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 ${\displaystyle n}$ input variables of a decision tree that need to be assigned a value in order to definitely establish the value of the Boolean function ${\displaystyle f}$.

The notion of certificate is used to define semi-decidability: a formal language ${\displaystyle L}$ is semi-decidable if there is a two-place predicate relation${\displaystyle R\subseteq \Sigma ^{*}\times \Sigma ^{*}}$ such that ${\displaystyle R}$ is computable, and such that for all ${\displaystyle x\in \Sigma ^{*}}$:

Certificates also give definitions for some complexity classes which can alternatively be characterised in terms of nondeterministic Turing machines. A language ${\displaystyle L}$ is in NP if and only if there exists a polynomial ${\displaystyle p}$ and a polynomial-time bounded Turing machine ${\displaystyle M}$ such that every word ${\displaystyle x\in \Sigma ^{*}}$ is in the language ${\displaystyle L}$ precisely if there exists a certificate ${\displaystyle c}$ of length at most ${\displaystyle p(|x|)}$ such that ${\displaystyle M}$ accepts the pair ${\displaystyle (x,c)}$. The class co-NP has a similar definition, except that there are certificates for the words not in the language.

The class NL has a certificate definition: a problem in the language has a certificate of polynomial length, which can be verified by a deterministic logarithmic-space bounded Turing machine that can read each bit of the certificate once only. Alternatively, the deterministic logarithmic-space Turing machine in the statement above can be replaced by a bounded-error probabilistic constant-space Turing machine that is allowed to use only a constant number of random bits.

The problem of determining, for a given graph ${\displaystyle G}$ and number ${\displaystyle k}$, if the graph contains an independent set of size ${\displaystyle k}$ is in NP. Given a pair ${\displaystyle (G,k)}$ in the language, a certificate is a set of ${\displaystyle k}$ vertices which are pairwise not adjacent (and hence are an independent set of size ${\displaystyle k}$).

A more general example, for the problem of determining if a given Turing machine accepts an input in a certain number of steps, is as follows: