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16 - 16 Turing Machines as Language Recognizers Hereafter...

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16. Turing Machines as Language Recognizers Hereafter, we shall ssume that the initial configuration of M on input w is ( s, w ). Let M = ( K, Σ , δ, s, H ) be a Turing Machine such that H = { y, n } where y means “yes”, n means “no”. Σ 0 Σ − { , } is the input alphabet of M . Let L Σ 0 be a language. On input w : If M halts at y , we say M accepts w . If M halts at n , we say M rejects w . If M loops forever, M neither accepts nor rejects w . 1
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Recursive Languages We say M decides L if for any string w Σ 0 , the following is true: if w L , then M accepts w (i.e. M halts at state y ). if w / L , then M rejects w (i.e. M halts at state n ). This means, M halts on all input and it correctly halts at either y or n depending on whether or not the input w is in L . Definition: A language L is said to be recursive (also known as Turing- decidable) if and only if there is a TM that decides it. Turing called such languages recursive because he believed that any calculation that could be defined recursively by algorithms could be performed by Turing machines. 2
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Recursive Languages Example: Prove that the following language is recursive. L = { a n b n c n : n 0 } d a b b c c d a d b c c d a d b d c L a, y : changes state to y n: changes state to n Two new basic machines: b c d d R R d R n y a b, c c, a,d b,d a a b b c c d R a, b c An informal description: 1. Start from the left end of the string. 2. Move to the right in search of an a ; if found, replace it by a d ; if b or c is found instead, reject .
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