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Games and full abstraction for nondeterministic languages
, 1999
"... Abstract Nondeterminism is a pervasive phenomenon in computation. Often it arises as an emergent property of a complex system, typically as the result of contention for access to shared resources. In such circumstances, we cannot always know, in advance, exactly what will happen. In other circumstan ..."
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Abstract Nondeterminism is a pervasive phenomenon in computation. Often it arises as an emergent property of a complex system, typically as the result of contention for access to shared resources. In such circumstances, we cannot always know, in advance, exactly what will happen. In other circumstances, nondeterminism is explicitly introduced as a means of abstracting away from implementation details such as precise command scheduling and control flow. However, the kind of behaviours exhibited by nondeterministic computations can be extremely subtle in comparison to those of their deterministic counterparts and reasoning about such programs is notoriously tricky as a result. It is therefore important to develop semantic tools to improve our understanding of, and aid our reasoning about, such nondeterministic programs. In this thesis, we extend the framework of game semantics to encompass nondeterministic computation. Game semantics is a relatively recent development in denotational semantics; its main novelty is that it views a computation not as a static entity, but rather as a dynamic process of interaction. This perspective makes the theory wellsuited to modelling many aspects of computational processes: the original use of game semantics in modelling the simple functional language PCF has subsequently been extended to handle more complex control structures such as references and continuations.
Projecting Sequential Algorithms on Strongly Stable Functions
 Annals of Pure and Applied Logic
, 1993
"... We relate two sequential models of PCF: the sequential algorithm model due to Berry and Curien and the strongly stable model due to Bucciarelli and the author. More precisely, we show that all the morphisms araising in the strongly stable model of PCF are sequential in the sense that they are the ..."
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Cited by 26 (2 self)
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We relate two sequential models of PCF: the sequential algorithm model due to Berry and Curien and the strongly stable model due to Bucciarelli and the author. More precisely, we show that all the morphisms araising in the strongly stable model of PCF are sequential in the sense that they are the "extensional projections" of some sequential algorithms. We define a model of PCF where morphisms are "extensional" sequential algorithms and prove that any equation between PCF terms which holds in this model also holds in the strongly stable model.
Correspondence between Operational and Denotational Semantics
 Handbook of Logic in Computer Science
, 1995
"... This course introduces the operational and denotational semantics of PCF and examines the relationship between the two. Topics: Syntax and operational semantics of PCF, Activity Lemma, undefinability of parallel or; Context Lemma (first principles proof) and proof by logical relations Denotational ..."
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Cited by 25 (0 self)
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This course introduces the operational and denotational semantics of PCF and examines the relationship between the two. Topics: Syntax and operational semantics of PCF, Activity Lemma, undefinability of parallel or; Context Lemma (first principles proof) and proof by logical relations Denotational semantics of PCF induced by an interpretation; (standard) Scott model, adequacy, weak adequacy and its proof (by a computability predicate) Domain Theory up to SFP and Scott domains; non full abstraction of the standard model, definability of compact elements and full abstraction for PCFP (PCF + parallel or), properties of orderextensional (continuous) models of PCF, Milner's model and Mulmuley's construction (excluding proofs) Additional topics (time permitting): results on pure simplytyped lambda calculus, Friedman 's Completeness Theorem, minimal model, logical relations and definability, undecidability of lambda definability (excluding proof), dIdomains and stable functions Homepa...
Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a ..."
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We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.
Bistable biorders: A sequential domain theory
 Logical Methods in Computer Science
"... Abstract. We give a simple ordertheoretic construction of a Cartesian closed category of sequential functions. It is based on bistable biorders, which are sets with a partial order — the extensional order — and a bistable coherence, which captures equivalence of program behaviour, up to permutation ..."
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Abstract. We give a simple ordertheoretic construction of a Cartesian closed category of sequential functions. It is based on bistable biorders, which are sets with a partial order — the extensional order — and a bistable coherence, which captures equivalence of program behaviour, up to permutation of top (error) and bottom (divergence). We show that monotone and bistable functions (which are required to preserve bistably bounded meets and joins) are strongly sequential, and use this fact to prove universality results for the bistable biorder semantics of the simplytyped lambdacalculus (with atomic constants), and an extension with arithmetic and recursion. We also construct a bistable model of SPCF, a higherorder functional programming language with nonlocal control. We use our universality result for the lambdacalculus to show that the semantics of SPCF is fully abstract. We then establish a direct correspondence between bistable functions and sequential algorithms by showing that sequential data structures give rise to bistable biorders, and that each bistable function between such biorders is computed by a sequential algorithm. 1.
