T. Ehrhard. Hypercoherences: a strongly stable model of linear logic. In Mathematical Structures in Computer Science, 3:365-385, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....N t . At higher levels this ceases even to make sense; at type 3 level the positions of e should be identifiable with equivalence classes of elements t . Recall that Ehrhard identified the extensional collapse of the sequential algorithms model of higher types with the hypercoherence model [5]. In other words if one starts with the identity relation on N t and defines the partial equivalence relations inductively in the usual realizability style (see for example [10] then the equivalence classes N t = are the sequentially realizable functionals and correspond to ....

T. Ehrhard. Hypercoherences: a strongly stable model of linear logic. Math. Struct. in Comp. Science, 3:365--385, 1993.

....classical linear logic. In Section 7 we consider bidomains, establishing the connection with bistructures (Theorem 2) In Section 8 we discuss possible variations and connections with other work; in particular we consider strengthenings of bistructures incorporating Ehrhard s hypercoherences (see [8]) thereby accounting for strong stability within our approach. In this paper, cpos are partial orders with a least element and lubs of all directed sets; continuous functions between cpos are those monotonic functions preserving all the directed lubs. For other domain theoretic terminology see, ....

....of strongly stable functions, which can be seen as an extensional (although not orderextensional) account of sequentiality. At first order, the strongly stable model contains exactly the sequential functions. At higher orders, it is the extensional collapse of the model of sequential algorithms [8, 9]. Generalisations encompassing both Girard s webs and hypercoherences have been proposed independently by Lamarche [18] based on quantale valued sets) and by Winskel [30] based on a notion of indexing inspired by logical relations) We believe that our biordering treatment can be applied to all ....

Ehrhard, T., Hypercoherences: a strongly stable model of linear logic. MSCS, Vol. 3, No. 4, pp. 365--385, 1993.

....e.g. 5] Chapter 8. Of course, it is hoped that bistructures have a more important role to play in understanding the extensional fully abstract model of PCF. Two rather independent lines of work on sequentiality appear central here: one is the work on coherences and hypercoherences [5, 6]; the other the successes in obtaining intensionally fully abstract models of PCF [1, 9] models which although not extensional do have the property that every finite element of the domains is definable by a PCF term. The work on hypercoherences combines smoothly with bistructures (as observed ....

Ehrhard, T., Hypercoherences: a strongly stable model of linear logic. Mathematical Structures in Computer Science, 1993.

.... historically introduced for approximate the notion of sequentiality in order to give fully abstract model for PCF (see [28, 14, 2] An important class of strongly stable models, intensively considered in this paper, is the class C hyp of reflexive hypercoherences introduced by Ehrhard in [15]. We respectively call continuous, stable or strongly stable models each object in respectively C cont ; C stab or C ststab . For an object in C hyp we speak about reflexive hypercoherence or hypercoherent model. The first incompleteness result has been given by Honsell and Ronchi in [19] for C ....

....are strongly stable or not with respect to the coherences. To do that we must quit the general case and investigate particular ones. Because of this lack of informations it is difficult to give reasonings in the general framework. A very elegant solution to these problems is given by Ehrhard [15], introducing the notion of hypercoherence. A hypercoherence H is a pair (D; Gamma) where D is a set and Gamma is a set of finite and non empty parts of D with one constraint: containing all singletons. A hypercoherence H induces (D(H) C(H) a DI domain with a strong and prime algebraic ....

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T. Ehrhard, Hypercoherence : a strongly stable model of linear logic, Mathematical Structure in Computer Science 2(1993) 365-385.

....similar to that of Honsell and Ronchi della Rocca [18] Gouy [16] proved the incompleteness of the stable semantics. Other more semantic proofs of incompleteness for the continuous, stable and hypercoherence semantics (that is a subclass of the strongly stable semantics introduced by Ehrhard [15]) can be found in [6, 7] and are briefly described in the following. The Park model P was first defined in the framework of continuous semantics. It is a variant of the Scott model D1 , but with a very different equational theory. This model has a stable analogue P s (which was defined by Honsell ....

T. Ehrhard, "Hypercoherences: a strongly stable model of linear logic", Mathematical Structures in Computer Science, 2 (1993), pp. 365-385

.... de facon uniforme (voir la section sur les r etractions universelle) Ehrhard a donn e un moyen el egant de palier a ce d efaut tout en conservant les bonnes propri et es sur les coh erences en introduisant une notion particuli erement bien adapt ee a la forte stablit e : les hypercoh erences [13]. Une hypercoh erence H est un couple (D; Gamma) o u D est un ensemble et Gamma est un ensemble de parties finies et non vides de D, avec une seule contrainte, celle de contenir tous les singletons. D une hypercoh erence H on d eduit (D(H) C(H) un DI domaine avec coh erence acceptable. ....

