Generic models for computational effects

A Freyd-category is a subtle generalisation of the notion of a category with finite products. It is suitable for modelling environments in call-by-value programming languages, such as the computational λ -calculus, with computational effects. We develop the theory of Freyd-categories with that in mi...

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Veröffentlicht in:Theoretical computer science Jg. 364; H. 2; S. 254 - 269
1. Verfasser: Power, John
Format: Journal Article Tagungsbericht
Sprache:Englisch
Veröffentlicht: Amsterdam Elsevier B.V 06.11.2006
Elsevier
Schlagworte:
Applied sciences
Canonical model
Computer science; control theory; systems
Conical colimit completion
Enriched Yoneda embedding
Exact sciences and technology
Freyd-category
Logic and foundations
Mathematical logic, foundations, set theory
Mathematics
Model theory
Programming languages
Programming theory
Sciences and techniques of general use
Software
Theoretical computing
Freyd-category
Enriched Yoneda embedding
Conical colimit completion
Canonical model
Computer theory
Programming language
Freyd category
Non determinism
Structural analysis
ISSN:0304-3975, 1879-2294
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Beschreibung
Zusammenfassung:A Freyd-category is a subtle generalisation of the notion of a category with finite products. It is suitable for modelling environments in call-by-value programming languages, such as the computational λ -calculus, with computational effects. We develop the theory of Freyd-categories with that in mind. We first show that any countable Lawvere theory, hence any signature of operations with countable arity subject to equations, directly generates a Freyd-category. We then give canonical, universal embeddings of Freyd-categories into closed Freyd-categories, characterised by being free cocompletions. The combination of the two constructions sends a signature of operations and equations to the Kleisli category for the monad on the category Set generated by it, thus refining the analysis of computational effects given by monads. That in turn allows a more structural analysis of the λ c -calculus. Our leading examples of signatures arise from side-effects, interactive input/output and exceptions. We extend our analysis to an enriched setting in order to account for recursion and for computational effects and signatures that inherently involve it, such as partiality, nondeterminism and probabilistic nondeterminism.
ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2006.08.006