The fixed point property in synthetic domain theory

Taylor, Paul

doi:10.1109/lics.1991.151640

Cited by 38 publications

(42 citation statements)

References 12 publications

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“…When we instantiate it to endomaps N ⊥ N → N ⊥ N we obtain a synthetic version of the Myhill-Shepherdson theorem [16], while the instance Σ N → Σ corresponds to Scott's principle from synthetic domain theory [22]. At last, let us reconcile Lawvere's and Tarski-Knaster fixed point theorems.…”

Section: Proof Supposementioning

confidence: 97%

On fixed-point theorems in synthetic computability

Bauer¹

2017

Tbilisi Math. J.

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Lawvere's fixed point theorem captures the essence of diagonalization arguments. Cantor's theorem, Gödel's incompleteness theorem, and Tarski's undefinability of truth are all instances of the contrapositive form of the theorem. It is harder to apply the theorem directly because non-trivial examples are not easily found, in fact, none exist if excluded middle holds. We study Lawvere's fixed-point theorem in synthetic computability, which is higher-order intuitionistic logic augmented with the Axiom of Countable Choice, Markov's principle, and the Enumeration axiom, which states that there are countably many countable subsets of N. These extra-logical principles are valid in the effective topos, as well as in any realizability topos built over Turing machines with an oracle, and suffice for an abstract axiomatic development of a computability theory. We show that every countably generated ω-chain complete pointed partial order (ωcppo) is countable, and that countably generated ωcppos are closed under countable products. Therefore, Lawvere's fixed-point theorem applies and we obtain fixed points of all endomaps on countably generated ωcppos. Similarly, the Knaster-Tarski theorem guarantees existence of least fixed points of continuous endomaps. To get the best of both theorems, namely that all endomaps on domains (ωcppos generated by a countable set of compact elements) have least fixed points, we prove a synthetic version of the Myhill-Shepherdson theorem: every map from an ωcpo to a domain is continuous. The proof relies on a new fixed-point theorem, the synthetic Recursion Theorem, so called because it subsumes the classic Kleene-Rogers Recursion Theorem. The Recursion Theorem takes the form of Lawvere's fixed point theorem for multi-valued endomaps.

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Section: Proof Supposementioning

confidence: 97%

On fixed-point theorems in synthetic computability

Bauer¹

2017

Tbilisi Math. J.

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“…This is proved in Theorem 9.6, so it is one of the points on which the logical proofs lag some way behind the topological intuitions. Of course, this result could have been used in place of Axiom 4.10, but I feel that I made an important point in [Tay91] by showing how sobriety (actually the slightly weaker notion of repleteness) transmits Scott continuity from the single object Σ N to the whole category.…”

Section: Sets Unions and Basesmentioning

confidence: 99%

Computably Based Locally Compact Spaces

Taylor¹

2006

Logical Methods in Computer Science

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Abstract. ASD (Abstract Stone Duality) is a re-axiomatisation of general topology in which the topology on a space is treated, not as an infinitary lattice, but as an exponential object of the same category as the original space, with an associated lambda-calculus. In this paper, this is shown to be equivalent to a notion of computable basis for locally compact sober spaces or locales, involving a family of open subspaces and accompanying family of compact ones. This generalises Smyth's effectively given domains and Jung's strong proximity lattices. Part of the data for a basis is the inclusion relation of compact subspaces within open ones, which is formulated in locale theory as the way-below relation on a continuous lattice. The finitary properties of this relation are characterised here, including the Wilker condition for the cover of a compact space by two open ones. The real line is used as a running example, being closely related to Scott's domain of intervals. ASD does not use the category of sets, but the full subcategory of overt discrete objects plays this role; it is an arithmetic universe (pretopos with lists). In particular, we use this subcategory to translate computable bases for classical spaces into objects in the ASD calculus.

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“…(We shall review the precise definition of Asm(A) in Section 2 below.) Categories of this form possess a very rich mathematical structure and have proved interesting from several points of view: for instance, as categorical models corresponding to Kleene-style realizability interpretations of intuitionistic logic [9]; as models for powerful polymorphic type systems [17]; or as universes within which certain 'sets' carry an intrinsic domain-like structure [10,21].…”

Section: Introductionmentioning

confidence: 99%

Computability structures, simulations and realizability

Longley¹

2013

Math. Struct. Comp. Sci.

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We generalize the standard construction of realizability models (specifically, of categories of assemblies) to a wide class of computability structures, broad enough to embrace models of computation such as labelled transition systems and process algebras. We consider a general notion of simulation between such computability structures, and show how these simulations correspond precisely to certain functors between the realizability models. Furthermore, we show that our class of computability structures has good closure properties -in particular, it is 'cartesian closed' in a slightly relaxed sense. Finally, we investigate some important subclasses of computability structures and of simulations between them. We suggest that our 2-category of computability structures and simulations may offer a useful framework for investigating questions of computational power, abstraction and simulability for a wide range of models.

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The fixed point property in synthetic domain theory

Cited by 38 publications

References 12 publications

On fixed-point theorems in synthetic computability

On fixed-point theorems in synthetic computability

Computably Based Locally Compact Spaces

Computability structures, simulations and realizability

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