....qui sont des objets plus simples a manipuler, est qu elles permettent de reconstruire l empilement de structures que constitue un DIC uniquement a partir d une trame (l ensemble D) Pour ces raisons ces objets constituent, a notre sens, le bon cadre pour utiliser la forte stabilit e. Dans [13] il est montr e que la cat egorie des hypercoh erences et des fonctions fortement stables est une c.c.c. En ce qui concerne le calcul non typ e on se propose ici d utiliser les i mod eles fortement stables Les applications que l on donne sont les suivantes : 1. On montre que toute hypercoh erence ....

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T. Ehrhard, Hypercoherences : a strongly stable model of linear logic, Advances in linear logic, Eds. J.Y. Girard, Y. Lafont & L. Regnier, London Mathematical Society, Lecture Note Series 222, Cambridge University Press, 1995, p. 83-108.

..... Y hy; y 0 i: Thus the tensor of coherence spaces is obtained using the tensor product of 3 and the linear function space of coherence spaces is similarly obtained from the linear function space . in 3. Consider a second example for a model of Linear Logic, namely that of hypercoherences [4]. Recall that a hypercoherence X is given by a set jXj (also called 4 the web ) and a subset (X) of the set of nite non empty subsets of jXj containing all singletons. This can be encoded as a function X : P fne jXj 3, where P fne denotes the ( nite, non empty) powerset functor: X maps ....

Thomas Ehrhard. Hypercoherences: a strongly stable model of linear logic. Math. Struct. in Comp. Science, 3:365-385, 1993.

....The strongly stable semantics introduced by Bucciarelly and Ehrhard in [11] is the class of the partially ordered models whose specialization order is a DI domain with coherence and the representable functions are all the strongly stable ones. The hypercoherence semantics introduced by Ehrhard [17] is a subclass of the strongly stable semantics. A class C of models of the lambda calculus represents a lambda theory T if there is a model in C whose theory is exactly T . A class of models is incomplete if it does not represent all lambda theories. The continuous, stable and strongly stable ....

Ehrhard T., Hypercoherences: a strongly stable model of linear logic, Mathematical Structures in Computer Science 2 (1993), 365--385.

....all the stable ones. The strongly stable semantics introduced by Bucciarelly and Ehrhard in [12] is the class of po models whose specialization order is a DI domain with coherence and the representable functions are all the strongly stable ones. The hypercoherence semantics introduced by Ehrhard [18] is a subclass of the strongly stable semantics. Stability and strong stability constitute restrictions of continuity to capture the notion of sequentiality. The first incompleteness result was given by Honsell and Ronchi della Rocca [25] for the continuous semantics. They proved that the ....

T. Ehrhard, "Hypercoherences: a strongly stable model of linear logic", Mathematical Structures in Computer Science, 2 (1993), pp. 365-385

....exposition of the categorical interpretation of calculus. Then we briefly recall Berry s stability (resp. Bucciarelli and Ehrhard s strong stability) in the framework of dI domains (resp. of dI domains with coherence) The second section is devoted to the hypercoherences, a notion due to Ehrhard [10]. They give rise to a particular class of dI domains with coherence. We adapt to this framework the model construction technique that Krivine introduced in the continuous and stable case [16] In section 3 we give the construction of Park s strongly stable model. The fourth section contains the ....

....from B to B defined by : f(a) V iff a = V or a = F . If we take P f (B) as coherence, then f is not strongly stable (for it does not preserve the glb of fV; Fg) On the other hand, one can easily show that it is strongly stable with respect to the canonical coherence. 2 Hypercoherences In [10], Ehrhard introduced the notion of hypercoherence, a class of dI domains with acceptable coherence which is stable under products and exponentials and can be described in a very simple way. A hypercoherence H is a pair (D; Gamma) where D is a set and Gamma is a set of finite and non empty ....

[Article contains additional citation context not shown here]

T. Ehrhard, Hypercoherence : a strongly stable model of linear logic, Mathematical Structure in Computer Science 2(1993) 365-385.

.... B is continuous if and only if # If a # b then F(a) # F(b) # If S # A is directed (that is, if a, b # S, then a # b # S too) then F( S S) S F(a) a # S . 81 Erhard s hypercoherences are a generalisation of coherence spaces which are richer than a graph represents [95]. In hypercoherences, a might be a coherent set without a # # a also being coherent. The category of hypercoherences is also a model of linear logic. 82 This shows how categories have a kind of flexibility unavailable to posets. In a poset, # = # only if the poset is trivial. In a ....

THOMAS EHRHARD. "Hypercoherences: a strongly stable model of linear logic". Mathematical Structures in Computer Science, 3:365--385, 1993.

