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DOI: 10.4230/LIPIcs.CALCO.2015.171
URN: urn:nbn:de:0030-drops-55335
URL: https://drops.dagstuhl.de/opus/volltexte/2015/5533/
Kissinger, Aleks ; Quick, David
A First-order Logic for String Diagrams
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Abstract
Equational reasoning with string diagrams provides an intuitive means of proving equations between morphisms in a symmetric monoidal category. This can be extended to proofs of infinite families of equations using a simple graphical syntax called !-box notation. While this does greatly increase the proving power of string diagrams, previous attempts to go beyond equational reasoning have been largely ad hoc, owing to the lack of a suitable logical framework for diagrammatic proofs involving !-boxes. In this paper, we extend equational reasoning with !-boxes to a fully-fledged first order logic with conjunction, implication, and universal quantification over !-boxes. This logic, called !L, is then rich enough to properly formalise an induction principle for !-boxes. We then build a standard model for !L and give an example proof of a theorem for non-commutative bialgebras using !L, which is unobtainable by equational reasoning alone.
BibTeX - Entry
@InProceedings{kissinger_et_al:LIPIcs:2015:5533,
author = {Aleks Kissinger and David Quick},
title = {{A First-order Logic for String Diagrams}},
booktitle = {6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)},
pages = {171--189},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-84-2},
ISSN = {1868-8969},
year = {2015},
volume = {35},
editor = {Lawrence S. Moss and Pawel Sobocinski},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/5533},
URN = {urn:nbn:de:0030-drops-55335},
doi = {10.4230/LIPIcs.CALCO.2015.171},
annote = {Keywords: string diagrams, compact closed monoidal categories, abstract tensor systems, first-order logic}
}
| Keywords: | string diagrams, compact closed monoidal categories, abstract tensor systems, first-order logic | |
| Seminar: | 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015) | |
| Issue date: | 2015 | |
| Date of publication: | 28.10.2015 |
