eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-02-27
3:1
3:16
10.4230/LIPIcs.STACS.2018.3
article
On the Positive Calculus of Relations with Transitive Closure
Pous, Damien
Binary relations are such a basic object that they appear in many
places in mathematics and computer science. For instance, when
dealing with graphs, program semantics, or termination guarantees,
binary relations are always used at some point.
In this survey, we focus on the relations themselves, and we
consider algebraic and algorithmic questions. On the algebraic side, we want to understand and characterise the laws governing the behaviour of the following standard operations on relations: union, intersection, composition, converse, and reflexive-transitive closure. On the algorithmic side, we look for decision procedures for equality or inequality of relations.
After having formally defined the calculus of relations, we recall
the existing results about two well-studied fragments of particular importance: Kleene algebras and allegories. Unifying those fragments yields a decidable theory whose axiomatisability remains an open problem.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol096-stacs2018/LIPIcs.STACS.2018.3/LIPIcs.STACS.2018.3.pdf
Relation Algebra
Kleene Algebra
Allegories
Automata
Graphs