G. Berry, Mod`eles compl`etement ad'equats et stables des -calculs typ'es. PhD thesis, Univ. of Paris7, Mar. 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....correspondence theorem between operational termination and denotational existence; it generally states that a program terminates according to the operational semantics if and only if its denotational semantics denotes a value. The first axiomatic version of such a result was provided by Berry in [Ber79] see also [BCL85] for PCF with respect to a class of models including both the standard one and the stable one. In this vein, Bra94] considered a term language for intuitionistic propositional linear logic extended with fixed point operators and provided a computational soundness and adequacy ....

G. Berry. Mod`eles Compl`etement Ad'equats et Stables des Lambda-Calculus Typ'es. PhD thesis, L'Universit'e Paris VII, 1979.

.... that Berry and Curien s category of concrete data structures and sequential algorithms [3] is got as a co Kleisli category from a games model of linear logic [3, 7] Bistructures were introduced in [10] as a generalisation of event structures to represent a full subcategory of Berry s bidomains [2]. Bidomains possess an intensional, stable ordering, based on the method of computation, and an extensional ordering, inherited from Scott s domain theory; their morphisms are functions which respect both, a property shared by functions definable in PCF. Here we show that, with a small ....

....spaces and linear functions. The stable functions are obtained via the co Kleisli construction associated with the model. Adding a partial order, we obtain event structures (E; which represent coherent prime algebraic domains [8] and, with a finiteness axiom added, coherent dI domains [2, 11, 12]. We can consider the corresponding categories of relations, again representing the linear stable functions; we must now add the condition: a ff b b ) 9a a: a ff b However, this only yields a model of intuitionistic linear logic [11, 13] By passing to bistructures we add enough ....

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Berry, G., Mod`eles compl`etement ad'equats et stables des lambdacalculs typ'es. Th`ese de Doctorat d'Etat, Universit'e de Paris VII, 1979.

....from E 0 to E 1 , associating to each stable function f its trace tr(f ) consisting of those pairs (x; e 1 ) such that e 1 2 f(x) and e 1 62 f(y) if y ae x. The inclusion of configurations determines an ordering on stable functions, refining the pointwise ordering and called the stable ordering [2]. The definition of E 0 E 1 is asymmetric in that configurations are paired with events, rather than events with events. This led Girard to two successive decompositions, each of which turned out to have deep logical significance. ffl First, E 0 E 1 can be obtained as ( E 0 ) E 1 , where, ....

....on the event e , in that the event e can only occur after e has occurred. Given this understanding it is reasonable to impose a finiteness axiom, expressing that an event has finite causes: eg is finite, for all events e. The event structures satisfying this axiom yield the dI domains [2] which are coherent, and therefore lead to a cartesian closed category of stably ordered stable functions. See [26] where an alternative description of event structures In [21] an axiom relating causal dependency and conflict is imposed; however it is inessential in that it does not affect the ....

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Berry, G., Mod`eles compl`etement ad'equats et stables des lambda-calculs typ'es. Th`ese de Doctorat d'Etat, Universit'e de Paris VII, 1979.

....models can be used to represent domains in the semantics of programming languages. For example, event Basic Research in Computer Science,Centre of the Danish National Research Foundation. structures give a representation of Berry s cartesian closed category of dIdomains and stable functions [2, 18], while Berry and Curien s category of sequential algorithms on Kahn and Plotkin s concrete datastructures provides a domain theory in which the method of computation is represented explicitly [3] These two domain theories can be used to give models of PCF. But they do not give order extensional ....

....on stable bistructure models of PCF, as a means to obtain order extensional models which at the same time take some account of the way a function is computed. Stable bistructures were introduced in [17] as a generalisation of event structures to represent a full subcategory of Berry s bidomains [2]. Bidomains possess an intensional, stable ordering, based on the method of computation, and an extensional, pointwise ordering, inherited from Scott s domain theory; their morphisms are functions which respect both, a property shared by functions definable in PCF. The represention of bidomains ....

Berry, G., Mod`eles compl`etement ad'equats et stables des lambdacalculs typ'es. Th`ese de Doctorat d'Etat, Universit'e de Paris VII, 1979.

....in order to obtain sequential functions. The notion of sequentiality comes in a natural way for a function whose domain is a product of flat domains. But, when we want to extend it to higher types, none of the existing definitions of sequentiality due to Vuillemin, Milner and Kahn Plotkin (cf. [4]) allow us to build a model of PCF: 1 the categories of complete partial orders with Milner or Vuilemin sequential functions, and concrete domains with Kahn Plotkin sequential functions, are not cartesian closed (cf. 9] If we want to stay in the framework of functional models i.e. ....

