In computational complexity theory, a certificate (also called a witness) is a string that certifies the answer to a computation, or certifies the membership of some string in a language. A certificate is often thought of as a solution path within a verification process, which is used to check whether a problem gives the answer "Yes" or "No".
In the decision tree model of computation, certificate complexity is the minimum number of the input variables of a decision tree that need to be assigned a value in order to definitely establish the value of the Boolean function .
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Definition
Certificate is generally used to prove semi-decidability as following:
L ? SD iff there is a two-place predicate R ? ?* × ?* such that R is computable, and such that for all x ? ?*:
x ? L <=> there exists y such that R(x, y)
and to prove NP as following:
L ? NP iff there is a polytime verifier V such that:
x ? L <=> there exists y such that |y| <= |x|c and V accepts (x, y)
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Example
L = {<<M>, x, w> | does <M> accept x in |w| steps?} Show L ? NP. verifier: gets string c = <M>, x, w such that |c| <= P(|w|) check if c is an accepting computation of M on x with at most |w| steps |c| <= O(|w|3) if we have a computation of a TM with k steps the total size of the computation string is k2 Thus, <<M>, x, w> ? L <=> there exists c <= a|w|3 such that <<M>, x, w, c> ? V ? P Source of the article : Wikipedia
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