John Longley

Interpreting Localized Computational Effects Using Operators of Higher Type

We outline a general approach to providing intensional models for languages with computational effects, whereby the problem of interpreting a given effect reduces to that of finding an operator of higher type satisfying certain equations. Our treatment consolidates and generalizes an idea that is already implicit in the literature on game semantics. As an example, we work out our approach in detail for the case of fresh name generation, and discuss some particular models to which it applies.

The Sequentially Realizable Functionals

In the previous chapter, we saw how the model PC offers a ‘maximal’ class of partial computable functionals strictly extending SF (in the sense of the poset \(\mathcal{J}(\mathbb{N}_\bot)\) of Subsection 3.6.4). In the present chapter, we show that SF can also be extended in a very different direction to yield another class SR of ‘computable’ functionals which is in some sense incompatible with PC. This class was first identified by Bucciarelli and Ehrhard [45] as the class of strongly stable functionals; later work by Ehrhard [69], van Oosten [294] and Longley [176] established the computational significance of these functionals, investigated their theory in some detail, and provided a range of alternative characterizations.

A uniform approach to domain theory in realizability models

We propose a uniform way of isolating a subcategory of predomains within the category of modest sets determined by a partial combinatory algebra (PCA). Given a divergence on a PCA (which determines a notion of partiality), we identify a candidate category of predomains, the well-complete objects. We show that, whenever a single strong completeness axiom holds, the category satisfies appropriate closure properties. We consider a range of examples of PCAs with associated divergences and show that in each case the axiom does hold. These examples encompass models allowing a ‘parallel’ style of computation (for example, by interleaving), as well as models that seemingly allow only ‘sequential’ computation, such as those based on term-models for the lambda-calculus. Thus, our approach provides a uniform approach to domain theory across a wide class of realizability models. We compare our treatment with previous approaches to domain theory in realizability models. It appears that no other approach applies across such a wide range of models.

Matching typed and untyped realizability

Abstract Realizability interpretations of logics are given by saying what it means for computational objects of some kind to realize logical formulae. The computational objects in question might be drawn from an untyped universe of computation, such as a partial combinatory algebra, or they might be typed objects such as terms of a PCF-style programming language. In some instances, one can show that a particular untyped realizability interpretation matches a particular typed one, in the sense that they give the same set of realizable formulae. In this case, we have a very good fit indeed between the typed language and the untyped realizability model — we refer to this condition as (constructive) logical full abstraction. We give some examples of this situation for a variety of extensions of PCF. Of particular interest are some models that are logically fully abstract for typed languages including non-functional features. Our results establish connections between what is computable in various programming languages and what is true inside various realizability toposes. We consider some examples of logical formulae to illustrate these ideas, in particular their application to exact real-number computability.

Partial Functions in a Total Setting

We discuss a scheme for defining and reasoning about partial recursive functions within a classical two-valued logic in which all terms denote. We show how a total extension of the partial function introduced by a recursive declaration may be axiomatized within a classical logic, and illustrate by an example the kind of reasoning that our scheme supports. By presenting a naive set-theoretic semantics, we show that the system we propose is logically consistent. Our work is motivated largely by the pragmatic issues arising from mechanical theorem proving – we discuss some of the practical benefits and limitations of our scheme for mechanical verification of software and hardware systems.

Constructive Data Refinement in Typed Lambda Calculus

A new treatment of data refinement in typed lambda calculus is proposed, phrased in terms of pre-logical relations [HS99] rather than logical relations, and incorporating a constructive element. Constructive data refinement is shown to have desirable properties, and a substantial example of refinement is presented.

Universal Types and What They are Good For

We discuss the standard notions of universal object and universal type, and illustrate the usefulness of these concepts via several examples from denotational semantics. The purpose of the paper is to provide a gentle introduction to these notions, and to advocate a particular point of view which makes significant use of them. The main ideas here are not new, though our expository slant is somewhat novel, and some of our examples lead to seemingly new results.

On the ubiquity of certain total type structures

It is an empirical observation arising 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 Ceff and the type structure HEO of the hereditarily effective operations. However, the proofs of the relevant equivalences are often non-trivial, and it is not immediately clear why these particular type structures should arise so ubiquitously. In this paper we present some new results that go some way towards explaining this phenomenon. Our results show that a large class of extensional collapse constructions always give rise to C, Ceff 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 three type structures under consideration are highly canonical mathematical objects.

Reasoning About CBV Functional Programs in Isabelle/HOL

We consider the old problem of proving that a computer program meets some specification. By proving, we mean machine checked proof in some formal logic. The programming language we choose to work with is a call by value functional language, essentially the functional core of Standard ML (SML). In future work we hope to add exceptions, then references and I/O to the language.

On the Ubiquity of Certain Total Type Structures

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 Kleene-Kreisel continuous functionals, its effective substructure C, and the type structure HEO of the hereditarily effective operations. However, the proofs of the relevant equivalences are often non-trivial, 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 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.