Operational domain theory and topology of sequential programming languages

Escardó, Martı ́n; Ho, Weng Kin

doi:10.1016/j.ic.2008.12.003

Cited by 15 publications

(7 citation statements)

References 23 publications

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“…Side remark: Here we must also mention the "rational chains" of Escardó and Ho [8]. These are ascending sequences of PCF terms that can be defined syntactically by a PCF procedure.…”

Section: Semantics Of Pcf: Non-complete Partial Order F-modelsmentioning

confidence: 99%

On Berry's conjectures about the stable order in PCF

Müller¹

2012

Logical Methods in Computer Science

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Abstract. PCF is a sequential simply typed lambda calculus language. There is a unique order-extensional fully abstract cpo-model of PCF, built up from equivalence classes of terms. In 1979, Gérard Berry defined the stable order in this model and proved that the extensional and the stable order together form a bicpo. He made the following two conjectures: 1) "Extensional and stable order form not only a bicpo, but a bidomain." We refute this conjecture by showing that the stable order is not bounded complete, already for finitary PCF of second-order types.2) "The stable order of the model has the syntactic order as its image: If a is less than b in the stable order of the model, for finite a and b, then there are normal form terms A and B with the semantics a, resp. b, such that A is less than B in the syntactic order." We give counter-examples to this conjecture, again in finitary PCF of second-order types, and also refute an improved conjecture: There seems to be no simple syntactic characterization of the stable order. But we show that Berry's conjecture is true for unary PCF.For the preliminaries, we explain the basic fully abstract semantics of PCF in the general setting of (not-necessarily complete) partial order models (f-models). And we restrict the syntax to "game terms", with a graphical representation.

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“…Side remark: Here we must also mention the "rational chains" of Escardó and Ho [8]. These are ascending sequences of PCF terms that can be defined syntactically by a PCF procedure.…”

Section: Semantics Of Pcf: Non-complete Partial Order F-modelsmentioning

confidence: 99%

On Berry's conjectures about the stable order in PCF

Müller¹

2012

Logical Methods in Computer Science

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“…In this paper, we export domain-theoretic tools directly to the operational setting of FPC. Such an enterprise can be seen as an extension of earlier programs taken on by Mason et al (1996) and Escardó and Ho (2009) for the language PCF. As a foundation for our operational treatment of recursive types, we have put in place an operational domain-theoretic toolkit in the style of Pitts (1997) to suit the setting of FPC.…”

Section: Introductionmentioning

confidence: 99%

An operational domain-theoretic treatment of recursive types

2013

Math. Struct. Comp. Sci.

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We develop an operational domain theory for treating recursive types with respect to contextual equivalence. The principal approach taken here deviates from classical domain theory in that we do not produce the recursive types via usual inverse limits constructions -we have it for free by working directly with the operational semantics. By extending type expressions to functors between some 'syntactic' categories, we establish algebraic compactness. To do this, we rely on an operational version of the minimal invariance property. In addition, we apply techniques developed therein to reason about FPC programs.

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“…We are interested in proving that (the semantics of) programs of first-order type R n → R enjoy first-order properties, such as continuity or differentiability, under their standard interpretation in calculus and real analysis. More specifically, our results do not cover notions of continuity and differentiability studied in fields such as (exact) real-number computation [57] or computable analysis [58], which have a strong domain-theoretical flavor, and higher-order generalizations of continuity and differentiability (see, e.g., [26,27,32,29]). We leave for future work the study of open logical relations in these settings.…”

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confidence: 99%

On the Versatility of Open Logical Relations

Barthe

Crubillé

Lago

et al. 2020

Lecture Notes in Computer Science

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Logical relations are one among the most powerful techniques in the theory of programming languages, and have been used extensively for proving properties of a variety of higher-order calculi. However, there are properties that cannot be immediately proved by means of logical relations, for instance program continuity and differentiability in higher-order languages extended with real-valued functions. Informally, the problem stems from the fact that these properties are naturally expressed on terms of non-ground type (or, equivalently, on open terms of base type), and there is no apparent good definition for a base case (i.e. for closed terms of ground types). To overcome this issue, we study a generalization of the concept of a logical relation, called open logical relation, and prove that it can be fruitfully applied in several contexts in which the property of interest is about expressions of first-order type. Our setting is a simply-typed λ-calculus enriched with real numbers and real-valued first-order functions from a given set, such as the one of continuous or differentiable functions. We first prove a containment theorem stating that for any collection of real-valued firstorder functions including projection functions and closed under function composition, any well-typed term of first-order type denotes a function belonging to that collection. Then, we show by way of open logical relations the correctness of the core of a recently published algorithm for forward automatic differentiation. Finally, we define a refinement-based type system for local continuity in an extension of our calculus with conditionals, and prove the soundness of the type system using open logical relations.

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Operational domain theory and topology of sequential programming languages

Cited by 15 publications

References 23 publications

On Berry's conjectures about the stable order in PCF

On Berry's conjectures about the stable order in PCF

An operational domain-theoretic treatment of recursive types

On the Versatility of Open Logical Relations

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