An Evaluation of Automated Theorem Proving in Regular Algebra ( Extended Abstract )

Introduction The Isabelle/HOL environment [8] combines the power of automated reasoning with higher-order features for theory engineering and proof management. Its built-in Sledgehammer tool integrates state of the art ATP and SMT tools, allowing for powerful automated reasoning in proofs [2]. Theory engineering features such as typeclasses and locales support the effective design of large theory hierarchies and allow for theorem propagation in these hierarchies [6]. They also allow for the connection between abstract algebras and concrete models. When using regular algebras such as ∗-continuous Kleene algebras or quantales, many proof obligations can be discharged using only first-order axioms. There are many first-order regular algebras, such as Kleene algebras, Pratt’s action algebras and Boffa’s algebras. Action algebras are particularly interesting as they have an axiomatisation based on Galois connections and an equivalent purely equational axiomatisation [9]. This result is interesting, as one might expect that purely equational axioms might be more amenable to ATP than other sets of axioms. However, this purely equational axiomatisation requires a large signature. This paper attempts to ascertain which of action or Kleene algebras are better from an ATP standpoint, providing insight into the trade-off between an equational axiomatisation and a larger signature. To achieve this, Isabelle’s built in benchmarking tool Mirabelle is used on facts from a large repository for algebraic methods in Isabelle which has been documented in previous papers [5]. Recently the repository has been extended to support higher-order regular algebra, and many higher-order order concepts such as Galois connections have been implemented in this setting [1]. We have also begun work on extending this repository with explicit carrier sets, allowing us to formalise concepts such as Galois connections between different partial orders, subalgebras and other concepts from universal algebra. Even in this higher-order setting, automated reasoning remains a valuable tool. However, a question remains—exactly how much does the carrier set based axiomatisation these features require impact the usefulness of ATP and SMT tools?