## Abstract

Given a p-order A over a universe of strings (i.e., a transitive, reflexive, antisymmetric relation such that if (x, y) ∈ A then |x| is polynomially bounded by |y|), an interval size function of A returns, for each string x in the universe, the number of strings in the interval between strings b(x) and t(x) (with respect to A), where b(x) and t(x) are functions that are polynomial-time computable in the length of x. By choosing sets of interval size functions based on feasibility requirements for their underlying p-orders, we obtain new characterizations of complexity classes. We prove that the set of all interval size functions whose underlying p-orders are polynomial-time decidable is exactly #P. We show that the interval size functions for orders with polynomial-time adjacency checks are closely related to the class FPSPACE(poly). Indeed, FPSPACE(poly) is exactly the class of all nonnegative functions that are an interval size function minus a polynomial-time computable function. We study two important functions in relation to interval size functions. The function #DIV maps each natural number n to the number of nontrivial divisors of n. We show that #DIV is an interval size function of a polynomial-time decidable partial p-order with polynomial-time adjacency checks. The function #MONSAT maps each monotone boolean formula F to the number of satisfying assignments of F. We show that #MONSAT is an interval size function of a polynomial-time decidable total p-order with polynomial-time adjacency checks. Finally, we explore the related notion of cluster computation.

## Publication Date

12-21-2006

## Document Type

Article

## Department, Program, or Center

Center for Advancing the Study of CyberInfrastructure

## Recommended Citation

Hemaspaandra Lane A., Homan Christopher M., Sven Kosub, Wagner Klaus W. (2007) The complexity of computing the size of an interval. SIAM Journal on Computing 36(5): 1264–1300; https://doi.org/10.1137/S0097539705447013

## Campus

RIT – Main Campus

## Comments

Copyright © 2006 Society for Industrial and Applied Mathematics

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