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 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.

Related talks
	The Equational Theory of Positive Relation Algebra MOVE in Marseille, March 2017.	More
	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
	Décidabilité des Treillis de Kleene sans identité JFLA in Val d'Ajol, January 2015 (in French) .	More
	Decidability of Identity-free Kleene Lattices LAC in Chambéry, November 2014.	More
	Kleene Algebra with Converse RAMiCS in Marienstatt, April 2014.	More
	Deciding Kleene Algebra with converse is PSpace-complete GeoCal in Bordeaux, March 2014.	More
	Deciding Kleene Algebra with converse is PSpace-complete 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 identity-free relational Kleene lattices (in JFLA 2015) with Damien Pous.	More
	Kleene algebra with converse (in RAMICS 2014) with Damien Pous.	More