Kleene Algebra with Converse

International audienceThe equational theory generated by all algebras of binary relations with operations of union, composition, converse and reflexive transitive closure was studied by Bernátsky, Bloom, Ésik, and Stefanescu in 1995. We reformulate some of their proofs in syntactic and elementary terms, and we provide a new algorithm to decide the corresponding theory. This algorithm is both simpler and more efficient; it relies on an alternative automata construction, that allows us to prove that the considered equational theory lies in the complexity class PSPACE. Specific regular languages appear at various places in the proofs. Those proofs were made tractable by considering appropriate automata recognising those languages, and exploiting symmetries in those automata

A Complete Axiomatisation of a Fragment of Language Algebra

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

Symbolic Algorithms for Language Equivalence and Kleene Algebra with Tests

We first propose algorithms for checking language equivalence of finite automata over a large alphabet. We use symbolic automata, where the transition function is compactly represented using a (multi-terminal) binary decision diagrams (BDD). The key idea consists in computing a bisimulation by exploring reachable pairs symbolically, so as to avoid redundancies. This idea can be combined with already existing optimisations, and we show in particular a nice integration with the disjoint sets forest data-structure from Hopcroft and Karp's standard algorithm. Then we consider Kleene algebra with tests (KAT), an algebraic theory that can be used for verification in various domains ranging from compiler optimisation to network programming analysis. This theory is decidable by reduction to language equivalence of automata on guarded strings, a particular kind of automata that have exponentially large alphabets. We propose several methods allowing to construct symbolic automata out of KAT expressions, based either on Brzozowski's derivatives or standard automata constructions. All in all, this results in efficient algorithms for deciding equivalence of KAT expressions

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

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

Algebraic coherent confluence and higher-dimensional globular Kleene algebras

We extend the formalisation of confluence results in Kleene algebras to a formalisation of coherent proofs by confluence. To this end, we introduce the structure of modal higher-dimensional globular Kleene algebra, a higher-dimensional generalisation of modal and concurrent Kleene algebra. We give a calculation of a coherent Church-Rosser theorem and Newman's lemma in higher-dimensional Kleene algebras. We interpret these results in the context of higher-dimensional rewriting systems described by polygraphs.Comment: Pre-print (second version

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

Algebraic coherent confluence and higher globular Kleene algebras

We extend the formalisation of confluence results in Kleene algebras to a formalisation of coherent confluence proofs. For this, we introduce the structure of higher globular Kleene algebra, a higher-dimensional generalisation of modal and concurrent Kleene algebra. We calculate a coherent Church-Rosser theorem and a coherent Newman's lemma in higher Kleene algebras by equational reasoning. We instantiate these results in the context of higher rewriting systems modelled by polygraphs

Higher Catoids, Higher Quantales and their Correspondences

We establish modal correspondences between omega-catoids and convolution omega-quantales. These are related to J\'onsson-Tarski style-dualities between relational structures and lattices with operators. We introduce omega-catoids as generalisations of (strict) omega-categories and in particular of the higher path categories generated by polygraphs (or computads) in higher rewriting. Convolution omega-quantales generalise the powerset omega-Kleene algebras recently proposed for algebraic coherence proofs in higher rewriting to weighted variants. We extend these correspondences to ({\omega},p)-catoids and convolution ({\omega},p)-quantales suitable for modelling homotopies in higher rewriting. We also specialise them to finitely decomposable ({\omega}, p)-catoids, an appropriate setting for defining ({\omega}, p)-semirings and ({\omega}, p)-Kleene algebras. These constructions support the systematic development and justification of higher quantale axioms relative to a previous ad hoc approach.Comment: 46 pages, 8 figure