:40:47.880 So let me write that out here. 00:40:47.880 --> 00:40:49.650 This is going to generate all strings that 00:40:49.650 --> 00:40:56.520 don't start with C start or C1, which is as specified. 00:40:56.520 --> 00:40:57.460 Looks like this. 00:40:57.460 --> 00:41:01.320 So any string that doesn't start with these symbols, doesn't 00:41:01.320 --> 00:41:06.610 start exactly like this, should be 00:41:06.610 --> 00:41:12.220 described by bad start, that regular expression. 00:41:12.220 --> 00:41:17.620 So that, in itself, is going to be further subdivided. 00:41:17.620 --> 00:41:22.600 And the reason for that is not that hard to understand. 00:41:22.600 --> 00:41:24.490 I'm going to-- bad start is going 00:41:24.490 --> 00:41:31.600 to accomplish its goal by saying, well, 00:41:31.600 --> 00:41:34.630 anything that doesn't start this way either 00:41:34.630 --> 00:41:38.260 doesn't start with a q0, or doesn't or doesn't 00:41:38.260 --> 00:41:40.570 have a w1 in the next place, or doesn't 00:41:40.570 --> 00:41:42.460 have a w2 in the next place. 00:41:42.460 --> 00:41:48.220 Or somewhere along the way, it has a wrong symbol. 00:41:48.220 --> 00:41:52.960 Each one of these guys is going to be about one 00:41:52.960 --> 00:41:58.973 of those symbols being wrong in some particular place. 00:41:58.973 --> 00:42:00.890 So I'm going to show you what those look like. 00:42:00.890 --> 00:42:07.750 So right now, I'm going to focus my attention 00:42:07.750 --> 00:42:12.960 on describing all strings except for this one. 00:42:12.960 --> 00:42:18.150 All strings that start with something except for this one. 00:42:20.970 --> 00:42:23.310 So just remember, delta is the alphabet 00:42:23.310 --> 00:42:26.070 for the competition histories. 00:42:26.070 --> 00:42:28.950 And some notation here, delta sub epsilon, 00:42:28.950 --> 00:42:30.450 we've seen this before, is you're 00:42:30.450 --> 00:42:35.490 going to add in epsilon as an allowed thing for delta. 00:42:35.490 --> 00:42:39.600 So it's all the symbols, or epsilon, now 00:42:39.600 --> 00:42:41.740 thought of as a set here. 00:42:41.740 --> 00:42:45.390 And furthermore, it's going to be convenient to talk about all 00:42:45.390 --> 00:42:49.080 of the symbols in delta, except for some symbol. 00:42:49.080 --> 00:42:51.570 So like at the very beginning, q0. 00:42:51.570 --> 00:42:55.092 I want to talk about all of the symbols except for q0 symbol. 00:42:55.092 --> 00:42:56.550 Because that's what I'm going to be 00:42:56.550 --> 00:43:02.130 using to start off R bad-start. 00:43:02.130 --> 00:43:05.590 It's going to be anything except for q0. 00:43:05.590 --> 00:43:07.370 So let's just see how that looks. 00:43:07.370 --> 00:43:11.890 So here is S0, the very first part of our bad start. 00:43:11.890 --> 00:43:14.420 It's going to say-- 00:43:14.420 --> 00:43:18.430 I'm trying to color the active ingredient here 00:43:18.430 --> 00:43:22.020 in the pink color. 00:43:22.020 --> 00:43:29.980 So delta, with q0 removed, followed by anything. 00:43:29.980 --> 00:43:31.570 So this little regular expression 00:43:31.570 --> 00:43:36.460 here describes all strings that don't start with a q0, 00:43:36.460 --> 00:43:38.110 as I'm indicating over here. 00:43:38.110 --> 00:43:40.930 All strings that don't start with a q0 00:43:40.930 --> 00:43:45.750 is what as S0 describes. 00:43:45.750 --> 00:43:47.782 You have to understand that, because it's just 00:43:47.782 --> 00:43:48.990 going to build up from there. 00:43:51.880 --> 00:43:53.820 So what do we want to say for S1? 00:43:53.820 --> 00:43:55.860 What's going to be all strings that don't 00:43:55.860 --> 00:43:58.990 have w1 in the second place? 00:43:58.990 --> 00:44:01.960 So I'm going to write that over here. 00:44:01.960 --> 00:44:06.520 S1 is anything in the first place-- 00:44:06.520 --> 00:44:09.400 I mean, if the first place was wrong, S0 took care of it. 00:44:09.400 --> 00:44:11.400 So I'm just going to keep my life simple. 00:44:11.400 --> 00:44:15.030 All I want to do is describe all of the places 00:44:15.030 --> 00:44:17.400 where the second symbol is wrong. 00:44:17.400 --> 00:44:19.530 Namely, it's not w1. 00:44:19.530 --> 00:44:22.600 So anything in the first place, something 00:44:22.600 --> 00:44:27.490 besides w1 in the next place, and then anything at all 00:44:27.490 --> 00:44:28.150 afterward. 00:44:28.150 --> 00:44:30.460 Those are all strings that don't have-- 00:44:30.460 --> 00:44:34.840 [AUDIO CUTS] 00:44:34.840 --> 00:44:37.090 So I'll write it over here like that. 00:44:37.090 --> 00:44:41.840 Now S2 similarly is going to d since I have exponentiation, 00:44:41.840 --> 00:44:45.100 let's use that for convenience. 00:44:45.100 --> 00:44:47.930 Delta delta, or just delta squared. 00:44:47.930 --> 00:44:53.645 So anything in the first two places, then not w2, and then 00:44:53.645 --> 00:44:55.640 the next place, and then anything. 00:44:55.640 --> 00:44:57.540 So that's going to capture this part. 00:44:57.540 --> 00:44:59.630 So this is what these S's do, and you 00:44:59.630 --> 00:45:01.622 can sort of get the idea. 00:45:01.622 --> 00:45:02.330 So dot, dot, dot. 00:45:02.330 --> 00:45:04.790 This Sn is going to describe everything 00:45:04.790 --> 00:45:08.675 except for wn in that location, which 00:45:08.675 --> 00:45:12.590 is going to be the n plus first location, actually. 00:45:12.590 --> 00:45:18.620 And now I have to continue on doing that for the blanks. 00:45:18.620 --> 00:45:24.380 So now, if you think with me, let's just 00:45:24.380 --> 00:45:26.120 take a look how that could go. 00:45:29.630 --> 00:45:34.790 The next symbol, which is skipping over the n plus 1 00:45:34.790 --> 00:45:38.440 that I've already taken care of, I 00:45:38.440 --> 00:45:41.890 want to say it's not a blank symbol in this very 00:45:41.890 --> 00:45:44.450 first location after the input. 00:45:44.450 --> 00:45:46.660 So again, I'm describing these non-- 00:45:46.660 --> 00:45:49.420 these strings which are not the start configuration. 00:45:49.420 --> 00:45:53.198 It could fail because there's not a blank where 00:45:53.198 --> 00:45:54.490 there's supposed to be a blank. 00:45:57.190 --> 00:45:59.380 Suppose I do that for each one of these guys. 00:46:02.790 --> 00:46:05.570 That would work. 00:46:05.570 --> 00:46:06.560 But. 00:46:06.560 --> 00:46:07.370 But what? 00:46:10.250 --> 00:46:13.060 Think. 