....(cf. 9] If we want to stay in the framework of functional models i.e. obtained in a category in which the objects are sets and the morphisms are functions we have to take functions that satisfy a weaker property than sequentiality. The notion of Stability, introduced by Berry (cf. [4]) is such a property. A continuous function f is stable if it satisfies : y f(x) 9x 0 x such that y f(x 0 ) and x 0 is minimum for that property. For example the parallel or function is not stable. In some structures, such as dI domains, stability is equivalent to the preservation of ....

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G. Berry, Mod`eles compl`etements ad'equats et stables des lambda-calculs typ'es, Th`ese de Doctorat d'Etat, Universit'e Paris VII, 1979.

....E be a PSS. If E admits a GSFR, then E is a sequential structure. Proof: Let (r n ) n2 be a GSFR on E. First, since (r n ) n2 is a stably increasing family of stable functions with finite images and having Id as lub, we know by standard considerations that E is a dI domain (see for instance [B1]) So we only have to prove that any isolated element of E answers only to a finite number of questions of E . So let u 2 E be isolated and let n 2 be such that r n (u) u. Let ff 2 E be such that u 2 ff. Then obviously ff 2 jr n j, which is finite by hypothesis, and we are done. From now ....

....and fix point combinators (for any type oe, we have a fix point combinator Y (oe oe) oe of type (oe oe) oe) Terms must be typable in Curry s system of simple types based on the only ground type . The notion of model of PCF we consider here is the one used by Berry in his thesis (see [B1], chapter 3.5) A model M consists essentially of a cartesian closed category (in fact, a category) also denoted by M, of the choice of an object [ M of M interpreting the type in the model, and of morphisms of M interpreting the basic constants of the language; if c is a constant of arity ....

G. Berry. Mod`eles compl`etement ad'equats et stables des lambda-calculs typ'es. Th`ese de Doctorat d'Etat, Universit'e Paris 7, 1979.

....is that from an abstract point of view, the largest class of operations which allows demand driven evaluation in dataflow is that of stable operations (which contain the sequential ones) It is rather interesting that the class of stable functions which was originally introduced by G. Berry (see [Ber76, Ber78a, Ber78b, Ber79]) has an important role in our work. What we actually show is that the following two properties are equivalent for a given set S of operations on streams of data tokens from flat domains: 1. Given any program P which is based on S and any demand D which P can satisfy, there exists a least legal ....

....filled. Basically, the above conjecture concerning sequential functions is proved in [Pin86] 9 . It turns out, however, that the condition of sequentiality is only sufficient for the existence of a least element in SP;D . A larger class of functions will in fact do: that of stable functions ([Ber76, Ber78a, Ber78b, Ber79]) Definition 3.4. Let D 1 and D 2 be cpos and let f : D 1 D 2 be a continuous function. We call f stable iff for every xfflD 1 and yfflD 2 where y f(x) there exists M(f; x; y)fflD 1 such that 8z x, y f(z) M(f; x; y) z. It is easy to show that sequentiality implies stability in the ....

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Berry G.: Mod`eles compl`etement ad'equateet Stable des -calculus typ'es. Th`ese de Doctorat d` ' Etat, Universit'e Paris VII, 1979.

....(t u; t[ u] t 0 ; u[ t] u 0 ; v 1 [ u 0 ] v 0 ; v 2 [ t 0 ] v 0 ) 9v: v[ t] v 1 ; v[ u] v 2 ) This axiom may seem incredibly complicated. It appears quite naturally in the uniqueness proof. It means that the characteristic function of a redex t is stable in the sense of [12]. Graphically, v u u t v 1 v 2 = u u t v 1 v 2 v v There is an extra problem; the nesting ordering is not total. There is no notion of left to right or rightto left. So it is hopeless to achieve a unique standard reduction in a permutation class, but standard reductions will be ....

G. Berry, Mod`eles compl`etement ad'equats et stables des -calculs typ'es. PhD thesis, Univ. of Paris7, Mar. 1979.

....s equentiel pour calculer les el ements finis des S domaines. Mots cl e : S fonctions, S domains, order enrichie categorie A THEORY OF SEQUENTIAL FUNCTIONS 3 1 Introduction The purpose of this paper is twofold: first it aims at generalising the notion of stable functions introduced by Berry in [Ber79], in order to construct a category whose homsets are extensionally ordered (ie: using a pointwise order) like in [Gam92] and secondly at obtaining models for languages like PCF (a typed calculus together with arithmetic and boolean operators, augmented with a fixpoint operator) whose terms are ....

