2016-03-17: Nicolas Behr (Edinburgh)

Tue, 14 Oct 2025 16:51:42 +0000

Nicolas Behr (Edinburgh)

On the rule algebraic reformulation of graph rewriting

Abstract

Motivated by the desire to understand the combinatorics of graph rewriting systems, it proved necessary to invent a formulation of graph rewriting itself that is not based on category theoretic structures as in the traditional framework, but on the concept of diagrammatic combinatorial Hopf algebras and reductions thereof. In this talk, I will present how the classical example of the Heisenberg-Weyl algebra of creation and annihilation of indistinguishable particles, which can alternatively be interpreted as the algebra of discrete graph rewriting, gave the initial clues for this novel framework. In hindsight, to pass from the special case of discrete graph rewriting to the case of general graph rewriting required every aspect of the framework of diagrammatic combinatorial Hopf algebras as a guideline for the construction, yet none of the traditional category theoretic ideas, whence one might indeed consider this reformulation as an independent formulation of graph rewriting.

The new framework results in a number of surprising results even directly from the formulation itself: besides the two main variants of graph rewriting known in the literature (DPO and SPO rewriting), there exist two more natural variants in the new framework. For all four variants, graph rewriting rules are encoded in so-called rule diagrams, with their composition captured in the form of diagrammatic compositions followed by one of four variants of reduction operations. Besides the general structure theory of the resulting algebras aka the rule algebras, one of the most important results to date of this framework in view of applications is the possibility to formulate stochastic graph rewriting systems in terms of the canonical representations of the rule algebras. Notably, this is closely analogous to the formulation of chemical reaction systems in terms of the canonical representation of the Heisenberg-Weyl algebra aka the bosonic Fock space. The presentation will not assume any prior knowledge of the audience on the particular mathematics required for this construction, and will be given on the whiteboard. The work presented is the result of a collaboration with Vincent Danos and Ilias Garnier (ENS Paris/LFCS University of Edinburgh), and (in an earlier phase) with Tobias Heindel (University of Copenhagen).

2016-03-17 11:00:00 UTC
LT1310