## Does co-NP have short interactive proofs ?(English)Zbl 0653.68037

L. Babai [Proc. 17th ACM Symp. Theor. Comput., 421-429 (1985)] and S. Goldwasser, S. Micali and C. Rackoff [ibid., 291-304 (1985)] introduced two probabilistic extensions of the complexity class NP. The two complexity classes, denoted AM[Q] and $${\mathbb{P}}[Q]$$, respectively, are defined using randomized interactive proofs between a prover and a verifier. S. Goldwasser and M. Sipser [Proc. 18th ACM Symp. Theor. Comput., 59-68 (1986)] proved that the two classes are equal. We prove that if the complexity class co-NP is contained in $${\mathbb{P}}[k]$$ for some constant k (i.e., if every language in co-NP has a short interactive proof), then the polynomial-time hierarchy collapses to the second level. As a corollary, we show that if the graph isomorphism problem is NP-complete, then the polynomial-time hierarchy collapses.

### MSC:

 68Q25 Analysis of algorithms and problem complexity 03D15 Complexity of computation (including implicit computational complexity)
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### References:

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