When Less Is More: Consequence-Finding in a Weak Theory of Arithmetic

TL;DR

This paper introduces a weakly-axiomatized non-linear arithmetic theory with effective consequence-finding algorithms, enabling loop invariant generation for non-linear safety property proofs in programs.

Contribution

It develops a novel non-linear arithmetic theory with algorithmic properties similar to linear arithmetic, and provides a consequence-finding method for invariant generation.

Findings

Effective consequence-finding in the new theory

Loop invariants generated are useful for safety proofs

Algorithms perform well in experiments

Abstract

This paper presents a theory of non-linear integer/real arithmetic and algorithms for reasoning about this theory. The theory can be conceived as an extension of linear integer/real arithmetic with a weakly-axiomatized multiplication symbol, which retains many of the desirable algorithmic properties of linear arithmetic. In particular, we show that the conjunctive fragment of the theory can be effectively manipulated (analogously to the usual operations on convex polyhedra, the conjunctive fragment of linear arithmetic). As a result, we can solve the following consequence-finding problem: given a ground formula F, find the strongest conjunctive formula that is entailed by F. As an application of consequence-finding, we give a loop invariant generation algorithm that is monotone with respect to the theory and (in a sense) complete. Experiments show that the invariants generated from the consequences are effective for proving safety properties of programs that require non-linear reasoning.