Other Models of Computation

Other Models of Computation - CS311 Computational...

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CS311 Computational Structures Other models of Computation 1 Lecture 13 Andrew Black Andrew Tolmach
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What is Computation? What does “computable” mean? A computer can calculate it? There is some (formally described execution) process and a (formally described) set of instructions — an algorithm — that describes how to get the answer 2
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Examples: Generating strings from a grammars: derivation is the process; the algorithm is encoded in the rules of the grammar Accepting a string in a state machine Executing an ML program These are all Models of Computation 3
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We know that some models of computation are more powerful than others: CFG are more powerful than Regular Grammars DFAs have the same power as NFAs Turing machines are more powerful than PDAs Is there a “most powerful model” 4 The Power of a Model
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Turing ʼ s Thesis Our intuitive notion of “computation” is precisely captured by the formal device known as a Turing Machine There is no model of computation more powerful than a Turing machine 5
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Steam-powered Turing machine Sieg Hall, 1987
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Recall: Church-Turing Thesis Thesis: the problems that can be decided by an algorithm are exactly those that can be decided by a Turing machine. This cannot be proved; it is essentially a defnition of the word “algorithm.” But there ʼ s lots of evidence that our intuitive notion of algorithm is equivalent to a TM, and no convincing counter- examples have been found yet. 7
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What about Alonzo Church? The Turing thesis is usually called the Church-Turing Thesis, in honor of Alonzo Church (1903–1995) Working with his students (J. Barkley Rosser, Steven C. Kleene, and Alan M. Turing) Church established the equivalence of the Lambda calculus , recursive function theory , and Turing machines They all capture the notion of computability 8
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Other Notions of Computability Many other notions of computability have been proposed, e.g. Grammars Partial Recursive Functions Lambda calculus Markov Algorithms Post Algorithms Post Canonical Systems Simple programming language with while loops All have been shown equivalent to Turing machines by simulation proofs 9
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Simple (Hein p 776) A Simple program is defned as Follows: 1. V, an infnite set oF variables that take values in Nat 0 , and are initially 0 2. S, statements, which are either 2.1. While statements: while V 0 do S od 2.2. Assignments: V := 0, V 1 := succ(V 2 ), V 1 := pred(V 2 ), or 2.3. a sequence oF statements separated by ; 3. S is a simple program 10
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Note that pred(0) = 0, to ensure that we stay in 0 Can we compute anything interesting with this language? yes!
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Other Models of Computation - CS311 Computational...

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