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URN: urn:nbn:de:0030-drops-55335
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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: 2015


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