Pattern graph rewrite systems

Aleks Kissinger
(University of Oxford)
Alex Merry
(University of Oxford)
Matvey Soloviev
(University of Cambridge)

String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. Dixon, Duncan and Kissinger introduced string graphs, which are a combinatoric representations of string diagrams, amenable to automated reasoning about diagrammatic theories via graph rewrite systems. In this extended abstract, we show how the power of such rewrite systems can be greatly extended by introducing pattern graphs, which provide a means of expressing infinite families of rewrite rules where certain marked subgraphs, called !-boxes ("bang boxes"), on both sides of a rule can be copied any number of times or removed. After reviewing the string graph formalism, we show how string graphs can be extended to pattern graphs and how pattern graphs and pattern rewrite rules can be instantiated to concrete string graphs and rewrite rules. We then provide examples demonstrating the expressive power of pattern graphs and how they can be applied to study interacting algebraic structures that are central to categorical quantum mechanics.

In Benedikt Löwe and Glynn Winskel: Proceedings 8th International Workshop on Developments in Computational Models (DCM 2012), Cambridge, United Kingdom, 17 June 2012, Electronic Proceedings in Theoretical Computer Science 143, pp. 54–66.
Published: 29th March 2014.

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