Certificate (complexity)

String that certifies the answer to a computation

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 {\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 f {\displaystyle f} .

## Use in definitions

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The notion of certificate is used to define semi-decidability:[1] a formal language L {\displaystyle L} is semi-decidable if there is a two-place predicate relation R ⊆ Σ ∗ × Σ ∗ {\displaystyle R\subseteq \Sigma ^{*}\times \Sigma ^{*}} such that  R {\displaystyle R} is computable, and such that for all  x ∈ Σ ∗ {\displaystyle x\in \Sigma ^{*}} :

x ∈ L ⇔ there exists y such that R(x, y)

Certificates also give definitions for some complexity classes which can alternatively be characterised in terms of nondeterministic Turing machines. A language  L {\displaystyle L} is in NP if and only if there exists a polynomial  p {\displaystyle p} and a polynomial-time bounded Turing machine M {\displaystyle M} such that every word  x ∈ Σ ∗ {\displaystyle x\in \Sigma ^{*}} is in the language  L {\displaystyle L} precisely if there exists a certificate  c {\displaystyle c} of length at most  p ( | x | ) {\displaystyle p(|x|)} such that  M {\displaystyle M} accepts the pair  ( x , c ) {\displaystyle (x,c)} .[2] 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.[3] 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.[4]

## Examples

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The problem of determining, for a given graph  G {\displaystyle G} and number  k {\displaystyle k} , if the graph contains an independent set of size  k {\displaystyle k} is in NP. Given a pair  ( G , k ) {\displaystyle (G,k)} in the language, a certificate is a set of  k {\displaystyle k} vertices which are pairwise not adjacent (and hence are an independent set of size  k {\displaystyle k} ).[5]

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:

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

## See also

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

## References

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1. ^ Cook, Stephen. "Computability and Noncomputability" (PDF). Retrieved 7 February 2013.
2. ^ Arora, Sanjeev; Barak, Boaz (2009). "Definition 2.1". Complexity Theory: A Modern Approach. Cambridge University Press. ISBN 978-0-521-42426-4.
3. ^ Arora, Sanjeev; Barak, Boaz (2009). "Definition 4.19". Complexity Theory: A Modern Approach. Cambridge University Press. ISBN 978-0-521-42426-4.
4. ^ A. C. Cem Say, Abuzer Yakaryılmaz, "Finite state verifiers with constant randomness," Logical Methods in Computer Science, Vol. 10(3:6)2014, pp. 1-17.
5. ^ Arora, Sanjeev; Barak, Boaz (2009). "Example 2.2". Complexity Theory: A Modern Approach. Cambridge University Press. ISBN 978-0-521-42426-4.

## External links

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* 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