Analyzing restricted fragments of the theory of linear arithmetic

Abstract

The theory of linear arithmetic (TLA), also known as the Theory of Rationals, is an extremely well-studied theory. It has been widely applied to a number of domains, including program verification and constraint specification. This paper discusses the computational complexities of two broad fragments of TLA, namely Quantified Linear Programs (QLPs) and Quantified Linear Implications (QLIs). These fragments are ideal for expressing specifications in real-time scheduling, and for modeling reactive systems. In this paper, we study the computational complexities of several variants of QLPs and QLIs. Our principal result shows that there exists a one-to-one correspondence between alternations in a class of QLIs and the complexity classes comprising the polynomial hierarchy PH. In other words, for each class of the PH, there exists a class of QLIs that is complete for that class. Our work mirrors the work in [L.J. Stockmeyer, The Polynomial-Time Hierarchy, Theoretical Computer Science, 3:1-22, 1977] which established the connection between the classes of PH and quantifier alternations in a Quantified Boolean Formula. Our results are surprising, since the variables in the fragments that we consider are continuous, as opposed to discrete.