Reducibility

Reducibility - Harvard CS 121 and CSCI E-207 Lecture 16...

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Unformatted text preview: Harvard CS 121 and CSCI E-207 Lecture 16: Reductions Harry Lewis November 5, 2009 • Reading: Sipser Ch. 5 Harvard CS 121 & CSCI E-207 November 5, 2009 “Co-X” • For any property X that a set might have, a set S is co-X iff S has property X. • For example, a co-finite set of natural numbers is a set that is missing only a finite number of elements. • A co-regular language is . . . ? • A co-recursive language is . . . ? • What about a co-CF language? • Proved last time: • A language is recursive if and only if it is both r.e. and co-r.e. 1 Harvard CS 121 & CSCI E-207 November 5, 2009 Non-r.e. Languages Theorem: The following co-r.e. languages are not r.e.: • A TM = {h M,w i : M does not accept w } • HALT TM = {h M,w i : M does not halt on w } • HALT ε TM = {h M i : M does not halt on ε } Proof: If these languages were r.e., then A TM , HALT TM , and HALT ε TM would be both r.e. and co-r.e. and hence recursive. 2 Harvard CS 121 & CSCI E-207 November 5, 2009 Is it possible to determine, given a TM M , whether M accepts a finite or infinite set? • Let A infinite = {h M i : L ( M ) is infinite } . Is A infinite recursive?...
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Reducibility - Harvard CS 121 and CSCI E-207 Lecture 16...

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