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Definability and full abstraction
 GDP FESTSCHRIFT
"... 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 sin ..."
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Cited by 17 (2 self)
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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.
Degrees of Parallelism in the Continuous Type Hierarchy
, 1995
"... A degree of parallelism is an equivalence class of Scottcontinuous functions which are relatively definable each other with respect to the language PCF (a paradigmatic sequential language). We introduce an infinite ("bidimensional") hierarchy of degrees. This hierarchy is inspired by a representat ..."
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Cited by 8 (1 self)
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A degree of parallelism is an equivalence class of Scottcontinuous functions which are relatively definable each other with respect to the language PCF (a paradigmatic sequential language). We introduce an infinite ("bidimensional") 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 PCFterm 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...
Relative Definability of Boolean Functions via Hypergraphs
"... 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 turn ..."
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Cited by 3 (1 self)
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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 supsemilattice. 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 supsemilattice 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
Course Notes in Typed Lambda Calculus
, 1998
"... this paper is clearly stated, after recalling how the logical connectives can be explained in term of the Sheffer connective: "We are led to the idea, which at first glance certainly appears extremely bold of attempting to eliminate by suitable reduction the remaining fundamental notions, those of p ..."
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Cited by 2 (0 self)
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this paper is clearly stated, after recalling how the logical connectives can be explained in term of the Sheffer connective: "We are led to the idea, which at first glance certainly appears extremely bold of attempting to eliminate by suitable reduction the remaining fundamental notions, those of proposition, propositional function, and variable, from those contexts in which we are dealing with completely arbitrary, logical general propositions . . . To examine this possibility more closely and to pursue it would be valuable not only from the methodological point of view that enjoins us to strive for the greatest possible conceptual uniformity but also from a certain philosophic, or if you wish, aesthetic point of view."
Linear L"auchli semantics
 Annals Pure Appl. Logic
, 1996
"... Dedicated to the memory of Moez Alimohamed ..."
Hypergraphs and degrees of parallelism: A completeness result, in: I. Walukiewicz (Ed
 Proceedings of the 7th International Conference of Foundations of Software Science and Computation Structures – FOSSACS 2004
, 2004
"... Abstract. In order to study relative PCFdefinability 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 PCFdefin ..."
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Cited by 1 (0 self)
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Abstract. In order to study relative PCFdefinability 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 PCFdefinable relatively to g. – Complete for subsequential functions: if f is PCFdefinable 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 NPcomplete. 1
Contents
, 2013
"... We apply Andy Pitts’s methods of defining relations over domains to several classical results in the literature. We show that the Y combinator coincides with the domaintheoretic fixpoint operator, that parallelor and the Plotkin existential are not definable in PCF, that the continuation semantics ..."
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We apply Andy Pitts’s methods of defining relations over domains to several classical results in the literature. We show that the Y combinator coincides with the domaintheoretic fixpoint operator, that parallelor and the Plotkin existential are not definable in PCF, that the continuation semantics for PCF coincides with the direct semantics, and that our domaintheoretic semantics for PCF is adequate for reasoning about contextual equivalence in an operational semantics. Our version of PCF is untyped and has both strict and nonstrict function abstractions. The development is carried out in HOLCF.