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Understanding Reducibility of Turing Computable Functions and Polynomial Time Complexity

This text explores the concepts of computable functions, polynomial-time functions, and the mapping reducibility of languages. It defines a computable function as one where a Turing machine halts with a specific output on its tape. The discussion includes polynomial-time reducibility, decidability theorems, and recognizability theorems, illustrating relationships between languages A and B. Furthermore, it defines NP-completeness and outlines the criteria that a language must satisfy to be classified as NP-complete, emphasizing the implications of reductions in complexity theory.

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Understanding Reducibility of Turing Computable Functions and Polynomial Time Complexity

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  1. Patterns of Reducability

  2. Turing computable functions • A function *  * is a computable function if some Turing Machine M, in every input w, halts with just f(w) on its tape.

  3. Polynomial time function • A function f: *  * is a polynomial time computable function if some polynomial time TM, M, exists that halts with just F(w) on its tape, when started on any input w.

  4. Mapping reducability • A language A is mapping reducable to language B, written A ≤m B, if there is a computable function f : *  * , where for every w  *, w  A  F(w)  B

  5. Polynomial time reducability • A language, A, is polynomial time mapping reducible (or simply polynomial time reducible) to a language, B, written A ≤p B ,if a polynomial time computable function f: : *  * exists, where for every w, w  A  f(w)  B • The function f is called the polynomial time reduction of A to B

  6. Decidability Theorems • A ≤m B and B is decidable then A is decidable • A ≤m B and A is undecidable, then B is undecidable

  7. Recognizability Theorems • A ≤m B and B is Turing recognizable then A is Turing recognizable • A ≤m B and A is not Turing recognizable then B is not Turing recognizable • Typically we let A be ATMthe complement of ATM

  8. Definition • A language B is NP-complete if it satisfies 2 conditions • B is n NP, and • Every A in NP is polynomial time reducable to B Forall A  NP . A ≤p B

  9. P or NP-complete Theorems • To show a language is in P • A ≤p B and BP then AP • To Show a language is NP-complete • If B is NP-complete and B ≤p C, for C  NP, then C in NP-complete • The most common “B” is the language booleansatifiability

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