....and sequential functions (as defined by Kahn and Plotkin) is not cartesian closed. Notice however that subsituting the notion of sequential algorithm for that of function and relaxing the extensional order, allow to provide PCF with a fully abstract but non order extensional model(see [Cur86] and [Ber79]) Attacks on the problem led Berry to define the notion of stable function (see [Ber79] Intuitively, stable functions are continuous functions together with a particular minimality property. Formally, a function f is stable if for any element x and for any approximant b of f(x) there exists ....

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G.Berry Mod`eles compl`etement ad'equats et stables des lambda-calculs types. Th`ese de doctorat d"etat, U.Paris VII INRIA A THEORY OF SEQUENTIAL FUNCTIONS 13

....Trees, and Berarducci Trees) by infinitary rewriting. In this section we will apply this rewrite system to obtain a perspicuous proof of an important theorem due to G. Berry that establishes the inherently sequential nature of evaluation in calculus. Other proofs can be found in Berry [Ber78] [Ber79], Barendregt [Bar84] Curien [Cur93] We will restrict ourselves to the case of Bohm trees, but we expect that the same analysis can also be applied to the other two kinds of trees (LLT and BeT) Analogous to the sections 7 and 8 we will start from the normal form BT (M ) and then trace back ....

G. Berry. Mod`eles compl`etement ad'equats et stables des -calculs typ'es. PhD thesis, University of Paris 7, 1979.

....from E 0 to E 1 , associating to each stable function f its trace tr(f ) consisting of those pairs (x; e 1 ) such that e 1 2 f(x) and e 1 62 f(y) if y ae x. The inclusion of configurations determines an ordering on stable functions, refining the pointwise ordering and called the stable ordering [2]. 4 The definition of E 0 E 1 is asymmetric in that configurations are paired with events, rather than events with events. This led Girard to two successive decompositions, each of which turned out to have deep logical significance. ffl First, E 0 E 1 can be obtained as ( E 0 ) E 1 , ....

....e 0 , in that the event e can only occur after e 0 has occurred. Given this understanding it is reasonable to impose a finiteness axiom, expressing that an event has finite causes: fe 0 j e 0 eg is finite, for all events e. The event structures satisfying this axiom yield the dI domains [2] which are coherent, and therefore lead to a cartesian closed category of stably ordered stable functions. See [26] where an alternative description of event structures 3 In [21] an axiom relating causal dependency and conflict is imposed; however it is inessential in that it does not affect ....

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Berry, G., Mod`eles compl`etement ad'equats et stables des lambda-calculs typ'es. Th`ese de Doctorat d'Etat, Universit'e de Paris VII, 1979.

....seem that in order to get a cartesian closed category of sequential functions , one of the two criteria has to give. One major effort consisted in relaxing the constraints of sequentiality but staying within the framework of functions. This led Berry to the notion of stability [ Berry, 1978b; Berry, 1979 ] The appropriate morphisms are stable functions which are continuous functions that preserve greatest lower bounds of consistent (or upper bounded ) subsets; and the objects are dI domains Scott domains which satisfy a distributivity property and axiom (I) which says that every compact ....

....[ Meyer and Cosmadakis, 1988 ] The gist of the Context Lemma 304 C. H. L. Ong is this: terms of the language pcf are determined by their applicative behaviour. Hence, it is quite appropriate to call pcf an applicative or functional programming language. The proof we have presented is due to Berry [ Berry, 1979 ] Milner [ Milner, 1977 ] has a similar result for a family of simply typed pcf like languages expressed in the formalism of combinatory logic. An interesting line of research is to establish a general Context Lemma. This is the problem of finding conditions under which the Context Lemma is ....

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G. Berry. Mod`eles compl`etement ad'equats et stables des lambda calculs typ'es. PhD thesis, Universit'e Paris VII, 1979.

....that although the space of continuous functions can be used to provide elegant mathematical models of programming languages, the construction is too general to serve as the basis for a practical formulation of domain theory. Fortunately, there are more restricted function constructions [KP93, Ber79, Cur93, CCF94, CF92] that accommodate all conventional programming languages. To date, they have not been used as the basis for a universal language and domain. In this thesis, we present a new formulation of domain theory that captures the terminating behavior of arbitrary sequential ....

....model is extensional if the following properties hold. environment extensional) 8ae : eae = e 0 ae ) e = e 0 (value extensional) 8d 00 : apply(d; d 00 ) apply(d 0 ; d 00 ) d = d 87 The classical model defined in [Sto81] is an extensional model. The stable function models as in [Ber79] based on the stable ordering is also extensional. However, the sequential algorithms model is not extensional. By adding the error elements, it can be made order extensional [Cur92] Cartwright et. al [CCF94] consider various categories that can be obtained by adding 0, 1, and 2 errors to the ....

G. Berry. Mod`eles compl`etement ad'equats et stables des lambda-calculus typ'e. PhD thesis, Universit'e Paris VII, 1979.

