G. Berry, "Stable models of typed -calculi", in C. Bohm, G. Ausiello, eds, Third colloquium on automata, languages and programming (Udine, 1978). Lecture Notes in Computer Science 62, Springer, Berlin, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....types, for instance, or used in a call by value setting. More importantly, it represents a link between the intensional notion of sequentiality represented by sequential algorithms, and the extensional notion of stability. The basis for our models are biordered sets similar to Berry s bidomains [1], except that the stable order is replaced with a bistable order, and functions are required to be bistable preserving binary lubs and glbs of bounded nite sets. This is similar to the property of additivity or linearity for stable functions, although the bistable and stable orders are di erent ....

....it is languages (such as SPCF and the simply typed calculus) in which such a function is not denotable, which can be modelled fully abstractly in bistable bidomains. 2 Bistability We shall introduce the notion of bistability in the context of biordered sets along the lines of Berry s bidomains [1]. This is not necessary bistability and the bistable ordering can be de ned independently, but it is only by combining them with the extensional order that we can build sequential models of the calculus. De nition 2.1 A bistable biorder is a tuple hD; v; i consisting of a set with two ....

[Article contains additional citation context not shown here]

G. Berry. Stable models of typed -calculi. In Proceedings of the 5th International Colloquium on Automata, Languages and Programming, number 62 in LNCS, pages 72{ 89. Springer, 1978.

.... are an elementary phrasing of the following condition: consider the binary meet completion of the type order, and extend # by union (i.e. for a new element x set # (x) y x # (y) then, we require that # is stable, that is, that it preserves the meet of elements with a common upper bound [Ber78]. The reader should note that there is another possibility for condition (3) we could allow more than one relationship, but always with the same element. However, this would give rise to a rather clumsy definition, and we see no motivation for such a change. 3.2 Down To SQL As we mentioned ....

G. Berry. Stable models of typed #-calculi. In G. Ausiello and C. Bhm, editors, Proceedings of the 5th Colloquium on Automata, Languages and Programming, volume 62 of Lecture Notes in Computer Science, pages 72--89, Udine, Italy, 1978. Springer--Verlag.

.... of a unique subtype F of E such that x # (F) but x # (G) for any proper subtype G of F ; such an F The last two conditions are an elementary phrasing of the fact that the map from the binarymeet completion of the type order to the chosen universe of sets induced by # must be stable [Ber78]. is called the type of x # (E) An entity is now a pair E , x # (E) such that E is the type of x . Note that we did not restrain entity sets to be disjoint, so an x of type E and an x of type F are actually distinct entities. The definition we gave of cardinality constraints is also valid ....

G. Berry. Stable models of typed #-calculi. In G. Ausiello and C. Bhm, editors, Proceedings of the 5th Colloquium on Automata, Languages and Programming, volume 62 of Lecture Notes in Computer Science, pages 72--89, Udine, Italy, 1978. Springer-- Verlag.

....is to show that stability is not a topological property. That is, that coherence spaces except for the so called flat ones cannot be equipped with a topology such that the notions of continuity and stability coincide. 1 Introduction 1 Coherence spaces were introduced in a restricted form by Berry [Ber 1978] to characterize sequential algorithms, and subsequently used by Girard [Gir 1986] Gir 1989] as a refinement of Scott domains for the semantics of lambda calculi. At first sight, coherence spaces are just a very special class of Scott domains. However, looking at the construction of their ....

G.Berry. Stable Models of Typed -Calculi, ICALP, Springer LNCS 62 (1978) 72-89.

....are unique [2, Corollary to Theorem II.13] this implies x = y contradicting x y. By the second requirement on divisibility monoids, the partial order (M; can be seen as the compact elements of a Scott domain. The lemma above ensures that it is even the set of compacts of a dI domain (cf. [1, 24]) Thus, we have in particular (x y) z = x z) y z) whenever the left hand side is defined. Let x = x 1 x 2 : x n with x i 2 T . Then f1; x 1 ; x 1 x 2 ; xg is a maximal chain in the finite distributive lattice # x. Since maximal chains in such finite distributive lattices have ....

G. Berry. Stable models of typed -calculi. In 5th ICALP, Lecture Notes in Comp. Science vol. 62, pages 72--89. Springer, 1978.

