Abstract
Gap-definability and the gap-closure operator were defined in [FFK91]. Few complexity classes were known at that time to be gap-definable. In this paper, we give simple characterizations of both gap-definability and the gap-closure operator, and we show that many complexity classes are gap-definable, including P#P, \(P^{\# P_{[1]} }\), PSPACE, EXP, NEXP, MP, and BP·⊕P. If a class is closed under union, intersection and contains λ and Σ*, then it is gap-definable if and only if it contains SPP; its gap-closure is the closure of this class together with SPP under union and intersection. On the other hand, we give some examples of classes which are reasonable gap-definable but not closed under union (resp. intersection, complement). Finally, we show that a complexity class such as PP or PSPACE, if it is not equal to SPP, contains a maximal proper gap-definable subclass which is closed under many-one reductions.
A full version of the paper is available from lilide@cs.uchicago.edu
Partially Supported by NSF Grant CCR-9209833.
Partially Supported by NSF Grant CCR-9009936 and CCR-9253582.
Partially Supported by NSF Grant CCR-9253582.
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© 1993 Springer-Verlag Berlin Heidelberg
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Fermer, S., Fortnow, L., Li, L. (1993). Gap-definability as a closure property. In: Enjalbert, P., Finkel, A., Wagner, K.W. (eds) STACS 93. STACS 1993. Lecture Notes in Computer Science, vol 665. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56503-5_48
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DOI: https://doi.org/10.1007/3-540-56503-5_48
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