Confluence of algebraic rewriting systems

Cyrille Chenavier (1) , Benjamin Dupont (2) , Philippe Malbos (3, 2)

1 JKU - University of Linz - Johannes Kepler Universität Linz
2 ICJ - Institut Camille Jordan
3 AGL - Algèbre, géométrie, logique

Convergent rewriting systems on algebraic structures give methods to solve decision problems, to prove coherence results, and to compute homological invariants. These methods are based on higher-dimensional extensions of the critical branching lemma that proves local confluence from confluence of the critical branchings. The analysis of local confluence of rewriting systems on algebraic structures, such as groups or linear algebras, is complicated because of the underlying algebraic axioms. This article introduces the structure of algebraic polygraph modulo that formalizes the interaction between the rules of an algebraic rewriting system and the inherent algebraic axioms, and we show a critical branching lemma for algebraic polygraphs. We deduce a critical branching lemma for rewriting systems on algebraic models whose axioms are specified by convergent modulo rewriting systems. We illustrate our constructions for string, linear, and group rewriting systems.

Mots clés: Term rewriting modulo; algebraic polygraphs; linear rewriting; string rewriting; group rewriting

Domaines: Mathématiques [math]; Algorithme et structure de données [cs.DS]; Calcul formel [cs.SC]; Catégories et ensembles [math.CT]

Cyrille Chenavier, Benjamin Dupont, Philippe Malbos. Confluence of algebraic rewriting systems. Mathematical Structures in Computer Science, 2021, 32 (7), pp.870-897. ⟨10.1017/S0960129521000426⟩. ⟨hal-02945665v2⟩