WEBVTT 00:00:00.429 --> 00:00:05.259 There are many other models of computation, each of which describes a class of integer 00:00:05.259 --> 00:00:10.210 functions where a computation is performed on an integer input to produce an integer 00:00:10.210 --> 00:00:11.210 answer. 00:00:11.210 --> 00:00:16.650 Kleene, Post and Turing were all students of Alonzo Church at Princeton University in 00:00:16.650 --> 00:00:18.990 the mid-1930's. 00:00:18.990 --> 00:00:22.860 They explored many other formulations for modeling computation: 00:00:22.860 --> 00:00:28.820 recursive functions, rule-based systems for string rewriting, and the lambda calculus. 00:00:28.820 --> 00:00:34.120 They were all particularly intrigued with proving the existence of problems unsolvable 00:00:34.120 --> 00:00:36.220 by realizable machines. 00:00:36.220 --> 00:00:41.700 Which, of course, meant characterizing the problems that could be solved by realizable 00:00:41.700 --> 00:00:43.900 machines. 00:00:43.900 --> 00:00:48.850 It turned out that each model was capable of computing *exactly* the same set of integer 00:00:48.850 --> 00:00:50.350 functions! 00:00:50.350 --> 00:00:54.870 This was proved by coming up with constructions that translated the steps in a computation 00:00:54.870 --> 00:00:57.420 between the various models. 00:00:57.420 --> 00:01:02.460 It was possible to show that if a computation could be described by one model, an equivalent 00:01:02.460 --> 00:01:06.530 description exists in the other model. 00:01:06.530 --> 00:01:11.390 This lead to a notion of computability that was independent of the computation scheme 00:01:11.390 --> 00:01:13.590 chosen. 00:01:13.590 --> 00:01:19.650 This notion is formalized by Church's Thesis, which says that every discrete function computable 00:01:19.650 --> 00:01:24.789 by any realizable machine is computable by some Turing Machine. 00:01:24.789 --> 00:01:30.610 So if we say the function f(x) is computable, that's equivalent to saying that there's a 00:01:30.610 --> 00:01:40.200 TM that given x as an input on its tape will write f(x) as an output on the tape and halt. 00:01:40.200 --> 00:01:45.890 As yet there's no proof of Church's Thesis, but it's universally accepted that it's true. 00:01:45.890 --> 00:01:51.440 In general "computable" is taken to mean "computable by some TM". 00:01:51.440 --> 00:01:56.590 If you're curious about the existence of uncomputable functions, please see the optional video at 00:01:56.590 --> 00:01:58.960 the end of this lecture. 00:01:58.960 --> 00:02:05.580 Okay, we've decided that Turing Machines can model any realizable computation. 00:02:05.580 --> 00:02:10.229 In other words for every computation we want to perform, there's a (different) Turing Machine 00:02:10.229 --> 00:02:12.169 that will do the job. 00:02:12.169 --> 00:02:16.170 But how does this help us design a general-purpose computer? 00:02:16.170 --> 00:02:22.200 Or are there some computations that will require a special-purpose machine no matter what? 00:02:22.200 --> 00:02:25.900 What we'd like to find is a universal function U: 00:02:25.900 --> 00:02:32.930 it would take two arguments, k and j, and then compute the result of running T_k on 00:02:32.930 --> 00:02:34.290 input j. 00:02:34.290 --> 00:02:40.319 Is U computable, i.e., is there a universal Turing Machine T_U? 00:02:40.319 --> 00:02:47.780 If so, then instead of many ad-hoc TMs, we could just use T_U to compute the results 00:02:47.780 --> 00:02:50.409 for any computable function. 00:02:50.409 --> 00:02:52.349 .249 Surprise! 00:02:52.349 --> 00:02:55.389 U is computable and T_U exists. 00:02:55.389 --> 00:03:01.110 If fact there are infinitely many universal TMs, some quite simple. 00:03:01.110 --> 00:03:07.129 The smallest known universal TM has 4 states and uses 6 tape symbols. 00:03:07.129 --> 00:03:12.269 A universal machine is capable of performing any computation that can be performed by any 00:03:12.269 --> 00:03:15.480 TM! 00:03:15.480 --> 00:03:17.099 What's going on here? 00:03:17.099 --> 00:03:25.500 k encodes a "program" - a description of some arbitrary TM that performs a particular computation. 00:03:25.500 --> 00:03:29.859 j encodes the input data on which to perform that computation. 00:03:29.859 --> 00:03:37.799 T_U "interprets" the program, emulating the steps T_k will take to process the input and 00:03:37.799 --> 00:03:40.329 write out the answer. 00:03:40.329 --> 00:03:46.819 The notion of interpreting a coded representation of a computation is a key idea and forms the 00:03:46.819 --> 00:03:51.879 basis for our stored program computer. 00:03:51.879 --> 00:03:57.439 The Universal Turing Machine is the paradigm for modern general-purpose computers. 00:03:57.439 --> 00:04:02.790 Given an ISA we want to know if it's equivalent to a universal Turing Machine. 00:04:02.790 --> 00:04:10.439 If so, it can emulate every other TM and hence compute any computable function. 00:04:10.439 --> 00:04:13.849 How do we show our computer is Turing Universal? 00:04:13.849 --> 00:04:18.870 Simply demonstrate that it can emulate some known Universal Turing Machine. 00:04:18.870 --> 00:04:24.780 The finite memory on actual computers will mean we can only emulate UTM operations on 00:04:24.780 --> 00:04:29.840 inputs up to a certain size but within this limitation we can show our 00:04:29.840 --> 00:04:35.020 computer can perform any computation that fits into memory. 00:04:35.020 --> 00:04:40.960 As it turns out this is not a high bar: so long as the ISA has conditional branches 00:04:40.960 --> 00:04:45.020 and some simple arithmetic, it will be Turing Universal. 00:04:45.020 --> 00:04:49.789 This notion of encoding a program in a way that allows it to be data to some other program 00:04:49.789 --> 00:04:52.110 is a key idea in computer science. 00:04:52.110 --> 00:04:58.340 We often translate a program Px written to run on some abstract high-level machine (eg, 00:04:58.340 --> 00:05:05.060 a program in C or Java) into, say, an assembly language program Py that can be interpreted 00:05:05.060 --> 00:05:06.740 by our CPU. 00:05:06.740 --> 00:05:10.389 This translation is called compilation. 00:05:10.389 --> 00:05:14.800 Much of software engineering is based on the idea of taking a program and using it as as 00:05:14.800 --> 00:05:19.110 component in some larger program. 00:05:19.110 --> 00:05:24.159 Given a strategy for compiling programs, that opens the door to designing new programming 00:05:24.159 --> 00:05:29.720 languages that let us express our desired computation using data structures and operations 00:05:29.720 --> 00:05:34.139 particularly suited to the task at hand. 00:05:34.139 --> 00:05:39.229 So what have learned from the mathematicians' work on models of computation? 00:05:39.229 --> 00:05:44.280 Well, it's nice to know that the computing engine we're planning to build will be able 00:05:44.280 --> 00:05:49.030 to perform any computation that can be performed on any realizable machine. 00:05:49.030 --> 00:05:55.419 And the development of the universal Turing Machine model paved the way for modern stored-program 00:05:55.419 --> 00:05:56.729 computers. 00:05:56.729 --> 00:06:00.589 The bottom line: we're good to go with the Beta ISA!