### 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 language | English (US) |
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Pages (from-to) | 222-245 |

Number of pages | 24 |

Journal | Theory of Computing Systems |

Volume | 46 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2010 |

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### Keywords

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

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*Theory of Computing Systems*,

*46*(2), 222-245. https://doi.org/10.1007/s00224-008-9127-9