00:46:13.060 --> 00:46:17.260 This is actually not going to be a good solution for us. 00:46:17.260 --> 00:46:21.700 Because there are exponentially many blanks over here. 00:46:21.700 --> 00:46:24.652 This is a hugely long configuration. 00:46:24.652 --> 00:46:26.360 And so there's exponentially many blanks. 00:46:26.360 --> 00:46:30.890 If I do it this way, I'm going to end up with an exponentially 00:46:30.890 --> 00:46:32.990 large regular expression. 00:46:32.990 --> 00:46:35.970 And that's not doable in polynomial time. 00:46:35.970 --> 00:46:39.380 So I have a more complicated way of getting the same effect. 00:46:39.380 --> 00:46:40.970 Which is-- I don't really expect you 00:46:40.970 --> 00:46:43.010 to fully parse through this right 00:46:43.010 --> 00:46:46.250 now, in real time in lecture, but let me try to help you. 00:46:46.250 --> 00:46:50.030 What I'm going to do is skip over these first initial n 00:46:50.030 --> 00:46:53.120 plus 1 places, and then a variable number 00:46:53.120 --> 00:46:58.310 of places, which is indicated by the next piece here. 00:46:58.310 --> 00:47:00.040 And the way that works is-- 00:47:00.040 --> 00:47:02.110 these are all strings of length n 00:47:02.110 --> 00:47:09.370 plus 1 through the end of the configuration. 00:47:09.370 --> 00:47:13.930 And to understand that, it's almost a little 00:47:13.930 --> 00:47:16.630 too technical to even try, but let's see. 00:47:16.630 --> 00:47:19.840 If I put delta to the 7, that's all strings of length 7. 00:47:19.840 --> 00:47:22.840 But if I put delta sub epsilon to the 7, 00:47:22.840 --> 00:47:24.400 if you think about what that means, 00:47:24.400 --> 00:47:28.120 that's all strings of length between 0 and 7. 00:47:31.530 --> 00:47:33.660 Because I can either have it as epsilon 00:47:33.660 --> 00:47:37.110 as my variable or a symbol from delta. 00:47:37.110 --> 00:47:39.360 And so that's what I'm doing over here. 00:47:39.360 --> 00:47:45.240 I'm getting a variable length space, spacer of deltas, 00:47:45.240 --> 00:47:48.420 that are going to then end up at a certain location-- 00:47:48.420 --> 00:47:50.670 I'm going to say at that place. 00:47:50.670 --> 00:47:53.430 Then I have a non-blank. 00:47:53.430 --> 00:47:56.220 Because all I need to do is describe 00:47:56.220 --> 00:48:02.440 the strings that fail to have a blank somewhere in this range. 00:48:02.440 --> 00:48:04.680 So we've got to sort have a variable spacer 00:48:04.680 --> 00:48:10.800 out to that spot, where that missing blank might be. 00:48:10.800 --> 00:48:12.850 So that's what this describes. 00:48:12.850 --> 00:48:15.420 If you didn't get that, don't worry. 00:48:15.420 --> 00:48:17.010 That is a technical point and you can 00:48:17.010 --> 00:48:19.620 try to think about it offline. 00:48:19.620 --> 00:48:25.380 And then at the very end, I'm going to describe what happens. 00:48:25.380 --> 00:48:27.600 Describe the strings that fail to have 00:48:27.600 --> 00:48:30.510 a hashtag in that location. 00:48:30.510 --> 00:48:36.180 It's how I describe all strings that don't start right. 00:48:36.180 --> 00:48:39.330 That's a lot of work, just to do that little piece. 00:48:39.330 --> 00:48:42.540 Fortunately, the next two pieces are easier, surprisingly. 00:48:46.110 --> 00:48:50.160 You can jump in with a question, but maybe I 00:48:50.160 --> 00:48:53.580 should move, push on. 00:48:53.580 --> 00:48:58.580 So now I'm going to describe the bad move and bad reject pieces. 00:48:58.580 --> 00:49:02.150 And bad reject generates all strings 00:49:02.150 --> 00:49:06.200 that don't contain the q reject symbol. 00:49:06.200 --> 00:49:07.850 So that's going to certainly describe 00:49:07.850 --> 00:49:11.580 all of the strings that don't end correctly. 00:49:11.580 --> 00:49:15.870 And that's just simply the delta with the q 00:49:15.870 --> 00:49:19.730 reject symbol removed, and then any string of those. 00:49:19.730 --> 00:49:22.140 That's all strings that don't have q reject. 00:49:22.140 --> 00:49:26.360 So that's going to describe all strings that 00:49:26.360 --> 00:49:28.713 don't end with a q reject, plus some other junk 00:49:28.713 --> 00:49:29.630 strings along the way. 00:49:29.630 --> 00:49:33.500 But that's all that's never a problem, 00:49:33.500 --> 00:49:36.737 to put in other strings that you might be capturing 00:49:36.737 --> 00:49:38.570 in some other part of the regular expression 00:49:38.570 --> 00:49:40.220 that you know are bad strings. 00:49:40.220 --> 00:49:43.400 You just want to make sure you don't put in that one uniquely 00:49:43.400 --> 00:49:45.500 good string, which is the rejecting computation 00:49:45.500 --> 00:49:47.700 history, good string. 00:49:47.700 --> 00:49:51.470 And lastly, we're going to use the notion 00:49:51.470 --> 00:49:54.512 of the neighborhoods. 00:49:54.512 --> 00:49:56.220 You might think this is the hardest part, 00:49:56.220 --> 00:49:57.820 but in fact not that hard. 00:49:57.820 --> 00:50:01.650 So these are all of the strings that 00:50:01.650 --> 00:50:08.160 have somewhere along the way a violation according 00:50:08.160 --> 00:50:09.480 to M's rules. 00:50:09.480 --> 00:50:11.580 You want to describe all of those as well. 00:50:11.580 --> 00:50:13.720 I'm going to do that in terms of the neighborhoods. 00:50:13.720 --> 00:50:19.320 But the neighborhoods are going to be stretched out. 00:50:19.320 --> 00:50:22.460 We don't have a tableau anymore, so they're not 00:50:22.460 --> 00:50:25.910 so easily visualizable, but it's the same idea, 00:50:25.910 --> 00:50:26.820 the neighborhood. 00:50:26.820 --> 00:50:28.700 So this is abc and def. 00:50:28.700 --> 00:50:32.630 But now it's an illegal neighborhood. 00:50:32.630 --> 00:50:34.745 def does not follow from abc. 00:50:39.170 --> 00:50:41.330 If all the neighborhoods are legal, 00:50:41.330 --> 00:50:45.710 then the whole computation is a legitimate computation, 00:50:45.710 --> 00:50:48.060 provided it starts and ends correctly. 00:50:48.060 --> 00:50:50.030 So if it's not a legitimate computation, 00:50:50.030 --> 00:50:52.520 there's got to be an illegal neighborhood somewhere. 00:50:52.520 --> 00:50:55.250 And I'm going to just describe all strings that 00:50:55.250 --> 00:50:57.530 have an illegal neighborhood. 