Using Proofs by Coinduction to Find “Traditional” Proofs

Abstract

Abstract. In the specific situation of formal reasoning concerned with "regular expression equivalence" we address instances of more general questions such as: how can coinductive argumentation be formalised logically and be applied effectively, as well as how is it linked to traditional forms of proof. For statements expressing that two regular expressions are language equivalent, we demonstrate that proofs by coinduction can be formulated in a proof system based on equational logic, where effective proof-search is possible. And we describe a proof-theoretic method for translating derivations in this proof system into a "traditional" axiom system: namely, into a "reverse form" of the axiomatisation of "regular expression equivalence" due to Salomaa. Hereby we obtain a coinductive completeness proof for the traditional proof system.

Proving language inclusion and equivalence by coinduction

“…Of course, one can reason about regular expressions in Kleene algebra; this is however a fundamentally different approach than the coinductive techniques of the present paper. In [13] a proof system for equivalence of regular expressions is presented, based on bisimulations but not on bisimulation-up-to. In [15] a general coinductive axiomatization of regular expression containment is given, based on an interpretation of regular expressions as types.…”

“…In [15] a general coinductive axiomatization of regular expression containment is given, based on an interpretation of regular expressions as types. The authors of [15] instantiate their axiomatization with the main coinductive rule from [13]. The focus of [15] is on constructive proofs based on parse trees of regular expressions; instead, we base ourselves on bisimulations between languages.…”

A Parametric Abstract Domain for Lattice-Valued Regular Expressions

“…By memorizing each pair of regular expressions and testing memo tbl for membership up to ACI equivalence the algorithm is guaranteed to terminate. As such, the algorithm is an inclusion (or containment ) analogue of Grabmayer's co-inductive axiomatization of regular expression equivalence [14,17]. There are different opportunities for optimizing the ordering algorithm.…”

“…A line of work has concerned axiomatizing equivalence (and containment) of regular expressions (REs) and of the more general Kleene algebras [27,19,14,17]. We refer to Henglein and Nielsen [17] for a historical account of such developments.…”

“…We refer to Henglein and Nielsen [17] for a historical account of such developments. Grabmayer [14] gave a co-inductive axiomatization of RE equivalence based on Brzozowski derivatives and connects it to an earlier axiomatization of Salomaa [27]. In particular, our nullable function corresponds to Grabmayer's o-function, and his COMP/FIX proof system rule concludes that two REs E and F are equivalent if o(E) = o(F ) and if all derivatives D a (E) = D a (F ) are equivalent much like our co-inductive leq test for deciding containment checks for nullable and queries all derivatives for containment.…”