The Equational Theory of Positive Relation Algebra
PACE
in Shanghai,
November 2016
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 talk was to export known results from Kleene algebra to some of its extensions.
We first consider a known extension: Kleene algebras with converse. Decidability of these algebras is already known, but the algorithm witnessing this result was too complicated to be practical. We propose a simpler algorithm, which is more efficient, and whose correctness is easier to establish. It allows us to prove that this problem lies in the complexity class PSpacecomplete.
Then we study Kleene allegories. Few results were known about this extension. Following results about closely related algebras, we establish the equivalence between equality in Kleene allegories and the equality of certain sets of graphs. We then develop a new automaton model (socalled Petri automata), based on Petri nets. We prove the equivalence between the original problem and comparing these automata. In the restricted setting of identityfree Kleene lattices, we also provide an algorithm performing this comparison. This algorithm uses exponential space. However, we prove that the problem of comparing Petri automata lies in the class ExpSpace complete.
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Related papers  
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with Damien Pous. 
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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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