Inductive Definition and Domain Theoretic Properties of Fully Abstract Models for PCF and PCF+
 LOGICAL METHODS IN COMPUTER SCIENCE 3(3:7), 1–50 (2007)
, 2007
"... A construction of fully abstract typed models for PCF and PCF+ (i.e., PCF+ “parallel conditional function”), respectively, is presented. It is based on general notions of sequential computational strategies and wittingly consistent nondeterministic strategies introduced by the author in the sevent ..."
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A construction of fully abstract typed models for PCF and PCF+ (i.e., PCF+ “parallel conditional function”), respectively, is presented. It is based on general notions of sequential computational strategies and wittingly consistent nondeterministic strategies introduced by the author in the seventies. Although these notions of strategies are old, the definition of the fully abstract models is new, in that it is given levelbylevel in the finite type hierarchy. To prove full abstraction and nondcpo domain theoretic properties of these models, a theory of computational strategies is developed. This is also an alternative and, in a sense, an analogue to the later game strategy semantics approaches of Abramsky, Jagadeesan, and Malacaria; Hyland and Ong; and Nickau. In both cases of PCF and PCF+ there are definable universal (surjective) functionals from numerical functions to any given type, respectively, which also makes each of these models unique up to isomorphism. Although such models are nonomegacomplete and therefore not continuous in the traditional terminology, they are also proved to be sequentially complete (a weakened form of omegacompleteness), “naturally” continuous (with respect to existing directed “pointwise”, or “natural” lubs) and also “naturally” omegaalgebraic and “naturally” bounded complete—appropriate generalisation of the ordinary notions of domain theory to the case of nondcpos.
Sequentiality in Bounded Biorders
 FUNDAMENTA INFORMATICAE
, 2005
"... We study a notion of bounded stable biorder, showing that the monotone and stable functions on such biorders are sequential. We construct bounded biorder models of a range of sequential, higherorder functional calculi, including unary PCF, (typed and untyped) callbyvalue and lazy λcalculi, and ..."
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We study a notion of bounded stable biorder, showing that the monotone and stable functions on such biorders are sequential. We construct bounded biorder models of a range of sequential, higherorder functional calculi, including unary PCF, (typed and untyped) callbyvalue and lazy λcalculi, and nondeterministic SPCF. We prove universality and full abstraction results for these models by reduction to the case of unary PCF, for which we give a simple new argument to show that any orderextensional and sequential model is universal.
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.
Realizability Models for Sequential Computation
, 1998
"... We give an overview of some recently discovered realizability models that embody notions of sequential computation, due mainly to Abramsky, Nickau, Ong, Streicher, van Oosten and the author. Some of these models give rise to fully abstract models of PCF; others give rise to the type structure of seq ..."
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We give an overview of some recently discovered realizability models that embody notions of sequential computation, due mainly to Abramsky, Nickau, Ong, Streicher, van Oosten and the author. Some of these models give rise to fully abstract models of PCF; others give rise to the type structure of sequentially realizable functionals, also known as the strongly stable functionals of Bucciarelli and Ehrhard. Our purpose is to give an accessible introduction to this area of research, and to collect together in one place the definitions of these new models. We give some precise definitions, examples and statements of results, but no full proofs. Preface Over the last two years, researchers in various places (principally Abramsky, Nickau, Ong, Streicher, van Oosten and the present author) have come up with a number of new realizability models that embody some notion of "sequential" computation. Many of these give rise to fully abstract and universal models for PCF and related languages. Alth...
Under consideration for publication in Math. Struct. in Comp. Science On the Reification of Semantic Linearity
, 2013
"... 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 domains. We propose a PCFlike language imposing linear constr ..."
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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 domains. We propose a PCFlike language imposing linear constraints on the use of variable to program only linear functions. To entail a full abstraction result, we introduce some higherorder operators related to exception handling and parallel evaluation. We study several notions of operational equivalence and we show them to coincide on our language. Last, we present a new operational evaluation of the language that provides the base for a real implementation. It exploit the denotational linearity to provide an efficient evaluation semantics SECDlike, that avoid the use of closures. 1.