WEBVTT 00:00:00.000 --> 00:00:04.910 [SQUEAKING] [RUSTLING] [CLICKING] 00:00:25.270 --> 00:00:29.350 MICHAEL SIPSER: OK, why don't we begin. 00:00:29.350 --> 00:00:30.250 Hi, everybody. 00:00:30.250 --> 00:00:36.040 Let's see how many we got here, most of you. 00:00:36.040 --> 00:00:39.490 I'm sure the others will show up, hopefully, soon enough. 00:00:39.490 --> 00:00:43.540 So welcome back. 00:00:43.540 --> 00:00:45.760 We have today's lecture 4. 00:00:45.760 --> 00:00:51.400 And let's just remember what we've been doing. 00:00:51.400 --> 00:00:54.640 In the last few lectures, we were 00:00:54.640 --> 00:01:01.180 exploring the regular languages, as described by finite automata 00:01:01.180 --> 00:01:03.190 and regular expressions. 00:01:03.190 --> 00:01:06.310 We showed how to convert them back and forth, those two 00:01:06.310 --> 00:01:08.780 models, to one another. 00:01:08.780 --> 00:01:12.100 And we also showed how to prove certain languages 00:01:12.100 --> 00:01:13.090 are not regular. 00:01:13.090 --> 00:01:18.520 Now remember, finite automata are a very weak model 00:01:18.520 --> 00:01:19.330 of computation. 00:01:19.330 --> 00:01:22.600 They only have a limited memory, finite memory. 00:01:22.600 --> 00:01:26.290 And they still aren't able to do certain things 00:01:26.290 --> 00:01:30.805 with their finite memory, but they are-- 00:01:33.670 --> 00:01:37.120 if you can pair them with a general purpose 00:01:37.120 --> 00:01:39.790 computer, at least the way we think about it, 00:01:39.790 --> 00:01:45.580 is their capabilities are just extremely limited. 00:01:45.580 --> 00:01:49.960 And so we're going, to over the next few lectures, 00:01:49.960 --> 00:01:52.390 explore some more powerful models. 00:01:52.390 --> 00:01:55.932 We started doing that last time, the context-free grammars. 00:01:55.932 --> 00:01:58.390 And as we'll see, there are certain things that you can do. 00:01:58.390 --> 00:01:59.800 Well, I think we saw that last time as well. 00:01:59.800 --> 00:02:02.620 There are some things you can do with context-free grammars that 00:02:02.620 --> 00:02:05.800 you cannot do with finite automata. 00:02:05.800 --> 00:02:09.169 But they still have their limitations, as we'll see. 00:02:09.169 --> 00:02:13.720 So today what we're going to do, we're 00:02:13.720 --> 00:02:15.820 going to continue that discussion 00:02:15.820 --> 00:02:19.480 by looking at the definition of context-free grammars in a more 00:02:19.480 --> 00:02:20.830 formal way. 00:02:20.830 --> 00:02:23.560 One of the things that we do in this course 00:02:23.560 --> 00:02:27.520 is develop a practice with formalism, 00:02:27.520 --> 00:02:30.460 so that's going to be in the spirit of that. 00:02:30.460 --> 00:02:33.700 We also are going to look at their associated languages, 00:02:33.700 --> 00:02:35.410 called the context-free languages. 00:02:35.410 --> 00:02:37.210 So they are going to be the counterpart 00:02:37.210 --> 00:02:40.270 for context-free grammars of what the regular languages are 00:02:40.270 --> 00:02:43.780 for the finite automata regular expressions. 00:02:43.780 --> 00:02:48.700 And then we're going to look at an automaton-based model, which 00:02:48.700 --> 00:02:53.620 is the counterpart to the grammars called the pushdown 00:02:53.620 --> 00:02:54.340 automata. 00:02:54.340 --> 00:02:57.830 And we'll see that those are equivalent in power. 00:02:57.830 --> 00:03:01.030 And finally-- well, and as part of that, 00:03:01.030 --> 00:03:03.700 we will show how to convert the context-free grammars 00:03:03.700 --> 00:03:05.500 to the pushdown automata. 00:03:05.500 --> 00:03:07.880 And that's what we're going to do today. 00:03:07.880 --> 00:03:10.990 So we're going to move on then and return 00:03:10.990 --> 00:03:14.290 to our topic of context-free grammars 00:03:14.290 --> 00:03:15.760 that we began last time. 00:03:15.760 --> 00:03:18.520 And just to refresh your memory, so here 00:03:18.520 --> 00:03:20.650 was that example of a context-free grammar 00:03:20.650 --> 00:03:22.810 that we gave last time. 00:03:22.810 --> 00:03:27.790 And it has-- the way we're going to be writing 00:03:27.790 --> 00:03:31.570 context-free grammars is using a bit of a shorthand, which 00:03:31.570 --> 00:03:32.620 looks like this. 00:03:32.620 --> 00:03:35.800 When you have multiple rules that 00:03:35.800 --> 00:03:39.710 have the same variable on the left-hand side, 00:03:39.710 --> 00:03:42.760 you can combine them into one line. 00:03:42.760 --> 00:03:47.410 So these two rules over here, S goes to 0S1 and S goes to R, 00:03:47.410 --> 00:03:51.370 can be written in one line as a little bit more compact way, 00:03:51.370 --> 00:03:55.990 this is standard, as S goes to 0S1 or R. 00:03:55.990 --> 00:03:57.500 That's the way you would read this. 00:03:57.500 --> 00:04:02.600 This is really two rules but written on one line. 00:04:02.600 --> 00:04:07.400 So as you recall from last time, a context-free grammar 00:04:07.400 --> 00:04:10.160 has terminals, variables, and rules. 00:04:10.160 --> 00:04:13.100 Those are the parts that we speak of, 00:04:13.100 --> 00:04:15.270 as well as one of the variables being 00:04:15.270 --> 00:04:17.954 designated as a starting variable, which 00:04:17.954 --> 00:04:19.079 gets the whole thing going. 00:04:19.079 --> 00:04:23.420 So I'll remind you about how the computation goes. 00:04:23.420 --> 00:04:25.970 So the variables are the symbols that 00:04:25.970 --> 00:04:28.960 appear in the left-hand side of the rules. 00:04:28.960 --> 00:04:31.480 The terminals are the other symbols 00:04:31.480 --> 00:04:33.910 that appear in the grammar. 00:04:33.910 --> 00:04:43.090 And the-- we take the grammar, and we 00:04:43.090 --> 00:04:47.410 use it to generate strings, according to a certain system. 00:04:47.410 --> 00:04:52.300 And the system is that you start up by writing down the starting 00:04:52.300 --> 00:04:53.020 variable. 00:04:53.020 --> 00:04:57.007 And then once you've written down that variable, 00:04:57.007 --> 00:04:58.840 or whatever variables you have written down, 00:04:58.840 --> 00:05:00.640 you're allowed to substitute them 00:05:00.640 --> 00:05:03.700 according to the rules, the substitution 00:05:03.700 --> 00:05:05.710 rules, that are in the grammar. 00:05:05.710 --> 00:05:08.170 So you can keep on replacing the variables 00:05:08.170 --> 00:05:11.120 that you have with the corresponding right-hand sides. 00:05:11.120 --> 00:05:14.470 And then you do that over and over again 00:05:14.470 --> 00:05:16.420 until you don't have any variables left, 00:05:16.420 --> 00:05:18.340 only terminal symbols remain. 00:05:18.340 --> 00:05:22.778 And at that point, you have generated a string that's 00:05:22.778 --> 00:05:24.070 in the language of the grammar. 00:05:26.610 --> 00:05:31.410 So the grammar's language is going 00:05:31.410 --> 00:05:37.320 to be a language over strings whose alphabet are the terminal 00:05:37.320 --> 00:05:38.380 symbols. 00:05:38.380 --> 00:05:40.620 So the terminal symbols in a certain sense 00:05:40.620 --> 00:05:44.490 play the same role as the input alphabet, 00:05:44.490 --> 00:05:47.700 say, for the finite automata. 00:05:47.700 --> 00:05:52.410 The variables are internal-working symbols 00:05:52.410 --> 00:05:53.190 for the grammar. 00:05:53.190 --> 00:05:59.940 The terminals are the symbols of which the language is written. 00:05:59.940 --> 00:06:01.610 We'll make that more precise in a minute 00:06:01.610 --> 00:06:03.780 when I give the formal definition. 00:06:03.780 --> 00:06:05.660 So the result is the generated string. 00:06:05.660 --> 00:06:07.820 And the language of the grammar is the language 00:06:07.820 --> 00:06:12.470 of all generated strings that you can get using that grammar. 00:06:12.470 --> 00:06:15.740 And the important thing is that we call that language 00:06:15.740 --> 00:06:19.620 a context-free language. 00:06:19.620 --> 00:06:22.470 So that's like what we get from-- 00:06:22.470 --> 00:06:25.440 that's the analogous thing to the regular languages, 00:06:25.440 --> 00:06:27.930 but here we call them context-free languages, 00:06:27.930 --> 00:06:30.270 the things that you can get from a context-free grammar. 00:06:30.270 --> 00:06:35.020 Again, just a quick recap of that example we did last time. 00:06:35.020 --> 00:06:39.580 So you start out by writing the start variable. 00:06:39.580 --> 00:06:43.110 And then I'm going to give you two views of that, 00:06:43.110 --> 00:06:45.630 either in terms of the tree of substitutions, which we call 00:06:45.630 --> 00:06:48.510 the parse tree, or in terms of the resulting string 00:06:48.510 --> 00:06:50.190 as you do the substitutions. 00:06:50.190 --> 00:06:54.420 So here is the parse tree, here is the resulting strings, 00:06:54.420 --> 00:06:57.610 here are the substitutions that you make. 00:06:57.610 --> 00:07:03.220 And now we have R coming from S, and we have 00R11. 00:07:03.220 --> 00:07:08.920 And now we have R in turn becomes an empty string. 00:07:08.920 --> 00:07:13.020 And then the string that we generated is 0011. 00:07:13.020 --> 00:07:16.490 That's in the language of the grammar. 00:07:16.490 --> 00:07:19.980 And now if you play with this a little bit, 00:07:19.980 --> 00:07:21.800 you'll see that the language of the grammar 00:07:21.800 --> 00:07:26.090 is all strings that look like runs of zeros followed 00:07:26.090 --> 00:07:26.885 by runs of ones. 00:07:29.410 --> 00:07:32.660 So is that clear? 00:07:32.660 --> 00:07:34.610 I think we're going to have a-- 00:07:34.610 --> 00:07:39.310 I think the next slide is going to have a check-in, 00:07:39.310 --> 00:07:43.360 and so hopefully that'll get us all together 00:07:43.360 --> 00:07:44.830 on the same page with this. 00:07:44.830 --> 00:07:46.300 Anyway. 00:07:46.300 --> 00:07:48.910 So here's our formal definition anyway. 00:07:48.910 --> 00:07:53.050 We have a context-free grammar is a 4-tuple. 00:07:53.050 --> 00:07:55.480 There are four parts to a context-free grammar. 00:07:55.480 --> 00:07:58.900 These are the parts we've already been discussing, 00:07:58.900 --> 00:08:03.070 the variables, the terminal symbols, the rules. 00:08:03.070 --> 00:08:04.570 The rules are always that they form 00:08:04.570 --> 00:08:09.040 a variable followed with an arrow 00:08:09.040 --> 00:08:12.070 to a string of variables and terminals. 00:08:12.070 --> 00:08:13.710 That's the way we just write that down. 00:08:13.710 --> 00:08:16.180 So this is the form of the rule, and then we 00:08:16.180 --> 00:08:17.620 have the special start variable. 00:08:17.620 --> 00:08:21.190 And we all wrap that up into a package, this 4-tuple. 00:08:21.190 --> 00:08:23.260 That's what the context-free grammar is. 00:08:26.430 --> 00:08:29.625 Now we have here-- 00:08:29.625 --> 00:08:32.470 and now, maybe a little bit overkill, 00:08:32.470 --> 00:08:37.620 but let's talk about, formally speaking, the way the grammar 00:08:37.620 --> 00:08:42.840 actually processes and produces strings. 00:08:42.840 --> 00:08:45.480 So we're going to write-- 00:08:45.480 --> 00:08:47.700 the standard notation for this is 00:08:47.700 --> 00:08:52.910 that if you have two strings of variables and terminals-- 00:08:52.910 --> 00:08:55.490 so imagine you have an intermediate string that 00:08:55.490 --> 00:09:00.350 you've generated in the grammar so far, 00:09:00.350 --> 00:09:04.230 which might be like 00S11 from the previous line, 00:09:04.230 --> 00:09:07.100 so that's an intermediate string that is so far what 00:09:07.100 --> 00:09:11.330 you've generated, you're going to say maybe that's u, 00:09:11.330 --> 00:09:15.270 and v might be the next line down. 00:09:15.270 --> 00:09:18.770 So that means we're going to write u arrow v-- 00:09:18.770 --> 00:09:21.650 and that arrow is-- 00:09:21.650 --> 00:09:23.810 the word we're going to use is "yields." 