00:50:57.530 --> 00:51:00.350 And the interesting part is that you have to describe-- 00:51:00.350 --> 00:51:07.430 you have to place that separator between abc and def. 00:51:07.430 --> 00:51:10.550 So this is another place where we're going to critically use 00:51:10.550 --> 00:51:17.780 the exponentiation, and the fact that all of the configurations 00:51:17.780 --> 00:51:19.020 are the same length. 00:51:19.020 --> 00:51:20.420 That's what we're using there. 00:51:20.420 --> 00:51:23.720 We know exactly how far apart the bottom 00:51:23.720 --> 00:51:26.960 of the 2 by 3 neighborhood is from the top of the 2 00:51:26.960 --> 00:51:28.640 by 3 neighborhood. 00:51:28.640 --> 00:51:33.110 So we're going to take a union over all illegal 2 00:51:33.110 --> 00:51:35.900 by 3 neighborhoods. 00:51:35.900 --> 00:51:37.520 Neighborhood settings, I should say. 00:51:37.520 --> 00:51:39.620 And there's only a fixed number of those, for the same reason 00:51:39.620 --> 00:51:41.360 that we had in the Cook-Levin theorem. 00:51:41.360 --> 00:51:44.210 There's a fixed number of those, depending upon the machine. 00:51:44.210 --> 00:51:45.920 And now we're going to have, say, we're 00:51:45.920 --> 00:51:48.050 going to start with anything. 00:51:48.050 --> 00:51:49.730 Here's the top of the neighborhood. 00:51:49.730 --> 00:51:53.480 Here is the separator that separates the top 00:51:53.480 --> 00:51:57.020 from the bottom in the two consecutive configurations, 00:51:57.020 --> 00:52:03.410 here's Ci going C i plus 1 inside my computation history. 00:52:03.410 --> 00:52:06.350 And then after that separator, I put 00:52:06.350 --> 00:52:09.365 in the second part of the neighborhood, which is the def. 00:52:13.020 --> 00:52:16.870 You have to really be comfortable with the way we've 00:52:16.870 --> 00:52:19.060 been presenting these other reductions up 00:52:19.060 --> 00:52:22.090 till now, to really get this. 00:52:22.090 --> 00:52:25.240 Anyway, I think we're at the break. 00:52:25.240 --> 00:52:27.940 So we can just take questions during the break, 00:52:27.940 --> 00:52:29.730 if you have any. 00:52:29.730 --> 00:52:35.910 And I will, otherwise, see you in five minutes. 00:52:35.910 --> 00:52:38.760 In my description back here-- 00:52:38.760 --> 00:52:40.020 let me just take this off. 00:52:42.810 --> 00:52:47.910 For bad reject, it looks like I'm doing kind of overkill, 00:52:47.910 --> 00:52:50.160 and maybe doing something wrong here. 00:52:50.160 --> 00:52:53.310 I'm describing all strings that don't have a reject anywhere. 00:52:53.310 --> 00:52:57.930 But as long as I don't describe the legitimate rejecting 00:52:57.930 --> 00:53:02.200 computation history, I do describe all strings that 00:53:02.200 --> 00:53:07.490 don't end correctly, I'm good. 00:53:07.490 --> 00:53:10.670 I could go through more effort to make sure 00:53:10.670 --> 00:53:15.380 that I'm only describing the very last configuration here 00:53:15.380 --> 00:53:17.270 as not having the reject. 00:53:17.270 --> 00:53:22.465 But that would just be more work, 00:53:22.465 --> 00:53:23.840 and I don't need to do that work. 00:53:23.840 --> 00:53:25.673 So maybe it would be good just to understand 00:53:25.673 --> 00:53:28.172 why this is sufficient, what I've described here, 00:53:28.172 --> 00:53:30.005 and it's not going to cause me any problems. 00:53:37.010 --> 00:53:41.270 I'm getting a note from one of my TAs, Thomas, 00:53:41.270 --> 00:53:43.910 saying that the notion "bad" perhaps 00:53:43.910 --> 00:53:46.610 is confusing, because bad sounds like rejecting. 00:53:46.610 --> 00:53:47.720 Yes. 00:53:47.720 --> 00:53:52.490 I mean bad in the sense of not describing a legal computation 00:53:52.490 --> 00:53:53.150 history. 00:53:53.150 --> 00:53:54.525 If you can think of another name, 00:53:54.525 --> 00:53:59.090 I'm happy to switch that for future years. 00:53:59.090 --> 00:54:00.020 Too late for now. 00:54:00.020 --> 00:54:00.950 But, yeah. 00:54:00.950 --> 00:54:03.290 I don't mean that rejecting, I mean that it's-- 00:54:07.050 --> 00:54:08.910 well, I don't know what the right term is. 00:54:08.910 --> 00:54:11.040 Illegal? 00:54:11.040 --> 00:54:14.515 Or-- I'm not sure what a good-- 00:54:14.515 --> 00:54:16.140 How are the neighborhoods defined here? 00:54:16.140 --> 00:54:18.980 What is the tableau here? 00:54:18.980 --> 00:54:22.790 I think you do need to think about it after lecture. 00:54:22.790 --> 00:54:25.580 But the tableau, you can think of the tableau 00:54:25.580 --> 00:54:28.030 now here just written out linearly. 00:54:28.030 --> 00:54:30.240 There are all of the rows now, instead of 00:54:30.240 --> 00:54:32.760 nicely organized into a table. 00:54:32.760 --> 00:54:35.557 They just appear consecutively, because I'm just 00:54:35.557 --> 00:54:36.390 trying to describe-- 00:54:36.390 --> 00:54:38.070 I need to do it to describe a string, 00:54:38.070 --> 00:54:40.080 whether my regular expression doesn't really 00:54:40.080 --> 00:54:41.520 make sense to think about. 00:54:41.520 --> 00:54:43.950 I mean you can fold it up into a tableau, if you like. 00:54:43.950 --> 00:54:47.220 And then abc and def will line up. 00:54:47.220 --> 00:54:50.280 But here, if you think about them written consecutively, 00:54:50.280 --> 00:54:52.740 this is exactly how far apart they end up being. 00:54:56.750 --> 00:55:02.570 Are there only polynomially many illegal neighborhoods? 00:55:02.570 --> 00:55:04.360 That's why I kind of corrected myself. 00:55:04.360 --> 00:55:07.030 It's not illegal neighborhoods that we're 00:55:07.030 --> 00:55:10.300 talking-- because the number of neighborhoods in this picture 00:55:10.300 --> 00:55:11.410 is vast. 00:55:11.410 --> 00:55:14.650 But the number of neighborhood settings, 00:55:14.650 --> 00:55:17.050 the way to set these values to abc, def. 00:55:20.950 --> 00:55:22.810 I mean these are symbols that can 00:55:22.810 --> 00:55:29.790 appear in a configuration of the machine. 00:55:29.790 --> 00:55:34.740 There's only a fixed number of symbols that can appear here, 00:55:34.740 --> 00:55:38.310 that depend upon the definition of the machine. 00:55:38.310 --> 00:55:39.720 So it's not only polynomial. 00:55:39.720 --> 00:55:42.420 There's a constant number of these things, that 00:55:42.420 --> 00:55:45.190 only depends on the machine. 00:55:45.190 --> 00:55:47.560 So you have to think about what's going on. 00:55:47.560 --> 00:55:49.830 There's a lot-- this is a lot on the slide. 00:55:54.480 --> 00:55:56.040 Bad history for reject. 