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In this paper we specialize the notion of abstract computational procedure previously introduced for intensionally presented structures to those which are extensionally given. This is provided by a form of generalized recursion theory which uses schemata for explicit definition, conditional definition and least fixed point (LFP) recursion in functionals of type level ≤ 2 over any appropriate structure. It is applied here to the case of potentially infinite (and more general partial) streams as an abstract data type. 1

A degree of parallelism is an equivalence class of Scott-continuous functions which are relatively definable each other with respect to the language PCF (a paradigmatic sequential language). We introduce an infinite ("bi-dimensional") hierarchy of degrees. This hierarchy is inspired by a representation of first order continuous functions by means of a class of hypergraphs. We assume some familiarity with the language PCF and with its continuous model. Keywords: sequentiality, stability, strong stability, logical relations, sequentiality relations. 1 Introduction A natural notion of relative definability in the continuous type hierarchy is given by the following definition: Definition 1 Given two continuous functions f and g, we say that f is less parallel than g (f par g) if there exists a PCF-term M such that [jM j]g = f . A degree of parallelism is a class of the equivalence relation associated to the preorder par . In this paper we deal with degrees of parallelism of first ord...

The aim of this work is to show how hypergraphs can be used as a sistematic tool in the classication of continous boolean functions according to their degree of parallelism. Intuitively f is \less parallel" than g if it can be dened by a sequential program using g as its only free variable. It turns out that the poset induced by this preorder is (as for the degrees of recursion) a sup-semilattice. Although hypergraphs had already been used in [6] as a tool for studying degrees of parallelism, no general results relating the former to the latter have been proved in that work. We show that the sup-semilattice of degrees has a categorical counterpart: we dene a category of hypergraphs such that every object \represents" a monotone boolean function; nite coproducts in this category correspond to lubs of degrees. Unlike degrees of recursion, where every set has a recursive upper bound, monotone boolean functions may have no sequential upper bound. However the ones which do have a sequential upper bound can be nicely characterised in terms of hypergraphs. These subsequential functions play a major role in the proof of our main result, namely that f is less parallel than g if there exists a morphism between their associated hypergraps. 1

Abstract. In order to study relative PCF-definability of boolean functions, we associate a hypergraph Hf to any boolean function f (following [3, 5]). We introduce the notion of timed hypergraph morphism and show that it is: – Sound: if there exists a timed morphism from Hf to Hg then f is PCF-definable relatively to g. – Complete for subsequential functions: if f is PCF-definable relatively to g, and g is subsequential, then there exists a timed morphism from Hf to Hg. We show that the problem of deciding the existence of a timed morphism between two given hypergraphs is NP-complete. 1

The focus of this thesis is the study of relative definability of first-order boolean functions with respect to the language PCF, a paradigmatic typed, higher-order language based on the simply-typed λ-calculus. The basic core language is sequential. We study the effect of adding construct that embody various notions of parallel execution. The resulting set of equivalence classes with respect to relative definability forms a supsemilattice analoguous to the lattice of degrees in recursion theory. Recent results of Bucciarelli show that the lattice of degrees of parallelism has both infinite chains and infinite antichains. By considering a very simple subset of Sieber’s sequentiality relations, we identify levels in the lattice and derive inexpressiblity results concerning functions on different levels. This allows us to explore further the structure of the lattice of degrees of parallelism and show the existence of new infinite hierarchies. We also identify four subsemilattices of this structure, all characterized by a simple property. ii Résumé Dans ce mémoire nous nous concentrons sur l’étude de la definition relative de fonctions