# On the amount of nondeterminism and the power of verifying

## 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*), *II*_{h}^{0}), *h* ≥ 1, characterize precisely the fixed-parameter intractability of *NP*-hard optimization problems. In particular, the (2*h*)th level *W*[2*h*] 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*).

## Keywords

Turing Machine Computation Path Input Length Fixed Parameter Tractability Circuit Family## Preview

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