50 Slides--Decidability

50 Slides--Decidability - CS103 HO#50 Slides-Decidability...

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CS103 HO#50 Slides--Decidability May 17, 2010 1 Definitions Turing-recognizable languages are also called recursively enumerable languages. Turing-decidable are also called recursive languages. Turing- recognizable Turing- decidable input w input w Yes, w L Yes, w L No , w L q b b a b b Multi-tape Turing Machines 0 1 0 0 0 1 1 0 x y x z z z (q 3 , h 1 , h 2 , . .., h k ) = (q 4 , (w 1 , m 1 ), (w 2 , m 2 ), . .., (w k , m k )) q b b a b b Multi-tape Turing Machines 0 1 0 0 0 1 1 0 x y x z z z # b b a b b # 0 1 0 0 0 1 1 0 # x y x z z z # q Every multi-tape Turing machine has an equivalent single-tape Turing machine. Nondeterministic Turing Machines : Q   P (Q  {L, R}) C 0 = q 0 w C 1 C 2 C 12 C 11 C 21 C 3 C 31 C 32 C 33 . . . . . . . . . Alfred North Whitehead (1861-1947) and Bertrand Russell (1872-1970) David Hilbert (1862 – 1943)
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CS103 HO#50 Slides--Decidability May 17, 2010 2 Wilhelm Ackermann (1896 – 1962) Algorithms An important milestone in mathematics was the publication, in 1910, of Principia Mathematica by Alfred North Whitehead and Bertrand Russell. It was an attempt to formalize mathematics by basing it on logic and set theory. For this approach to succeed, two questions had to be answered in the affirmative: 1. Is it possible to axiomatize all mathematical structures in such a way that every true statement becomes a theorem? 2. Does there exist an algorithm to decide, given a set of axioms, whether a given statement is a theorem? Kurt Gödel (1906 – 1978) Algorithms An important milestone in mathematics was the publication, in 1910, of Principia Mathematica by Alfred North Whitehead and Bertrand Russell. It was an attempt to formalize mathematics by basing it on logic and set theory. For this approach to succeed, two questions had to be answered in the affirmative: 1. Is it possible to axiomatize all mathematical structures in such a way that every true statement becomes a theorem? 2. Does there exist an algorithm to decide, given a set of axioms, whether a given statement is a theorem? In his paper on the Incompleteness Theorem , Gödel showed in 1931 that the answer to the first question was no . No formal system powerful enough to include arithmetic can be both consistent and complete. Question 2 was called the
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50 Slides--Decidability - CS103 HO#50 Slides-Decidability...

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