00:09:23.810 --> 00:09:28.910 We'll say u yields v if it can go from u to v just 00:09:28.910 --> 00:09:30.320 with one substitution step. 00:09:33.260 --> 00:09:37.310 And then we'll write u yields v in some number of steps-- 00:09:37.310 --> 00:09:40.910 or actually, we say u derives v if it 00:09:40.910 --> 00:09:43.610 can go to u to v with some number of substitutions instead 00:09:43.610 --> 00:09:44.470 of just one. 00:09:44.470 --> 00:09:49.010 And that's used with the yields arrow with the star above it. 00:09:49.010 --> 00:09:52.130 It means "some number of." 00:09:52.130 --> 00:09:54.590 Another way of writing that is you can say, 00:09:54.590 --> 00:09:58.760 u goes to v if there are a bunch of one-step moves 00:09:58.760 --> 00:10:01.460 that you can make which take you from u to v. 00:10:01.460 --> 00:10:05.830 And that whole sequence is called a derivation of v 00:10:05.830 --> 00:10:08.590 from u. 00:10:08.590 --> 00:10:10.090 That's a sequence of steps that you 00:10:10.090 --> 00:10:14.090 go through doing these substitutions one by one 00:10:14.090 --> 00:10:16.190 to take you from u to v, according 00:10:16.190 --> 00:10:18.080 to the rules of the grammar. 00:10:18.080 --> 00:10:22.100 And lastly, if u is a starting variable, 00:10:22.100 --> 00:10:26.450 then we call that sequence just the derivation of v. It 00:10:26.450 --> 00:10:28.610 could be the derivation from the start variable, 00:10:28.610 --> 00:10:31.640 but that's the assumed if you don't say it's 00:10:31.640 --> 00:10:33.020 a derivation from anything. 00:10:33.020 --> 00:10:35.180 The derivation of v in the grammar 00:10:35.180 --> 00:10:37.440 is the derivation of v from the start variable. 00:10:37.440 --> 00:10:39.290 It's just the sequence of substitutions 00:10:39.290 --> 00:10:45.590 that you make, kind of what I think you would expect. 00:10:45.590 --> 00:10:47.900 Now, the language of the grammar is the set 00:10:47.900 --> 00:10:53.930 of all strings of terminal symbols 00:10:53.930 --> 00:10:58.015 that you can get from starting at the starting variable 00:10:58.015 --> 00:10:58.640 of the grammar. 00:11:01.410 --> 00:11:03.160 And that's called a context-free language, 00:11:03.160 --> 00:11:04.320 as I mentioned before. 00:11:04.320 --> 00:11:06.090 So it's a context-free language. 00:11:06.090 --> 00:11:10.740 It's the language of the grammar for some grammar. 00:11:10.740 --> 00:11:13.560 So let's have a little check in here, again, nothing too hard, 00:11:13.560 --> 00:11:15.060 nothing to get worried about anyway. 00:11:15.060 --> 00:11:17.230 We're not counting correct here. 00:11:17.230 --> 00:11:23.910 So let's just see, I'm going to give you two things that 00:11:23.910 --> 00:11:26.060 look like grammars. 00:11:26.060 --> 00:11:28.550 Which of them are actually grammars? 00:11:28.550 --> 00:11:32.820 And let me just pull that poll up here. 00:11:32.820 --> 00:11:36.720 So which of these are valid grammars here? 00:11:36.720 --> 00:11:37.540 Are they both? 00:11:37.540 --> 00:11:38.040 Neither? 00:11:40.910 --> 00:11:43.712 I mean, you could kind of make an argument either way 00:11:43.712 --> 00:11:44.420 for both of them. 00:11:44.420 --> 00:11:47.160 But both of them are a little-- have their own-- 00:11:47.160 --> 00:11:49.280 a bit of weirdness to them in a way 00:11:49.280 --> 00:11:52.010 if you study them for a second. 00:11:56.400 --> 00:12:00.020 That's pretty much converged. 00:12:00.020 --> 00:12:00.830 Share the results. 00:12:07.800 --> 00:12:11.550 So, in fact, the correct answer is b. 00:12:14.060 --> 00:12:18.920 And why is only C2-- 00:12:18.920 --> 00:12:20.255 well, first of all-- 00:12:20.255 --> 00:12:23.390 well, what's wrong with C1? 00:12:23.390 --> 00:12:25.760 The problem with C1 is that the rules 00:12:25.760 --> 00:12:31.470 have things besides a single variable on the left-hand side. 00:12:31.470 --> 00:12:33.650 So having a B1 on the left-hand side 00:12:33.650 --> 00:12:37.160 is not legal in a context-free grammar. 00:12:37.160 --> 00:12:40.400 In fact, there are other kinds of grammars. 00:12:40.400 --> 00:12:43.040 There's a kind of grammar called the context-sensitive grammar. 00:12:43.040 --> 00:12:46.970 The term "context-free" means you can replace the variable 00:12:46.970 --> 00:12:50.720 independent of its context in the intermediate string, 00:12:50.720 --> 00:12:52.550 so independent of what's around it. 00:12:52.550 --> 00:12:56.150 But here, this substitution is going to-- you can replace B, 00:12:56.150 --> 00:13:01.078 but it depends on there being a 1 next to it. 00:13:01.078 --> 00:13:02.870 This is called a context-sensitive grammar, 00:13:02.870 --> 00:13:05.328 but it's not the kind of grammar we're going to be using, 00:13:05.328 --> 00:13:06.870 which are only context-free grammars. 00:13:06.870 --> 00:13:08.570 So C1 is out. 00:13:08.570 --> 00:13:12.800 That's not a legit context-free grammar. 00:13:12.800 --> 00:13:17.090 C2, the thing that's a little weird about C2 00:13:17.090 --> 00:13:19.520 is if you try to generate a string in C2, 00:13:19.520 --> 00:13:24.800 you'll see that there's no way to get rid of the variables, 00:13:24.800 --> 00:13:27.600 that you're always going to be stuck with a variable. 00:13:27.600 --> 00:13:30.350 Now, that doesn't violate the definition 00:13:30.350 --> 00:13:31.880 of a context-free grammar. 00:13:31.880 --> 00:13:34.552 So this is a context-free grammar, 00:13:34.552 --> 00:13:36.260 but it's not going to be able to generate 00:13:36.260 --> 00:13:39.260 any strings of only terminals. 00:13:39.260 --> 00:13:41.710 So this is a context-free grammar 00:13:41.710 --> 00:13:44.320 whose language happens to be the empty language, 00:13:44.320 --> 00:13:47.200 but that's totally OK. 00:13:47.200 --> 00:13:51.820 So the correct answer here is B, that only C2 here is 00:13:51.820 --> 00:13:53.860 a valid context-free grammar. 00:13:58.290 --> 00:13:59.310 Common-- let's just see. 00:13:59.310 --> 00:14:03.525 Common question, does a string u derive itself? 00:14:03.525 --> 00:14:06.660 Yes, a string u derives itself. 00:14:06.660 --> 00:14:07.942 That's a little bit of a-- 00:14:07.942 --> 00:14:09.900 little bit of an esoteric question there for us 00:14:09.900 --> 00:14:12.270 right now, but yes. 00:14:12.270 --> 00:14:14.580 A string u-- in this definition here, 00:14:14.580 --> 00:14:18.420 u arrow star u is legit-- is legal. 00:14:18.420 --> 00:14:22.820 Maybe it isn't according to the way I've written it down here, 00:14:22.820 --> 00:14:24.982 but it is a legal thing. 00:14:24.982 --> 00:14:26.940 It's not going to really matter for you anyway, 00:14:26.940 --> 00:14:28.250 but it is legal. 00:14:28.250 --> 00:14:31.190 OK, let's continue. 00:14:31.190 --> 00:14:34.040 Let's do another somewhat interesting example 00:14:34.040 --> 00:14:36.110 of a context-free grammar. 00:14:36.110 --> 00:14:41.180 This is a grammar that can generate 00:14:41.180 --> 00:14:46.360 arithmetical expressions involving pluses and times. 00:14:46.360 --> 00:14:47.460 So here it is. 00:14:47.460 --> 00:14:49.740 It has how many rules? 00:14:49.740 --> 00:14:51.120 Well, there are six rules here. 00:14:51.120 --> 00:14:52.890 Each line represents two rules. 00:14:52.890 --> 00:14:57.090 So E goes to E plus T or T; T goes to T times F or F; 00:14:57.090 --> 00:15:02.990 and F goes to parentheses E parentheses or a. 00:15:02.990 --> 00:15:06.210 And so the variables are going to be the symbols that 00:15:06.210 --> 00:15:08.400 appear on the left-hand side, E, T, and F. 00:15:08.400 --> 00:15:10.110 The terminal symbols, which are going 00:15:10.110 --> 00:15:12.360 to be the symbols of the language 00:15:12.360 --> 00:15:15.060 that you're going to be generating, is going to be 00:15:15.060 --> 00:15:16.830 the plus, the time symbols. 00:15:16.830 --> 00:15:20.200 The parentheses are just terminal symbols here. 00:15:20.200 --> 00:15:23.010 So they're not playing any special role besides that. 00:15:23.010 --> 00:15:25.650 And then you have the a, which is representing 00:15:25.650 --> 00:15:30.840 the upper end on which those operators would 00:15:30.840 --> 00:15:33.700 be working if there was actually an expression you would use. 00:15:33.700 --> 00:15:37.890 But they're just symbols from the perspective of the grammar. 00:15:37.890 --> 00:15:39.840 And lastly, the start variable is 00:15:39.840 --> 00:15:42.540 going to be, as normally appears, 00:15:42.540 --> 00:15:46.950 on the upper left-hand side of the grammar, 00:15:46.950 --> 00:15:49.990 in terms of the way you write it down. 00:15:49.990 --> 00:15:52.830 So sometimes, you might specify a different start variable, 00:15:52.830 --> 00:15:55.620 but otherwise-- if it's not specified, 00:15:55.620 --> 00:15:58.170 it's the one in this corner here. 00:15:58.170 --> 00:16:02.400 So let's just see some examples of using the grammar 00:16:02.400 --> 00:16:06.440 to generate a string. 00:16:06.440 --> 00:16:09.590 So here is a string in the language, a plus a times a. 00:16:09.590 --> 00:16:13.780 And this example will reveal some other interesting features 00:16:13.780 --> 00:16:17.270 of the grammar, but let's just see it in operation first. 00:16:17.270 --> 00:16:19.330 So, again, I'll try to write it to you 00:16:19.330 --> 00:16:21.760 in both ways in terms of the parse tree and the resulting 00:16:21.760 --> 00:16:24.670 string as you're doing the substitutions. 00:16:24.670 --> 00:16:30.010 So the-- so first we start with the E, 00:16:30.010 --> 00:16:33.340 then we substitute E plus T. And we see the resulting string's E 00:16:33.340 --> 00:16:37.330 plus T. But now as we're doing additional substitutions, 00:16:37.330 --> 00:16:38.800 the resulting string that you get 00:16:38.800 --> 00:16:41.960 is going to evolve accordingly. 00:16:41.960 --> 00:16:48.040 And so I hope it comes across that this tree here 00:16:48.040 --> 00:16:51.220 picture on the left shows you the structure 00:16:51.220 --> 00:16:53.687 of the various substitutions, whereas on the right, 00:16:53.687 --> 00:16:55.270 it just shows you the strings that you 00:16:55.270 --> 00:16:57.400 get as a result of those substitutions. 00:17:00.410 --> 00:17:04.609 So now you can generate this particular string, 00:17:04.609 --> 00:17:06.500 which is now in the language of this grammar. 00:17:06.500 --> 00:17:08.510 You can generate all sorts of other strings 00:17:08.510 --> 00:17:13.670 as well, such as parentheses a plus a parentheses times a 00:17:13.670 --> 00:17:14.880 and so on. 00:17:14.880 --> 00:17:21.290 And in fact, this might be a piece of a programming language 00:17:21.290 --> 00:17:24.230 that you're trying to describe. 00:17:24.230 --> 00:17:28.700 And one application of context-free grammars 00:17:28.700 --> 00:17:32.660 is to describe the syntax of programming languages. 00:17:32.660 --> 00:17:37.370 What are the legal programs that you can write 00:17:37.370 --> 00:17:39.020 in that programming language? 00:17:39.020 --> 00:17:43.250 And not only that, the grammar can 00:17:43.250 --> 00:17:46.040 be used to automatically generate 00:17:46.040 --> 00:17:50.450 the part of the compiler for that programming language, 00:17:50.450 --> 00:17:53.315 which will interpret the-- 00:17:53.315 --> 00:17:57.080 which will interpret the structure 00:17:57.080 --> 00:18:01.310 of the input, the so-called parser, which will figure out 00:18:01.310 --> 00:18:05.240 the meaning of the input to the compiler 00:18:05.240 --> 00:18:07.820 so that the compiler then can generate the code, 00:18:07.820 --> 00:18:12.530 or if it's an interpreter, it can interpret the resulting 00:18:12.530 --> 00:18:14.360 code that you've given it. 00:18:14.360 --> 00:18:16.700 But the very first step in both of those 00:18:16.700 --> 00:18:18.290 is to figure out the meaning. 