....structures and coherent, finitary prime algebraic domains are equivalent; one can be used to represent the other. Such domains are familiar in another guise. Recall that the dI domains of Berry are distributive algebraic cpos in which every finite element only dominates finitely many elements [8]. Theorem 25 The finitary, prime algebraic domains are precisely the dI domains of Berry. Proof: See [98] or [93] Following Girard, call a function linear iff it is stable in the sense of Berry (i.e. preserves bounded meets) and preserves joins when they exist. The covering relation between ....

Berry, G., Mod`eles completement ad'equats et stables des -calculs typ'ees, Th`ese de Doctorat d'Etat, Universit'e Paris VII, 1979.

....[22] recognized this problem nearly twenty years ago and identified the construction of sequential denotational language models as an important research problem. Early work in the search for sequential models focused on the typed calculus with constants for arithmetic and recursion (PCF) [32, 19, 22, 17, 3, 4, 6, 11, 20]. While this strategy avoids many arbitrary language design decisions and simplifies the investigation, it also eliminates programming facilities that are essential for understanding the sequential behavior of programs. The missing facilities include error values, which permit a programmer to ....

Berry, G. Mod`eles compl`etement ad'equats et stables des lambda-calculus typ'e. Ph.D. dissertation, Universit'e Paris VII, 1979.

....(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 ....

.... PCF augmented with the parallel or operation, is the continuous function model, which is inequationally fully abstract and order extensional [Plo] In fact, a result of Plotkin Milner Berry s shows that this model is the unique inequationally fully abstract, extensional model of parallel PCF [Ber][BCL] Mil] Plo] But does parallel PCF have extensional models that are not inequationally fully abstract or not even equationally fully abstract What about (necessarily non inequationally fully abstract) extensional models that are not order extensional The purpose of this paper is to answer ....

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G. Berry. Mod`eles Compl`etement Ad'equats et Stables des Lambda-calculs Typ'es. Th`ese de Doctorat d'Etat, Universit'e Paris VII, 1979.

....between the continuous function model and the procedures definable in the language. Unfortunately, neither Milner s nor Plotkin s result showed how to construct fully abstract denotational models for sequential languages. In a subsequent investigation of sequential languages, Berry and Curien [3, 4, 6, 8] constructed models for PCF with more restrictive domains of procedure denotations. Berry eliminated many parallel functions from these domains by forcing functions to be stable. This construction eliminated some of the spurious distinctions between phrases in the conventional model, but it ....

Berry, G. Mod`eles compl`etement ad'equats et stables des lambda-calculus typ'e. Ph.D. dissertation, Universit'e Paris VII, 1979.

.... Delta Delta Delta Cm [t] and so, since s app t, we have tC 1 [t] Delta Delta Delta Cm [t] j C[t] v. The Context Lemma tells us that applicative contexts alone are enough to determine the observational preorder (between closed terms) The proof we have presented is due to Berry [2]. Milner [15] has a similar result for a family of languages expressed in the formalism of Combinatory Logic. 4.3 Another proof of the Context Lemma We present a proof due to Martin Hyland. Fix hole variables X and Y . We define by simultaneous recursion: C Y : Y j c j x j (x A :C Y ) j (C Y ....

G. Berry. Mod`eles compl`etement ad'equats et stables des lambda calculs typ'es. Technical report, Universit'e Paris VII, 1979. Th`ese de Doctorat d'Etat.

....continuous function model and the procedures definable in the language. Unfortunately, neither Milner s nor Plotkin s result showed how to construct fully abstract denotational models for sequential languages. In a later effort to understand the semantics of sequential languages, Berry and Curien [2, 3, 5, 8] constructed models for PCF with more restrictive domains of procedure denotations. Berry eliminated many parallel functions from the domain of procedure denotations by forcing functions to be stable. This construction eliminated some of the spurious distinctions between phrases in the ....

Berry, G. Mod`eles compl`etement ad'equats et stables des lambda-calculus typ'e. Ph.D. dissertation, Universit'e Paris VII, 1979.

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G. Berry, Mod`eles compl`etement ad'equats et stables des -calculs typ'es. PhD thesis, Univ. of Paris7, Mar. 1979.

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G. Berry. Mod`eles compl`etement ad'equats et stables des -calculs typ'es. PhD thesis, University of Paris 7, 1979.

No context found.

G. Berry. Mod`eles compl`etement ad'equats et stables des lambda-calculs typ'es. Th`ese de Doctorat d'Etat, Universit'e Paris 7, 1979.

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G.Berry. Mod`eles compl`etement ad'equats et stables des lambda-calculs typ'es. Technical report. Universit'e Paris VII, Th`ese de Doctorat d'Etat, 1979.

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