LectureNotes21 - ECS 120 Lesson 21 Mapping Reducibility...

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ECS 120 Lesson 21 – Mapping Reducibility Oliver Kreylos Friday, May 18th, 2001 In the last two lectures, we have successfully used reduction techniques to prove several problems undecidable. Today we are going to refine our un- derstanding of reducibility, to be able to apply these techniques to a broader class of problems. 1 Computable Functions Until now, we only looked at Turing Machines as acceptors of languages: When given an input word w Σ * , a machine M either accepts the word or rejects (or loops). In other words, we viewed Turing Machines as computing functions f M : Σ * → { accept , reject } . In doing so, we ignored the fact that Turing Machines leave some string on their work tape when they finish. This becomes obvious from our definition of computation: A Turing Machine M accepts a word w , if and only if the initial configuration ( ±, q 0 , w ) is related to an accepting configuration ( u, q accept , v ), where u, v Γ * are arbitrary strings. By shifting our focus from the acceptance to the string left behind on the work tape, we can view Turing Machines as transducers (“computers”) by defining their output as follows: If a Turing Machine M halts (either accepting or rejecting), and the current tape content is a string 1 γ tt . .. , where x Σ * , γ Γ * , and γ 1 Γ \ Σ is a character not in the input alphabet, then the output of M is the string x . In other words, we consider as output the longest prefix of the current tape contents on halting that consists only of input characters. This definition of output is similar to the definition of input of a Turing Machine: The input word w Σ * consists only of input characters, and is delimited by the first blank character, which is not in Σ. Using this definition, the tape content of M on halting can directly be fed 1
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into another Turing Machine, and the second machine’s input would exactly be the first machine’s output. Since a (deterministic) Turing Machine M always produces the same out- put when given the same input, the input and output strings are related by a function f M : Σ * Σ * : On input w , machine M produces f M ( w ) as output. This leads to the following definition: Definition 1 (Computable Functions) A function f : Σ * Σ * is called computable , if and only if there exists a Turing Machine M that halts on any input word w Σ * and produces f ( w ) Σ * as output. We have already encountered transducing Turing Machines and com- putable functions several times, without actually pointing them out as such. For example, most reduction proofs seen so far consisted of an algorithm
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LectureNotes21 - ECS 120 Lesson 21 Mapping Reducibility...

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