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Algorithms

Algorithms. Sipser 3.3 (pages 154-159). 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. . Algorithms. Intuitively:

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Algorithms

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  1. Algorithms Sipser 3.3 (pages 154-159)

  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(x1,x2,...,xn) has an integral root.

  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

  4. Church-Turing Thesis Turing machine = Algorithm

  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

  6. D = {p | p is a polynomial with an integral root} • Turing-recognizable • M = ”On input p, where p is a polynomial p(x1,x2,...,xn). • Lexicographically generate integer values for (x1,x2,...,xn). • Evaluate p as each set of values is generated. • If, at any point, the polynomial evaluates to 0, accept."

  7. Hierarchy of languages All languages Turing-recognizable D Turing-decidable Context-free languages anbncn Regular languages 0n1n 0*1*

  8. Describing Turing machines

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

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