00:18:18.290 --> 00:18:22.550 And the meaning is embedded within the structure 00:18:22.550 --> 00:18:24.690 of the parse tree. 00:18:24.690 --> 00:18:27.450 Now in the case of this particular tree, just 00:18:27.450 --> 00:18:31.080 to give you some sense of what meaning I have in mind, 00:18:31.080 --> 00:18:34.650 this parse tree, due to the structure of this grammar, 00:18:34.650 --> 00:18:38.670 has the precedents for times over plus. 00:18:38.670 --> 00:18:41.930 So normally when we write down a plus a times a, 00:18:41.930 --> 00:18:44.782 you assume you're going to do the multiplication before you 00:18:44.782 --> 00:18:46.740 do the addition, even though it appears second. 00:18:46.740 --> 00:18:49.050 That's just the way we tend to write things. 00:18:49.050 --> 00:18:52.770 And this grammar has grouped it that way for you. 00:18:52.770 --> 00:18:56.880 It groups the times lower down in the tree than the plus. 00:18:56.880 --> 00:19:01.050 So the times is going to be done before the plus, 00:19:01.050 --> 00:19:03.510 if you imagine doing this in terms of the way the tree is 00:19:03.510 --> 00:19:04.380 guiding you. 00:19:04.380 --> 00:19:06.810 So the tree, as you can see, has a certain amount 00:19:06.810 --> 00:19:08.277 of meaning built into it. 00:19:08.277 --> 00:19:10.860 Now, we're not actually going to be using that in this course, 00:19:10.860 --> 00:19:16.620 but I just want to describe that as an illustration of how 00:19:16.620 --> 00:19:18.120 this material can get applied. 00:19:20.710 --> 00:19:23.320 So here is what I'm saying, that the tree contains 00:19:23.320 --> 00:19:25.430 additional information. 00:19:25.430 --> 00:19:30.910 Now, that's also relevant if you happen 00:19:30.910 --> 00:19:34.420 to have a grammar which might allow multiple parse 00:19:34.420 --> 00:19:35.890 trees for the same string. 00:19:38.600 --> 00:19:41.210 That can happen. 00:19:41.210 --> 00:19:44.990 And this particular grammar does not allow that, 00:19:44.990 --> 00:19:46.800 but you might write other grammars, 00:19:46.800 --> 00:19:49.220 as we'll see in a minute, that could generate 00:19:49.220 --> 00:19:52.010 the same string in multiple ways with multiple different parse 00:19:52.010 --> 00:19:53.180 trees. 00:19:53.180 --> 00:19:57.110 Now that might be undesirable when you have a programming 00:19:57.110 --> 00:19:59.390 language because typically you want 00:19:59.390 --> 00:20:03.110 it to be only a single meaning for your code. 00:20:03.110 --> 00:20:05.210 You don't want it to be ambiguous and have 00:20:05.210 --> 00:20:06.500 multiple meanings. 00:20:06.500 --> 00:20:14.150 But ambiguity does occur, and it's not 00:20:14.150 --> 00:20:16.130 necessarily something we're always 00:20:16.130 --> 00:20:18.880 going to see as a bad thing. 00:20:18.880 --> 00:20:21.560 So I think as I mentioned last time, 00:20:21.560 --> 00:20:26.090 a lot of this subject originated with linguistics, 00:20:26.090 --> 00:20:28.700 and that's where the terminology comes from, grammar, 00:20:28.700 --> 00:20:34.170 and languages, and so on. 00:20:34.170 --> 00:20:35.850 The terminology for the subject really 00:20:35.850 --> 00:20:37.440 comes out of linguistics. 00:20:37.440 --> 00:20:39.270 In fact, one of the key players for that 00:20:39.270 --> 00:20:42.090 is an emeritus faculty member at MIT, Noam Chomsky. 00:20:42.090 --> 00:20:45.480 He was instrumental in setting a lot of this stuff up. 00:20:45.480 --> 00:20:52.970 But the-- you can think of grammars 00:20:52.970 --> 00:20:55.710 as applying to natural human languages as well. 00:20:55.710 --> 00:20:58.670 So let me give you a little example as a pop-up. 00:20:58.670 --> 00:21:00.790 It's not directly a pop-up-- 00:21:00.790 --> 00:21:03.710 a check-in, not directly relevant to the material 00:21:03.710 --> 00:21:07.820 I just presented, but just a curiosity. 00:21:07.820 --> 00:21:11.240 If you take the English sentence "The boy saw the girl 00:21:11.240 --> 00:21:16.310 with the mirror," does that-- 00:21:16.310 --> 00:21:20.450 is there only one natural interpretation 00:21:20.450 --> 00:21:22.130 for that sentence, or are there perhaps 00:21:22.130 --> 00:21:24.680 other natural interpretations for that sentence? 00:21:24.680 --> 00:21:30.270 So let me pose that to you as another poll here. 00:21:30.270 --> 00:21:36.450 And so I ask you to think about how many different meanings you 00:21:36.450 --> 00:21:41.210 might find for a-- reasonable different meanings-- 00:21:41.210 --> 00:21:42.725 I mean, you can-- 00:21:42.725 --> 00:21:45.350 you're going to go wild, you can think of zillions of meanings. 00:21:45.350 --> 00:21:47.183 But I think in terms of reasonable meanings, 00:21:47.183 --> 00:21:50.030 how many meanings might you get for the sentence? 00:21:50.030 --> 00:21:55.310 People are seeing more meanings than I'm seeing, but that's OK. 00:21:55.310 --> 00:22:02.440 So this is a quick-- why don't we just give this another 10 00:22:02.440 --> 00:22:03.400 seconds here. 00:22:05.950 --> 00:22:12.460 And then most of you are in agreement with me. 00:22:12.460 --> 00:22:19.830 I can see here that you are seeing 00:22:19.830 --> 00:22:21.310 that there were two meanings. 00:22:21.310 --> 00:22:24.898 The two meanings that I see here for this sentence are, 00:22:24.898 --> 00:22:26.940 when you say the boy saw the girl with the mirror 00:22:26.940 --> 00:22:28.890 is, who has the mirror? 00:22:28.890 --> 00:22:32.040 Is it the boy seeing the girl through the mirror, 00:22:32.040 --> 00:22:34.080 or is it the girl that has the mirror 00:22:34.080 --> 00:22:35.730 and the boy just happens to see her? 00:22:35.730 --> 00:22:39.090 So two very different meanings for the same sentence. 00:22:39.090 --> 00:22:44.440 And that's in the nature of English is just the way-- 00:22:44.440 --> 00:22:46.980 it's an ambiguous structure there. 00:22:46.980 --> 00:22:49.440 And often we resolve that ambiguity in English 00:22:49.440 --> 00:22:52.570 with other information that we might have. 00:22:52.570 --> 00:22:55.380 But typically you don't want there to be ambiguity 00:22:55.380 --> 00:22:59.020 when you have a grammar, say, describing a programming 00:22:59.020 --> 00:22:59.520 language. 00:23:02.610 --> 00:23:06.300 So let's continue on that. 00:23:06.300 --> 00:23:08.370 So talking a little bit more about ambiguity, 00:23:08.370 --> 00:23:10.530 I promised you an example where you might 00:23:10.530 --> 00:23:12.840 have an ambiguous grammar. 00:23:12.840 --> 00:23:17.280 So if you take these two grammars, G2 and G3, and G2 00:23:17.280 --> 00:23:21.730 from the last slide and G3 is a similar grammar, in fact, 00:23:21.730 --> 00:23:24.000 it's the grammar that has the very same language-- 00:23:26.407 --> 00:23:27.990 that gives you the very same language. 00:23:27.990 --> 00:23:30.260 So L of G2 equals L of G3. 00:23:30.260 --> 00:23:34.550 Both of them are describing these arithmetical expressions. 00:23:34.550 --> 00:23:38.150 But whereas G2 has a unique parse tree 00:23:38.150 --> 00:23:40.070 for every string that you generate, 00:23:40.070 --> 00:23:44.180 G3 can have multiple parse trees for the same string. 00:23:44.180 --> 00:23:46.187 So I'm just going to illustrate that here. 00:23:46.187 --> 00:23:48.020 So here is the same string that we generated 00:23:48.020 --> 00:23:50.750 last time, a plus a times a. 00:23:50.750 --> 00:23:56.340 In G3, the parse tree is actually even simpler here. 00:23:56.340 --> 00:23:58.220 So here I'm showing you the-- 00:23:58.220 --> 00:23:59.810 there's just the two substituents 00:23:59.810 --> 00:24:03.500 that you need to make starting from E. 00:24:03.500 --> 00:24:07.610 And then to get the string a plus a times a, 00:24:07.610 --> 00:24:10.130 it's a simpler grammar in a sense. 00:24:10.130 --> 00:24:11.930 But there's another parse tree that'll 00:24:11.930 --> 00:24:14.870 give you the same result. And I've 00:24:14.870 --> 00:24:19.270 written that down below here upside down. 00:24:19.270 --> 00:24:24.970 So the upper parse tree groups the times before the plus, 00:24:24.970 --> 00:24:28.510 more inside than the plus. 00:24:28.510 --> 00:24:31.780 But the lower parse tree doesn't have that precedence built 00:24:31.780 --> 00:24:34.765 into it and can alternatively interpret the plus 00:24:34.765 --> 00:24:37.510 as being of higher precedence than the times. 00:24:37.510 --> 00:24:41.260 And so in that sense, we have here a grammar 00:24:41.260 --> 00:24:47.710 which has two interpretations for this same string. 00:24:47.710 --> 00:24:49.300 And we call that-- whoops, we call 00:24:49.300 --> 00:24:56.650 that an ambiguous derivation, an ambiguously-derived string. 00:24:56.650 --> 00:24:59.140 And the grammar itself is called an ambiguous grammar. 00:25:05.210 --> 00:25:08.835 So let us continue on from that. 00:25:08.835 --> 00:25:11.210 By the way, I think there's a question here that came in. 00:25:13.910 --> 00:25:17.030 Like, for example, a plus a, is that ambiguous in G2? 00:25:17.030 --> 00:25:18.740 No. 00:25:18.740 --> 00:25:24.020 If you try to apply it, you'll see the way that G2 can produce 00:25:24.020 --> 00:25:29.420 a plus a-- a plus a plus a is going to group the first two 00:25:29.420 --> 00:25:31.200 and then the second one-- 00:25:31.200 --> 00:25:32.750 then the last one. 00:25:32.750 --> 00:25:36.620 You can't derive things in multiple ways. 00:25:36.620 --> 00:25:40.030 Addition is associative, but the grammar doesn't-- 00:25:40.030 --> 00:25:41.270 it doesn't-- the grammar-- 00:25:41.270 --> 00:25:44.450 for the grammar, it's going to have 00:25:44.450 --> 00:25:49.970 a prescribed order for the way things get interpreted there. 00:25:49.970 --> 00:25:51.890 So that's ambiguity. 00:25:51.890 --> 00:25:56.210 So let's introduce pushdown automata, which 00:25:56.210 --> 00:25:58.850 is going to be our automata counterpart 00:25:58.850 --> 00:26:01.880 for context-free languages. 00:26:05.120 --> 00:26:08.210 So the way I'm going to introduce pushdown automata, 00:26:08.210 --> 00:26:13.550 sort of shifting gears here now, is 00:26:13.550 --> 00:26:19.040 by first giving a new view of finite automata. 00:26:19.040 --> 00:26:22.190 Remember before when we presented a finite automaton, 00:26:22.190 --> 00:26:24.560 we gave it in terms of a state diagram, which 00:26:24.560 --> 00:26:27.380 I've shown here in miniature form on the picture. 00:26:31.100 --> 00:26:33.350 We could do that for pushdown automata, 00:26:33.350 --> 00:26:36.180 but the picture tends to be very complicated. 00:26:36.180 --> 00:26:40.790 So I'm going to take a bit of a higher level 00:26:40.790 --> 00:26:43.070 description for pushdown automata, which 00:26:43.070 --> 00:26:46.700 is I'm calling a schematic view or a schematic diagram. 00:26:46.700 --> 00:26:48.800 And there I'm really not going to be showing you 00:26:48.800 --> 00:26:51.050 the individual states, but I'm going to be showing you 00:26:51.050 --> 00:26:54.800 the individual components of the machine, more 00:26:54.800 --> 00:26:56.270 of an abstraction-- 00:26:56.270 --> 00:26:58.590 from a more abstract perspective. 00:26:58.590 --> 00:27:03.950 And so from that perspective, a finite automaton 00:27:03.950 --> 00:27:07.790 has here what I'm going to call the "finite control." 00:27:07.790 --> 00:27:09.560 So I'm going to be suppressing the details 00:27:09.560 --> 00:27:12.590 of the states in this picture. 