J. M. E. Hyland

On Full Abstraction for PCF

We present an order-extensional, order (or inequationally) fully abstract model for Scotts language pcf. The approach we have taken is very concrete and in nature goes back to S. C Kleene (1978, in “General Recursion Theory II, Proceedings of the 1977 Oslo Symposium,” North-Holland, Amsterdam) and R. O. Gandy (1993, “Dialogues, Blass Games, Sequentiality for Objects of Finite Type,” unpublished manuscript) in one tradition, and to G. Kahn and G. D. Plotkin (1993, Theoret. Comput. Sci.121, 187?278) and G. Berry and P.-L. Curien (1982, Theoret. Comput. Sci.20, 265?321) in another. Our model of computation is based on a kind of game in which each play consists of a dialogue of questions and answers between two players who observe the following principles of civil conversation: 1.Justification. A question is asked only if the dialogue at that point warrants it. An answer is proffered only if a question expecting it has already been asked. 2.Priority. Questions pending in a dialogue are answered on a last-asked-first-answered basis. This is equivalent to Gandys no-dangling-question-mark condition. We analyze pcf-style computations directly in terms of partial strategies based on the information available to each player when he or she is about to move. Our players are required to play an innocent strategy: they play on the basis of their view which is that part of the history that interests them currently. Views are continually updated as the play unfolds. Hence our games are neither history-sensitive nor history-free. Rather they are view-dependent. These considerations give expression to what seems to us to be the nub of pcf-style higher-type sequentiality in a (dialogue) game-semantical setting.

The Effective Topos

Publisher Summary This chapter describes the most accessible of the series of toposes that can be constructed from notions of realizability: it is that based on the original notion of recursive realizability and presents the abstract approach to recursive realizability in some detail. The chapter introduces effective topos and discusses the notion of a negative formula that arises naturally in the theory of sheaves. The chapter presents features of effective topos, where the power-set matters: uniformity principles and properties of j-operators. The chapter recommends that while constructing a topos from a tripos, one must add new subobjects of the sets one has started with to represent the nonstandard predicates and take quotients of these by the nonstandard equivalence relations.

A small complete category

This paper is concerned with a remarkable fact. The effective topos contains a small complete subcategory, essentially the familiar category of partial equivalence realtions. This is in contrast to the category of sets (indeed to all Grothendieck toposes) where any small complete category is equivalent to a (complete) poset. Note at once that the phrase ‘a small complete subcategory of a topos’ is misleading. It is not the subcategory but the internal (small) category which matters. Indeed for any ordinary subcategory of a topos there may be a number of internal categories with global sections equivalent to the given subcategory. The appropriate notion of subcategory is an indexed (or better fibred) one, see 0.1. Another point that needs attention is the definition of completeness (see 0.2). In my talk at the Church’s Thesis meeting, and in the first draft of this paper, I claimed too strong a form of completeness for the internal category. (The elementary oversight is described in 2.7.) Fortunately during the writing of [13] my collaborators Edmund Robinson and Giuseppe Rosolini noticed the mistake. Again one needs to pay careful attention to the ideas of indexed (or fibred) categories. The idea that small (sufficiently) complete categories in toposes might exist, and would provide the right setting in which to discuss models for strong polymorphism (quantification over types), was suggested to me by Eugenio Moggi. And he first realized that the effective topos did indeed contain a small complete category. When, led by Moggi’s suggestion, I first came to consider the matter, I realized that the ‘result’ was staring me in the face. It is just a matter of putting together some well-known facts. The effective topos is the world of realizability (Kleene [15]) extended from arithmetic to general constructive mathematics. Details are in [ll], and the general context in [12] and [21]. The relevant subcategory, called the category of effective objects in [ll], is already in Kreisel [16]. Briefly the problem is to show * Paper presented at the conference “Church’s Thesis after fifty years”, Zeist, The Netherlands, June 14, 15, 1986.

Algebraic types in PER models

Huet has conjectured that the interpretations of a class of types (the “algebraic types”) in the PER model on the natural numbers for the second-order lambda calculus are in a certain sense the initial algebras. In this paper we examine several different PER models, and show that Huets conjecture holds in each.

Modified Realizability Toposes and Strong Normalization Proofs

This paper is motivated by the discovery that an appropriate quotient SN of the strongly normalising untyped λ*-terms (where * is just a formal constant) forms a partial applicative structure with the inherent application operation. The quotient structure satisfies all but one of the axioms of a partial combinatory algebra (pca). We call such partial applicative structures conditionally partial combinalory algebras (c-pca). Remarkably, an arbitrary rightabsorptive c-pca gives rise to a tripos provided the underlying intuitionistic predicate logic is given an interpretation in the style of Kreisels modified realizabilily, as opposed to the standard Kleene-style realizability. Starting from an arbitrary right-absorptive C-PCA U, the tripos-to-topos construction due to Hyland et al. can then be carried out to build a modified realizability topos TOPm(U) of non-standard sets equipped with an equality predicate. Churchs Thesis is internally valid in TOP m (K1) (where the pca k1 is “Kleenes first model” of natural numbers) but not Markovs Principle. There is a topos inclusion of SET-the “classical” topos of sets-into TOPm(U); the image of the inclusion is just sheaves for the ⌝⌝-topology. Separated objects of the ⌝⌝-topology are characterized. We identify the appropriate notion of PERs (partial equivalence relations) in the modified realizability setting and state its completeness properties. The topos TOP m (U) has enough completeness property to provide a category-theoretic semantics for a family of higher type theories which include Girards System F and the Calculus of Constructions due to Coquand and Huet. As an important application, by interpreting type theories in the topos TOP m (SN.), a clean semantic explanation of the Tait-Girard style strong normalization argument is obtained. We illustrate how a strong normalization proof for an impredicative and dependent type theory may be assembled from two general “stripping arguments” in the framework of the topos TOP m (SN.). This opens up the possibility of a “generic” strong normalization argument for an interesting class of type theories.

Applications of Constructivity

Publisher Summary This chapter provides a sketch of some proof-theoretical results concerning constructive mathematics and indications how results may be used both to understand the results in pure mathematics and as a guide in discovering the physically significant results. The proof-theoretic results are concerned with the notions of (local) continuity in parameters. It highlights that attempts to model many physical problems gives rise to differential equations whose dynamics depends sensitively on the initial conditions, for example, every trajectory in some bounded region may be Liapunov unstable. The term chaos has been applied to extreme situations of this sort and they are the subject of much research at this time. Some results are known concerning the topological structure of attracting sets that occur in particular situations and much more is plausibly indicated based on computer simulation. The detailed structure of the flow is too complicated for any useful description. Certainly, for particular dynamical systems, there would be interest in regions where there were “approximately periodic orbits” of “approximately the same period”, if this qualitative situation is stable.