00:55:56.040 --> 00:55:57.900 It's a bad history for rejecting, 00:55:57.900 --> 00:55:59.310 somebody's suggesting. 00:55:59.310 --> 00:56:01.765 Yeah, it's a bad history. 00:56:04.930 --> 00:56:05.590 Fake news. 00:56:05.590 --> 00:56:07.360 Maybe we should be fake. 00:56:07.360 --> 00:56:09.010 Fake would be a good term. 00:56:09.010 --> 00:56:10.360 No, that's not so good. 00:56:10.360 --> 00:56:10.960 I don't know. 00:56:17.600 --> 00:56:18.740 Yeah, 2 by 3. 00:56:18.740 --> 00:56:21.380 The reason 2 by 3, is the right-- 00:56:21.380 --> 00:56:22.880 Somebody's asking why 2 by 3. 00:56:22.880 --> 00:56:25.670 2 by 3 is exactly the size you need 00:56:25.670 --> 00:56:28.280 to say that, if all the 2 by 3 neighborhoods 00:56:28.280 --> 00:56:33.620 are correct everywhere in the computation history, 00:56:33.620 --> 00:56:36.590 then the whole history is going to be 00:56:36.590 --> 00:56:38.480 consistent with the rules of M. It's 00:56:38.480 --> 00:56:41.030 going to be a legal representation of a computation 00:56:41.030 --> 00:56:42.350 of M. 00:56:42.350 --> 00:56:48.500 So if the string, which is allegedly a computation 00:56:48.500 --> 00:56:51.710 history, has a bad neighborhood somewhere, 00:56:51.710 --> 00:56:55.140 bad 2 by 3 neighborhood somewhere, then-- 00:56:55.140 --> 00:56:57.470 well if it's not a legal computation history, 00:56:57.470 --> 00:57:00.920 it's got to have a bad neighborhood, 2 00:57:00.920 --> 00:57:02.406 by 3 neighborhood somewhere. 00:57:06.390 --> 00:57:10.130 OK, let's move on. 00:57:10.130 --> 00:57:13.970 Because I think we're out of time here. 00:57:13.970 --> 00:57:16.220 Our timer is up. 00:57:16.220 --> 00:57:18.440 We're going to shift gears now anyway. 00:57:18.440 --> 00:57:22.803 So if you got a little lost in the previous proof, 00:57:22.803 --> 00:57:24.720 we're going to talk about something different. 00:57:24.720 --> 00:57:25.820 And in some ways, a little bit, I 00:57:25.820 --> 00:57:27.500 think a little lighter, a little less technical. 00:57:27.500 --> 00:57:28.542 And that's about oracles. 00:57:36.570 --> 00:57:37.710 What are oracles? 00:57:37.710 --> 00:57:39.000 Oracles are a simple thing. 00:57:41.700 --> 00:57:48.008 But they are a useful concept for a number of reasons. 00:57:48.008 --> 00:57:49.800 Especially because they're going to tell us 00:57:49.800 --> 00:57:52.500 something interesting about methods, which may or may not 00:57:52.500 --> 00:57:55.560 be useful for proving the P versus NP question, 00:57:55.560 --> 00:57:57.990 when someday somebody hopefully does that. 00:58:01.170 --> 00:58:03.030 What is an oracle? 00:58:03.030 --> 00:58:05.340 An oracle is free information you're 00:58:05.340 --> 00:58:07.050 going to give to a Turing machine, which 00:58:07.050 --> 00:58:11.443 might affect the difficulty of solving problems. 00:58:11.443 --> 00:58:13.860 And the way we're going to represent that free information 00:58:13.860 --> 00:58:17.400 is, we're going to allow the Turing machine to test 00:58:17.400 --> 00:58:21.740 membership in some specified language, 00:58:21.740 --> 00:58:27.230 without charging for the work involved. 00:58:30.970 --> 00:58:34.090 I'm going to allow you have any language at all, some language 00:58:34.090 --> 00:58:38.810 A. And say a Turing machine with an oracle for A 00:58:38.810 --> 00:58:43.085 is written this way, M with a superscript A. It's 00:58:43.085 --> 00:58:46.370 a machine that has a black box that can answer questions. 00:58:46.370 --> 00:58:50.540 Is some string, which the machine can choose, 00:58:50.540 --> 00:58:52.680 in A or not? 00:58:52.680 --> 00:58:56.340 And it gets that answer in one step, effectively for free. 00:59:01.020 --> 00:59:04.320 So you can imagine, depending upon the language that you're 00:59:04.320 --> 00:59:07.515 providing to the machine, that may or may not be useful. 00:59:11.250 --> 00:59:15.800 For example, suppose I give you an oracle for the SAT language. 00:59:15.800 --> 00:59:17.880 That can be very useful. 00:59:17.880 --> 00:59:21.440 It could be very useful for deciding SAT, for example. 00:59:21.440 --> 00:59:24.110 Because now you don't have to go through a brute force search 00:59:24.110 --> 00:59:25.310 to solve SAT. 00:59:25.310 --> 00:59:26.720 You just ask the oracle. 00:59:26.720 --> 00:59:29.630 And the oracle is going to say, yes it's satisfiable, 00:59:29.630 --> 00:59:31.430 or no it's not satisfiable. 00:59:31.430 --> 00:59:36.230 But you can use that to solve other languages too, quickly. 00:59:36.230 --> 00:59:40.670 Because anything that you can do in NP, you can reduce to SAT. 00:59:40.670 --> 00:59:43.383 So you can convert it to a SAT question, which you can then 00:59:43.383 --> 00:59:45.050 ship up to the oracle, and the oracle is 00:59:45.050 --> 00:59:47.540 going to tell you the answer. 00:59:47.540 --> 00:59:51.480 The word "oracle" already sort of conveys something magical. 00:59:51.480 --> 00:59:53.268 We're not really going to be concerned 00:59:53.268 --> 00:59:55.310 with the operation of the oracle, so don't ask me 00:59:55.310 --> 00:59:57.602 how does the oracle work, or what does it correspond to 00:59:57.602 --> 00:59:58.320 in reality. 00:59:58.320 --> 00:59:59.090 It doesn't. 00:59:59.090 --> 01:00:01.280 It's just a mathematical device which 01:00:01.280 --> 01:00:03.770 provides this free information to the Turing machine, which 01:00:03.770 --> 01:00:05.870 enables it to compute certain things. 01:00:05.870 --> 01:00:07.657 It turns out to be a useful concept. 01:00:07.657 --> 01:00:09.740 It's used in cryptography, where you might imagine 01:00:09.740 --> 01:00:13.250 the oracle could provide the factors to some number, 01:00:13.250 --> 01:00:16.340 or the password to some system or something. 01:00:16.340 --> 01:00:17.510 Free information. 01:00:17.510 --> 01:00:19.680 And then what can you do with that? 01:00:19.680 --> 01:00:22.475 So this is a notion that comes up in other places. 01:00:27.660 --> 01:00:34.370 If we have an oracle, we can think of all of the things 01:00:34.370 --> 01:00:37.220 that you can compute in polynomial time 01:00:37.220 --> 01:00:39.020 relative to that oracle. 01:00:39.020 --> 01:00:41.600 So that's what we-- 01:00:41.600 --> 01:00:43.445 the terminology that people usually use 01:00:43.445 --> 01:00:46.490 is sometimes called relativism, or computation 01:00:46.490 --> 01:00:49.020 relative to having this extra information. 