Petri automata for Kleene allegories
Midlands Graduate School
in Sheffield,
April 2015
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 identityfree 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 graphlanguage equivalence in the full model remains open.
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Related papers  
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Cardinalities of finite relations in Coq
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Algorithms for Kleene algebra with converse
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Algebras of relations: from algorithms to formal proofs (in Université de Lyon 2016)  More  
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Decidability of identityfree relational Kleene lattices
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Kleene algebra with converse
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