We present an equational framework for term graph rewriting with cycles. The usual notion of homomorphism is phrased in terms of the notion of bisimulation, which is well-known in process algebra and concurrency theory. Specifically, a homomorphism is a functional bisimulation. We prove that the bisimilarity class of a term graph, partially ordered by functional bisimulation, is a complete lattice. It is shown how Equational Logic induces a notion of copying and substitution on term graphs, or systems of recursion equations, and also suggests the introduction of hidden or nameless nodes in a term graph. Hidden nodes can be used only once. The general framework of term graphs with copying is compared with the more restricted copying facilities embodied in the µ-rule, and translations are given between term graphs and µ-expressions. Using these, a proof system is given for µ-expressions that is complete for the semantics given by infinite tree unwinding. Next, orthogonal term graph rewrite ...

We introduce the new framework of Labeled Terms Rewriting Systems (T l RS), a general framework to express sharing in Term Rewriting Systems (TRS). For Orthogonal T l RS, an important subclass of T l RS, we characterize optimal derivations. This result is applied to weak -calculi, showing the optimality of the lazy strategy, that is, the call-by-name with sharing strategy. The result is also valid in the presence of ffi -rules, as in PCF. Orthogonal T l RS is also useful as a calculus for proving syntactic properties of functional languages. 1 Compilation of the -calculus Most compilers for functional languages translate their source language into some enriched -calculus [17], and then, compile this intermediate language to a low-level language, such as mutually recursive supercombinators, as in LML [2, 10], or categorical combinators, as in CAML [4]. These low-level languages define different forms of weak fi-reduction. We now describe two of these low-level languages, superc...

We relate two sequential models of PCF: the sequential algorithm model due to Berry and Curien and the strongly stable model due to Bucciarelli and the author. More precisely, we show that all the morphisms araising in the strongly stable model of PCF are sequential in the sense that they are the "extensional projections" of some sequential algorithms. We define a model of PCF where morphisms are "extensional" sequential algorithms and prove that any equation between PCF terms which holds in this model also holds in the strongly stable model.

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...

We propose a definition by reducibility of sequentiality for the interpretations of higher-order programs and prove the equivalence between this notion and strong stability.

A comprehensive semantics for functional programs is presented, which generalizes the well-known call-by-value and call-by-name semantics. By permitting a separate choice between call-by value and call-by-name for every argument position of every function and parameterizing the semantics by this choice we abstract from the parameter-passing mechanism. Thus common and distinguishing features of all instances of the &-semantics, especially call-by-value and call-byname semantics, are highlighted. Furthermore, a property can be validated for all instances of the &-semantics by a single proof. This is employed for proving the equivalence of the given denotational (fixed-point based) and two operational (reduction based) definitions of the &-semantics. We present and apply means for very simple proofs of equivalence with the denotational &-semantics for a large class of reduction-based &-semantics. Our basis are simple first-order constructorbased functional programs with patterns. Keyw...