01:00:49.020 --> 01:00:52.850 So P with an A oracle is all of the language 01:00:52.850 --> 01:00:54.740 that you can decide in polynomial time 01:00:54.740 --> 01:00:59.450 if you have an oracle for A. Let's see. 01:01:05.190 --> 01:01:05.785 Yeah. 01:01:05.785 --> 01:01:07.410 Somebody's asking me, is it really free 01:01:07.410 --> 01:01:10.326 or does it cost one unit? 01:01:10.326 --> 01:01:12.947 Even just setting up the oracle and writing down the question 01:01:12.947 --> 01:01:15.280 to the oracle is going to take you some number of steps. 01:01:15.280 --> 01:01:17.447 So you're not going to be able do an infinite number 01:01:17.447 --> 01:01:19.690 of oracle calls in zero time. 01:01:19.690 --> 01:01:22.198 So charging one step or zero steps, 01:01:22.198 --> 01:01:23.490 not going to make a difference. 01:01:23.490 --> 01:01:25.532 Because you still have to formulate the question. 01:01:32.230 --> 01:01:34.608 As I pointed out, P with a SAT oracle-- 01:01:34.608 --> 01:01:36.400 so all the things you do in polynomial time 01:01:36.400 --> 01:01:39.530 with a SAT oracle includes NP. 01:01:39.530 --> 01:01:42.770 Does it perhaps include other stuff? 01:01:42.770 --> 01:01:46.192 Or does it equal NP? 01:01:46.192 --> 01:01:47.900 Would have been a good check-in question, 01:01:47.900 --> 01:01:50.330 but I'm not going to ask that. 01:01:50.330 --> 01:01:54.320 In fact, it seems like it contains other things too. 01:01:54.320 --> 01:02:00.590 Because co-NP is also contained within P, given a SAT oracle. 01:02:00.590 --> 01:02:04.310 Because the SAT oracle answer is both yes or no, 01:02:04.310 --> 01:02:07.050 depending upon the answer. 01:02:07.050 --> 01:02:09.890 So if the formula is unsatisfiable, 01:02:09.890 --> 01:02:12.980 the oracle is going to say no, it's not in the language. 01:02:12.980 --> 01:02:16.640 And now you can do the complement of the SAT problem 01:02:16.640 --> 01:02:17.190 as well. 01:02:17.190 --> 01:02:18.580 The unsatisfiability problem. 01:02:18.580 --> 01:02:21.620 So you can do all of co-NP in the same way. 01:02:21.620 --> 01:02:25.840 You can also define NP relative to some oracle. 01:02:25.840 --> 01:02:28.600 So all the things you can do with a non-deterministic Turing 01:02:28.600 --> 01:02:30.910 machine, where all of the branches 01:02:30.910 --> 01:02:33.400 have separately access. 01:02:33.400 --> 01:02:35.710 And they can ask multiple questions, by the way, 01:02:35.710 --> 01:02:38.840 of the oracle. 01:02:38.840 --> 01:02:39.770 Independently. 01:02:43.450 --> 01:02:46.690 Let's do another, a little bit of a more challenging example. 01:02:46.690 --> 01:02:49.363 The MIN-FORMULA language, which I hope you 01:02:49.363 --> 01:02:50.530 remember from your homework. 01:02:53.510 --> 01:02:57.710 So those are all of the formulas that do not have 01:02:57.710 --> 01:03:01.850 a shorter equivalent formula. 01:03:01.850 --> 01:03:03.360 They are minimal. 01:03:03.360 --> 01:03:06.030 You cannot make a smaller formula that's equivalent that 01:03:06.030 --> 01:03:09.950 gives you the same Boolean function. 01:03:09.950 --> 01:03:14.170 So you showed, for example, that that language is in P space, 01:03:14.170 --> 01:03:15.430 as I recall. 01:03:15.430 --> 01:03:17.440 And there was some other-- you had another two 01:03:17.440 --> 01:03:18.648 problems about that language. 01:03:23.200 --> 01:03:26.580 The complement of the MIN-FORMULA problem 01:03:26.580 --> 01:03:28.800 is in NP with a SAT oracle. 01:03:31.640 --> 01:03:34.310 So mull that over for a second and then we'll see why. 01:03:42.040 --> 01:03:49.900 Here's an algorithm, in NP with a SAT oracle algorithm. 01:03:49.900 --> 01:03:57.470 So in other words, now I want to kind of implement 01:03:57.470 --> 01:03:59.810 that strategy, which I argued in the homework problem 01:03:59.810 --> 01:04:00.710 was not legal. 01:04:00.710 --> 01:04:02.330 But now that I have the SAT oracle, 01:04:02.330 --> 01:04:05.600 it's going to make it possible where before it 01:04:05.600 --> 01:04:06.980 was not possible. 01:04:10.230 --> 01:04:12.950 So let's just understand what I mean by that. 01:04:15.730 --> 01:04:20.920 If I'm trying to do the non minimal formulas, 01:04:20.920 --> 01:04:29.050 namely the formulas that do have a shorter equivalent formula. 01:04:29.050 --> 01:04:31.630 I'm going to guess that shorter formula, called psi. 01:04:35.050 --> 01:04:38.300 The challenge before was testing whether that shorter formula 01:04:38.300 --> 01:04:41.400 was actually equivalent to phi. 01:04:41.400 --> 01:04:46.040 Because that's not obviously doable in polynomial time. 01:04:46.040 --> 01:04:51.620 But the equivalence problem for formulas is a co-NP problem. 01:04:51.620 --> 01:04:53.630 Or if you like to think about it the other way, 01:04:53.630 --> 01:04:57.440 any formula in equivalence is an NP problem, 01:04:57.440 --> 01:04:59.360 because you just have to-- the witness 01:04:59.360 --> 01:05:01.460 is the assignment on which they disagree. 01:05:05.370 --> 01:05:09.940 So two formulas are equivalent if they never disagree. 01:05:09.940 --> 01:05:11.295 And so that's a co-NP problem. 01:05:15.560 --> 01:05:18.320 A SAT oracle can solve a co-NP problem. 01:05:18.320 --> 01:05:22.310 Namely, the equivalence of the two formulas, the input formula 01:05:22.310 --> 01:05:25.820 and the one that you now deterministically guessed. 01:05:25.820 --> 01:05:28.240 And if it turns out that they are equivalent, 01:05:28.240 --> 01:05:30.820 a smaller formula is equivalent to the input formula, 01:05:30.820 --> 01:05:34.568 you know the input formula is not minimal. 01:05:34.568 --> 01:05:35.485 And so you can accept. 01:05:39.210 --> 01:05:40.710 And if it gets the wrong formula, 01:05:40.710 --> 01:05:43.350 it turns out not to be equivalent, 01:05:43.350 --> 01:05:45.630 then you reject on that branch of the non-determinism, 01:05:45.630 --> 01:05:47.200 just like we did before. 01:05:47.200 --> 01:05:51.810 And if the formula really was minimal, none of the branches 01:05:51.810 --> 01:05:54.340 is going to find a shorter equivalent formula. 01:05:54.340 --> 01:05:58.920 So that's why this problem here is in NP with a SAT oracle. 01:06:04.510 --> 01:06:08.920 So now we're going to try to investigate this on my-- 01:06:08.920 --> 01:06:13.040 we're getting near the end of the lecture. 