Computing Certificates of Regular Expression Equivalence

Deciding the equivalence of regular expressions is a fundamental problem of theoretical computer science. On one hand, there are several decision procedures to solve this problem. On the other hand, there are several axiomatisations of the equivalence of regular expression. The axiomatisations are proved to be complete and the proofs consists of showing that each steps of a decision procedure correspond to using certain axioms from the axiomatisation. In principle, any axiomatisation that is proved to be complete, should correspond to an algorithm computing formal certificates. We present the design of a program to compute certificates of equivalence of regular expressions based on Salomaa’s axiomatisation. It includes a language to express certificates and a simple proof system to validate certificates. We implemented two decision procedures producing certificates. One based on Salomaa’s original proof of completeness and the other, more efficient, based on nondeterministic finite automata and their determinisation. We report on experiments producing and validating certificates for these two decision procedures. We hope that the same methodology would apply to other problems and axiomatisations related to Kleene Algebras.