Much Ado about Functions

  • Authors:

  • Alan L. Selman

  • Affiliations:

  • -

  • Venue:
  • CCC '96 Proceedings of the 11th Annual IEEE Conference on Computational Complexity
  • Year:
  • 1996

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Abstract

This paper surveys basic results on complexity classes of partial multivalued functions. We stress basic inclusion relations, interesting hierarchies, and results that demonstrate that hierarchies are extant. We relate NPMV and similar function classes with the difficulty of computing NP-search functions and of inverting functions in PF. We compare the complexity classes {\rm PF}^{\rm NP}, {\rm PF}_{tt}^{\rm NP}, and {\rm PF}^{\rm NP}(O(\log n)). We study exact parallels to partial multivalued functions of well-known hierarchies of languages, the polynomial hierarchy, the bounded query hierarchies, and the difference hierarchy. It is known that each of these hierarchies collapses if and only if the corresponding hierarchy for languages collapses. The difference hierarchy for partial multivalued functions does not interleave with the bounded query hierarchies. We define one new hierarchy that is based on the number of distinct outputs of a function in NPMV, and we show that this hierarchy collapses only if the polynomial hierarchy collapses.