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Commutative monads, diagrams and knots

Published:31 August 2009Publication History
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Abstract

There is certain diverse class of diagram that is found in a variety of branches of mathematics and which all share this property: there is a common scheme for translating all of these diagrams into useful functional code. These diagrams include Bayesian networks, quantum computer circuits [1], trace diagrams for multilinear algebra [2], Feynman diagrams and even knot diagrams [3]. I will show how a common thread lying behind these diagrams is the presence of a commutative monad and I will show how we can use this fact to translate these diagrams directly into Haskell code making use of do-notation for monads. I will also show a number of examples of such translated code at work and use it to solve problems ranging from Bayesian inference to the topological problem of untangling tangled strings. Along the way I hope to give a little insight into the subjects mentioned above and illustrate how a functional programming language can be a valuable tool in mathematical research and experimentation.

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Supplemental Material

commutativemonadsdiagramsandknotsonvimeo.mp4

References

  1. Coecke, Bob. Kindergarten Quantum Mechanics. http://arxiv.org/abs/quant-ph/0510032Google ScholarGoogle Scholar
  2. Kauffman, L.H.: Knots and Physics, 3rd edn. World Scientific (2001).Google ScholarGoogle Scholar
  3. Steven Morse and Elisha Peterson,Trace Diagrams, Matrix Minors, and Determinant Identities, http://arxiv.org/abs/0903.1373Google ScholarGoogle Scholar

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  1. Commutative monads, diagrams and knots

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    • Published in

      cover image ACM SIGPLAN Notices
      ACM SIGPLAN Notices  Volume 44, Issue 9
      ICFP '09
      September 2009
      343 pages
      ISSN:0362-1340
      EISSN:1558-1160
      DOI:10.1145/1631687
      Issue’s Table of Contents
      • cover image ACM Conferences
        ICFP '09: Proceedings of the 14th ACM SIGPLAN international conference on Functional programming
        August 2009
        364 pages
        ISBN:9781605583327
        DOI:10.1145/1596550

      Copyright © 2009 Copyright is held by the author/owner(s)

      Publisher

      Association for Computing Machinery

      New York, NY, United States

      Publication History

      • Published: 31 August 2009

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