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On the Positive Calculus of Relations with Transitive Closure

Author Damien Pous

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Damien Pous

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Damien Pous. On the Positive Calculus of Relations with Transitive Closure. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.STACS.2018.3

Abstract

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.

Subject Classification

Keywords
  • Relation Algebra
  • Kleene Algebra
  • Allegories
  • Automata
  • Graphs

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