01:06:13.040 --> 01:06:20.270 We're going to look at problems like, well, 01:06:20.270 --> 01:06:22.700 suppose I compare P with a SAT oracle 01:06:22.700 --> 01:06:25.980 and NP with a SAT oracle. 01:06:25.980 --> 01:06:28.810 Could those be the same? 01:06:28.810 --> 01:06:31.440 Well, there's reasons to believe those are not the same. 01:06:34.050 --> 01:06:41.330 But could there be any A where P with A oracle 01:06:41.330 --> 01:06:43.610 is the same as NP with an A oracle? 01:06:43.610 --> 01:06:48.050 It seems like no, but actually that's wrong. 01:06:48.050 --> 01:06:51.800 There is a language, there are languages for which 01:06:51.800 --> 01:06:55.460 NP with that oracle and P with that oracle 01:06:55.460 --> 01:06:57.260 are exactly the same. 01:06:57.260 --> 01:06:59.660 And that actually is an interest-- it's not just 01:06:59.660 --> 01:07:03.860 a curiosity, it actually has relevance to strategies 01:07:03.860 --> 01:07:05.450 for solving P versus NP. 01:07:08.330 --> 01:07:11.720 Hopefully I'll be able to have time to get to. 01:07:11.720 --> 01:07:15.130 Can we think of an oracle like a hash table? 01:07:15.130 --> 01:07:17.335 I think hashing is somehow different in spirit. 01:07:20.920 --> 01:07:23.080 I understand there's some similarity there, 01:07:23.080 --> 01:07:24.190 but I don't see the-- 01:07:24.190 --> 01:07:33.550 hashing is a way of finding sort of a short name for objects, 01:07:33.550 --> 01:07:36.460 which has a variety of different purposes 01:07:36.460 --> 01:07:38.340 why you might want to do that. 01:07:38.340 --> 01:07:40.520 So I don't really think it's the same. 01:07:40.520 --> 01:07:43.360 Let's see, an oracle question, OK, let's see. 01:07:43.360 --> 01:07:46.030 How do we use SAT oracle to solve whether two formulas are 01:07:46.030 --> 01:07:47.128 equivalent? 01:07:51.195 --> 01:07:52.820 OK, this is getting back to this point. 01:07:52.820 --> 01:07:55.790 How can we use a SAT oracle to solve whether two formulas are 01:07:55.790 --> 01:07:58.690 equivalent? 01:07:58.690 --> 01:08:03.010 Well, we can use a SAT oracle to solve any NP problem, 01:08:03.010 --> 01:08:04.480 because it's reducible to SAT. 01:08:07.080 --> 01:08:08.470 In other words-- 01:08:08.470 --> 01:08:12.407 P with a SAT oracle contains all of NP, 01:08:12.407 --> 01:08:14.490 so you have to make sure you understand that part. 01:08:21.010 --> 01:08:23.569 If you have the clique problem, you can reduce. 01:08:23.569 --> 01:08:27.010 If I give you a clique problem, which is an NP problem, 01:08:27.010 --> 01:08:31.510 and I want to use the oracle to test if the formula-- 01:08:31.510 --> 01:08:34.960 if the graph has a clique, I reduce that problem 01:08:34.960 --> 01:08:37.029 to a SAT problem using the Cook-Levin theorem. 01:08:41.240 --> 01:08:43.520 And knowing that a clique of a certain size 01:08:43.520 --> 01:08:46.500 is going to correspond to having a formula which is satisfiable, 01:08:46.500 --> 01:08:49.770 now I can ask the oracle. 01:08:49.770 --> 01:08:52.319 And if I can do NP problems, I can do co-NP problems, 01:08:52.319 --> 01:08:54.527 because P is a deterministic class. 01:08:54.527 --> 01:08:56.819 Even though it has an oracle, it's still deterministic. 01:08:56.819 --> 01:08:58.830 It can invert the answer. 01:08:58.830 --> 01:09:03.520 Something that non-deterministic machines cannot necessarily do. 01:09:03.520 --> 01:09:05.229 So I don't know, maybe that's-- 01:09:05.229 --> 01:09:08.109 let's move on. 01:09:08.109 --> 01:09:11.939 So there's an oracle where P to the A 01:09:11.939 --> 01:09:15.827 equals NP to the A, which kind of seems kind of amazing 01:09:15.827 --> 01:09:16.410 at some level. 01:09:16.410 --> 01:09:19.740 Because here's an oracle where the non-determinism-- if I 01:09:19.740 --> 01:09:23.520 give you that oracle, non-determinism doesn't help. 01:09:26.399 --> 01:09:28.939 And it's actually a language we've seen. 01:09:28.939 --> 01:09:29.594 It's TQBF. 01:09:32.470 --> 01:09:34.021 Why is that? 01:09:34.021 --> 01:09:35.229 Well, here's the whole proof. 01:09:38.220 --> 01:09:41.294 If I have NP with a TQBF oracle. 01:09:44.840 --> 01:09:46.429 Let's just check each of these steps. 01:09:49.380 --> 01:09:54.860 I claim I can do that with a non-deterministic polynomial 01:09:54.860 --> 01:10:02.760 space machine, which no longer has an oracle. 01:10:02.760 --> 01:10:06.210 The reason is that, if I have polynomial space, 01:10:06.210 --> 01:10:10.170 I can answer questions about TQBF without needing an oracle. 01:10:10.170 --> 01:10:12.510 I have enough space just to answer the question directly 01:10:12.510 --> 01:10:14.370 myself. 01:10:14.370 --> 01:10:15.990 And I use my non-determinism here 01:10:15.990 --> 01:10:20.610 to simulate the non-determinism of the NP machine. 01:10:20.610 --> 01:10:24.450 So every time the NP machine branches non-deterministically, 01:10:24.450 --> 01:10:26.850 so do I. Every time one of those branches 01:10:26.850 --> 01:10:30.060 asks the oracle a TQBF question, I just 01:10:30.060 --> 01:10:34.810 do my polynomial space algorithm to solve that question myself. 01:10:34.810 --> 01:10:38.700 But now NPSPACE equals PSPACE by Savitch's theorem. 01:10:38.700 --> 01:10:43.170 And because TQBF is PSPACE complete, 01:10:43.170 --> 01:10:47.070 for the very same reason that a SAT oracle allows 01:10:47.070 --> 01:10:50.190 me to do every NP problem, a TQBF problem 01:10:50.190 --> 01:10:51.780 allows me to do every PSPACE problem. 01:10:54.400 --> 01:10:59.310 And so I get NP contained within P for a TQBF oracle. 01:10:59.310 --> 01:11:02.190 And of course, you get the containment the other way 01:11:02.190 --> 01:11:03.340 immediately. 01:11:03.340 --> 01:11:06.290 So they're equal. 01:11:06.290 --> 01:11:08.035 What does that have to do with-- 01:11:11.470 --> 01:11:14.707 somebody said-- well, I'll just-- 01:11:14.707 --> 01:11:16.040 I don't want to run out of time. 01:11:16.040 --> 01:11:19.810 So I'll take any questions at the end. 01:11:19.810 --> 01:11:24.050 What does this got to do with the P versus NP problem? 01:11:24.050 --> 01:11:27.140 OK, so this is a very interesting connection. 01:11:35.380 --> 01:11:39.600 Remember, we just showed through a combination 01:11:39.600 --> 01:11:41.850 of today's lecture and yesterday's lecture, 01:11:41.850 --> 01:11:49.470 and I guess Thursday's lecture, that this problem, 01:11:49.470 --> 01:11:52.530 this equivalence problem, is not in PSPACE, 01:11:52.530 --> 01:11:56.140 and therefore it's not in P, and therefore it's intractable. 