Cook'sTheorem

# Cook'sTheorem - Cook's Theorem Definiton of NP: A decision...

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Cook's Theorem Definiton of NP: A decision problem Π is in the set NP if and only if there exists a Non Deterministic Turing Machine (NDTM) M that, when given an arbitrary instance x of Π on the input tape of M, will: (1) Non deterministically place a string w Σ * on M's tape just to the left of instance x. If x Y( Π ), w will be the "answer" to instance x – that is, sufficient information, along with x, to verify (in step (2) below) that x Y( Π ). Otherwise, x N( Π ) and the string w will be a random set of characters, or empty (we need not verify instances in N( Π ). If we can – fine, but it is not necessary); and (2) If x Y( Π ), deterministically (and in polynomial time with respect to n = |x|), verify that x Y( Π ), and finally halt in state q Y . If x Y( Π ) (i. e., x N( Π )), the machine M may halt in state q N , or it may compute forever – we are only guaranteed that M will not halt in state q Y . Notes We may interpret this as two Turing Machines acting sequentially. The first is non deterministic and simply writes characters on the tape. The second is totally deterministic and can, but need not, utilize the result of the first. For certain problems in NP, TM's exist which ignore the "answer" given by the non deterministic phase. For example, if Π P there is no need to examine the answer, because a DTM exists which can compute the answer within the required time. In these case, "no" instances also can be correctly identified in deterministic polynomial time. Further, we can not simply terminate execution (manually, or automatically, after some number of state transitions) of an apparently "looping" computation. We usually don't know the exact polynomial p(n) and, therefore, can never know when p(n) transitions have been exceeded. Completeness We first describe Completeness in terms of languages, since that is the setting in which the terminology originated. For any set of languages, say X, a particular language L X may "embody" the properties of all languages in X in the sense that for every language L' X all strings x Σ * can be transformed into strings f(x) so that x L' if and only if f(x) L. We call L the "completion of the set X," that L is "complete in X," or that L is,

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## This note was uploaded on 07/14/2011 for the course COT 4610 taught by Professor Dutton during the Fall '10 term at University of Central Florida.

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Cook'sTheorem - Cook's Theorem Definiton of NP: A decision...

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