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...

A number of authors have exported domain-theoretic techniques from denotational semantics to the operational study of contextual equivalence and order. We further develop this, and, moreover, we additionally export topological techniques. In particular, we work with an operational notion of compact set and show that total programs with values on certain types are uniformly continuous on compact sets of total elements. We apply this and other conclusions to prove the correctness of non-trivial programs that manipulate infinite data. What is interesting is that the development applies to sequential programming languages, in addition to languages with parallel features. 1

We call language L 1 intensionally more expressive than L 2 if there are functions which can be computed faster in L 1 than in L 2 . We study the intensional expressiveness of several languages: the Berry-Curien programming language of sequential algorithms, CDS0, a deterministic parallel extension to it, named CDSP, and various parallel extensions to the functional programming language PCF. The paper consists of two parts. In the first part, we show that CDS0 can compute the minimum of two numbers n and p in unary representation in time O(min(n; p)). However, it cannot compute a "natural" version of this function. CDSP allows us to compute this function, as well as functions like parallel-or. This work can be seen as an extension of the work of Colson [7, 8] with primitive recursive algorithms to the setting of sequential algorithms. In the second part, we show that deterministic parallelism adds intensional expressiveness, settling a "folk" conjecture from the literature in the nega...

ion: [[H B x : u: M 0 : u ! v]]ae is the function from [[u]] to [[v]] given by d 7! [[H; x : u B M 0 : v]](ae[x 7! d]), that is, the function f defined by f(d) = [[H; x : u B M 0 : v]](ae[x 7! d]): 22 Carl A. Gunter ffl Application: [[H B L(N ) : t]]ae is the value obtained by applying the function [[H B L : s ! t]]ae to argument [[H B N : s]]ae where s is the unique type such that H ` L : s ! t and H ` N : s. It will save us quite a bit of ink to drop the parentheses that appear as part of expressions such as [[H; x : u B M 0 : v]](ae[x 7! d]) and simply write [[H; x : u B M 0 : v]]ae[x 7! d]. Doing so appears to violate the convention of associating applications to the left, but there is little chance of confusion in the case of expressions such as these. Hence, we will adopt the convention that the postfix update operator binds more tightly than general application. It can be shown that this assignment of meanings respects our equational rules. This is the soundness ...

We show that the continuous function model is the unique extensional (but not necessarily pointwise ordered) model of the variant of the applied typed lambda calculus PCF that includes the "parallel or" operation. 1 Introduction Several extensional models of the applied typed lambda calculus PCF are known to exist, including: (i) The continuous function model, which is order-extensional (pointwise ordered) but not equationally fully abstract [Plo]. (A model is equationally fully abstract when terms are identified in the model exactly when they are operationally equivalent.) (ii) The stable function model, which is neither order-extensional nor equationally fully abstract [Ber][BCL]. (iii) The terminal object of the category of equationally fully abstract, extensional models, which is inequationally fully abstract and order-extensional [Mil][Sto2]. (A model is inequationally fully abstract iff one term is less than another in the model exactly when the first is operationally less defin...

It is well-known that stable models (as dI-domains, qualitative domains and coherence spaces) are not fully abstract for the languagePCF. This fact is related to the existence of stable parallel functions and of stable functions that are not monotone with respect to the extensional order, which cannot be defined by programs ofPCF. In this paper, a paradigmatic programming language namedStPCF is proposed, which extends the languagePCF with two additional operators. The operational description of the extended language is presented in an effective way, although the evaluation of one of the new operators cannot be formalized in a PCF-like rewrite system. SinceStPCF can define all finite cliques of coherence spaces the above gap with stable models is filled, consequently stable models are fully abstract for the extended language. 1