....bilimits and there is an algebraic and a generalised topological description of its morphisms. 1 Introduction There are two kinds of morphism that are studied in classical domain theory, Scott continuous ones and stable ones. Berry introduced stable maps to model sequentiality in the calculus [Ber78] A stable map, in addition to being continuous (i.e. preserving directed suprema) also preserves bounded binary infima. So, for first order the stable functions are a subset of the continuous ones, but at higher order types the continuous and the stable function space become incomparable. ....

G. Berry. Stable models of typed -calculi. In Proceedings of the 5th International Colloquium on Automata, Languages and Programming, volume 62 of Lecture Notes in Computer Science, pages 72--89. Springer Verlag, 1978.

....arose in part from Girard s insights into particular categorical models for intuitionistic logic. In giving a domain theoretic semantics to his System F (the polymorphic calculus) Gir86] Girard re invented the stable domain theory of Berry based on the category dI domains and stable functions [Ber78], though for rather special domains, the coherence spaces. It turns out that Girard s construction for modelling polymorphic types can be carried through, with some modifications, in the more standard domain theory based on the category of Scott domains and continuous functions (see [CGW89] But ....

G. Berry. Stable models of typed -calculi. In Fifth International Colloquium on Automata, Languages and Programs, pages 72--89. Springer-Verlag (LNCS 62), 1978.

....This order can be extended to an order on the infinite computations. If one restricts the consideration to those computations that start in a given initial state, these orders are certain domains (algebraic cpo s) that are closely related with event domains [Wi87] concrete domains and dI domains [Be78] (see [Dr90, Dr92, Ku94a, Ku94b] Basic order theoretic definitions of lattices and distributivity will be given in Sect. 3. Proposition 2.8 ( Ku94b] Let A be a stably concurrent automaton and x 2 M(A) nf0g. Then (fy 2 M(A) j y xg; is a finite distributive lattice. In [Ku94b] Kuske even ....

Berry, G.: Stable models of typed -calculi. In: 5th ICALP, Lecture Notes in Computer Science vol. 62, Springer, 1978, 72-89.

....functions. It should be clear that our analysis can be adapted to other settings and other notions of intensional behavior where the necessary categorical and algebraic conditions hold. Various restricted notions of continuous function have been used elsewhere, including Berry s stable functions [Ber78] and Kahn and Plotkin s sequential functions [KP78] Various different kinds of semantic domains have been shown to be useful. We plan to investigate the possibility of emulating our algorithm construction when we vary the choice of underlying ccc, or the choice of ordering on paths, or even when ....

G. Berry. Stable models of typed -calculi. In Proc. 5th Coll. on Automata, Languages and Programming. LNCS 62, number 62 in LNCS, pages 72--89, Berlin, New-York, July 1978. Springer Verlag.

....compositional reasoning about behavior. It has turned out to be surprisingly difficult to give natural (i.e. language independent) constructions of fully abstract semantic models for sequential languages such as PCF [Plo77, BCL85] The known constructions of fully abstract models for PCF [Mil77, Ber78, Mul87] are not natural, yet there are natural fully abstract models for an extension of PCF with parallel facilities [Plo77] and, more recently, with control facilities [CF92, Cur92] There is currently no definition of sequential functions suitable for defining a natural extensional semantic ....

....and concrete domains, and defined sequential functions between concrete domains. However, the sequential functions between two concrete domains do not form a concrete domain (under either the pointwise or stable orders) Berry introduced dI domains, stable functions and the stable ordering [Ber78] the stable functions between two dI domains, ordered stably, form a dI domain. However, the stable functions do not provide the desired notion of sequential functions, since some stable functions are not sequential. Berry and Curien [BC82, Cur86] defined sequential algorithms between concrete ....

[Article contains additional citation context not shown here]

G. Berry. Stable models of typed -calculi. In Proc. 5 th Coll. on Automata, Languages and Programming, number 62 in Lecture Notes in Computer Science, pages 72--89. Springer-Verlag, July 1978.

