# Kolmogorov complexity, restricted nondeterminism and generalized spectra

## Abstract

This paper uses the technique of generalized spectra and expressibility of complexity classes in logic, developed by Fagin and Immerman, to give alternate characterizations of specific subclasses of *NP*. These characterizations serve to unify concepts that appear in seemingly different areas of complexity theory; namely, the restricted nondeterminism of Kintala and Fischer and the time bounded Kolmogorov complexity of Daley and Ko. As consequences of these characterizations we show that relatively easy subsets of *NP−P* can not be pseudorandomly generated, unless *UTIME*[*t*(*n*)]=*DTIME*[*t*(*n*)] for certain exponential functions *t*. Secondly, we show that no easy subset of the set of all satisfying assignments of satisfiable *g*(*n*)-easy formulas contains an assignment for each of these formulas, unless *NEXPTIME* = *EXPTIME*. The latter partially answers a question raised by Hartmanis.

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