00:27:12.590 --> 00:27:16.430 I'm going to represent those states as the control 00:27:16.430 --> 00:27:19.520 of the DFA or the NFA. 00:27:19.520 --> 00:27:23.150 They're really going to be the same from this pictorial point 00:27:23.150 --> 00:27:24.320 of view. 00:27:24.320 --> 00:27:29.180 The input is going to appear as a string that's written down 00:27:29.180 --> 00:27:30.965 on what we're calling a "tape." 00:27:30.965 --> 00:27:35.090 Again, this is somewhat of an anachronistic terminology. 00:27:35.090 --> 00:27:39.050 Back in my days, people actually did feed their inputs 00:27:39.050 --> 00:27:42.620 into computers on a tape sometimes. 00:27:42.620 --> 00:27:44.120 We don't do that so much anymore, 00:27:44.120 --> 00:27:46.490 but that terminology has stuck, and it's 00:27:46.490 --> 00:27:50.480 going to be a persisting later on in the course, 00:27:50.480 --> 00:27:52.850 too, so you might as well get used to it. 00:27:52.850 --> 00:27:54.920 So the input is going to appear on a tape, 00:27:54.920 --> 00:27:57.290 or sometimes called an input tape. 00:27:57.290 --> 00:28:01.820 And the way the machine actually will read that input-- 00:28:01.820 --> 00:28:05.240 whoops-- it's going to have a head, which 00:28:05.240 --> 00:28:08.600 is going to be starting at the left side 00:28:08.600 --> 00:28:12.020 and moving from left to right, reading the symbols that appear 00:28:12.020 --> 00:28:14.330 on the input tape one by one. 00:28:14.330 --> 00:28:19.700 So this is our picture of a finite automaton, 00:28:19.700 --> 00:28:23.570 just redone from last time, just a different way 00:28:23.570 --> 00:28:25.100 of picturing it. 00:28:25.100 --> 00:28:28.220 Now, that's going to be setting the stage for the picture 00:28:28.220 --> 00:28:31.610 of a pushdown automaton, because a pushdown automaton is 00:28:31.610 --> 00:28:35.360 like a finite automaton, but it has an extra feature, 00:28:35.360 --> 00:28:38.010 it has an extra device attached to it. 00:28:38.010 --> 00:28:41.590 And that's called a "stack." 00:28:41.590 --> 00:28:44.680 So here's the schematic diagram for a pushdown automaton, 00:28:44.680 --> 00:28:48.310 and that's going to be a stack, which 00:28:48.310 --> 00:28:55.750 is going to be basically a form of auxiliary storage. 00:28:55.750 --> 00:28:57.640 Now remember, part of the limitation 00:28:57.640 --> 00:29:00.070 for a finite automaton was that we 00:29:00.070 --> 00:29:05.910 had a limited amount of memory. 00:29:05.910 --> 00:29:09.560 So we were not able to do some very simple things, 00:29:09.560 --> 00:29:12.530 like counting, because we had a limited memory. 00:29:12.530 --> 00:29:14.450 So the pushdown automaton is going 00:29:14.450 --> 00:29:18.860 to be able to use its stack as a kind of unbounded memory, 00:29:18.860 --> 00:29:23.130 but a memory that's restricted in the way it can be used. 00:29:23.130 --> 00:29:28.600 So it's unlimited but still restricted, as we'll see. 00:29:28.600 --> 00:29:35.480 So the way the pushdown automaton uses its extra memory 00:29:35.480 --> 00:29:38.820 on what we're calling the stack or a pushdown stack 00:29:38.820 --> 00:29:44.070 is that you can write symbols instead of only reading 00:29:44.070 --> 00:29:44.790 symbols. 00:29:44.790 --> 00:29:49.200 But those symbols can only be read at the very-- 00:29:49.200 --> 00:29:53.610 written or read at the very top of this list of symbols. 00:29:53.610 --> 00:29:57.060 And every time you add a new symbol, the other symbols that 00:29:57.060 --> 00:30:00.570 are already there get pushed down, so hence the name. 00:30:00.570 --> 00:30:03.240 People also often refer to it as a stack 00:30:03.240 --> 00:30:07.560 of plates in a cafeteria, if you've ever seen those things, 00:30:07.560 --> 00:30:09.060 or you can remember back to the days 00:30:09.060 --> 00:30:12.300 when we went to cafeteria, which are getting further and further 00:30:12.300 --> 00:30:12.850 away. 00:30:12.850 --> 00:30:18.120 But even at the cafeteria, you had a stack of plates. 00:30:18.120 --> 00:30:21.450 And as you remove plates from them, they were on a spring, 00:30:21.450 --> 00:30:22.590 and they kept coming up. 00:30:22.590 --> 00:30:24.480 Or if you add more, they would go down. 00:30:24.480 --> 00:30:25.570 And it's the same idea. 00:30:25.570 --> 00:30:29.580 Imagine these symbols here are on a spring, 00:30:29.580 --> 00:30:32.400 and the more symbols you add, the more they go down. 00:30:32.400 --> 00:30:36.030 Or if you move them, and read them, and remove them, 00:30:36.030 --> 00:30:39.400 then they move back up. 00:30:39.400 --> 00:30:43.570 So a pushdown automaton operates like a finite-- 00:30:43.570 --> 00:30:46.600 like a non-deterministic finite automaton, as we'll see. 00:30:46.600 --> 00:30:49.810 Pushdown automata for us are always 00:30:49.810 --> 00:30:51.850 going to be allowed to be non-deterministic. 00:30:51.850 --> 00:30:54.970 So we're not going to be studying the pushdown 00:30:54.970 --> 00:30:59.168 automata that are restricted to be only deterministic. 00:30:59.168 --> 00:31:00.710 I'll say more about that in a second. 00:31:00.710 --> 00:31:07.390 But they operate like an NFA, except they can write or read 00:31:07.390 --> 00:31:10.040 symbols from the top of the stack. 00:31:10.040 --> 00:31:12.350 And when they write, they're adding the symbol on, 00:31:12.350 --> 00:31:14.185 pushing down that stack. 00:31:14.185 --> 00:31:15.560 And when they're reading, they're 00:31:15.560 --> 00:31:19.280 removing symbols from the stack, and thereby lifting 00:31:19.280 --> 00:31:21.830 up the stack. 00:31:21.830 --> 00:31:22.985 We give them special names. 00:31:22.985 --> 00:31:24.860 So those of you who have seen stacks already, 00:31:24.860 --> 00:31:28.160 this is, I'm sure, old hat for you. 00:31:28.160 --> 00:31:32.750 But I'm sure not everyone have seen stacks before. 00:31:32.750 --> 00:31:36.290 So the special name for writing onto a stack 00:31:36.290 --> 00:31:38.990 is called a "push operation," so that you're 00:31:38.990 --> 00:31:41.450 pushing a new symbol down on the top of the stack, 00:31:41.450 --> 00:31:43.250 and it pushes everything down. 00:31:43.250 --> 00:31:45.140 Whereas when you're reading a symbol 00:31:45.140 --> 00:31:46.940 and removing it from the top of the stack, 00:31:46.940 --> 00:31:48.600 that's called a "pop." 00:31:48.600 --> 00:31:50.720 So that's reading and removing. 00:31:50.720 --> 00:31:52.880 And we always think of those as going together. 00:31:52.880 --> 00:31:56.390 Writing and adding and reading and removing are combined. 00:31:56.390 --> 00:31:58.610 I mean, you might wonder, well, can't I 00:31:58.610 --> 00:32:03.390 just read it and leave it alone and not just remove it? 00:32:03.390 --> 00:32:03.890 No. 00:32:03.890 --> 00:32:07.820 You can get that effect by reading it, which removes it, 00:32:07.820 --> 00:32:10.790 and then putting it back if you really want it to stay there. 00:32:10.790 --> 00:32:13.130 But the way we're setting it up is 00:32:13.130 --> 00:32:15.500 that reading comes with removing, 00:32:15.500 --> 00:32:18.750 writing comes with adding. 00:32:18.750 --> 00:32:21.497 Again, they're called pushing and popping. 00:32:21.497 --> 00:32:22.455 So let's do an example. 00:32:25.590 --> 00:32:27.650 So we have here a pushdown automaton 00:32:27.650 --> 00:32:30.800 for a language we'll call D. We've 00:32:30.800 --> 00:32:33.350 seen that language before. 00:32:33.350 --> 00:32:37.050 It was-- actually, we used the same letter for it, 00:32:37.050 --> 00:32:39.830 strings of zeros followed by ones, 00:32:39.830 --> 00:32:43.010 where the numbers were the same of the two, 0 to the k 1 00:32:43.010 --> 00:32:44.210 to the k. 00:32:44.210 --> 00:32:46.310 We couldn't do that with a finite automaton. 00:32:46.310 --> 00:32:49.100 We will be able to do that with a pushdown automaton. 00:32:51.800 --> 00:32:52.912 And here I-- 00:32:55.804 --> 00:32:59.110 I thought I wrote down the input here, but OK. 00:32:59.110 --> 00:33:06.040 So the basic idea is I'm going to give you an input now, 00:33:06.040 --> 00:33:07.750 and the pushdown automaton is supposed 00:33:07.750 --> 00:33:11.050 to test whether that input is in the language, 00:33:11.050 --> 00:33:12.235 whether it's of this form. 00:33:14.740 --> 00:33:18.850 Now it has the ability to use the stack 00:33:18.850 --> 00:33:21.430 because it's going to have to count how many zeros it has. 00:33:21.430 --> 00:33:23.050 And so the way it's going to do it 00:33:23.050 --> 00:33:25.690 is it'll have a bunch of zeros, hopefully, and then 00:33:25.690 --> 00:33:27.430 a bunch of ones, and you want to see 00:33:27.430 --> 00:33:29.770 they are of the same number. 00:33:29.770 --> 00:33:32.950 It's going to take the zeros and store them on the stack 00:33:32.950 --> 00:33:34.690 until it sees a one. 00:33:34.690 --> 00:33:36.892 And then it's going to start to read the ones, 00:33:36.892 --> 00:33:38.350 and it's going to remove the zeros, 00:33:38.350 --> 00:33:43.620 matching them off one to one with the ones that it's seeing. 00:33:43.620 --> 00:33:48.720 So you initially first read the zeros and push them 00:33:48.720 --> 00:33:50.490 onto the stack until you read a one, 00:33:50.490 --> 00:33:54.870 and then you read the ones while popping zeros from the stack. 00:33:54.870 --> 00:33:59.310 And you enter the accept state if the stack is empty. 00:33:59.310 --> 00:34:02.340 Just like with a finite automaton, 00:34:02.340 --> 00:34:04.978 entering the accept state only counts when 00:34:04.978 --> 00:34:06.270 you're at the end of the input. 00:34:09.830 --> 00:34:13.370 So without even me needing to say anything, 00:34:13.370 --> 00:34:15.650 it's really saying you would enter the accept state 00:34:15.650 --> 00:34:18.409 if the stack is empty at the end of the input string. 00:34:18.409 --> 00:34:21.828 But that's implicit because it only takes effect 00:34:21.828 --> 00:34:23.120 at the end of the input string. 00:34:23.120 --> 00:34:25.429 If you enter an accept state alone in the middle somewhere, 00:34:25.429 --> 00:34:26.159 it doesn't matter. 00:34:26.159 --> 00:34:27.284 It doesn't affect anything. 00:34:32.139 --> 00:34:34.389 And with that we're going to take a little break. 00:34:34.389 --> 00:34:38.480 And then we will be back shortly to look at pushdown automata 00:34:38.480 --> 00:34:40.060 again in a more-- 00:34:40.060 --> 00:34:42.973 with a more formal definition. 00:34:42.973 --> 00:34:44.890 Let me put that it's going to be five minutes. 00:34:44.890 --> 00:34:47.989 Let me see if I can figure out how to get my timer screen up 00:34:47.989 --> 00:34:48.489 here. 00:34:48.489 --> 00:34:49.690 Yes. 00:34:49.690 --> 00:34:57.270 And we will-- when the candle burns down to nothing, 00:34:57.270 --> 00:34:59.595 we will return and continue. 00:35:07.600 --> 00:35:13.130 OK, our candle has burned down and has gone out. 00:35:13.130 --> 00:35:17.150 I never actually watched and see what would happened at the end. 00:35:17.150 --> 00:35:19.185 So we're good to go. 00:35:19.185 --> 00:35:19.810 Let's continue. 00:35:24.480 --> 00:35:25.320 Good. 00:35:25.320 --> 00:35:29.145 And let me put myself back in there. 00:35:29.145 --> 00:35:29.645 All righty. 00:35:32.390 --> 00:35:34.370 So we were doing pushdown automata. 00:35:34.370 --> 00:35:38.660 And we just did that example of 0 to the k, 1 to the k. 00:35:38.660 --> 00:35:40.850 Now that you have a stack, we can 00:35:40.850 --> 00:35:45.110 do all sorts of fancy things that finite automata could not 00:35:45.110 --> 00:35:49.580 do just with their limited memory. 00:35:49.580 --> 00:35:54.086 So let's take a look at how we define pushdown automata. 00:35:57.740 --> 00:36:02.420 So now pushdown automata is actually going to be a 6-tuple. 