Certificate (complexity)

Certificate (complexity)

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 .

Certificate is generally used to prove semi-decidability as following:

L ∈ SD iff there is a two-place predicate R ⊆ E∗ × E∗ such that R is computable, and such that for all x ∈ E∗: (E being capitalized sigma)

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)