LWPP and WPP are not uniformly gap-definable

AbstractResolving an issue open since Fenner, Fortnow, and Kurtz raised it in [S. Fenner, L. Fortnow, S. Kurtz, Gap-definable counting classes, J. Comput. System Sci. 48 (1) (1994) 116–148], we prove that LWPP is not uniformly gap-definable and that WPP is not uniformly gap-definable. We do so in the context of a broader investigation, via the polynomial degree bound technique, of the lowness, Turing hardness, and inclusion relationships of counting and other central complexity classes

Degree Bounds on Polynomials and Relativization Theory

We demonstrate the applicability of the polynomial degree bound technique to notions such as the nonexistence of Turing-hard sets in some relativized world, (non)uniform gap-definability, and relativized separations. Our contributions are as follows. (a) There is a relativized world where AWPP has no polynomial-time Turing hard set for UP \cap coUP. This implies that classes like SPP, LWPP, WPP and AWPP do not possess Turing complete sets robustly; thus we answer an open question of [HRZ95] and extend one of the main results of [HJV93]. (b) There is a relativized world where AWPP has no polynomial-time Turing hard set for ZPP. (c) There is a relativized world where UP \cap coUP is not low for LWPP as well as for WPP. As a consequence, we show that both LWPP and WPP are not uniformly gap-definable. This settles an open question of [FFK94] and gives a relativized answer to another question of [FFK94]. (d) There is a relativized world where NP \cap coNP \nsubseteq AWPP. (e) There is a relativized world where ZPP is not contained in WPP^{WPP}. Thus WPP differs considerably from its superclass C_{=}P, for which it is known that PH \subseteq C_{=}P^{C_{=}P} in every relativized world. (f) Finally, we demonstrate that our proof technique is applicable also to classes that are not known to be gap-definable. We construct a relativized world where MIP \cap coMIP has no polynomial-time Turing hard set for ZPP. This extends one of the main results of [HJV93]