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Spider Diagrams

Published online by Cambridge University Press:  01 February 2010

John Howse ,
Gem Stapleton  and
John Taylor
John Howse
Affiliation:
School of Computing, Mathematical and Information Sciences, University of Brighton, Watts Building, Brighton BN2 4GJ, United Kingdom, John.Howse@brighton.ac.uk, http://www.brighton.ac.uk/cmis/research/vmg/
Gem Stapleton
Affiliation:
School of Computing, Mathematical and Information Sciences, University of Brighton, Watts Building, Brighton BN2 4GJ, United Kingdom, G.E.Stapleton@brighton.ac.uk, http://www.brighton.ac.uk/cmis/research/vmg/
John Taylor
Affiliation:
School of Computing, Mathematical and Information Sciences, University of Brighton, Watts Building, Brighton BN2 4GJ, United Kingdom, John.Taylor@brighton.ac.uk, http://www.brighton.ac.uk/cmis/research/vmg/

Abstract

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The use of diagrams in mathematics has traditionally been restricted to guiding intuition and communication. With rare exceptions such as Peirce's alpha and beta systems, purely diagrammatic formal reasoning has not been in the mathematician's or logician's toolkit. This paper develops a purely diagrammatic reasoning system of “spider diagrams” that builds on Euler, Venn and Peirce diagrams. The system is known to be expressively equivalent to first-order monadic logic with equality. Two levels of diagrammatic syntax have been developed: an ‘abstract’ syntax that captures the structure of diagrams, and a ‘concrete’ syntax that captures topological properties of drawn diagrams. A number of simple diagrammatic transformation rules are given, and the resulting reasoning system is shown to be sound and complete.

Information

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 8 , 2005 , pp. 145 - 194
Copyright
Copyright © London Mathematical Society 2005

References

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