Computability structures, simulations and realizability

Longley, John

doi:10.1017/s0960129513000182

Cited by 6 publications

(11 citation statements)

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“…Related to the work of Boker and Dershowitz, is that of Cockett and Hofstra [9], and Longley [26] which are both concerned with category-theoretic descriptions of abstract computational models. In these frameworks model equivalence is interpreted by categorical isomorphism, and so akin to the strongest notion of equivalence considered by Boker and Dershowitz.…”

Section: Expressiveness Of Factorisation: Discussion and Related Workmentioning

confidence: 99%

Encoding the Factorisation Calculus

Rowe

2015

Electron. Proc. Theor. Comput. Sci.

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Jay and Given-Wilson have recently introduced the Factorisation (or SF-) calculus as a minimal fundamental model of intensional computation. It is a combinatory calculus containing a special combinator, F, which is able to examine the internal structure of its first argument. The calculus is significant in that as well as being combinatorially complete it also exhibits the property of structural completeness, i.e. it is able to represent any function on terms definable using pattern matching on arbitrary normal forms. In particular, it admits a term that can decide the structural equality of any two arbitrary normal forms. Since SF-calculus is combinatorially complete, it is clearly at least as powerful as the more familiar and paradigmatic Turing-powerful computational models of Lambda Calculus and Combinatory Logic. Its relationship to these models in the converse direction is less obvious, however. Jay and Given-Wilson have suggested that SF-calculus is strictly more powerful than the aforementioned models, but a detailed study of the connections between these models is yet to be undertaken. This paper begins to bridge that gap by presenting a faithful encoding of the Factorisation Calculus into the Lambda Calculus preserving both reduction and strong normalisation. The existence of such an encoding is a new result. It also suggests that there is, in some sense, an equivalence between the former model and the latter. We discuss to what extent our result constitutes an equivalence by considering it in the context of some previously defined frameworks for comparing computational power and expressiveness.Comment: In Proceedings EXPRESS/SOS 2015, arXiv:1508.0634

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Section: Expressiveness Of Factorisation: Discussion and Related Workmentioning

confidence: 99%

Encoding the Factorisation Calculus

Rowe

2015

Electron. Proc. Theor. Comput. Sci.

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“…This notion is rooted in previous work of Longley in[6][7][8], and is influenced by the work of Cockett and Hofstra in[3] and[4] (see[9], p. 52).…”

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confidence: 89%

“…Longley and Normann associated in a canonical way to a computability model 2 C its category of assemblies Asm(C), "the world of all datatypes that can be represented in" C. They also showed that the computability models C and D are equivalent if and only if the categories of assemblies Asm(C) and Asm(D) are equivalent. The category of computability models and the corresponding functor C → Asm(C) are studied extensively in [8].…”

Section: Introductionmentioning

confidence: 99%

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Computability Models over Categories and Presheaves

Petrakis¹

2021

Logical Foundations of Computer Science

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Generalising slightly the notions of a strict computability model and of a simulation between them, which were elaborated by Longley and Normann in [9], we define canonical computability models over certain categories and appropriate presheaves on them. We study the canonical total computability model over a category C and a covariant presheaf on C, and the canonical partial computability model over a category C with pullbacks and a pullback preserving, covariant presheaf on C. These computability models are shown to be special cases of a computability model over a category C with a so-called base of computability and a pullback preserving, covariant presheaf on C. In this way Rosolini's theory of dominions is connected with the theory of computability models. All our notions and results are dualised by considering certain (contravariant) presheaves on appropriate categories.

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“…The second reason for moving to this level of generality is a potential connection with more general forms of realizability. Longley has recently proposed a notion of 'computability structure' [Lon13] that encompasses partial combinatory algebras and is similar to the 'basic combinatorial objects' of Hofstra [Hof06]. Each of these is clearly trying very hard to be a category enriched in some sort of 2-category, and so one might wonder whether they are examples of a still more general notion of 'coefficient object' for realizability that encompasses the two, and whether the passage from a partial combinatory algebra to its associated tripos can be seen in the context of the equivalence between fibrations and categories enriched in certain 2-categories.…”

Section: 'Variation Through Enrichment'mentioning

confidence: 99%

Fibrations of predicates and bicategories of relations

Lawler

2015

Preprint

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For this version I have again made some small corrections and added a few notes, but the document remains the same in substance.i My thanks must go first of all to my supervisor, Hugh Gibbons, for his unfailing patience and support. When Hugh took me on, my intention was to write a thesis on logic and computation, so he must have been somewhat dismayed to watch me then veer sharply into category theory. But he did not despair, and in fact this meant that I had to make my ideas and explanations intelligible to a sympathetic but lay audience, forcing me to think more deeply and expound more clearly than I might otherwise have done. I hope that shows. Hugh's help in navigating the College bureaucracy, and indeed all the usual travails of post-graduate life, has been likewise invaluable.If Hugh has been my mainstay, then Arthur Hughes has been my binnacle. The two have together supported and guided me through the whole process of planning and writing this thesis. Though his status as my co-supervisor remains unofficial, Arthur has more than earned that acknowledgement by his willingness to listen, suggest and advise, particularly when it comes to category theory. His good nature and solicitousness make him a pleasure to work with too. I should also thank the other members of the Foundations and Methods Group of the Computer Science Department for their help and advice, especially Matthew Hennessy.Even despite Arthur's mathematical company, though, I would not have come to understand category theory at anywhere near the level this thesis required without the n-Lab (ncatlab.org). Of course, I can't possibly thank by name everyone involved, but I do want to make clear just how much I owe to all of them. My understanding of category theory, and of mathematics in general, has been immeasurably deepened and broadened by the material on the Lab, as well as by discussions on the associated n-Forum. I must thank Mike Shulman and Todd Trimble in particular for the latter.Writing a doctoral thesis is never easy, and this one too had a difficult birth. But when things looked grim, the College support services were there to help. I really cannot overstate how much it has meant to me to have their professional, compassionate and effective support available, or how important these services are in general. It reflects very well indeed on College that they employ so many staff whose sole concern is the well-being of students, and long may this continue. I don't suppose it would be appropriate to mention the people I have in mind here by name, but if I say that they are C. G., C. R. and Dr. N. F., then they will know who they are, and know that they have my immense gratitude. The system works, and this thesis is the proof. Likewise, the College administrative staff, in my dealings with them, have never failed to be friendly, efficient, flexible and eager to help. In particular, Helen Thornbury of the Graduate Studies Office is a sparkling exemplar of administrative excellence, who has personally dug me out of several holes.In the end,...

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Computability structures, simulations and realizability

Cited by 6 publications

References 20 publications

Encoding the Factorisation Calculus

Encoding the Factorisation Calculus

Computability Models over Categories and Presheaves

Fibrations of predicates and bicategories of relations

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