Gap-Definability as a Closure Property

Gap-definability and the gap closure operator were defined by S. Fenner, L. Fortnow and S. Kurth (J. Comput. System Sci. 48, 116 148 (1994)). Few complexity classes were known at that time to be gapdefinable. In this paper, we give simple characterizations of both gapdefinability and the gap-closure operator, and we show that many complexity classes are gap-definable, including P, P, PSPACE, EXP, NEXP, MP (Middle-bit P), and BP } P. If a class is closed under union and intersection and contains < and 7*, then it is gapdefinable 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 and gapdefinable but not closed under union (resp. intersection, complement). Finally, we show that a complexity class such as PSPACE or PP, if it is not equal to SPP, contains a maximal gap-definable many one reduction-closed subclass, which is properly between SPP and the class of all PSPACE-incomplete (PP-incomplete) sets with respect to containment. The gap-closure of the class of all incomplete sets in PSPACE (resp. PP) is PSPACE (resp. PP). 1996 Academic Press, Inc.