On Automating the Calculus of Relations

Abstract

Abstract. Relation algebras provide abstract equational axioms for the calculus of binary relations. They name an established area of mathematics with various applications in computer science. We prove more than hundred theorems of relation algebras with off-the-shelf automated theorem provers. This yields a basic calculus from which more advance applications can be explored. Here, we present two examples from the formal methods literature. Our experiments not only further underline the feasibility of automated deduction in complex algebraic structures and provide theorem proving benchmarks, they also pave the way for lifting established formal methods such as B or Z to a new level of automation.

Automated verification of refinement laws

“…In particular, this verification led to the simplification of some known proofs, to the discovery of new refinement laws and to the discovery of some errors in previous theorems and proofs (in the relational setting, an error in a soundness proof for data refinement in the book of de Roever and Engelhardt has also been discovered with the use of ATP and counterexample search [17]). This suggests that large parts of Back and von Wright's original refinement calculus can also effectively be automated and that automated deduction provides an innovative technology for program refinement.…”

“…Previous experiments with more than 500 theorems in Kleene algebras (e.g. [14,17]) suggest that an equational encoding is usually sufficient for finding ATP proofs of simple theorems. However, more complex theorems often succeed only with the inequational encoding.…”

“…Recent work already investigated the automation of computational logics [14], of relational reasoning [17], of rewriting and termination analysis [32,33], and of hybrid system analysis [13]. It has also been shown that ATP can be very helpful in developing consistent irredundant specifications and axiomatisations of algebraic theories [11,10].…”

Automated Verification of Relational While-Programs

“…Encouraged by the practicability and elegance of the latter results and the positive experiences of [16], the combination of assertion-based program verification and relation algebra was combined with automated theorem proving using Prover9; see [5]. This paper is a continuation as well as a step further of this idea.…”

“…Full automatisation of proofs can frequently be achieved by off-the-shelf automated theorem provers, such as Prover9 (see [31]). We refer to [16] for such an application. In the present paper, we will follow the latter approach and use Prover9 for automated program verification of while-programs.…”