Theory of Computing Systems

, Volume 46, Issue 2, pp 222–245 | Cite as

On the Autoreducibility of Functions

Article

Abstract

This paper studies the notions of self-reducibility and autoreducibility. Our main result regarding length-decreasing self-reducibility is that any complexity class \(\mathcal{C}\) that has a (logspace) complete language and is closed under polynomial-time (logspace) padding has the property that if all \(\mathcal{C}\) -complete languages are length-decreasing (logspace) self-reducible then \(\mathcal{C}\subseteq \mathrm {P}\) (\(\mathcal {C}\subseteq \mathrm {L}\) ). 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.

Keywords

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

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of RochesterRochesterUSA
  2. 2.Department of Computer ScienceUniversity of MiamiCoral GablesUSA

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