01:11:56.140 --> 01:11:57.140 That's what we just did. 01:12:01.700 --> 01:12:04.280 We showed it's complete for a class, which 01:12:04.280 --> 01:12:13.345 is provably outside of P, provably bigger than P. 01:12:13.345 --> 01:12:14.720 That's the kind of thing we would 01:12:14.720 --> 01:12:17.660 like to be able to do to separate P and NP. 01:12:17.660 --> 01:12:20.510 We would like to show that some other problem is not 01:12:20.510 --> 01:12:23.480 in P. Some other problem is intractable. 01:12:23.480 --> 01:12:24.380 Namely, SAT. 01:12:24.380 --> 01:12:27.560 If we could do SAT, then we're good. 01:12:27.560 --> 01:12:29.990 We've solved P and NP. 01:12:29.990 --> 01:12:33.260 So we already have an example of being able to do that. 01:12:33.260 --> 01:12:37.020 Could we use the same method? 01:12:37.020 --> 01:12:39.780 Which is something people did try to do many years ago, 01:12:39.780 --> 01:12:45.900 to show that SAT is not in P. So what is that method really? 01:12:45.900 --> 01:12:49.647 The guts of that method really comes from the hierarchy there. 01:12:49.647 --> 01:12:52.230 That's where you were actually proving problems that are hard. 01:12:52.230 --> 01:12:56.460 You're getting this problem with through the hierarchy 01:12:56.460 --> 01:13:02.830 construction that's provably outside of PSPACE. 01:13:02.830 --> 01:13:05.490 And outside of P. 01:13:05.490 --> 01:13:07.340 That's a diagonalization. 01:13:07.340 --> 01:13:10.840 And if you look carefully at what's going on there-- 01:13:10.840 --> 01:13:12.450 so we're going to say, no, we're not 01:13:12.450 --> 01:13:15.240 going to be able to solve SAT, show SAT's outside of P 01:13:15.240 --> 01:13:16.180 in the same way. 01:13:16.180 --> 01:13:19.290 And the reason is, suppose we could. 01:13:19.290 --> 01:13:22.450 Well the hierarchy theorems are proved by diagonalization. 01:13:25.550 --> 01:13:30.260 What I mean by that is that in the hierarchy theorem, 01:13:30.260 --> 01:13:34.430 there's a machine D, which is simulating some other machine, 01:13:34.430 --> 01:13:39.210 M. To remember what's going on there, 01:13:39.210 --> 01:13:41.700 remember that we made a machine which 01:13:41.700 --> 01:13:44.580 is going to make its language different from the language 01:13:44.580 --> 01:13:47.700 of every machine that's running with less space 01:13:47.700 --> 01:13:48.600 or with less time. 01:13:52.160 --> 01:13:54.680 That's how D was defined. 01:13:54.680 --> 01:14:00.360 It wants to make sure its language cannot be done in less 01:14:00.360 --> 01:14:01.090 space. 01:14:01.090 --> 01:14:04.180 So it makes sure that its language is different. 01:14:04.180 --> 01:14:07.860 It simulates the machines that use less space, 01:14:07.860 --> 01:14:12.080 and does something different from what they do. 01:14:12.080 --> 01:14:18.240 Well, that simulation-- if we had an oracle, 01:14:18.240 --> 01:14:22.650 if we're trying to show that if we provide 01:14:22.650 --> 01:14:25.860 both a simulator and the machine being simulated 01:14:25.860 --> 01:14:29.340 with the same oracle, the simulation still works. 01:14:29.340 --> 01:14:33.600 Every time the machine you're simulating asks a question, 01:14:33.600 --> 01:14:35.220 the simulator has the same oracle 01:14:35.220 --> 01:14:37.320 so it can also ask the same question, 01:14:37.320 --> 01:14:39.490 and can still do the simulation. 01:14:39.490 --> 01:14:43.830 So in other words, if you could prove P different from NP 01:14:43.830 --> 01:14:46.290 using basically a simulation, which 01:14:46.290 --> 01:14:48.990 is what a diagonalization is, then you 01:14:48.990 --> 01:14:51.570 would be able to prove that P is different from NP 01:14:51.570 --> 01:14:52.530 for every oracle. 01:14:55.820 --> 01:14:58.670 So if you can prove P different from NP by a diagonalization, 01:14:58.670 --> 01:15:00.290 that would also immediately prove 01:15:00.290 --> 01:15:03.620 that P is different from NP for every oracle. 01:15:03.620 --> 01:15:10.850 Because the argument is transparent to the oracle. 01:15:10.850 --> 01:15:15.360 If you just put the oracle down, everything still works. 01:15:15.360 --> 01:15:24.120 But-- here is the big but, it can't be that-- 01:15:24.120 --> 01:15:27.990 we know that P A is-- 01:15:27.990 --> 01:15:29.250 we know this is false. 01:15:29.250 --> 01:15:33.220 We just exhibit an oracle for which they're equal. 01:15:33.220 --> 01:15:35.280 It's not the case that P is different from NP 01:15:35.280 --> 01:15:36.120 for every oracle. 01:15:38.810 --> 01:15:41.480 Sometimes they're equal, for some oracles. 01:15:41.480 --> 01:15:44.180 So something that's just basically a very 01:15:44.180 --> 01:15:46.670 straightforward diagonalization, something that's 01:15:46.670 --> 01:15:48.860 at its core is a diagonalization, 01:15:48.860 --> 01:15:51.320 is not going to be enough to solve P and NP. 01:15:51.320 --> 01:15:53.210 Because otherwise it would prove that they're 01:15:53.210 --> 01:15:54.335 different for every oracle. 01:15:54.335 --> 01:15:57.320 And sometimes they're not different, for some oracles. 01:16:00.460 --> 01:16:04.240 That's an important insight for what kind of a method 01:16:04.240 --> 01:16:09.340 will not be adequate to prove P different from NP. 01:16:09.340 --> 01:16:12.490 And this comes up all the time. 01:16:12.490 --> 01:16:16.870 People who propose hypothetical solutions that they're trying 01:16:16.870 --> 01:16:18.220 to show P different from NP. 01:16:18.220 --> 01:16:21.430 One of the very first things people ask is, well, 01:16:21.430 --> 01:16:27.400 would that argument still work if you put an oracle there. 01:16:27.400 --> 01:16:31.750 Often it does, which points out there was a flaw. 01:16:31.750 --> 01:16:35.720 Anyway, last check in. 01:16:35.720 --> 01:16:38.830 So this is just a little test of your knowledge about oracles. 01:16:41.780 --> 01:16:43.655 Why don't we-- in our remaining minute here. 01:16:46.400 --> 01:16:48.230 Let's say 30 seconds. 01:16:48.230 --> 01:16:51.770 And then we'll do a wrap on this, 01:16:51.770 --> 01:16:54.290 and I'll point out which ones are right. 