Chapter 1 Partial Combinatory Algebras
This chapter explores the theories and definitions related to the partial-combinatory algebras. It distinguishes a few closed terms––pairing, booleans, and definition by cases. One of the main motivating constructions for the study of pcas is introduced––P(A)-valued predicates, conditional pcas, and...
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| Vydáno v: | Studies in Logic and the Foundations of Mathematics Ročník 152; s. 1 - 47 |
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| Médium: | Kapitola |
| Jazyk: | angličtina |
| Vydáno: |
2008
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| ISBN: | 9780444515841, 0444515844 |
| ISSN: | 0049-237X |
| On-line přístup: | Získat plný text |
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| Abstract | This chapter explores the theories and definitions related to the partial-combinatory algebras. It distinguishes a few closed terms––pairing, booleans, and definition by cases. One of the main motivating constructions for the study of pcas is introduced––P(A)-valued predicates, conditional pcas, and Longley's theorem. Some examples and important properties of pcas are also discussed and the role of recursion theory in pcas and the finite types are also focused. The best known pca isKleene's first model or κ1 and is the set N with partial recursive application. Kleene's second model κ2 or the pca for function realizability, the set of functions N N as an infinite product, with the product topology are considered. The sequential computations, the Scott's graph model P(ω), domain models, relativized model, term models, Pitt's model, and arithmetic models are also described. Morphisms and assemblies are chosen to organize pcas into a category. Applicative morphisms and S-functors among categories of assemblies are focused. Decidable applicative morphisms and adjoining partial functions to a pca proves the partial recursive application in F. The chapter also discusses order-pcas as a generalization of the notion of pca. |
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| AbstractList | This chapter explores the theories and definitions related to the partial-combinatory algebras. It distinguishes a few closed terms––pairing, booleans, and definition by cases. One of the main motivating constructions for the study of pcas is introduced––P(A)-valued predicates, conditional pcas, and Longley's theorem. Some examples and important properties of pcas are also discussed and the role of recursion theory in pcas and the finite types are also focused. The best known pca isKleene's first model or κ1 and is the set N with partial recursive application. Kleene's second model κ2 or the pca for function realizability, the set of functions N N as an infinite product, with the product topology are considered. The sequential computations, the Scott's graph model P(ω), domain models, relativized model, term models, Pitt's model, and arithmetic models are also described. Morphisms and assemblies are chosen to organize pcas into a category. Applicative morphisms and S-functors among categories of assemblies are focused. Decidable applicative morphisms and adjoining partial functions to a pca proves the partial recursive application in F. The chapter also discusses order-pcas as a generalization of the notion of pca. |
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| Copyright | 2008 Elsevier B.V. |
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