On the Amount of Nondeterminism and the Power of Verifying (Extended Abstract)

The relationship between nondeterminism and other computational resources is studied based on a special interactive-proof system model GC. Let s(n) be a function and C be a complexity class. Define GC(s(n), C) to be the class of languages that are accepted by verifiers in C that can make an extra O(s(n)) amount of nondeterminism. Our main results are (1) A systematic technique is developed to show that for many functions s(n) and for many complexity classes C, the class GC(s(n), C) has natural complete languages; (2) The class ∏ h 0 of languages accepted by log-time alternating Turing machines making h alternations is precisely the class of languages accepted by uniform families of circuits of depth h; (3) The classes GC(s(n), IIh0), h ≥ 1, characterize precisely the fixed-parameter intractability of NP-hard optimization problems. In particular, the (2h)th level W[2h] of W-hierarchy introduced by Downey and Fellows collapses if and only if \(GC(s(n),\prod _{2h}^0 ) \subseteq P\) for some s(n)=ω(log n).

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