The Complexity of Computing the Size of an Interval

Abstract

Given a p-order $A$ over a universe of strings (i.e., a transitive, reflexive, antisymmetric relation such that if $(x, y) \in A$, 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 mP. 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 mDIV maps each natural number $n$ to the number of nontrivial divisors of $n$. We show that mDIV is an interval size function of a polynomial-time decidable partial p-order with polynomial-time adjacency checks. The function mMONSAT maps each monotone boolean formula $F$ to the number of satisfying assignments of $F$. We show that mMONSAT 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.