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Rewriting modulo symmetric monoidal structure

Rewriting modulo symmetric monoidal structure
Rewriting modulo symmetric monoidal structure
String diagrams are a powerful and intuitive graphical syntax for terms of symmetric monoidal categories (SMCs). They find many applications in computer science and are becoming increasingly relevant in other fields such as physics and control theory.

An important role in many such approaches is played by equa- tional theories of diagrams, typically oriented and applied as rewrite rules. This paper lays a comprehensive foundation for this form of rewriting. We interpret diagrams combinatorially as typed hypergraphs and establish the precise correspondence between dia- gram rewriting modulo the laws of SMCs on the one hand and dou- ble pushout (DPO) rewriting of hypergraphs, subject to a soundness condition called convexity, on the other. This result rests on a more general characterisation theorem in which we show that typed hy- pergraph DPO rewriting amounts to diagram rewriting modulo the laws of SMCs with a chosen special Frobenius structure.

We illustrate our approach with a proof of termination for the theory of non-commutative bimonoids.
Bonchi, Filippo
3c53e89d-d280-4911-9938-eb861553d04e
Gadducci, Fabio
58494b0f-53f3-4751-8a6c-1738b1c14c79
Kissinger, Aleks
b41eacd9-1a0e-44f1-ab52-7ad1741ae6b3
Sobocinski, Pawel
439334ab-2826-447b-9fe5-3928be3fd4fd
Zanasi, Fabio
5bc03cd7-0fb6-4e14-bae8-8bf0d5d4be38
Bonchi, Filippo
3c53e89d-d280-4911-9938-eb861553d04e
Gadducci, Fabio
58494b0f-53f3-4751-8a6c-1738b1c14c79
Kissinger, Aleks
b41eacd9-1a0e-44f1-ab52-7ad1741ae6b3
Sobocinski, Pawel
439334ab-2826-447b-9fe5-3928be3fd4fd
Zanasi, Fabio
5bc03cd7-0fb6-4e14-bae8-8bf0d5d4be38

Bonchi, Filippo, Gadducci, Fabio, Kissinger, Aleks, Sobocinski, Pawel and Zanasi, Fabio (2016) Rewriting modulo symmetric monoidal structure , United States. 05 - 08 Jul 2016. 10 pp.

Record type: Conference or Workshop Item (Paper)

Abstract

String diagrams are a powerful and intuitive graphical syntax for terms of symmetric monoidal categories (SMCs). They find many applications in computer science and are becoming increasingly relevant in other fields such as physics and control theory.

An important role in many such approaches is played by equa- tional theories of diagrams, typically oriented and applied as rewrite rules. This paper lays a comprehensive foundation for this form of rewriting. We interpret diagrams combinatorially as typed hypergraphs and establish the precise correspondence between dia- gram rewriting modulo the laws of SMCs on the one hand and dou- ble pushout (DPO) rewriting of hypergraphs, subject to a soundness condition called convexity, on the other. This result rests on a more general characterisation theorem in which we show that typed hy- pergraph DPO rewriting amounts to diagram rewriting modulo the laws of SMCs with a chosen special Frobenius structure.

We illustrate our approach with a proof of termination for the theory of non-commutative bimonoids.

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More information

Accepted/In Press date: 4 April 2016
Published date: July 2016
Venue - Dates: LiCS 2016: 31st Annual ACM/IEEE Symposium on Logic in Computer Science, United States, 2016-07-05 - 2016-07-08
Organisations: Electronic & Software Systems

Identifiers

Local EPrints ID: 393991
URI: https://eprints.soton.ac.uk/id/eprint/393991
PURE UUID: b171d754-a1d0-45d0-8f6f-c63ee097bb86

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Date deposited: 26 May 2016 11:50
Last modified: 09 Jan 2018 18:08

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Contributors

Author: Filippo Bonchi
Author: Fabio Gadducci
Author: Aleks Kissinger
Author: Pawel Sobocinski
Author: Fabio Zanasi

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