Inductive Definition and Domain Theoretic Properties of Fully Abstract

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 non-deterministic 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 level-by-level in the finite type hierarchy. To prove full abstraction and non-dcpo 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 non-omega-complete and therefore not continuous in the traditional terminology, they are also proved to be sequentially complete (a weakened form of omega-completeness), "naturally" continuous (with respect to existing directed "pointwise", or "natural" lubs) and also "naturally" omega-algebraic and "naturally" bounded complete -- appropriate generalisation of the ordinary notions of domain theory to the case of non-dcpos.Comment: 50 page

On Berry's conjectures about the stable order in PCF

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, G\'erard 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.Comment: submitted to LMCS, 39 pages, 23 pstricks/pst-tree figures, main changes for this version: 4.1: proof of game term theorem corrected, 7.: the improved chain conjecture is made precise, more references adde

Extensional Collapse Situations I: non-termination and unrecoverable errors

We consider a simple model of higher order, functional computation over the booleans. Then, we enrich the model in order to encompass non-termination and unrecoverable errors, taken separately or jointly. We show that the models so defined form a lattice when ordered by the extensional collapse situation relation, introduced in order to compare models with respect to the amount of "intensional information" that they provide on computation. The proofs are carried out by exhibiting suitable applied {\lambda}-calculi, and by exploiting the fundamental lemma of logical relations

Preliminary draft

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Intensional and Extensional Semantics of Bounded and Unbounded Nondeterminism

We give extensional and intensional characterizations of nondeterministic functional programs: as structure preserving functions between biorders, and as nondeterministic sequential algorithms on ordered concrete data structures which compute them. A fundamental result establishes that the extensional and intensional representations of non-deterministic programs are equivalent, by showing how to construct a unique sequential algorithm which computes a given monotone and stable function, and describing the conditions on sequential algorithms which correspond to continuity with respect to each order. We illustrate by defining may and must-testing denotational semantics for a sequential functional language with bounded and unbounded choice operators. We prove that these are computationally adequate, despite the non-continuity of the must-testing semantics of unbounded nondeterminism. In the bounded case, we prove that our continuous models are fully abstract with respect to may and must-testing by identifying a simple universal type, which may also form the basis for models of the untyped lambda-calculus. In the unbounded case we observe that our model contains computable functions which are not denoted by terms, by identifying a further "weak continuity" property of the definable elements, and use this to establish that it is not fully abstract