Certificate (complexity)

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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 ${\displaystyle f}$.

Video Certificate (complexity)

Definition

The notion of certificate is used to define semi-decidability:

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)  

It can also be used to define the class NP:

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)  

Maps Certificate (complexity)

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  

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See also

• Witness (mathematics), an analogous concept in mathematical logic

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References

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External links

• Buhrman, Harry; de Wolf, Ronald (2002), Complexity Measures and Decision Tree Complexity:A Survey.
• Computational Complexity: a Modern Approach by Sanjeev Arora and Boaz Barak

Source of article : Wikipedia