Gap-Definable Counting Classes

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Gap-Definable Counting Classes (1992)

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by Stephen A. Fenner , Lance J. Fortnow , Stuart A. Kurtz , Stephen Fenner

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126 - 15 self

BibTeX

@MISC{Fenner92gap-definablecounting,
    author = {Stephen A. Fenner and Lance J. Fortnow and Stuart A. Kurtz and Stephen Fenner},
    title = {Gap-Definable Counting Classes},
    year = {1992}
}

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Abstract

The function class #P lacks an important closure property: it is not closed under subtraction. To remedy this problem, we introduce the function class GapP as a natural alternative to #P. GapP is the closure of #P under subtraction, and has all the other useful closure properties of #P as well. We show that most previously studied counting classes, including PP , C=P , and Mod k P , are "gap-definable," i.e., definable using the values of GapP functions alone. We show that there is a smallest gap-definable class, SPP , which is still large enough to contain Few. We also show that SPP consists of exactly those languages low for GapP, and thus SPP languages are low for any gap-definable class. These results unify and improve earlier disparate results of Cai & Hemachandra [7] and Kobler, Schoning, Toda, & Tor'an [15]. We show further that any countable collection of languages is contained in a unique minimum gap-definable class, which implies that the gap-definable classes form a lattic...

Keyphrases

gap-definable counting class gap-definable class natural alternative countable collection counting class unique minimum gap-definable class gapp function spp language smallest gap-definable class important closure property cai hemachandra function class gapp disparate result useful closure property function class

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