....i . Y hy; y 0 i: Thus the tensor of coherence spaces is obtained using the tensor product of 3 and the linear function space of coherence spaces is similarly obtained from the linear function space in 3. Consider a second example for a model of Linear Logic, namely that of hypercoherences [3]. Recall that a hypercoherence X is given by a set jXj (also called the web ) and a subset (X) of the set of nite non empty subsets of jXj containing all singletons. This can be encoded as a function X : P fne jXj 3, 4 where P fne denotes the ( nite, non empty) powerset functor: X maps a ....

Thomas Ehrhard. Hypercoherences: a strongly stable model of linear logic. Math. Struct. in Comp. Science, 3:365-385, 1993.

....ff X hx; x 0 i . ff Y hy; y 0 i. Thus the tensor of coherence spaces is obtained using the tensor product of 3 and the linear hom of coherence spaces is similarly obtained from the linear hom in 3. Consider a second example for a model of Linear Logic, namely that of hypercoherences [Ehr93]. Recall that a hypercoherence X is given by a set jX j (also called the web ) and a subset Gamma(X ) of the set of finite non empty subsets of jX j containing all singletons. This can be encoded as a function ff X : P fne jX j Gamma 3, where P fne denotes the (finite, non empty) powerset ....

Thomas Ehrhard. Hypercoherences: a strongly stable model of linear logic. Math. Struct. in Comp. Science, 3:365--385, 1993.

....product of 3 and the linear hom of coherence spaces is similarly obtained from the linear hom in 3. We say that the symmetric monoidal closed structure on 3 induces that of the category of coherence spaces. Now consider a second example of a model of linear logic, namely that of hypercoherences [Ehr93]. A hypercoherence X is given by a set jX j (also called the web ) and a subset Gamma (X) of the set of finite non empty subsets of jX j containing all singletons. This can be encoded as a function ff X : P fne jX j Gamma 3, where P fne denotes the (finite, non empty) powerset functor: ff X ....

Thomas Ehrhard. Hypercoherences: a strongly stable model of linear logic. Math. Struct. in Comp. Science, 3:365--385, 1993.

.... c 0 Omega a ) b Omega c 0 Omega a fl(c Omega a 0 Omega b ) c Omega a 0 Omega b fl(b Omega b 0 Omega b ) b Omega b 0 Omega b fl(c Omega c 0 Omega c ) c Omega c 0 Omega c Usual denotational semantics, including coherent spaces accept this function ; however Ehrhard [4] has been able to introduce hypercoherences, a beautiful Coherent Banach Spaces 23 generalization of coherent spaces, in which Gustave is not accepted as a clique. The question is therefore whether or not CBS accept this function. In other terms, let us implement A; B; by a non zero ....

T. Ehrhard. Hypercoherences : a strongly stable model of linear logic. In Girard, Lafont, and Regnier, editors, Advances in Linear Logic, pages 83--108, Cambridge, 1995. Cambridge University Press.

....classical linear logic. In Section 7 we consider bidomains, establishing the connection with bistructures (Theorem 2) In Section 8 we discuss possible variations and connections with other work; in particular we consider strengthenings of bistructures incorporating Ehrhard s hypercoherences (see [8]) thereby accounting for strong stability within our approach. In this paper, cpos are partial orders with a least element and lubs of all directed sets; continuous functions between cpos are those monotonic functions preserving all the directed lubs. For other domain theoretic terminology see, ....

....of strongly stable functions, which can be seen as an extensional (although not orderextensional) account of sequentiality. At first order, the strongly stable model contains exactly the sequential functions. At higher orders, it is the extensional collapse of the model of sequential algorithms [8, 9]. Generalisations encompassing both Girard s webs and hypercoherences have been proposed independently by Lamarche [18] based on quantale valued sets) and by Winskel [30] based on a notion of indexing inspired by logical relations) We believe that our biordering treatment can be applied to all ....

Ehrhard, T., Hypercoherences: a strongly stable model of linear logic. MSCS, Vol. 3, No. 4, pp. 365--385, 1993.

....the union of a clique in A B and a clique in B A ; but the union is not a clique, and this example is killed by On the meaning of logical rules I 17 coherent spaces 28 . The case of Gustave is much more delicate : Ehrhard found a beautiful generalization of coherent spaces, hypercoherences, [7], which manages to kill Gustave s function, but it seems that this is not the right way to full completeness. 5.6 Time Monastic life helps to put things together ; after weeks of meditation (on this problem) I realized that the completeness argument would anyway be trivial. So what is the main ....

T. Ehrhard. Hypercoherences : a strongly stable model of linear logic. In Girard, Lafont, and Regnier, editors, Advances in Linear Logic, pages 83--108, Cambridge, 1995. Cambridge University Press.

