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Computational Category-Theoretic Rewriting

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Graph Transformation (ICGT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13349))

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Abstract

We demonstrate how category theory provides specifications that can efficiently be implemented via imperative algorithms and apply this to the field of graph transformation. By examples, we show how this paradigm of software development makes it easy to quickly write correct and performant code. We provide a modern implementation of graph rewriting techniques at the level of abstraction of finitely-presented \(\mathcal C\)-sets and clarify the connections between \(\mathcal C\)-sets and the typed graphs supported in existing rewriting software. We emphasize that our open-source library is extensible: by taking new categorical constructions (such as slice categories, structured cospans, and distributed graphs) and relating their limits and colimits to those of their underlying categories, users inherit efficient algorithms for pushout complements and (final) pullback complements. This allows one to perform double-, single-, and sesqui-pushout rewriting over a broad class of data structures. Graph transformation researchers, scientists, and engineers can then use this library to computationally manipulate rewriting systems and apply them to their domains of interest.

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Notes

  1. 1.

    https://nbviewer.org/github/kris-brown/Computational-Category-Theoretic-Rewriting/blob/main/Computational_Category_Theoretic_Rewriting.ipynb.

  2. 2.

    \(\mathcal C\)-sets are also called copresheaves on \(\mathcal C\) or presheaves on \(\mathcal C\) \(^{op}\), and are what Löwe studied as graph structures or unary algebras.

  3. 3.

    When specialized to typed graphs, \(\mathcal {E} \xrightarrow {F}\mathcal {C}\) is a graph homomorphism and the graphs are regarded as their path categories.

  4. 4.

    The L of structured cospans should not be confused with the L of the rewrite rule \(L\leftarrow I \rightarrow R\).

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Authors and Affiliations

  1. University of Florida, Gainesville, FL, USA

    Kristopher Brown, Tyler Hanks & James Fairbanks

  2. Topos Institute, Berkeley, CA, USA

    Evan Patterson

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  1. Kristopher Brown
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Correspondence to Kristopher Brown .

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Editors and Affiliations

  1. Université Paris Cité, Paris, France

    Nicolas Behr

  2. Computer Science and Engineering, Chalmers | Gothenburg University, Gothenburg, Sweden

    Daniel Strüber

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Brown, K., Patterson, E., Hanks, T., Fairbanks, J. (2022). Computational Category-Theoretic Rewriting. In: Behr, N., Strüber, D. (eds) Graph Transformation. ICGT 2022. Lecture Notes in Computer Science, vol 13349. Springer, Cham. https://doi.org/10.1007/978-3-031-09843-7_9

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  • DOI: https://doi.org/10.1007/978-3-031-09843-7_9

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