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lecture23 - Notes on Complexity Theory Last updated...

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Notes on Complexity Theory Last updated: November, 2011 Lecture 23 Jonathan Katz 1 The Complexity of Counting 1.1 The Class # P P captures problems where we can efficiently find an answer; NP captures problems where we can efficiently verify an answer. Counting the number of answers gives rise to the class # P . Recall that L ∈ NP if there is a (deterministic) Turing machine M running in time polynomial in its first input such that x L ⇔ ∃ w M ( x, w ) = 1 . (1) The corresponding counting problem is: given x , determine the number of strings w for which M ( x, w ) = 1. (Note that x L iff this number is greater than 0.) An important point is that for a given L , there might be several (different) machines for which Eq. (1) holds; when specifying the counting problem, we need to fix not only L but also a specific machine M . Sometimes, however, we abuse notation when there is a “canonical” M for some L . We let # P denote the class of counting problems corresponding to polynomial-time M as above. The class # P can be defined as a function class or a language class; we will follow the book and speak about it as a function class. Let M be a (two-input) Turing machine M that halts on all inputs, and say M runs in time t ( n ) where n denotes the length of its first input. Let # M ( x ) def = fl fl ' w ∈ { 0 , 1 } t ( | x | ) | M ( x, w ) = 1 “fl fl . Then: 1 Definition 1 A function f : { 0 , 1 } * N is in # P if there is a Turing machine M running in time polynomial in its first input such that f ( x ) = # M ( x ) . We let FP denote the class of functions computable in polynomial time; this corresponds to the language class P .
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