Fuhrmann circle

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

Definition

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)

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

References

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• Computational Complexity: a Modern Approach by Sanjeev Arora and Boaz Barak