Gap-definable counting classes

Abstract

The function class Hash P lacks a crucial closure property: it is not closed under subtraction. To remedy this problem, the authors introduce the function class GapP as a natural alternative to Hash P. GapP is the closure of Hash P under subtraction, and has all the other useful closure properties of Hash P as well. It is shown that most previously studied counting classes are gap-definable, i.e., definable using the values of GapP functions alone. It is shown that there is a smallest gap-definable class, SPP, which is still large enough to contain Few. It is also shown that SPP consists of exactly those languages low for GapP, and thus SSP languages are low for any gap-definable class. It is further shown that any countable collection of languages is contained in a unique minimum gap-definable class, which implies that the gap-definable classes form a lattice under inclusion. Subtraction seems necessary for this result, since nothing similar is known for the Hash P-definable classes.<>