Algebraic coherent confluence and higher globular Kleene algebras

Abstract

We extend the formalisation of confluence results in Kleene algebras to a formalisation of coherent confluence proofs. For this, we introduce the structure of higher globular Kleene algebra, a higher-dimensional generalisation of modal and concurrent Kleene algebra. We calculate a coherent Church-Rosser theorem and a coherent Newman's lemma in higher Kleene algebras by equational reasoning. We instantiate these results in the context of higher rewriting systems modelled by polygraphs.

Catoids and modal convolution algebras

“…We finish with directions for future work: In combination with previous work on concurrent relational monoids and concurrent quantales [5], it seems interesting to build models for non-interleaving concurrent systems based on po(m)sets and graphs with interfaces and lift them to modal concurrent semirings and quantales. Along similar lines it seems possible to define higherdimensional globular catoids as generalisations of 2-categories and prove correspondence results with respect to the higher globular Kleene algebras introduced for higher rewriting [2]. Ultimately, this should lead to ω-catoids as generalisations of ω-categories and the corresponding ω-quantales, which would share features of modal and concurrent quantales, in combination with axioms for globular structure.…”

“…Algebras of weighted paths are generally important for quantitative analyses of systems or the design of algorithms. The construction of the modal powerset quantale over the path category of a digraph has recently been extended to higher-dimensional modal Kleene algebras and higher-dimensional (poly)graphs [2], which applications in higher rewriting.…”

Relating Kleene Algebras with Pseudo Uninorms

“…Let ⟨P, ≤⟩ be a poset and e ∈ P . Then, trivially, ⟨P e , ≤ e ⟩ and ⟨P e , ≤ e ⟩ are poset with a greater and least, respectively, element when P e = {x ∈ P ∶ x ≤ e}, P e = {x ∈ P ∶ e ≤ x} and ≤ e and ≤ e are the restriction of ≤ to P e and P e , respectively 3 . Let ∆ P = {a ∈ P ∶ for each x ∈ P a ≤ x or x ≤ a}.…”

“…On the basis of this field is the notion of Kleene Algebra [14], today accepted as the standard abstraction of a computational system. Among its examples, an algebraic framework for coherent confluence proofs, in rewriting theory, for a higher dimensional generalisation of modal Kleene algebra proposes in [3] and the algebra of the regular languages, traces of programs and the algebra of relations on which the program states transitions are modelled as binary relations on the set of states. For instance, by starting from the atomic programs represented in the transition systems of Fig.…”