Gap-definable counting classes

Abstract

The function claw #P lacks a crucial 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 mast previously studied counting classes, including PP, C,P, and ModkP, are "gapdefinable," i.e., definable using the values of GapP functions alone. We show that there is a smallest gapdefinable class, SPP, which is still large enough to contain Few. We also show that SPP consists of exactly thoee languages low for GapP, and thus SPP languages are low for any gapdefinable class. These results unify and improve earlier disparate results of Beigel & Gill [3], Cai & Hemachandra [5], and Kobler, et al. [9]. We show further that any countable collection of languages is contained in a unique minimum gapdefinable class, which implies that the g a p definable classes form a lattice under inclusion. Subtraction seems neceessry for this result, since nothing similar is known for the #P-definable classes.

NondeterministicNC1Computation

“…Observe that PP is defined analogously in terms of GapP functions in [18]. (a) Given a sequence of constant dimension matrices over the integers [ &1, 0, 1], determine whether a specific entry in the matrix product is zero (resp.…”

“…K Proof. The only nontrivial point is closure under binomial coefficients, but this follows from the other closure properties by expressing binomial coefficients involving negative numbers as in [18] using differences of binomial coefficients with only nonnegative numbers. K Corollary 4.11.…”

“…A number of such classes were first defined in the context of polynomial time computation [25,18]. Then, the logspace counting class *L was investigated [5], together with many logspace variants adapted from the polynomial time case.…”

Verifying the determinant in parallel

“…The basic idea of our reduction is the proof in [12], which shows that GapP = #P − FP. (a) We give an AC 0 many-one reduction from v-paths-difference to v-paths.…”

Looking for an analogue of Rice's Theorem in circuit complexity theory

“…Analogously to the way we defined the three types of counting problems we can define three types of counting classes: For a given subset A of N (Z, D) the absolute (gap, relative) counting class for A consists of the languages L for which there is a polynomial-time nondeterministic machine M such that a word x is in L iff the number of accepting paths (the difference of the number of accepting paths and non-accepting paths, the share of the accepting paths compared with the total number of paths) of M on input x is in A. This definition of gap countable classes equals the definition of nice gap definable classes by Fenner, Fortnow and Kurtz [13]. As a special case of the main result in [8,30] it follows that an absolute (gap, relative) counting problem is p-m-complete for the corresponding absolute (gap, relative) counting class.…”