00:36:02.420 --> 00:36:05.330 So it's a little bit-- got some fancier stuff here 00:36:05.330 --> 00:36:09.380 to deal with, not too much, but a little bit. 00:36:09.380 --> 00:36:12.402 And so it has-- 00:36:12.402 --> 00:36:14.360 let's look at these a little bit more carefully 00:36:14.360 --> 00:36:16.530 since there's some novelty here. 00:36:16.530 --> 00:36:22.190 We have the input alphabet, just as we had before, sigma, 00:36:22.190 --> 00:36:25.580 but we also have gamma, which is the alphabet 00:36:25.580 --> 00:36:28.580 for using the stack. 00:36:28.580 --> 00:36:32.510 Now, you might ask, why don't we just use the same alphabet? 00:36:32.510 --> 00:36:35.180 Well, it's really a matter of convenience 00:36:35.180 --> 00:36:41.210 that we would like to be able to have other symbols that 00:36:41.210 --> 00:36:43.010 could include the input alphabet, 00:36:43.010 --> 00:36:44.910 but could include other things as well. 00:36:44.910 --> 00:36:46.490 So it just gives you more flexibility 00:36:46.490 --> 00:36:49.010 in terms of what you're going to be writing on the stack. 00:36:52.440 --> 00:36:55.157 The transition function, more complicated. 00:36:55.157 --> 00:36:57.240 So I think-- I don't know if I'm going to even say 00:36:57.240 --> 00:36:59.823 what the other things are, but these are the accepting states, 00:36:59.823 --> 00:37:01.030 this is the starting state. 00:37:01.030 --> 00:37:04.060 So that's the same as before. 00:37:04.060 --> 00:37:08.340 But the transition function is a much different animal 00:37:08.340 --> 00:37:10.530 here in a pushdown automaton. 00:37:10.530 --> 00:37:13.260 So let's just try to unpack that and understand 00:37:13.260 --> 00:37:15.130 what it's saying. 00:37:15.130 --> 00:37:17.670 So the transition function tells us 00:37:17.670 --> 00:37:21.840 how the machine operates, how it goes from state to state, 00:37:21.840 --> 00:37:24.000 how it's going to read the input, how it reads it 00:37:24.000 --> 00:37:26.610 from the stack, and what am I to write on the stack, 00:37:26.610 --> 00:37:29.280 too, because that's going to all happen under program control. 00:37:31.800 --> 00:37:37.410 So what this means here is that when 00:37:37.410 --> 00:37:42.630 the machine is in a particular state, 00:37:42.630 --> 00:37:44.760 reading a particular input symbol-- 00:37:44.760 --> 00:37:48.330 let's ignore the empty string subscript for the moment. 00:37:48.330 --> 00:37:52.500 So it's in a particular state reading a particular input 00:37:52.500 --> 00:37:56.550 symbol and with a certain stack symbol appearing 00:37:56.550 --> 00:37:59.230 at the top of the stack. 00:37:59.230 --> 00:38:03.120 So that's all information that's available to the controller 00:38:03.120 --> 00:38:07.620 of this pushdown automaton, the transition 00:38:07.620 --> 00:38:12.210 function, the current state, the next input 00:38:12.210 --> 00:38:14.910 symbol, and the symbol at the top of the stack. 00:38:14.910 --> 00:38:20.430 And once we have that, we know what new state we can go into 00:38:20.430 --> 00:38:27.300 and what new symbol we can write on the top of the stack. 00:38:27.300 --> 00:38:32.430 So that's what the right-hand side of this function 00:38:32.430 --> 00:38:33.930 specification means. 00:38:33.930 --> 00:38:37.140 So this is where the input to the function, this 00:38:37.140 --> 00:38:41.310 is going to be the output of the function, state and new symbol 00:38:41.310 --> 00:38:42.910 to appear on the stack. 00:38:42.910 --> 00:38:46.988 So this is the popping symbol, this is the pushing symbol. 00:38:46.988 --> 00:38:48.530 So now there are two things that bear 00:38:48.530 --> 00:38:50.180 explanation here first of all. 00:38:50.180 --> 00:38:53.860 Now this is a power set. 00:38:53.860 --> 00:38:57.480 So this is going to be representing, as we did before, 00:38:57.480 --> 00:39:00.510 a non-deterministic machine. 00:39:00.510 --> 00:39:02.915 We may have several possibilities, 00:39:02.915 --> 00:39:04.290 and we're going to represent that 00:39:04.290 --> 00:39:07.800 as a set of possibilities for the machine 00:39:07.800 --> 00:39:12.260 that it could go to at any point. 00:39:12.260 --> 00:39:15.590 I will give an example of how a pushdown automaton uses 00:39:15.590 --> 00:39:18.840 its non-determinism in a minute. 00:39:18.840 --> 00:39:20.760 The other thing is these epsilons. 00:39:20.760 --> 00:39:23.390 So we have to understand why they are there. 00:39:23.390 --> 00:39:26.300 And we remember we had them for the NFAs, 00:39:26.300 --> 00:39:31.670 corresponding to when the NFA had an epsilon transition, 00:39:31.670 --> 00:39:32.790 the empty transition. 00:39:32.790 --> 00:39:34.770 So it could go along in that transition 00:39:34.770 --> 00:39:37.650 without reading any input. 00:39:37.650 --> 00:39:40.470 So this is going to play the same role here. 00:39:40.470 --> 00:39:42.470 So if you have-- 00:39:42.470 --> 00:39:46.010 instead of an input symbol from sigma 00:39:46.010 --> 00:39:52.232 appearing in this part of the-- 00:39:52.232 --> 00:39:55.190 for the transition function, instead you 00:39:55.190 --> 00:39:57.260 have an epsilon appearing. 00:39:57.260 --> 00:40:01.430 That means that the transition-- that move of the machine 00:40:01.430 --> 00:40:06.332 can happen without reading any input symbol, 00:40:06.332 --> 00:40:07.290 just like for the NFAs. 00:40:10.230 --> 00:40:13.110 Or if you have an epsilon appearing for the stack symbol, 00:40:13.110 --> 00:40:15.690 that means you can make that transition without reading 00:40:15.690 --> 00:40:17.040 any stack symbol. 00:40:17.040 --> 00:40:19.410 So whatever is sitting on the top of the stack, 00:40:19.410 --> 00:40:23.760 it doesn't matter, the machine can make that move. 00:40:23.760 --> 00:40:25.262 And it won't read anything either, 00:40:25.262 --> 00:40:26.470 or not going to pop anything. 00:40:26.470 --> 00:40:29.580 It's just going to be proceeding without looking 00:40:29.580 --> 00:40:31.200 at the stack at all. 00:40:31.200 --> 00:40:33.660 Or it might have both of them, in which case, it's 00:40:33.660 --> 00:40:35.580 going to go from one state to another state 00:40:35.580 --> 00:40:39.250 without looking at the input or at the top of the stack. 00:40:39.250 --> 00:40:43.560 So that's what the possibility of epsilon 00:40:43.560 --> 00:40:50.340 means for the transition function in those places. 00:40:50.340 --> 00:40:52.620 The epsilon appearing over here means 00:40:52.620 --> 00:40:54.930 something a little different, but very similar. 00:40:54.930 --> 00:41:01.060 What that means is that we won't write anything 00:41:01.060 --> 00:41:02.530 on the top of the stack. 00:41:02.530 --> 00:41:03.790 That's going to be-- 00:41:03.790 --> 00:41:07.210 we will go to a new state, but without doing any writing. 00:41:07.210 --> 00:41:10.380 So we'll leave the stack alone. 00:41:10.380 --> 00:41:13.020 So here it means we're not going to read anything 00:41:13.020 --> 00:41:15.600 if it's in this position, and this position 00:41:15.600 --> 00:41:19.230 means we're not going to write anything. 00:41:19.230 --> 00:41:22.410 So all of those things are valid and legal 00:41:22.410 --> 00:41:25.600 from the perspective of constructing a pushdown 00:41:25.600 --> 00:41:27.310 automaton. 00:41:27.310 --> 00:41:29.890 And I've illustrated here, just with a little bit 00:41:29.890 --> 00:41:35.890 of an example, if you have delta that applies to some state q, 00:41:35.890 --> 00:41:38.800 reading an input symbol a and popping 00:41:38.800 --> 00:41:42.540 a c from the top of the stack, then you might have, 00:41:42.540 --> 00:41:44.473 let's say, in this case, two possibilities 00:41:44.473 --> 00:41:45.765 that you might end up going to. 00:41:45.765 --> 00:41:50.460 You might end up going to states r1 or to state r2. 00:41:50.460 --> 00:41:53.430 And in the former case, you'll end up writing a d, 00:41:53.430 --> 00:41:55.590 pushing a d onto the top of the stack. 00:41:55.590 --> 00:41:58.080 And in the latter case, you would be pushing an e 00:41:58.080 --> 00:42:01.220 onto the top of the stack. 00:42:01.220 --> 00:42:04.625 So this is I'm trying to help you look at this notation. 00:42:07.370 --> 00:42:10.070 I hope that this is clear to you. 00:42:10.070 --> 00:42:14.360 I'm sure for some of you it's too slow, but others of you 00:42:14.360 --> 00:42:15.590 I'm trying to help along. 00:42:15.590 --> 00:42:17.690 But if you're really struggling with this notation 00:42:17.690 --> 00:42:20.750 at this point, you're going to have to dig in and make 00:42:20.750 --> 00:42:21.645 sure you follow it. 00:42:21.645 --> 00:42:23.270 It's only going to get harder from now. 00:42:23.270 --> 00:42:28.020 I'm going to stop going over these kinds of points. 00:42:28.020 --> 00:42:30.330 And if you're still struggling, you can't get it, 00:42:30.330 --> 00:42:32.630 this is not the right class for you, I'll be honest. 00:42:35.456 --> 00:42:39.040 So we're just going to be taking off like-- 00:42:39.040 --> 00:42:43.640 we're going to start to accelerate fairly quickly. 00:42:43.640 --> 00:42:46.490 So it's a non-deterministic machine. 00:42:46.490 --> 00:42:49.490 We accept, like we did before. 00:42:49.490 --> 00:42:52.430 There might be several different threads of the computation. 00:42:52.430 --> 00:42:55.670 You're going to end up accepting if some 00:42:55.670 --> 00:42:58.833 of the threads, at least one of the threads 00:42:58.833 --> 00:43:01.250 ends up in an accept state at the end of the input string. 00:43:04.290 --> 00:43:06.840 That's when the machine overall accepts. 00:43:06.840 --> 00:43:10.140 It's just the way we normally think of non-determinism. 00:43:10.140 --> 00:43:11.880 Again, you can use the models that we 00:43:11.880 --> 00:43:14.700 had before in terms of guessing or parallelism, 00:43:14.700 --> 00:43:16.060 whatever works for you. 00:43:16.060 --> 00:43:18.240 And sometimes different things work in different-- 00:43:18.240 --> 00:43:19.740 at different occasions. 00:43:19.740 --> 00:43:21.540 But that's how non-determinism works. 00:43:21.540 --> 00:43:23.340 We'll do an example of it now. 00:43:23.340 --> 00:43:27.540 Here is a pushdown automaton for a different language 00:43:27.540 --> 00:43:31.700 we haven't seen before, I don't think, well, maybe we have, 00:43:31.700 --> 00:43:36.790 which is going to be using it's non-determinism 00:43:36.790 --> 00:43:38.170 in an essential way. 00:43:38.170 --> 00:43:42.550 This is a language where non-determinism 00:43:42.550 --> 00:43:44.830 is going to be critical. 00:43:44.830 --> 00:43:47.260 Without it, you can't-- a deterministic pushdown 00:43:47.260 --> 00:43:50.590 automaton, which is something, by the way, that people study. 00:43:50.590 --> 00:43:52.750 And there's a section of my book about that, 00:43:52.750 --> 00:43:56.322 it's section 2.4, because it has relevance to applications. 00:43:56.322 --> 00:43:58.280 We're not going to address that in this course. 00:43:58.280 --> 00:44:00.190 So you can just skip section 2.4. 00:44:00.190 --> 00:44:04.120 It's pretty technical, I'll have to say, but still 00:44:04.120 --> 00:44:06.550 quite interesting and beautiful if you like that stuff. 00:44:06.550 --> 00:44:08.200 But it's technical. 00:44:08.200 --> 00:44:10.430 We won't do it. 00:44:10.430 --> 00:44:19.240 So here is this input string ww reverse for all possible w's 00:44:19.240 --> 00:44:21.040 over our alphabet 0, 1. 00:44:21.040 --> 00:44:25.040 And what w reversed, by the way, means is writing w backwards. 00:44:25.040 --> 00:44:28.660 So this is all strings followed by a reversal 00:44:28.660 --> 00:44:34.070 of the same string, the string written backwards. 00:44:34.070 --> 00:44:37.230 Really you can think of these as-- 00:44:37.230 --> 00:44:40.190 well, so these are strings that-- 00:44:40.190 --> 00:44:45.090 well, here's an example, 0, 1, 1, 1, 1, 0, the string 00:44:45.090 --> 00:44:45.840 written backwards. 