Coming to terms with quantified reasoning
Journal article, 2017

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 inductively defined 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 alge- bras 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 inductive datatype 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 inductive data types.

Program analysis and verification

first-order theorem proving

automated reasoning

algebraic data types

superposition proving


Laura Kovacs

Vienna University of Technology

Simon Robillard

Software Technology, Group C2

Andrei Voronkov

Chalmers, Computer Science and Engineering (Chalmers), Formal methods

University of Manchester


1523-2867 (ISSN)

Vol. 52 1 260-270

Subject Categories

Algebra and Logic

Computer Science

Mathematical Analysis



More information

Latest update