....(the space of points of E) to qD(X) the qualitative domain induced by X. The function is required to be linear, strongly stable (with respect to both the linear coherence induced by E on E , as well as the coherence induced by the hypercoherence X on qD(X) and onto. Hypercoherence (see [ Ehrhard, 1994a ] is a simplified framework for dealing with strong stability which also gives rise to a model of linear logic. A hypercoherence is a hypergraph which naturally gives rise to a qualitative domain equipped with a coherence. The intuition is this: E is the space of sequential algorithms, qD(X) ....

T. Ehrhard. Hypercoherences: a strongly stable model of linear logic, 1994. Preprint.

....We prove that the main isomorphisms of linear logic are satisfied by this interpretation of formulae. 1 Preliminaries If A is a set, we denote by #A its cardinality. We first recall some basic definitions on coherence spaces and hypercoherences. For more informations on these topics, we refer to [Gir95, Ehr93]. Definition 1 A coherence space is a symmetric and reflexive graph. More precisely, it is a pair E = jEj; E ) where jEj is a set (the web of E, its elements are called atoms or vertices) and E is a symmetric and reflexive binary relation on jEj. Two elements of jEj which are related by ....

....(X) This entails that U is not a singleton, otherwise, U = fyg where y 2 qD(X ) and then all finite sections of U belong to Gamma(X ) as in that case all sections of U are subsets of y. For more details on hypercoherences and the hypercoherent semantics of linear logic, we refer to [Ehr93]. 4 A family of cliques of a hypercoherence X is said to be coherent if it is finite and non empty, and if all its finite and non empty sections belong to Gamma(X ) 6 2 A motivating example The goal of this section is to motivate the forthcoming definitions and constructions by a detailed ....

Thomas Ehrhard. Hypercoherences: a strongly stable model of linear logic. Mathematical Structures in Computer Science, 3:365--385, 1993.

....i n and ff i = and, for any fi 2 N n , if fi ff and f(fi) j 6= then fi i 6= In this definition, the integer i is called sequentiality index of f for j at ff . 2 The elements of N n are sequences indexed on the set fi j 0 i ng 3 3 Hypercoherences In a recent paper (see [E]) the second author has introduced the notion of hypercoherence as a simplified framework where strong stability makes sense. We recall here the basic definitions and the properties of this model that we use in the sequel. Definition 3.1 A hypercoherence X is a pair (jXj ; Gamma (X) where jXj ....

....a qualitative domain, and its web is jXj by our only requirement about hypercoherences. The morphisms between hypercoherences that we shall consider in this paper are the strongly stable functions. There is also a notion of linear morphisms between hypercoherences; their theory is developped in [E]. Definition 3.3 Let X and Y be hypercoherences. A strongly stable function from X to Y (we shall also write f : X Y ) is a function f : qD (X) qD (Y ) which is continuous and satisfies 8A 2 C (X) f(A) 2 C (Y ) and f( A) f(A) We have chosen this notation f : X Y because it ....

T. Ehrhard. Hypercoherences: a strongly stable model of linear logic. To appear in Mathematical Structures in Computer Science.

....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. Introduction In previous works ([BE1, BE2, BE4, E]) we introduced the notion of strong stability as an alternative way to deal with sequentiality. Our first observation was that the definition of sequentiality that Kahn and Plotkin have proposed in [KP] could be expressed as a preservation property. To express sequentiality of functions, one ....

....A morphism between two domains equipped with a coherence is a continuous function which sends coherent sets on coherent sets and commutes to the glb s of coherent sets. More recently, we have found a simplified framework for dealing with strong stability, namely the framework of hypercoherences [E] which also gives rise to a new model of linear logic, and we shall use this theory of strongly stable functions in the sequel. A hypercoherence is a very simple structure (a hypergraph) which naturally gives rise to a qualitative domain equipped with a coherence. All the constructions that we ....

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T. Ehrhard. Hypercoherences: a strongly stable model of linear logic. Mathematical Structures in Computer Science 3(1993), pp. 365-385.

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T. Ehrhard. Hypercoherences: a strongly stable model of linear logic. In Mathematical Structures in Computer Science, 3:365-385, 1993.

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T. Ehrhard. Hypercoherences: a strongly stable model of linear logic. Mathematical Structures in Computer Science, 3:365-385, 1993.

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T. Ehrhard. Hypercoherences: A strongly stable model of linear logic. Mathematical Structures in Computer Science, 3:365--386, 1993.

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Ehrhard, T.: Hypercoherences: a strongly stable model of linear logic. Mathematical Structures in Computer Science 3 (1993) 365--385

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