00:44:45.840 --> 00:44:49.055 So this is a string in the language appearing on a tape 00:44:49.055 --> 00:44:51.680 as I described. 00:44:51.680 --> 00:44:55.640 So how is the machine going to recognize this line, which 00:44:55.640 --> 00:44:58.550 is going to be somewhat similar to before, but with one 00:44:58.550 --> 00:45:01.990 important difference. 00:45:01.990 --> 00:45:03.360 And if you imagine-- 00:45:03.360 --> 00:45:07.500 and I like to use this kind of anthropomorphizing 00:45:07.500 --> 00:45:10.500 these things, putting yourself in the place of the machine, 00:45:10.500 --> 00:45:13.900 and thinking how you would do it. 00:45:13.900 --> 00:45:17.555 So if you imagine getting these symbols one by one, 0, 1, 1, 00:45:17.555 --> 00:45:18.930 you don't know what's coming next 00:45:18.930 --> 00:45:21.030 as you're getting the symbols. 00:45:21.030 --> 00:45:23.340 You have to figure out how to match off 00:45:23.340 --> 00:45:24.960 the second half with the first half. 00:45:24.960 --> 00:45:28.110 So you're going to put the first half on the stack, 00:45:28.110 --> 00:45:31.890 and then you're going to remove the first half 00:45:31.890 --> 00:45:36.360 and match it off with the second half. 00:45:36.360 --> 00:45:39.540 Conveniently, the first half comes out backwards. 00:45:39.540 --> 00:45:42.670 The stack is a first in last out kind of thing. 00:45:42.670 --> 00:45:46.390 So it comes out in reverse order. 00:45:46.390 --> 00:45:48.735 So that's perfect for matching off with the second half. 00:45:51.320 --> 00:45:54.650 But the tricky part with this language 00:45:54.650 --> 00:45:58.250 is, how do you know when you're at the middle 00:45:58.250 --> 00:46:02.510 because you don't get to see the rest. 00:46:02.510 --> 00:46:04.760 You only get to see what you've seen so far. 00:46:04.760 --> 00:46:06.140 You don't know what's coming. 00:46:06.140 --> 00:46:13.000 So when you read that second one-- well, at this point, 00:46:13.000 --> 00:46:16.540 you're reading 0, 1, 1, now you're reading that second one, 00:46:16.540 --> 00:46:18.430 you don't know that perhaps it's just 00:46:18.430 --> 00:46:21.770 going to be a 0 following that and it's going to be all. 00:46:21.770 --> 00:46:25.610 So maybe you should be deciding so that this point here 00:46:25.610 --> 00:46:31.680 that I've marked is the midpoint. 00:46:31.680 --> 00:46:35.450 And you put 0, 1 on the tape and then 00:46:35.450 --> 00:46:39.400 start popping the second one and matching it off 00:46:39.400 --> 00:46:41.270 with the first one. 00:46:41.270 --> 00:46:45.960 That would be a tempting thing to do, but you just don't know. 00:46:45.960 --> 00:46:47.750 And that's where the non-determinism 00:46:47.750 --> 00:46:48.950 is going to be essential. 00:46:48.950 --> 00:46:50.370 So let me write down more of this. 00:46:50.370 --> 00:46:51.870 So what you're going to do is you're 00:46:51.870 --> 00:46:53.660 going to read and push input symbols, 00:46:53.660 --> 00:46:55.490 but non-deterministically guessing 00:46:55.490 --> 00:46:56.820 that you're at the middle. 00:46:56.820 --> 00:46:58.487 So you're going to non-deterministically 00:46:58.487 --> 00:47:01.490 either repeat that and continue to read and push symbols 00:47:01.490 --> 00:47:03.290 onto the stack, or you're going to go 00:47:03.290 --> 00:47:08.960 to 2 deciding that, or guessing that you're at the midpoint, 00:47:08.960 --> 00:47:12.320 and now it's time to start reading and popping instead 00:47:12.320 --> 00:47:13.718 of reading and pushing. 00:47:13.718 --> 00:47:15.260 So you're going to read input symbols 00:47:15.260 --> 00:47:18.740 and popping the stack symbols, comparing 00:47:18.740 --> 00:47:22.405 the two, the symbols that you're reading at the top, 00:47:22.405 --> 00:47:24.620 the symbols you're removing from the stack. 00:47:24.620 --> 00:47:28.370 If they ever disagree, then this thread of the non-determinism 00:47:28.370 --> 00:47:32.900 rejects because either the input is not in the language 00:47:32.900 --> 00:47:35.360 or at least you have made a wrong choice 00:47:35.360 --> 00:47:36.740 as to where the midpoint is. 00:47:40.192 --> 00:47:42.150 And then you're going to enter the accept state 00:47:42.150 --> 00:47:44.910 if the stack is empty. 00:47:44.910 --> 00:47:47.670 And ignore this part for the moment, this software 00:47:47.670 --> 00:47:48.930 reference. 00:47:48.930 --> 00:47:52.120 Let's just-- I'll speak to that in a second. 00:47:52.120 --> 00:47:54.750 But I just want to make sure we understand 00:47:54.750 --> 00:47:57.780 that at an intuitive level how this machine is using 00:47:57.780 --> 00:48:03.130 it's non-determinism to recognize 00:48:03.130 --> 00:48:05.470 this language because the non-determinism is critical. 00:48:05.470 --> 00:48:08.450 And it's important that you understand it. 00:48:08.450 --> 00:48:10.148 So let me just make some side comments, 00:48:10.148 --> 00:48:12.190 and then we'll come back to this software remark. 00:48:12.190 --> 00:48:16.705 So first of all, one question that comes up a lot-- 00:48:19.445 --> 00:48:22.168 well, I'm not paying attention to the chat here, sorry. 00:48:22.168 --> 00:48:23.960 So if you're not getting an answer from me, 00:48:23.960 --> 00:48:25.010 you try the TAs. 00:48:25.010 --> 00:48:26.750 But one of the-- 00:48:29.720 --> 00:48:31.758 one of the questions that comes up 00:48:31.758 --> 00:48:33.800 a lot when they're thinking about non-determinism 00:48:33.800 --> 00:48:38.530 for pushdown automata is, what happens to the stack? 00:48:38.530 --> 00:48:43.620 The stack gets replicated in the non-determinism every time 00:48:43.620 --> 00:48:46.230 the machine forks, just like everything else 00:48:46.230 --> 00:48:47.010 gets replicated. 00:48:47.010 --> 00:48:48.810 So an entire-- every time there's 00:48:48.810 --> 00:48:52.050 a fork in the non-determinism and the machine branches 00:48:52.050 --> 00:48:55.980 into multiple possibilities, the entire machine 00:48:55.980 --> 00:48:58.650 replicates the current state, the current position 00:48:58.650 --> 00:49:02.520 of the head, the stack and its contents. 00:49:02.520 --> 00:49:04.920 All of that gets replicated. 00:49:04.920 --> 00:49:10.230 And the two sides of the-- the two branches or the two sides 00:49:10.230 --> 00:49:13.170 of the fork each go on independently 00:49:13.170 --> 00:49:17.150 in their merry way, doing their own thing independently. 00:49:17.150 --> 00:49:18.770 And then if any one of them accepts, 00:49:18.770 --> 00:49:21.410 that's the only way they sort of-- a kind of a communication 00:49:21.410 --> 00:49:23.780 because the one that accepts raises the flag, 00:49:23.780 --> 00:49:28.250 and then the overall machine is set to accept. 00:49:28.250 --> 00:49:30.783 So the non-determinism forks replicate the stack. 00:49:30.783 --> 00:49:31.700 [INAUDIBLE] saying it. 00:49:31.700 --> 00:49:34.220 I just want to make sure you got that. 00:49:34.220 --> 00:49:36.680 And then this language requires non-determinism, 00:49:36.680 --> 00:49:41.180 that I said earlier, so our PDA, pushdown automaton model is 00:49:41.180 --> 00:49:42.470 going to be non-deterministic. 00:49:42.470 --> 00:49:44.803 I mean, you might have examples which are deterministic, 00:49:44.803 --> 00:49:48.020 but the model is going to always allow non-determinism. 00:49:48.020 --> 00:49:50.600 What's this bit about the software? 00:49:50.600 --> 00:49:53.230 So if you look at this formal definition here, 00:49:53.230 --> 00:49:56.260 it doesn't have anywhere in it the ability 00:49:56.260 --> 00:49:58.930 to test if the stack is empty. 00:49:58.930 --> 00:50:01.720 That's not part of the hardware specification, 00:50:01.720 --> 00:50:04.570 at least as we are describing it for a pushdown automaton. 00:50:04.570 --> 00:50:07.690 You can might imagine somebody else describes pushdown 00:50:07.690 --> 00:50:10.218 automata in some other way which gives that as a primitive, 00:50:10.218 --> 00:50:11.510 but we're not going to do that. 00:50:11.510 --> 00:50:12.160 Why? 00:50:12.160 --> 00:50:13.990 Because we don't need a primitive for that. 00:50:13.990 --> 00:50:17.410 You can get the effect of testing 00:50:17.410 --> 00:50:21.070 if there's an empty stack, even if you don't have that 00:50:21.070 --> 00:50:26.590 as a primitive for the machine because what you can do 00:50:26.590 --> 00:50:28.750 is you can start the machine off when it-- 00:50:28.750 --> 00:50:32.020 and the very first thing it does is it writes a special symbol 00:50:32.020 --> 00:50:34.420 to mark the bottom-- 00:50:34.420 --> 00:50:36.670 what's going to eventually be the bottom of the stack. 00:50:36.670 --> 00:50:39.170 There's going to be some special symbol, maybe a dollar sign 00:50:39.170 --> 00:50:39.955 symbol. 00:50:39.955 --> 00:50:41.830 That's the very first thing the machine does. 00:50:41.830 --> 00:50:43.122 And then it proceeds as before. 00:50:43.122 --> 00:50:45.970 And if it ever sees that dollar sign symbol again, 00:50:45.970 --> 00:50:50.720 it knows the stack is effectively empty. 00:50:50.720 --> 00:50:53.180 So you can get the effect of testing 00:50:53.180 --> 00:50:55.217 for the static being empty, even if you don't 00:50:55.217 --> 00:50:56.300 have a primitive for that. 00:50:56.300 --> 00:51:00.960 And we're not going to actually fuss about details like that. 00:51:00.960 --> 00:51:03.950 So you can use-- when you're writing your homework sets, 00:51:03.950 --> 00:51:06.020 you can just use the assumption that you 00:51:06.020 --> 00:51:07.458 can test for an empty stack, which 00:51:07.458 --> 00:51:08.750 is what I'm going to do myself. 00:51:13.380 --> 00:51:15.540 So let's continue on. 00:51:19.150 --> 00:51:19.720 Yeah. 00:51:19.720 --> 00:51:22.842 So now what we're going to do, we're going to prove our one-- 00:51:22.842 --> 00:51:24.550 so far we really haven't proved anything, 00:51:24.550 --> 00:51:27.580 we've just given some definitions and some examples. 00:51:27.580 --> 00:51:30.160 Today was going to-- now we're going to come to our big 00:51:30.160 --> 00:51:34.510 theorem, which actually is important and has some meat 00:51:34.510 --> 00:51:38.110 to it, and that is, how do we convert-- 00:51:38.110 --> 00:51:41.080 I claim that context-free grammars and pushdown 00:51:41.080 --> 00:51:42.920 automata are equivalent. 00:51:42.920 --> 00:51:45.940 Well, we're going to prove that equivalence in one direction, 00:51:45.940 --> 00:51:49.750 converting the grammars to pushdown automata. 00:51:49.750 --> 00:51:53.860 So let me show you how that goes. 00:51:53.860 --> 00:51:57.460 In some ways-- and it's a nice proof, 00:51:57.460 --> 00:52:02.270 not super complicated, but it has some meat to it. 00:52:02.270 --> 00:52:04.810 So if I give you a grammar here, what 00:52:04.810 --> 00:52:07.630 I'm going to tell you how to do is convert that grammar 00:52:07.630 --> 00:52:10.195 into pushdown automaton, which does the same language. 00:52:12.750 --> 00:52:15.850 So if you were checked out for a minute, 00:52:15.850 --> 00:52:18.810 please come back because we're starting this topic now, 00:52:18.810 --> 00:52:22.770 that you can think about this as a good re-entry 00:52:22.770 --> 00:52:26.280 point if you've been doing something else, which 00:52:26.280 --> 00:52:30.270 I can't tell, a good thing. 00:52:30.270 --> 00:52:34.440 So converting a given grammar to a pushdown automaton, 00:52:34.440 --> 00:52:36.070 how is that going to work? 00:52:36.070 --> 00:52:38.130 So the idea is-- 00:52:38.130 --> 00:52:41.720 OK-- actually, before I tell you the idea, 00:52:41.720 --> 00:52:45.100 let's just think about it together. 