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 ⊆ Σ∗ × Σ∗ such that R is computable, and such that for all x ∈ Σ∗:
and to prove NP as following:
L ∈ NP iff there is a polytime verifier V such that:
Witness (mathematics), an analogous concept in mathematical logic
Buhrman, Harry; Wolf, Ronald (2002), Complexity Measures and Decision Tree Complexity:A Survey .