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A Fresh Look at Commutativity: Free Algebraic Structures via Fresh Lists

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Programming Languages and Systems (APLAS 2023)

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Abstract

We show how types of free idempotent commutative monoids and free commutative monoids can be constructed in ordinary dependent type theory, without the need for quotient types or setoids, and prove that these constructions realise finite sets and multisets, respectively. Both constructions arise as generalisations of C. Coquand’s data type of fresh lists. We also show how many other free structures also can be realised by other instantiations. All of our results have been formalised in Agda.

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Acknowledgements

We thank Guillaume Allais for interesting suggestions, Ezra Schoen for the idea to consider reflexive partial monoids in Sect. 6.4, and the referees for insightful comments and improvements. This work was supported by the Engineering and Physical Sciences Research Council [EP/W52394X/1]; the National Physical Laboratory; and the Leverhulme Trust [RPG-2020-232].

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  1. University of Strathclyde, Glasgow, UK

    Clemens Kupke, Fredrik Nordvall Forsberg & Sean Watters

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  1. Clemens Kupke
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  2. Fredrik Nordvall Forsberg
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Correspondence to Sean Watters .

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  1. Seoul National University, Seoul, Korea (Republic of)

    Chung-Kil Hur

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Kupke, C., Forsberg, F.N., Watters, S. (2023). A Fresh Look at Commutativity: Free Algebraic Structures via Fresh Lists. In: Hur, CK. (eds) Programming Languages and Systems. APLAS 2023. Lecture Notes in Computer Science, vol 14405. Springer, Singapore. https://doi.org/10.1007/978-981-99-8311-7_7

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  • DOI: https://doi.org/10.1007/978-981-99-8311-7_7

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