00:52:45.100 --> 00:52:47.710 Again, I like to think about the pushdown automaton, 00:52:47.710 --> 00:52:51.210 building a pushdown automaton the way you would do it. 00:52:51.210 --> 00:52:54.860 So a grammar is a generation device. 00:52:54.860 --> 00:52:56.420 It generates strings. 00:52:56.420 --> 00:52:59.480 A pushdown automaton-- or thinking about it as you, 00:52:59.480 --> 00:53:00.755 you are recognizing it. 00:53:00.755 --> 00:53:02.630 You are given an input, and you want to know, 00:53:02.630 --> 00:53:03.770 is it in the language? 00:53:03.770 --> 00:53:07.850 So you want to know, is it possible for that grammar 00:53:07.850 --> 00:53:10.290 to generate that input you're given? 00:53:10.290 --> 00:53:11.750 So how are you going-- 00:53:11.750 --> 00:53:14.270 how are you going to do that? 00:53:14.270 --> 00:53:22.480 And how are you going to test if the input is 00:53:22.480 --> 00:53:25.408 in the language of the grammar? 00:53:25.408 --> 00:53:27.200 Well, the thing that you would naturally do 00:53:27.200 --> 00:53:30.178 is you'd say, well, can I derive that string using 00:53:30.178 --> 00:53:31.220 the rules of the grammar? 00:53:31.220 --> 00:53:33.320 Let me start with the start string 00:53:33.320 --> 00:53:36.650 and try to do substitutions and see 00:53:36.650 --> 00:53:39.620 if I get the string I'm given. 00:53:39.620 --> 00:53:43.540 And if I can get it, then I know it's in the language. 00:53:43.540 --> 00:53:44.990 That's a natural thing to do. 00:53:44.990 --> 00:53:48.430 You're just going to try to do the substitution see 00:53:48.430 --> 00:53:49.670 if you can get to the string. 00:53:49.670 --> 00:53:52.930 Now the thing is, there might be many different substitutions 00:53:52.930 --> 00:53:54.170 that you could make. 00:53:54.170 --> 00:53:57.413 And that seems like a really challenging, hard thing 00:53:57.413 --> 00:53:59.080 to figure out which substitutions to use 00:53:59.080 --> 00:54:00.640 among the many possibilities. 00:54:00.640 --> 00:54:02.260 That's where non-determinism is going 00:54:02.260 --> 00:54:05.720 to come in because you can think of yourself as guessing 00:54:05.720 --> 00:54:06.930 which substitutions to make. 00:54:06.930 --> 00:54:09.090 And you're always going to make the right guess. 00:54:09.090 --> 00:54:12.030 So the choices of which substitutions to make, 00:54:12.030 --> 00:54:14.220 that's not going to be a problem for you. 00:54:14.220 --> 00:54:16.360 That's going to be managed by the non-determinism. 00:54:16.360 --> 00:54:19.260 So imagine you're always going to make the right substitution, 00:54:19.260 --> 00:54:21.390 but now the challenge is, how do you 00:54:21.390 --> 00:54:24.660 keep track of the intermediate results 00:54:24.660 --> 00:54:26.490 as you're doing those substitutions? 00:54:30.310 --> 00:54:33.910 And that's where the stack is going to come in. 00:54:33.910 --> 00:54:37.360 The machine is going to write down those intermediate results 00:54:37.360 --> 00:54:39.480 on the stack. 00:54:39.480 --> 00:54:42.210 But even there, there's a subtlety. 00:54:42.210 --> 00:54:45.160 That's an important subtlety that you have to look at. 00:54:45.160 --> 00:54:47.710 So let's try pulling that together 00:54:47.710 --> 00:54:51.090 so far before I get to that subtlety. 00:54:51.090 --> 00:54:57.280 So as I mentioned, the pushdown automaton 00:54:57.280 --> 00:54:59.500 is going to start out with a starting variable 00:54:59.500 --> 00:55:02.110 and is going to be guessing the substitutions to make. 00:55:02.110 --> 00:55:04.390 It's going to keep the intermediate results 00:55:04.390 --> 00:55:05.650 on the stack. 00:55:05.650 --> 00:55:07.790 When it's done doing all the substitutions 00:55:07.790 --> 00:55:10.180 and it has only terminal strings on the stack, 00:55:10.180 --> 00:55:11.770 it can compare with the input and see 00:55:11.770 --> 00:55:15.100 if it got the right thing, so if it made all the right guesses. 00:55:15.100 --> 00:55:17.590 So you think of it as guessing, doing 00:55:17.590 --> 00:55:19.060 the right guesses, but in the end 00:55:19.060 --> 00:55:21.730 you have to check to make sure that you got the right-- 00:55:21.730 --> 00:55:23.147 that you did all the right things, 00:55:23.147 --> 00:55:26.950 and you will accept when things have matched up, 00:55:26.950 --> 00:55:28.497 and you made all the right guesses. 00:55:28.497 --> 00:55:30.580 So in the end, you have to check that you actually 00:55:30.580 --> 00:55:37.230 got that input from doing those substitutions. 00:55:37.230 --> 00:55:41.940 So let's try to see this operating in action. 00:55:41.940 --> 00:55:47.835 And then you'll see the subtlety, the delicacy, 00:55:47.835 --> 00:55:50.010 a problem that's going to arise. 00:55:50.010 --> 00:55:52.140 Hopefully you're following, at least in part, 00:55:52.140 --> 00:55:54.270 what I'm saying so far. 00:55:54.270 --> 00:55:56.520 So here is the input. 00:55:56.520 --> 00:55:59.040 We do know that that's an input in the language 00:55:59.040 --> 00:56:01.538 that-- we've been seeing this example multiple times. 00:56:01.538 --> 00:56:03.330 So here is the input appearing on the input 00:56:03.330 --> 00:56:06.480 tape, a plus a times a. 00:56:06.480 --> 00:56:08.510 And now the pushdown automaton is 00:56:08.510 --> 00:56:11.088 supposed to be accepting that input because it's 00:56:11.088 --> 00:56:12.380 in the language of the grammar. 00:56:14.920 --> 00:56:18.160 So it's going to operate by first writing, 00:56:18.160 --> 00:56:22.690 to start off, the starting variable on the stack, 00:56:22.690 --> 00:56:25.960 and then doing the substitutions as we're going along. 00:56:25.960 --> 00:56:33.020 So we're going to substitute E goes to E plus T. 00:56:33.020 --> 00:56:35.990 So we do that first substitution. 00:56:35.990 --> 00:56:40.740 And then we do the next substitution here, the E-- 00:56:40.740 --> 00:56:43.640 so if you're looking at this tree here, 00:56:43.640 --> 00:56:48.450 this is the right tree for that input. 00:56:48.450 --> 00:56:52.790 So we substitute E by T. So far so good. 00:56:52.790 --> 00:56:55.760 The automaton can make that substitution. 00:56:55.760 --> 00:56:59.330 Then the next substitution is going to be a little-- 00:56:59.330 --> 00:57:02.630 so where E plus T, we substitute here. 00:57:02.630 --> 00:57:06.410 We got T plus T. And now we're going to substitute the T times 00:57:06.410 --> 00:57:08.630 F, which is this T over here. 00:57:08.630 --> 00:57:10.340 We want to substitute that. 00:57:10.340 --> 00:57:15.410 And that appears as T times as F now on the stack. 00:57:15.410 --> 00:57:17.680 Now, if you're following me, you should 00:57:17.680 --> 00:57:25.260 be suddenly getting nervous because we just cheated. 00:57:25.260 --> 00:57:30.150 It's OK doing substitution-- doing these replacements 00:57:30.150 --> 00:57:33.570 at the very top of the stack because the pushdown automaton 00:57:33.570 --> 00:57:35.190 has access to the top. 00:57:35.190 --> 00:57:37.570 That's how stacks work. 00:57:37.570 --> 00:57:41.140 But it does not have access deep down within this deck. 00:57:41.140 --> 00:57:44.540 That is not how-- 00:57:44.540 --> 00:57:45.830 that is not how stacks work. 00:57:45.830 --> 00:57:47.640 So that's cheating. 00:57:47.640 --> 00:57:50.460 But ignoring the cheating for the minute, 00:57:50.460 --> 00:57:54.300 if you could replace those-- 00:57:54.300 --> 00:57:57.180 do those substitutions deep down within the stack, 00:57:57.180 --> 00:57:58.110 this would all work. 00:57:58.110 --> 00:57:58.920 We would be good. 00:57:58.920 --> 00:58:01.980 You would do the substitutions one after another 00:58:01.980 --> 00:58:05.790 until you ended up with no variables, 00:58:05.790 --> 00:58:07.350 and then you have the string here, 00:58:07.350 --> 00:58:09.720 and you're going to match it off and compare it with the input. 00:58:09.720 --> 00:58:11.970 It's all done in the right way so that the things are 00:58:11.970 --> 00:58:16.800 in the right order so after all the substitutions, 00:58:16.800 --> 00:58:19.567 you'd have a plus a times a sitting here in the stack, 00:58:19.567 --> 00:58:21.900 you compare that with the input, it's going to match up, 00:58:21.900 --> 00:58:24.120 and you'll end up accepting, all good. 00:58:24.120 --> 00:58:26.235 So how do we deal with that problem here? 00:58:26.235 --> 00:58:30.870 The problem, access below the top of stack is cheating. 00:58:30.870 --> 00:58:33.580 What are we going to do instead? 00:58:33.580 --> 00:58:35.520 So the idea is actually pretty simple. 00:58:35.520 --> 00:58:39.180 Well, if you understood what I've said so far, fixing 00:58:39.180 --> 00:58:40.890 that is actually not too bad. 00:58:45.060 --> 00:58:47.715 Sort of fading out here. 00:58:47.715 --> 00:58:52.150 I'll put some more light on my image. 00:58:52.150 --> 00:58:55.480 So how do we do that? 00:58:55.480 --> 00:58:57.720 How do we get the effect of the access 00:58:57.720 --> 00:58:59.980 below the top of the stack? 00:58:59.980 --> 00:59:03.250 And the way we're going to do that is 00:59:03.250 --> 00:59:06.820 by making the-- what we're going to do, we're only going to do 00:59:06.820 --> 00:59:09.208 substitutions that we can make at the top of the stack. 00:59:09.208 --> 00:59:11.500 So whenever there's a variable at the top of the stack, 00:59:11.500 --> 00:59:12.917 we're going to do the substitution 00:59:12.917 --> 00:59:16.000 because the top we can access. 00:59:16.000 --> 00:59:18.430 Now what happens if we have a terminal symbol sitting 00:59:18.430 --> 00:59:23.590 at the top blocking our way from accessing variables below? 00:59:23.590 --> 00:59:25.810 Well, actually, that's an easy case 00:59:25.810 --> 00:59:28.270 to handle because we have a terminal symbol sitting 00:59:28.270 --> 00:59:30.680 at the top, they're never going to change anyway, 00:59:30.680 --> 00:59:34.120 so you might as well match them with the input at that time. 00:59:37.340 --> 00:59:40.330 So when you have a terminal sitting at the top, 00:59:40.330 --> 00:59:45.130 we'll just read another input symbol and match it off there. 00:59:45.130 --> 00:59:47.470 And we just keep reading the terminal symbols off 00:59:47.470 --> 00:59:49.690 until we have a variable sitting on the top, 00:59:49.690 --> 00:59:53.593 then we do a substitution. 00:59:53.593 --> 00:59:55.010 And we keep substituting variables 00:59:55.010 --> 00:59:56.760 until we have a terminal, then we read it, 00:59:56.760 --> 00:59:58.260 then we compare it with the input. 00:59:58.260 --> 01:00:02.120 And in so doing, we will end up getting the same effect 01:00:02.120 --> 01:00:04.130 that I described before without ever 01:00:04.130 --> 01:00:08.400 needing to dig down into the interior of the stack 01:00:08.400 --> 01:00:09.650 and doing substitutions there. 01:00:09.650 --> 01:00:11.960 They're all going to rise up to the top. 01:00:11.960 --> 01:00:16.590 And we can always do them at the top. 01:00:16.590 --> 01:00:19.780 So anyway, I forgot to do that here. 01:00:19.780 --> 01:00:21.930 So instead, only substitute variables 01:00:21.930 --> 01:00:24.090 when they're at the top of the stack. 01:00:24.090 --> 01:00:27.090 If a terminal's on the top, pop it and compare with the input, 01:00:27.090 --> 01:00:28.510 reject it if they're not equal. 01:00:28.510 --> 01:00:30.450 So if you ever have something which is not 01:00:30.450 --> 01:00:31.867 matching the way it's supposed to, 01:00:31.867 --> 01:00:35.070 I mean, that thread's just going to fail. 01:00:35.070 --> 01:00:39.437 Then there was not a-- a bad non-deterministic choice was 01:00:39.437 --> 01:00:41.770 made, or maybe the input was not in the language anyway, 01:00:41.770 --> 01:00:44.980 and there were no good choices. 