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 pla...
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| Veröffentlicht in: | Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science S. 710 - 719 |
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| Format: | Tagungsbericht |
| Sprache: | Englisch |
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New York, NY, USA
ACM
05.07.2016
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| Schriftenreihe: | ACM Conferences |
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| ISBN: | 9781450343916, 1450343910 |
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| 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 equational 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 diagram rewriting modulo the laws of SMCs on the one hand and double 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 hypergraph 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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| AbstractList | 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 equational 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 diagram rewriting modulo the laws of SMCs on the one hand and double 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 hypergraph 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. |
| Author | Gadducci, Fabio Bonchi, Filippo Sobociński, Paweł Kissinger, Aleks Zanasi, Fabio |
| Author_xml | – sequence: 1 givenname: Filippo surname: Bonchi fullname: Bonchi, Filippo email: filippo.bonchi@gmail.com organization: CNRS, ENS Lyon – sequence: 2 givenname: Fabio surname: Gadducci fullname: Gadducci, Fabio email: gadducci@di.unipi.it organization: U. Pisa – sequence: 3 givenname: Aleks surname: Kissinger fullname: Kissinger, Aleks email: aleks@cs.ru.nl organization: Radboud U. Nijmegen – sequence: 4 givenname: Paweł surname: Sobociński fullname: Sobociński, Paweł email: ps@ecs.soton.ac.uk organization: U. Southampton – sequence: 5 givenname: Fabio surname: Zanasi fullname: Zanasi, Fabio email: fzanasi@cs.ru.nl organization: Radboud U. Nijmegen |
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| Snippet | String diagrams are a powerful and intuitive graphical syntax for terms of symmetric monoidal categories (SMCs). They find many applications in computer... |
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| SubjectTerms | Theory of computation -- Design and analysis of algorithms |
| Title | Rewriting modulo symmetric monoidal structure |
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