Abstract
We develop rules for coalgebras in type theory, and give meaning explanations for them. We show that elements of coalgebras are determined by their elimination rules, whereas the introduction rules can be considered as derived. This is in contrast with algebraic data types, for which the opposite is true: elements are determined by their introduction rules, and the elimination rules can be considered as derived. In this sense, the function type from the logical framework is more like a coalgebraic data type, the elements of which are determined by the elimination rule. We illustrate why the simplest form of guarded recursion is nothing but the introduction rule originating from the formulation of coalgebras in category theory. We discuss restrictions needed in order to preserve decidability of equality.
Supported by EPSRC grant EP/G033374/1
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Acknowledgements
We want to thank the anonymous referee for valuable comments on earlier version of this articles. We want to thank as well our PhD student Fredrik Nordvall Forsberg for diligent proof reading and valuable remarks.
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Dedicated to Per Martin-Löf on the occasion of his retirement.
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Setzer, A. (2012). Coalgebras as Types Determined by Their Elimination Rules. In: Dybjer, P., Lindström, S., Palmgren, E., Sundholm, G. (eds) Epistemology versus Ontology. Logic, Epistemology, and the Unity of Science, vol 27. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4435-6_16
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