An Automation-Friendly Set Theory for the B Method

Abstract

We propose an automation-friendly set theory for the B method. This theory is expressed using first order logic extended to polymorphic types and rewriting. Rewriting is introduced along the lines of deduction modulo theory, where axioms are turned into rewrite rules over both propositions and terms. We also provide experimental results of several tools able to deal with polymorphism and rewriting over a benchmark of problems in pure set theory (i.e. without arithmetic).

Goéland: A Concurrent Tableau-Based Theorem Prover (System Description)

“…On SET, Goéland+DMT obtains significantly better results than other tableau-based provers. This confirms the previous results on the performance of deduction modulo theory for set theory [6,7].…”

“…However, we have implemented an extension that implements deduction modulo theory [9], i.e., transforms axioms into rewrite rules over propositions and terms. Deduction modulo theory has proved very useful to improve proof search when integrated into usual automated proof techniques [5], and also produces excellent results with manually-defined rewrite rules [6,7]. In Goéland, deduction modulo theory selects some axioms on the basis of a simple syntactic criterion and replaces them by rewrite rules.…”

Proof Automation in the Theory of Finite Sets and Finite Set Relation Algebra

“…Our work follows this approach as {log} can be seen as a specialized prover for polymorphic set theory. The empirical assessment presented in this paper confirms the results reported by other polymorphic provers [23,24,25,26,27].…”

First-Order Automated Reasoning with Theories: When Deduction Modulo Theory Meets Practice

“…Currently, we are testing an experimental prototype of a superposition-based ATP where rewriting over terms and propositions has been integrated. More precisely, this prototype is built on top of the Zipperposition tool and is tested over a small benchmark of B set problems (about 300 problems), for which it obtains very promising results [43].…”

“…A rewrite system is then a set of axioms with triggers (inserted manually or automatically), and no change in the architecture of the SMT solver is actually required. An experiment is currently being carried out using the ArchSAT SMT solver [42] over the same benchmark of B set problems introduced previously for Zipperposition, and ArchSAT obtains quite promising results as well [43].…”