12 research outputs found
A complete axiomatisation of reversible Kleene lattices
We consider algebras of languages over the signature of reversible Kleene lattices, that is the regular operations (empty and unit languages, union, concatenation and Kleene star) together with intersection and mirror image. We provide a complete set of axioms for the equational theory of these algebras. This proof was developed in the proof assistant Coq
Words-to-Letters Valuations for Language Kleene Algebras with Variable Complements
We investigate the equational theory of Kleene algebra terms with variable
complements -- (language) complement where it applies only to variables --
w.r.t. languages. While the equational theory w.r.t. languages coincides with
the language equivalence (under the standard language valuation) for Kleene
algebra terms, this coincidence is broken if we extend the terms with
complements. In this paper, we prove the decidability of some fragments of the
equational theory: the universality problem is coNP-complete, and the
inequational theory t <= s is coNP-complete when t does not contain
Kleene-star. To this end, we introduce words-to-letters valuations; they are
sufficient valuations for the equational theory and ease us in investigating
the equational theory w.r.t. languages. Additionally, we prove that for words
with variable complements, the equational theory coincides with the word
equivalence.Comment: In Proceedings AFL 2023, arXiv:2309.0112
Decidability of Identity-free Relational Kleene Lattices
National audienceFamilies of binary relations are important interpretations of regular expressions, and the equivalence of two regular expressions with respect to their relational interpretations is decidable: the problem reduces to the equality of the denoted regular languages.Putting together a few results from the literature, we first make explicit a generalisation of this reduction, for regular expressions extended with converse and intersection: instead of considering sets of words (i.e., formal languages), one has to consider sets of directed and labelled graphs.We then focus on identity-free regular expressions with intersection—a setting where the above graphs are acyclic—and we show that the corresponding equational theory is decidable. We achieve this by defining an automaton model, based on Petri Nets, to recognise these sets of acyclic graphs, and by providing an algorithm to compare such automata
Completeness for Identity-free Kleene Lattices
We provide a finite set of axioms for identity-free Kleene lattices, which we prove sound and complete for the equational theory of their relational models. Our proof builds on the completeness theorem for Kleene algebra, and on a novel automata construction that makes it possible to extract axiomatic proofs using a Kleene-like algorithm
Petri automata for Kleene allegories
International audienceKleene algebra axioms are complete with respect to both language models and binary relation models. In particular, two regular expressions recognise the same language if and only if they are universally equivalent in the model of binary relations.We consider Kleene allegories, i.e. Kleene algebra with two additional operations which are natural in binary relation models: intersection and converse. While regular languages are closed under those operations, the above characterisation breaks. Instead, we give a characterisation in terms of languages of directed and labelled graphs. We then design a finite automata model allowing to recognise such graphs, by taking inspiration from Petri nets.This model allows us to obtain decidability of identity-free relational Kleene lattices, i.e., the equational theory generated by binary relations on the signature of regular expressions with intersection, but where one forbids unit. This restriction is used to ensure that the corresponding graphs are acyclic. The decidability of graph-language equivalence in the full model remains open