Paul Brunet, Tobias Kappé, Alexandra Silva, and Fabio Zanasi.
Concurrent Kleene Algebra (CKA) was introduced by Hoare, Moeller, Struth and Wehrman in 2009 as a framework to reason about concurrent programs. We prove that the semantics proposed in the original paper is the free model of CKA with bounded parallel iteration, meaning the completeness of these axioms. This result settles a conjecture of Hoare and collaborators. Moreover, it allows us to establish a Kleene Theorem for CKA, extending an earlier Kleene Theorem for a fragment of CKA.
Paul Brunet, Damien Pous and Georg Struth.
Concurrent Kleene algebras support equational reasoning about computing systems with concurrent behaviours. Their natural semantics is given by series(-parallel) rational pomset languages, a standard true concurrency semantics, which is often associated with processes of Petri nets. We use constructions on Petri nets to provide two decision procedures for such pomset languages motivated by the equational and the refinement theory of concurrent Kleene algebra. The contribution to the first problem lies in a much simpler algorithm and an ExpSpace complexity bound. Decidability of the second, more interesting problem is new and, in fact, ExpSpace-complete.
We investigate the equational theory of reversible Kleene lattices, that is algebras of languages with the regular operations (union, composition and Kleene star), together with the intersection and mirror image. Building on results by AndrÃ©ka, MikulÃ¡s and NÃ©meti from 2011, we construct the free representation of this algebra. We then provide an automaton model to compare representations. These automata are adapted from Petri automata, which we introduced with Pous in 2015 to tackle a similar problem for algebras of binary relations. This allows us to show that testing the validity of equations in this algebra is decidable, and in fact ExpSpace-complete.
Paul Brunet, Tobias Kappé, Bas Luttik, Alexandra Silva, and Fabio Zanasi.
Concurrent Kleene Algebra (CKA) is a mathematical formalism to study programs that exhibit concurrent behaviour. As with previous extensions of Kleene Algebra, characterizing the free model is crucial in order to develop the foundations of the theory and potential applications. For CKA, this has been an open question for a few years and this paper makes an important step towards an answer. We present a new automaton model and a Kleene-like theorem for pomset languages, which are a natural candidate for the free model. There are two substantial differences with previous work: from expressions to automata, we use Brzozowski derivatives, which enable a direct construction of the automaton; from automata to expressions, we provide a syntactic characterization of the automata that denote valid CKA behaviours.
Paul Brunet, Damien Pous.
Kleene 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 algebras with two additional operations and a constant which are natural in binary relation models: intersection, converse, and the full relation. While regular languages are closed under those operations, the above characterisation breaks. Putting together a few results from the literature, we give a characterisation in terms of languages of directed and labelled graphs. By taking inspiration from Petri nets, we design a finite automata model, Petri automata, allowing to recognise such graphs. We prove a Kleene theorem for this automata model: the sets of graphs recognisable by Petri automata are precisely the sets of graphs definable through the extended regular expressions we consider. Petri automata allow 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. We actually show that this decision problem is EXPSPACE-complete.
Algebras of relations appear naturally in many contexts, in computer science as well as in mathematics. They constitute a framework well suited to the semantics of imperative programs. Kleene algebras are a starting point: these algebras enjoy very strong decidability properties, and a complete axiomatisation. The goal of this thesis was to export known results from Kleene algebra to some of its extensions.We first considered a known extension: Kleene algebras with converse. Decidability of these algebras was already known, but the algorithm witnessing this result was too complicated to be practical. We proposed a simpler algorithm, which is more efficient, and whose correctness is easier to establish. It allowed us to prove that this problem lies in the complexity class PSpace-complete.Then we studied Kleene allegories. Few results were known about this extension. Following results about closely related algebras, we established the equivalence between equality in Kleene allegories and the equality of certain sets of graphs. We then developed a new automaton model (so-called Petri automata), based on Petri nets. We proved the equivalence between the original problem and comparing these automata. In the restricted setting of identity-free Kleene lattices, we also provided an algorithm performing this comparison. This algorithm uses exponential space. However, we proved that the problem of comparing Petri automata lies in the class ExpSpace- complete.Finally, we studied Nominal Kleene algebras. We realised that existing descriptions of these algebra were not suited to relational semantics of programming languages. We thus modified them accordingly, and doing so uncovered several natural variations of this model. We then studied formally the bridges one could build between these variations, and between the existing model and our new version of it. This study was conducted using the proof assistant Coq.
An axiomatisation of Nominal Kleene Algebra has been proposed by Gabbay and Ciancia, and then shown to be complete and decidable by Kozen et al. However, one can think of at least four different formulations for a Kleene Algebra with names: using freshness conditions or a presheaf structure (types), and with explicit permutations or not. We formally show that these variations are all equivalent.Then we introduce an extension of Nominal Kleene Algebra, motivated by relational models of programming languages. The idea is to let letters (i.e., atomic programs) carry a set of names, rather than being reduced to a single name. We formally show that this extension is at least as expressive as the original one, and that it may be presented with or without a presheaf structure, and with or without syntactic permutations. Whether this extension is strictly more expressive remains open.All our results were formally checked using the Coq proof assistant.
Paul Brunet, Damien Pous and Insa Stucke.
In this paper we present an extension of a Coq library for relation algebras and related algebraic structures. So far, the library did not provide any tools about the cardinalities of relations. Thus we add an algebraic axiomatisation of cardinalities. Its point-free nature makes it possible to reason about cardinal purely algebraically, which is well- suited for mechanisation. We present several applications, in the area of graph theory and program verification.
The 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. In particular, they obtained its decidability by using a particular automata construction.We show that deciding this equational theory is PSpace-complete, by providing a PSpace algorithm (the problem is easily shown to be PSpace-hard). We obtain other algorithms that are time-efficient in practice, despite not being PSpace.Our results use an alternative automata construction, inspired by the one from Bloom, Ésik and Stefanescu. We relate those two constructions by exhibiting a bisimulation between the resulting deterministic automata, and by showing how our construction results in more sharing between states, thus producing smaller automata.
Kleene 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 algebras 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.
Families 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.
The 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.