01:16:54.290 --> 01:16:56.780 Oh boy, we're all over the place on this one. 01:17:00.820 --> 01:17:02.170 You're liking them all. 01:17:02.170 --> 01:17:08.100 Well, I guess the ones that are false are lagging slightly. 01:17:15.630 --> 01:17:16.950 OK, let's conclude. 01:17:21.310 --> 01:17:23.800 Did I give you enough time there? 01:17:23.800 --> 01:17:24.430 Share results. 01:17:30.160 --> 01:17:31.020 So, in fact-- 01:17:37.510 --> 01:17:41.427 Yeah, so having an oracle for the complement 01:17:41.427 --> 01:17:42.760 is the same as having an oracle. 01:17:42.760 --> 01:17:46.160 So this is certainly true. 01:17:46.160 --> 01:17:50.220 NP SAT equal coNP SAT, we have no reason 01:17:50.220 --> 01:17:53.808 to believe that would be true, and we 01:17:53.808 --> 01:17:54.850 don't know it to be true. 01:17:54.850 --> 01:18:00.660 So B is not a good choice and that's the laggard here. 01:18:00.660 --> 01:18:05.700 MIN-FORMULA, well, is in PSPACE, and anything in PSPACE 01:18:05.700 --> 01:18:09.660 is reducible to TQBF, so this is certainly true. 01:18:09.660 --> 01:18:14.370 And same thing for NP with TQBF and coNP with TQBF. 01:18:14.370 --> 01:18:22.640 Once you have TQBF, you're going to get all of PSPACE. 01:18:22.640 --> 01:18:27.450 And as we pointed out, this is going to be equal as well. 01:18:27.450 --> 01:18:30.500 So why don't we end here. 01:18:30.500 --> 01:18:34.910 And I think that's my last slide. 01:18:34.910 --> 01:18:36.800 Oh no, there's my summary here. 01:18:36.800 --> 01:18:38.150 This is what we've done. 01:18:38.150 --> 01:18:43.888 And I will send you all off on your way. 01:18:43.888 --> 01:18:45.680 How does the interaction between the Turing 01:18:45.680 --> 01:18:46.847 machine and the oracle look? 01:18:46.847 --> 01:18:49.490 Yeah, I didn't define exactly how the machine interacts 01:18:49.490 --> 01:18:50.510 with an oracle. 01:18:50.510 --> 01:18:52.580 You can imagine having a separate tape 01:18:52.580 --> 01:18:54.110 where it writes the oracle question 01:18:54.110 --> 01:18:56.960 and then goes into a special query state. 01:18:56.960 --> 01:18:59.212 You can formalize it however you like. 01:18:59.212 --> 01:19:01.670 They're all going to be-- any reasonable way of formalizing 01:19:01.670 --> 01:19:06.500 it is going to come up with the same notion in the end. 01:19:06.500 --> 01:19:10.070 It does show that P with a TQBF oracle equals PSPACE. 01:19:10.070 --> 01:19:11.330 Yes, that is correct. 01:19:11.330 --> 01:19:13.970 Good point. 01:19:13.970 --> 01:19:16.220 Why do we need the oracle to be TQBF? 01:19:16.220 --> 01:19:19.170 Wouldn't SAT also work because it could solve any NP problem? 01:19:19.170 --> 01:19:23.010 So you're asking, does P with a SAT oracle 01:19:23.010 --> 01:19:25.170 equal NP with a SAT oracle? 01:19:25.170 --> 01:19:26.190 Not known. 01:19:26.190 --> 01:19:29.670 And believed not to be true, but we don't have 01:19:29.670 --> 01:19:32.460 a compelling reason for that. 01:19:32.460 --> 01:19:34.290 No one has any idea how to do that. 01:19:34.290 --> 01:19:42.630 Because, for example, we showed the complement of MIN-FORMULA 01:19:42.630 --> 01:19:44.235 is in NP with a SAT oracle. 01:19:47.050 --> 01:19:48.820 But no one knows how to do-- 01:19:48.820 --> 01:19:51.070 because there's sort of two levels of non-determinism 01:19:51.070 --> 01:19:51.570 there. 01:19:51.570 --> 01:19:54.160 There's guessing the smaller formula, 01:19:54.160 --> 01:19:59.070 and then guessing again to check the equivalence. 01:19:59.070 --> 01:20:01.320 And they really can't be combined, because one of them 01:20:01.320 --> 01:20:03.750 is sort of an exist type guessing, 01:20:03.750 --> 01:20:05.970 the other one is kind of a for all type guessing. 01:20:11.010 --> 01:20:13.320 No one knows how to do that in polynomial time 01:20:13.320 --> 01:20:14.190 with a SAT oracle. 01:20:18.085 --> 01:20:19.960 Generally believed that they're not the same. 01:20:23.250 --> 01:20:24.990 In the check-in, why was B false? 01:20:24.990 --> 01:20:29.900 B is the same question. 01:20:29.900 --> 01:20:35.750 Does NP with a SAT oracle equal coNP with a SAT oracle? 01:20:35.750 --> 01:20:39.820 I'm not saying it's false, it's just not known to be true. 01:20:39.820 --> 01:20:43.870 It doesn't follow from anything that we've shown so far. 01:20:43.870 --> 01:20:47.500 And I think that would be something that-- 01:20:47.500 --> 01:20:49.510 well, I guess it doesn't immediately 01:20:49.510 --> 01:20:51.565 imply any famous open problem. 01:20:54.790 --> 01:20:57.280 I wouldn't necessarily expect you 01:20:57.280 --> 01:21:00.430 to know that it's an unsolved problem, but it is. 01:21:04.870 --> 01:21:06.850 Could we have oracles for undecidable language? 01:21:06.850 --> 01:21:08.410 Absolutely. 01:21:08.410 --> 01:21:09.370 Would it be helpful? 01:21:09.370 --> 01:21:12.160 Well, if you're trying to solve an undecidable problem, 01:21:12.160 --> 01:21:13.690 it would be helpful. 01:21:13.690 --> 01:21:16.240 But people do study that. 01:21:16.240 --> 01:21:22.690 In fact, the original concept of oracles was presented, 01:21:22.690 --> 01:21:27.670 was derived in the computability theory. 01:21:27.670 --> 01:21:31.413 And a side note, you can talk about reducibility. 01:21:31.413 --> 01:21:32.830 No, I don't want to even go there. 01:21:32.830 --> 01:21:33.910 Too complicated. 01:21:37.130 --> 01:21:41.000 What is not known to be true? 01:21:41.000 --> 01:21:44.210 What is not known to be true is that NP with a SAT oracle 01:21:44.210 --> 01:21:48.980 equals coNP with a SAT oracle, or equals P with a SAT oracle. 01:21:48.980 --> 01:21:53.120 Nothing is known except the obvious relations among those. 01:21:53.120 --> 01:21:57.485 Those are all unknown, and just not known to be true or false. 01:22:05.020 --> 01:22:08.730 Is NP with a SAT oracle equal to NP? 01:22:08.730 --> 01:22:10.140 Probably not. 01:22:10.140 --> 01:22:15.050 NP with a SAT oracle, for one thing, contains coNP. 01:22:15.050 --> 01:22:16.700 Because it's even more powerful. 01:22:16.700 --> 01:22:21.050 We pointed out that P with a SAT oracle contains coNP. 01:22:21.050 --> 01:22:24.870 And so NP with a SAT oracle is going to be at least as good. 01:22:24.870 --> 01:22:27.350 And so it's going to contain coNP. 01:22:27.350 --> 01:22:29.810 And so, probably not going to be equal to 01:22:29.810 --> 01:22:34.380 NP unless shockingly unexpected things 01:22:34.380 --> 01:22:37.460 happen in our complexity world.