# Ch11 - Chapters 11 and 12 Decision Problems and...

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Chapters 11 and 12 Decision Problems and Undecidability

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2 11.1 Decision Problems A decision problem consists of a set of questions whose answers are either yes or no is undecidable if no algorithm that can solve the problem; otherwise, it is decidable The Church-Turing thesis asserts that a decision problem P has a solution if, and only if, there exists a TM that determines the answer for every p P if no such TM exists, the problem is said to be undecidable An unsolvable problem is a problem such that there does not exist any TM that can solve the problem
3 Decision Problems Algorithm L that solves a decision problem should be effective , i.e., Complete : L produces the correct answer (yes/or) to each question Mechanistic : L consists of a finite sequence of instructions Deterministic : L produces the same result for the same input The Church-Turing Thesis for Computable Functions : A function f is effective, i.e., effectively computable, if, and only if, there is a TM that computes f .

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4 11.2 Recursive Languages Defn . A recursive language L is a formal language for which there exists a TM that will halt and accept an input string in L , and halt and reject , otherwise. Example 11.2.1 The decision problem of determining whether a natural number is a perfect square (represented by using the string a n ) is decidable. Example 11.2.2 The decision problem of determining whether there is a path P from node v i to a node v j in a directed graph G (with nodes v 1 , …, v n ) using a NTM M with 2-tape is decidable. G
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