Tight Polynomial Bounds for Loop Programs in Polynomial Space

Abstract

We consider the following problem: given a program, find tight asymptotic bounds on the values of some variables at the end of the computation (or at any given program point) in terms of its input values. We focus on the case of polynomially-bounded variables, and on a weak programming language for which we have recently shown that tight bounds for polynomially-bounded variables are computable. While their computability has been settled, the complexity of this program-analysis problem remained open. In this paper, we establish its complexity class to be PSPACE. Intuitively, we show that it is possible to compute these tight bounds by interpreting the program using a novel, compact abstract representation which nevertheless achieves completeness. One of the keys to this reduction in size is the restriction to univariate bounds. Then, a solution for multivariate bounds is achieved by reducing this problem to the univariate case; this reduction is orthogonal to the solution of the univariate problem and uses a different technique altogether. Another auxiliary result is the proof of a bound on the degree of polynomial bounds for such programs; we show that the degree is at most exponential in the program size.