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Algorithms Sipser 3.3 (pages 154-159)

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CS 311 Fall 2008 2 Computability Hilbert's Tenth Problem: Find a process according to which it can be determined by a finite number of operations whether a given a polynomial p(x 1,x 2,...,x n ) has an integral root.

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CS 311 Fall 2008 3 Algorithms Intuitively: –An algorithm is a finite sequence of operations, each chosen from a finite set of well-defined operations, that halts in a finite time. –Sometimes also called procedures or recipes

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CS 311 Fall 2008 4 Church-Turing Thesis Algorithm Turing machine =

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CS 311 Fall 2008 5 Languages and Problems Let D = {p | p is a polynomial with an integral root} Hilbert's Tenth Problem: Determine whether D is Turing-decidable

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CS 311 Fall 2008 6 D = {p | p is a polynomial with an integral root} Turing-recognizable M = On input p, where p is a polynomial p(x 1,x 2,...,x n ). 1.Lexicographically generate integer values for (x 1,x 2,...,x n ). 2.Evaluate p as each set of values is generated. 3.If, at any point, the polynomial evaluates to 0, accept."

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CS 311 Fall 2008 7 Hierarchy of languages All languages Turing-recognizable Turing-decidable Context-free languages Regular languages 0n1n0n1n anbncnanbncn D 0*1*0*1*

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CS 311 Fall 2008 8 Describing Turing machines

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CS 311 Fall 2008 9 Describing Turing machines Formal: Implementation: – M = "On input string w. 1.Sweep across tape, crossing off every other 0. 2.If tape contained one 0, accept. 3.Else, if number of 0's is odd, reject. 4.Return head to left-hand end of tape. 5.Go to step 1. High-level: repeat until n=1 exit if n mod 2 != 0 set n = n / 2

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