On the autoreducibility of functions

Piotr Faliszewski, Mitsunori Ogihara

Research output: Contribution to journalArticle

4 Scopus citations

Abstract

This paper studies the notions of self-reducibility and autoreducibility. Our main result regarding length-decreasing self-reducibility is that any complexity class C that has a (logspace) complete language and is closed under polynomial-time (logspace) padding has the property that if all C-complete languages are length-decreasing (logspace) self-reducible then C ⊆ P(C ⊆ L). In particular, this result applies to NL, NP and PSPACE. We also prove an equivalent of this theorem for function classes (for example, for #P). We also show that for several hard function classes, in particular for #P, it is the case that all their complete functions are deterministically autoreducible. In particular, we show the following result. Let f be a #P parsimonious function with two preimages of 0. We show that there are two FP functions h and t such that for all inputs x we have f(x)=t(x)+f(h(x)), h(x)≠x, and t(x)∈{0,1}. Our results regarding single-query autoreducibility of #P functions can be contrasted with random self-reducibility for which it is known that if a #P complete function were random self-reducible with one query then the polynomial hierarchy would collapse.

Original languageEnglish (US)
Pages (from-to)222-245
Number of pages24
JournalTheory of Computing Systems
Volume46
Issue number2
DOIs
StatePublished - Feb 1 2010

Keywords

  • Autoreducibility
  • Complete functions
  • Function classes
  • Length-decreasing self-reducibility
  • Reductions

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics

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