Coming to Terms with Quantified Reasoning

TL;DR

This paper develops two first-order theorem proving methods for reasoning about properties of programs manipulating finite term algebras, enabling automated analysis of algebraic data types with improved performance over existing solvers.

Contribution

It introduces two novel approaches for full first-order reasoning about term algebras, implemented in Vampire, enhancing automated verification of algebraic data types.

Findings

Successfully proved many previously unsolvable problems

Vampire outperforms existing SMT solvers on algebraic data type benchmarks

Methods are effective for complex program analysis involving term algebras

Abstract

The theory of finite term algebras provides a natural framework to describe the semantics of functional languages. The ability to efficiently reason about term algebras is essential to automate program analysis and verification for functional or imperative programs over algebraic data types such as lists and trees. However, as the theory of finite term algebras is not finitely axiomatizable, reasoning about quantified properties over term algebras is challenging. In this paper we address full first-order reasoning about properties of programs manipulating term algebras, and describe two approaches for doing so by using first-order theorem proving. Our first method is a conservative extension of the theory of term algebras using a finite number of statements, while our second method relies on extending the superposition calculus of first-order theorem provers with additional inference rules. We implemented our work in the first-order theorem prover Vampire and evaluated it on a large number of algebraic data type benchmarks, as well as game theory constraints. Our experimental results show that our methods are able to find proofs for many hard problems previously unsolved by state-of-the-art methods. We also show that Vampire implementing our methods outperforms existing SMT solvers able to deal with algebraic data types.