# 38.pdf - CMPSCI 250 Introduction to Computation Lecture#38...

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CMPSCI 250: Introduction to Computation Lecture #38: Turing Machine Semantics David Mix Barrington 8 December 2017

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Turing Machine Semantics Review: A Turing Machine Example Turing Recognizable Languages Turing Decidable Languages The TR/TD Theorem Running Two Machines in Parallel Multitape Turing Machines Nondeterministic Turing Machines
A Turing Machine Example Here is a machine that solves a problem that a DFA cannot. When started in configuration i w 1 w 2 ...w n , it will halt if and only if w is in the language {a n b n : n 0} -- otherwise it will hang. With input aabb we get i aabb, paabb, ☐☐ qabb, ☐☐ aqbb, ☐☐ abqb, ☐☐ abbq , ☐☐ abrb, ☐☐ asb, ☐☐ sab, s ab, ☐☐ pab, ☐☐☐ qb, ☐☐☐ bq , ☐☐☐ rb, ☐☐ s , ☐☐☐ p , ☐☐☐ h . The string aabb is accepted.

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A Turing Machine Example In i: Move R and go to p. In p: On , go to h. On b, move L and go to z. On a, print , move R, and go to q. In q: On a or b, move R and stay in q. On , move L and go to r. In r: On a or , move L and go to z. On b, print , move L, and go to s. In s: On a or b, move L and stay in s. On , move R and go to p. In h: Halt (final state). In z: Move left and stay in z.
The Church-Turing Thesis The Church-Turing Thesis says that any “reasonable” general-purpose model of computation will be able to compute exactly the same functions from strings to strings as Turing machines or the lambda calculus. (More precisely, they compute the same set of partial functions , because a general computation always has the possibility of not returning an output.)

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The Church-Turing Thesis We can’t mathematically prove this thesis, only amass evidence for it. In fact it actually serves as an implicit definition of “reasonable”. Serious people have argued against the thesis -- for example physicist Roger Penrose argues that quantum effects in the brain compute in ways that a Turing machine could not. (He’s wrong.) For more on this see Turing’s article On Minds and Machines or almost anything by Douglas Hofstadter.
The Church-Turing Thesis You probably believe that we could simulate a Turing machine in Java, given unlimited memory. Could a Turing machine simulate any Java program? We know that Java can be compiled into machine language, so we would have to believe that any machine language program could be simulated by a TM.

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Turing Recognizable Languages We’ve seen that a Turing machine, when started on a string, may or may not ever reach a final state. We define the language of the machine M , called L(M), to be the set of strings on which M eventually halts. If a language X is equal to L(M) for some Turing machine M, we say that X is Turing recognizable . The idea is that the machine “recognizes” strings in the language by halting, but gives no output on strings not in the language.
Turing Recognizable Languages In the example earlier, the language of our TM is {a n b n : n 0}, a language that we now know is not regular.

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• Fall '09
• Turing Machines, multitape Turing machines, Turing decidable languages

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