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19
EXHAUSTIBLE SETS IN HIGHERTYPE COMPUTATION
, 2008
"... We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no example. The C ..."
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We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no example. The Cantor space of infinite sequences of binary digits is known to be searchable. Searchable sets are exhaustible, and we show that the converse also holds for sets of hereditarily total elements in the hierarchy of continuous functionals; moreover, a selection functional can be constructed uniformly from a quantification functional. We prove that searchable sets are closed under intersections with decidable sets, and under the formation of computable images and of finite and countably infinite products. This is related to the fact, established here, that exhaustible sets are topologically compact. We obtain a complete description of exhaustible total sets by developing a computational version of a topological Arzela–Ascoli type characterization of compact subsets of function spaces. We also show that, in the nonempty case, they are precisely the computable images of the Cantor space. The emphasis of this paper is on the theory of exhaustible and searchable sets, but we also briefly sketch applications.
Infinite sets that admit fast exhaustive search
 In Proceedings of the 22nd Annual IEEE Symposium on Logic In Computer Science
, 2007
"... Abstract. Perhaps surprisingly, there are infinite sets that admit mechanical exhaustive search in finite time. We investigate three related questions: What kinds of infinite sets admit mechanical exhaustive search in finite time? How do we systematically build such sets? How fast can exhaustive sea ..."
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Cited by 15 (8 self)
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Abstract. Perhaps surprisingly, there are infinite sets that admit mechanical exhaustive search in finite time. We investigate three related questions: What kinds of infinite sets admit mechanical exhaustive search in finite time? How do we systematically build such sets? How fast can exhaustive search over infinite sets be performed? Keywords. Highertype computability and complexity, Kleene–Kreisel functionals, PCF, Haskell, topology. 1.
A stable programming language
 I&C
"... It is wellknown that stable models (as dIdomains, qualitative domains and coherence spaces) are not fully abstract for the languagePCF. This fact is related to the existence of stable parallel functions and of stable functions that are not monotone with respect to the extensional order, which cann ..."
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Cited by 10 (2 self)
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It is wellknown that stable models (as dIdomains, qualitative domains and coherence spaces) are not fully abstract for the languagePCF. This fact is related to the existence of stable parallel functions and of stable functions that are not monotone with respect to the extensional order, which cannot be defined by programs ofPCF. In this paper, a paradigmatic programming language namedStPCF is proposed, which extends the languagePCF with two additional operators. The operational description of the extended language is presented in an effective way, although the evaluation of one of the new operators cannot be formalized in a PCFlike rewrite system. SinceStPCF can define all finite cliques of coherence spaces the above gap with stable models is filled, consequently stable models are fully abstract for the extended language. 1
On the ubiquity of certain total type structures
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2007
"... It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel co ..."
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Cited by 4 (2 self)
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It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel continuous functionals, its effective substructure C eff, and the type structure HEO of the hereditarily effective operations. However, the proofs of the relevant equivalences are often nontrivial, and it is not immediately clear why these particular type structures should arise so ubiquitously. In this paper we present some new results which go some way towards explaining this phenomenon. Our results show that a large class of extensional collapse constructions always give rise to C, C eff or HEO (as appropriate). We obtain versions of our results for both the “standard” and “modified” extensional collapse constructions. The proofs make essential use of a technique due to Normann. Many new results, as well as some previously known ones, can be obtained as instances of our theorems, but more importantly, the proofs apply uniformly to a whole family of constructions, and provide strong evidence that the above three type structures are highly canonical mathematical objects.
Computability structures, simulations and realizability
, 2011
"... We generalize the standard construction of realizability models (specifically, of categories of assemblies) to a very wide class of computability structures, broad enough to embrace models of computation such as labelled transition systems and process algebras. We also discuss a general notion of si ..."
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We generalize the standard construction of realizability models (specifically, of categories of assemblies) to a very wide class of computability structures, broad enough to embrace models of computation such as labelled transition systems and process algebras. We also discuss a general notion of simulation between such computability structures, and show that such 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. We also investigate some important subclasses of computability structures and of simulations between them. We suggest that our 2category of computability structures and simulations may offer a framework for a general investigation of questions of computational power, abstraction and simulability for a wide range of computation models from across computer science.
Interpreting localized computational effects using operators of higher type (extended abstract)
"... ..."
Linearity and PCF: a semantic insight
 Proceeding of the 16th ACM SIGPLAN international conference on Functional Programming, ICFP 2011
, 2011
"... Linearity is a multifaceted and ubiquitous notion in the analysis and the development of programming language concepts. We study linearity in a denotational perspective by picking out programs that correspond to linear functions between coherence spaces. We introduce a language, named SPCF, that in ..."
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Linearity is a multifaceted and ubiquitous notion in the analysis and the development of programming language concepts. We study linearity in a denotational perspective by picking out programs that correspond to linear functions between coherence spaces. We introduce a language, named SPCF, that increases the higherorder expressivity of a linear core of PCF by means of new operators related to exception handling and parallel evaluation. SPCF allows us to program all the finite elements of the model and, consequently, it entails a full abstraction result that makes the reasoning on the equivalence between programs simpler. Denotational linearity provides also crucial information for the operational evaluation of programs. We formalize two evaluation machineries for the language. The first one is an abstract and concise operational semantics designed with the aim of explaining the new operators, and is based on an infinitebranching search of the evaluation space. The second one is more concrete and it prunes such a space, by exploiting the linear assumptions. This can also be regarded as a base for an implementation.
On the calculating power of Laplace’s demon (Part I)
, 2006
"... We discuss several ways of making precise the informal concept of physical determinism, drawing on ideas from mathematical logic and computability theory. We outline a programme of investigating these notions of determinism in detail for specific, precisely articulated physical theories. We make a s ..."
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We discuss several ways of making precise the informal concept of physical determinism, drawing on ideas from mathematical logic and computability theory. We outline a programme of investigating these notions of determinism in detail for specific, precisely articulated physical theories. We make a start on our programme by proposing a general logical framework for describing physical theories, and analysing several possible formulations of a simple Newtonian theory from the point of view of determinism. Our emphasis throughout is on clarifying the precise physical and metaphysical assumptions that typically underlie a claim that some physical theory is ‘deterministic’. A sequel paper is planned, in which we shall apply similar methods to the analysis of other physical theories. Along the way, we discuss some possible repercussions of this kind of investigation for both physics and logic. 1
On the calculating power of Laplace’s demon
"... Abstract. We discuss some of the choices that arise when one tries to make the idea of physical determinism more precise. Broadly speaking, ‘ontological ’ notions of determinism are parameterized by one’s choice of mathematical ideology, whilst ‘epistemological ’ notions of determinism are parameter ..."
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Abstract. We discuss some of the choices that arise when one tries to make the idea of physical determinism more precise. Broadly speaking, ‘ontological ’ notions of determinism are parameterized by one’s choice of mathematical ideology, whilst ‘epistemological ’ notions of determinism are parameterized by the choice of an appropriate notion of computability. We present some simple examples to show that these choices can indeed make a difference to whether a given physical theory is ‘deterministic’ or not. Keywords: Laplace’s demon, physical determinism, philosophy of mathematics, notions of computability. 1