01:00:44.980 --> 01:00:48.900 So my animation broke here, so let me just put the whole thing 01:00:48.900 --> 01:00:50.260 up in front of you. 01:00:50.260 --> 01:00:54.790 So here is the actual construction. 01:00:54.790 --> 01:00:57.070 Push the start symbol on the stack. 01:00:57.070 --> 01:00:59.650 If the top of the stack is a variable, 01:00:59.650 --> 01:01:02.320 replace it with a corresponding right-hand side, 01:01:02.320 --> 01:01:04.090 doing a non-deterministic choice among 01:01:04.090 --> 01:01:05.980 the various possibilities. 01:01:05.980 --> 01:01:07.780 If it's a terminal, you pop it and match it 01:01:07.780 --> 01:01:08.905 with the next input symbol. 01:01:08.905 --> 01:01:11.450 And if the stack is empty, you accept. 01:01:11.450 --> 01:01:13.640 So here is how the stack would actually 01:01:13.640 --> 01:01:16.613 look for this particular input. 01:01:16.613 --> 01:01:17.780 It would start off the same. 01:01:17.780 --> 01:01:20.330 You'd have E, and then substitute with E plus T. 01:01:20.330 --> 01:01:23.750 And then we're going to always do 01:01:23.750 --> 01:01:25.670 the substitutions at the top. 01:01:25.670 --> 01:01:28.670 So E gets substituted by F 01:01:28.670 --> 01:01:29.870 Is that right? 01:01:29.870 --> 01:01:30.920 No. 01:01:30.920 --> 01:01:32.630 This slide I messed up. 01:01:32.630 --> 01:01:33.360 I apologize. 01:01:33.360 --> 01:01:38.210 So E gets substituted by T, which gets substituted by F. 01:01:38.210 --> 01:01:40.070 And the point is that when you get 01:01:40.070 --> 01:01:42.170 to an A sitting at the top-- 01:01:42.170 --> 01:01:46.070 forgive the typos here, now we have a terminal symbol, 01:01:46.070 --> 01:01:50.760 and that's going to get matched off with the next input symbol 01:01:50.760 --> 01:01:51.780 and get removed. 01:01:51.780 --> 01:01:54.450 And now we have just the plus and the T left. 01:01:54.450 --> 01:01:56.393 And then the plus is also a terminal symbol. 01:01:56.393 --> 01:01:58.560 That's going to get matched off with the next thing. 01:01:58.560 --> 01:02:00.130 We just have a T sitting on the top, 01:02:00.130 --> 01:02:02.800 and now we can do a substitution. 01:02:02.800 --> 01:02:06.250 So that's how it works. 01:02:06.250 --> 01:02:08.445 That's all I wanted to say, I think. 01:02:08.445 --> 01:02:10.870 Oh, yeah, there's one just remark. 01:02:10.870 --> 01:02:12.380 So we're not going to prove this, 01:02:12.380 --> 01:02:16.300 but I do need to say this, that actually you 01:02:16.300 --> 01:02:20.290 can do the conversion in the other direction, too. 01:02:20.290 --> 01:02:22.030 You can convert a-- 01:02:25.010 --> 01:02:28.640 so a is a context-free language if and only 01:02:28.640 --> 01:02:32.360 if some pushdown automaton recognizes a. 01:02:32.360 --> 01:02:37.850 And if you haven't seen if and only if, it's an expression 01:02:37.850 --> 01:02:41.930 I'm going to use, again, over and over. 01:02:41.930 --> 01:02:43.310 So you should get used to it. 01:02:43.310 --> 01:02:45.500 It stands for "if and only if," and which just 01:02:45.500 --> 01:02:47.700 means the implication goes both ways. 01:02:47.700 --> 01:02:51.260 So "a is a context-free language" 01:02:51.260 --> 01:02:53.900 implies that some pushdown automaton recognizes 01:02:53.900 --> 01:02:56.910 a and vice versa. 01:02:56.910 --> 01:02:59.315 So there's really two things you need to prove whenever 01:02:59.315 --> 01:03:00.440 you have an if and only if. 01:03:00.440 --> 01:03:03.360 You have to prove both directions. 01:03:03.360 --> 01:03:07.610 So thinking about it that way, splitting them in half, 01:03:07.610 --> 01:03:10.490 the forward direction we've already proved, 01:03:10.490 --> 01:03:12.110 converting a pushdown-- 01:03:12.110 --> 01:03:16.280 a context-free grammar to a pushdown automaton. 01:03:16.280 --> 01:03:18.320 The reverse direction we're not going to prove. 01:03:18.320 --> 01:03:20.480 It's in the book if you're curious, 01:03:20.480 --> 01:03:23.390 and you are responsible knowing that the fact is true, 01:03:23.390 --> 01:03:26.668 but you don't have to know the proof, which is a somewhat-- 01:03:26.668 --> 01:03:27.710 a little bit complicated. 01:03:27.710 --> 01:03:30.740 And I think it would take us too long to go through it, 01:03:30.740 --> 01:03:34.830 so you're not responsible for it. 01:03:34.830 --> 01:03:36.620 So there's a last check in here that I 01:03:36.620 --> 01:03:40.670 have for you, which is just a question, which 01:03:40.670 --> 01:03:42.350 you can answer based on the material 01:03:42.350 --> 01:03:44.000 that we presented so far. 01:03:44.000 --> 01:03:49.150 Is every regular language also a context-free language? 01:03:49.150 --> 01:03:52.300 Just "yes," "no," or you're not sure. 01:03:52.300 --> 01:03:56.040 So let me launch that as a poll here. 01:03:56.040 --> 01:03:57.075 OK, about to close. 01:03:59.900 --> 01:04:04.010 A new polling, and sharing results. 01:04:06.600 --> 01:04:08.970 This one, I think, you pretty much-- 01:04:08.970 --> 01:04:10.170 most of you got. 01:04:10.170 --> 01:04:14.550 Some of you are not sure. 01:04:14.550 --> 01:04:19.560 Every language is in fact a context-free language. 01:04:19.560 --> 01:04:22.800 And the way to see that is that every regular language 01:04:22.800 --> 01:04:27.600 can be done by a DFA or an NFA, as we already showed. 01:04:27.600 --> 01:04:32.400 And a DFA or an NFA is really just a pushdown automaton 01:04:32.400 --> 01:04:35.430 that never uses its stack. 01:04:35.430 --> 01:04:40.080 So you can always think of a DFA as a pushdown automaton. 01:04:40.080 --> 01:04:43.050 And we already argue that pushdown automata are 01:04:43.050 --> 01:04:45.120 equivalent to context-free grammars, 01:04:45.120 --> 01:04:48.170 and so they do the context-free languages. 01:04:48.170 --> 01:04:50.360 So anything that you can do with a DFA 01:04:50.360 --> 01:04:52.490 you can also do with a pushdown automaton, 01:04:52.490 --> 01:04:57.380 and so therefore all the regular languages are also 01:04:57.380 --> 01:04:59.540 context-free languages. 01:04:59.540 --> 01:05:04.220 So with that, let's just pull things together, 01:05:04.220 --> 01:05:06.050 a little quick recap as to what we've 01:05:06.050 --> 01:05:07.640 been doing so far in the class. 01:05:07.640 --> 01:05:10.970 We have the regular languages and the context-free languages. 01:05:10.970 --> 01:05:13.550 We had the two forms of getting at them, 01:05:13.550 --> 01:05:16.340 the recognizer form, which is like the automata-based 01:05:16.340 --> 01:05:19.250 perspective, like either a DFA or an NFA 01:05:19.250 --> 01:05:21.620 in the case of the regular languages, the pushdown 01:05:21.620 --> 01:05:24.410 automaton for the context-free languages. 01:05:24.410 --> 01:05:28.400 And for the generators, we had the regular expression 01:05:28.400 --> 01:05:31.220 for the regular languages and the context-free grammars-- 01:05:31.220 --> 01:05:35.390 for the context-free languages. 01:05:35.390 --> 01:05:39.440 And as we just pointed out in our last check-in, 01:05:39.440 --> 01:05:42.560 the regular languages form a subset and, in fact, 01:05:42.560 --> 01:05:46.130 a proper subset of the context-free languages, 01:05:46.130 --> 01:05:48.650 as shown in this Venn diagram because we have already 01:05:48.650 --> 01:05:51.860 exhibited languages that are context-free but not regular. 01:05:55.330 --> 01:05:56.650 So a quick review. 01:05:56.650 --> 01:05:58.750 We've defined the context-free grammars 01:05:58.750 --> 01:06:01.690 and their associated languages, the context-free languages; 01:06:01.690 --> 01:06:04.180 we defined pushdown automata; and we 01:06:04.180 --> 01:06:06.430 showed how to convert context-free grammars 01:06:06.430 --> 01:06:09.290 to pushdown automata. 01:06:09.290 --> 01:06:11.842 And that's all I have for you today. 01:06:11.842 --> 01:06:13.800 But here's a question I'll answer to everybody. 01:06:13.800 --> 01:06:15.467 Why do we restrict ourselves to a stack? 01:06:15.467 --> 01:06:17.170 Why don't we use random access memory? 01:06:17.170 --> 01:06:21.570 We will use random access memory for the next model, 01:06:21.570 --> 01:06:22.650 called a Turing machine. 01:06:22.650 --> 01:06:26.380 And we're going to introduce that I think the next lecture. 01:06:26.380 --> 01:06:29.160 So that's going to be the model that we're 01:06:29.160 --> 01:06:30.930 going to stick with throughout the term. 01:06:30.930 --> 01:06:33.480 But we have not-- 01:06:33.480 --> 01:06:38.790 we were introducing weaker models as a kind of prelude 01:06:38.790 --> 01:06:44.940 to the more general-purpose computational model really 01:06:44.940 --> 01:06:52.110 to get ourselves warmed up and also because, for the weaker 01:06:52.110 --> 01:06:57.775 models, you can fully analyze them in a way that you cannot 01:06:57.775 --> 01:07:00.520 for Turing machines. 01:07:00.520 --> 01:07:06.940 As you will see, you can determine properties 01:07:06.940 --> 01:07:09.760 of languages for the weaker models that you cannot 01:07:09.760 --> 01:07:11.680 for the more general models. 01:07:11.680 --> 01:07:15.940 And so I think that's helpful to have that perspective, 01:07:15.940 --> 01:07:20.590 that for some cases, you can get a full analysis, 01:07:20.590 --> 01:07:23.110 in some other cases you cannot. 01:07:23.110 --> 01:07:27.100 But anyway, that's the reason why we restrict it 01:07:27.100 --> 01:07:29.410 to the stack, besides the fact that these models have 01:07:29.410 --> 01:07:36.160 applications that I think are worth people seeing. 01:07:36.160 --> 01:07:39.610 Why-- yeah, for some reason we chose a stack? 01:07:39.610 --> 01:07:41.680 Well, why did we choose a stack and not 01:07:41.680 --> 01:07:45.070 some other data structure for our temporary-- 01:07:45.070 --> 01:07:47.590 for our extra storage? 01:07:47.590 --> 01:07:51.460 And the reason for a stack, for one thing, the stack 01:07:51.460 --> 01:07:53.720 is exactly what you need to get you the correspondence 01:07:53.720 --> 01:07:54.845 with context-free grammars. 01:07:57.750 --> 01:08:03.280 If we used some other storage, like a queue, 01:08:03.280 --> 01:08:05.740 for example, instead of a stack, in fact, 01:08:05.740 --> 01:08:07.540 you get a very different outcome. 01:08:07.540 --> 01:08:11.545 And it's an actually interesting exercise to see what happens. 01:08:11.545 --> 01:08:15.010 What do you get if you use a queue as an external storage 01:08:15.010 --> 01:08:17.042 instead of as a stack? 01:08:17.042 --> 01:08:18.250 It's a good homework problem. 01:08:18.250 --> 01:08:21.189 Maybe I'll assign it. 01:08:21.189 --> 01:08:21.984 Let's see. 01:08:24.660 --> 01:08:26.859 And if a-- 01:08:26.859 --> 01:08:28.840 OK. 01:08:28.840 --> 01:08:35.109 We showed-- so we showed that non-determinism can be 01:08:35.109 --> 01:08:36.800 eliminated for finite automata. 01:08:36.800 --> 01:08:39.729 So NFAs and DFAs are equivalent. 01:08:39.729 --> 01:08:42.432 What about for pushdown automata? 01:08:42.432 --> 01:08:44.140 The answer is no, they're not equivalent. 01:08:44.140 --> 01:08:45.515 I think I mentioned that earlier, 01:08:45.515 --> 01:08:47.529 but I don't mind repeating it. 01:08:47.529 --> 01:08:50.710 There are certain languages that can be done only with 01:08:50.710 --> 01:08:53.680 non-deterministic pushdown automata and cannot be done 01:08:53.680 --> 01:08:55.750 with deterministic pushdown automata, 01:08:55.750 --> 01:08:59.800 for example that language ww reverse that requires 01:08:59.800 --> 01:09:02.620 the non-determinism in order for the machine to be able to guess 01:09:02.620 --> 01:09:04.060 where the middle is. 01:09:04.060 --> 01:09:07.149 So, OK, I'm going to head off. 01:09:07.149 --> 01:09:10.529 Thank you, guys, and see you on Tuesday.