The Equational Theory of Positive Relation Algebra
MOVE
in Marseille,
March 2017
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 ExpSpacecomplete.
Related talks  
The Equational Theory of Positive Relation Algebra
PACE
in Shanghai,
November 2016
.

More  
A Kleene Theorem for Petri automata
Highlights
in Brussels,
September 2016
.

More  
Une introduction aux algèbres de Kleene
Inter'Actions
in Lyon,
May 2016
(in French)
.

More  
Petri automata for Kleene Allegories
LiCS
in Kyoto,
July 2015
.

More  
Petri automata for Kleene allegories
Rapido
in Paris,
June 2015
.

More  
Petri automata for Kleene allegories
Midlands Graduate School
in Sheffield,
April 2015
.

More  
Décidabilité des Treillis de Kleene sans identité
JFLA
in Val d'Ajol,
January 2015
(in French)
.

More  
Decidability of Identityfree Kleene Lattices
LAC
in Chambéry,
November 2014
.

More  
Kleene Algebra with Converse
RAMiCS
in Marienstatt,
April 2014
.

More  
Deciding Kleene Algebra with converse is PSpacecomplete
GeoCal
in Bordeaux,
March 2014
.

More  
Deciding Kleene Algebra with converse is PSpacecomplete
PACE
in Lyon,
February 2014
.

More  
Algèbres de Relations, Étude des algèbres de Kleene avec converse
Internship defence
in Paris,
September 2013
(Internship defence, in French)
.

More  
Equivalence of regular expressions with converse on relations
PiCoq
in Lyon,
June 2013
.

More  
Related papers  
Petri automata
(in LMCS 2017)
with Damien Pous. 
More  
Cardinalities of finite relations in Coq
(in ITP 2016)
with Insa Stucke and Damien Pous. 
More  
Algorithms for Kleene algebra with converse
(in JLAMP 2016)
with Damien Pous. 
More  
Algebras of relations: from algorithms to formal proofs (in Université de Lyon 2016)  More  
Petri automata for Kleene allegories
(in LICS 2015)
with Damien Pous. 
More  
Decidability of identityfree relational Kleene lattices
(in JFLA 2015)
with Damien Pous. 
More  
Kleene algebra with converse
(in RAMICS 2014)
with Damien Pous. 
More 