. We give a new characterization of lambda definability in Henkin models using logical relations defined over ordered sets with varying arity. The advantage of this over earlier approaches by Plotkin and Statman is its simplicity and universality. Yet, decidability of lambda definability for hereditarily finite Henkin models remains an open problem. But if the variable set allowed in terms is also restricted to be finite then our techniques lead to a decision procedure. 1 Introduction An applicative structure consists of a family (A oe ) oe2T of sets, one for each type oe, together with a family (app oe;ø ) oe;ø 2T of application functions, where app oe;ø maps A oe!ø \Theta A oe into A ø . For an applicative structure to be a model of the simply typed lambda calculus (in which case we call it a Henkin model, following [4]), one requires two more conditions to hold. It must be extensional which means that the elements of A oe!ø are uniquely determined by their behavior under app oe;ø...

ing with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from Publications Dept, ACM Inc., fax +1 (212) 869-0481, or permissions@acm.org. Relational Parametricity and Units of Measure Andrew J. Kennedy LIX, ' Ecole Polytechnique 91128 Palaiseau cedex, France Abstract Type systems for programming languages with numeric types can be extended to support the checking of units of measure. Quantification over units then introduces a new kind of parametric polymorphism with a corresponding Reynolds-style representation independence principle: that the behaviour of programs is invariant under changes to the units used. We prove this `dimensional invariance' result and describe four consequences. The first is that the type of an expression can be used to derive equations which describe its properties with respect to scaling (akin to Wadler's `theorems for free' for ...

The question of the decidability of the observational ordering of finitary PCF was raised [5] to give mathematical content to the full abstraction problem for PCF [9, 14]. We show that the ordering is in fact undecidable. This result places limits on how explicit a representation of the fully abstract model can be. It also gives a slight strengthening of the author’s earlier result on typed λ-definability [6].

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 order-extensional (continuous) models of PCF, Milner's model and Mulmuley's construction (excluding proofs) Additional topics (time permitting): results on pure simply-typed lambda calculus, Friedman 's Completeness Theorem, minimal model, logical relations and definability, undecidability of lambda definability (excluding proof), dI-domains and stable functions Homepa...

Game semantics has renewed denotational semantics. It offers among other things an attractive classification of programming features, and has brought a bunch of new definability results. In parallel, in the denotational semantics of proof theory, several full completeness results have been shown since the early nineties. In this note, we review the relation between definability and full abstraction, and we put a few old and recent results of this kind in perspective.

We present a denotational semantics for an Algol-like language Alg which is fully abstract for the second order subset of Alg. This constitutes the first significant full abstraction result for a block structured language with local variables. In this preliminary report we concentrate on the construction of the denotational model and on the main ideas of the full abstraction proof. For more background information about (problems involved with) the semantics of local variables, especially for further interesting examples of observational congruences we refer the reader to [MS88, OT93b].

We construct a model for FPC, a purely functional, sequential, call-by-value language. The model is built from partial continuous functions, in the style of Plotkin, further constrained to be uniform with respect to a class of logical relations. We prove that the model is fully abstract.

. Lambda definability is characterized in categorical models of simply typed lambda calculus with type variables. A category-theoretic framework known as glueing or sconing is used to extend the Jung-Tiuryn characterization of lambda definability [JuT93], first to ccc models, and then to categorical models of the calculus with type variables. Logical relations are now a well-established tool for studying the semantics of various typed lambda calculi. The main lines of research are focused in two areas, the first of which strives for an understanding of Strachey's notion of parametric polymorphism. The main idea is that a parametricly polymorphic function acts independently from the types to which its type variables are instantiated, and that this uniformity may be captured by imposing a relational structure on the types [OHT93, MSd93, MaR91, Wad89, Rey83, Str67]. The other line of research concerns lambda definability and the full abstraction problem for various models of languag...

Models of Lambda Calculi and Linear Logic: Structural, Equational and Proof-Theoretic Characterisations Ralph Loader, of St. Hugh's College, Oxford. Thesis submitted for the Degree of D.Phil. Michaelmas term, 1994. T his thesis is an investigation into models of typed -calculi and of linear logic. The models we investigate are denotational in nature; we construct various categories, in which types (or formulae) are interpreted by objects, and terms (proofs) by morphisms. The results we investigate compare particular properties of the syntax and the semantics of a calculus, by trying to use syntax to characterise features of a model, or vice versa. There are four chapters in the thesis, one each on linear logic and the simply typed -calculus, and two on inductive datatypes. In chapter one, we look at some models of linear logic, and prove a full completeness result for multiplicative linear logic. We form a model, the linear logical predicates , by abstracting a little the structure ...

A model theory for proving correctness of abstract data types is developed within the framework of the behavior-realization adjunction. To allow for incomplete specifications, proof-of-correctness ...