....on T 2 D such that d 0 . dN Gamma1 d N v T 2 D d 0 . dN Gamma1 t if t 2 T 2 D dN vD t 0 : This ordering is based on the prefix ordering on sequences, but adjusted to take appropriate account of the underlying order on data values. The order v T 2 D is actually the stable ordering [2] on T 2 D, when we regard the elements of T 2 D as (strictly increasing, possibly eventually constant) stable functions from VNat to D. Note that every continuous function from VNat to D is also stable. ffl For a continuous function f : D D 0 we define T 2 f to be the function which applies ....

....structures [5] It would be interesting to see if the Berry Curien sequential algorithms category could be embedded in the Kleisli category of a suitable comonad over a sequential functions category. We intend to investigate notions of computation on the category of dI domains and stable functions [2], and on the category of qualitative domains and linear functions [7] Acknowledgements The diagrams in this paper were drawn using macros designed by John Reynolds. ....

G. Berry. Stable models of typed -calculi. In Proc. 5 th Coll. on Automata, Languages and Programming, number 62 in Lecture Notes in Computer Science, pages 72--89. Springer-Verlag, July 1978.

....concrete domains, and gave a definition of sequential function between concrete domains. However, the sequential functions between two concrete domains do not form a concrete domain (under both the pointwise and stable orders) Berry introduced dI domains, stable functions and the stable ordering [Ber78] the stable functions between two dI domains, ordered stably, form a dI domain. The stable functions do not provide the desired notion of sequential functions, since some stable functions are not sequential. Berry and Curien [BC82, Cur86] defined sequential algorithms between concrete domains, ....

....functions between Scott domains, induced by differing notions of open sets: Scott opens, stable opens and sequential opens, respectively. Scott opens and continuous functions are well known [Sco72, GHK 80] and may be considered classical by now. Stable functions were introduced by Berry [Ber78] and our presentation here generalizes Zhang s presentation of stable functions between dI domains [Zha89] We take D and E to be generic Scott domains, unless stated otherwise. We present a number of examples in the running text. In addition, an appendix presents all Scott opens, stable opens ....

[Article contains additional citation context not shown here]

G. Berry. Stable models of typed -calculi. In Proc. 5 th Coll. on Automata, Languages and Programming, number 62 in Lecture Notes in Computer Science, pages 72--89. Springer-Verlag, July 1978.

....f; g from D to E, f is below g in the stable ordering iff f is pointwise below g and, for each d 2 D and e f(d) M(f; d; e) M(g; d; e) ffl A dI domain is a distributive Scott domain that has property (I) It is well known that the category of dI domains and stable functions is a ccc. See [Ber78] for a fuller treatment, as well as alternative (but equivalent) definitions of stability and the stable ordering. We qualify the states introduced so far as being ct states, and use D ct (M) for the domain of ct states of M , ordered by set inclusion we call this the ct domain generated by ....

G. Berry. Stable models of typed -calculi. In Proc. 5 th Coll. on Automata, Languages and Programming, number 62 in Lecture Notes in Computer Science, pages 72--89. SpringerVerlag, July 1978.

....these arguments, so that their notions of sequentiality are not general enough. Kahn and Plotkin [KP78] defined concrete data structures, or CDSs, together with their order theoretic counterparts, concrete domains, which made possible a more general definition of sequentiality of functions. Berry [Ber78] introduced the notion of stability, a property of functions intermediate between sequentiality and continuity. However, Berry and Curien [BC82, Cur86] showed that the category of concrete domains fails to be cartesian closed when the morphisms in the category are taken to be the continuous ....

....at least the same output as a. We regard intensional strictness as a natural generalization to the intensional setting of the standard extensional ordering on continuous functions. In contrast, the set inclusion ordering on algorithms used by Berry and Curien corresponds to the stable ordering [Ber78] on sequential functions. We show that, at first order types, with suitable countability assumptions, the intensional strictness order is a directed complete algebraic pre order on parallel algorithms. We show that application and currying are continuous with respect to the new ordering. This ....

[Article contains additional citation context not shown here]

G. Berry. Stable models of typed -calculi. In Proc. 5th Coll. on Automata, Languages and Programming, number 62 in Lecture Notes in Computer Science, pages 72--89, Berlin, New-York, July 1978. Springer-Verlag.

....a subset of needed redexes will spring to the reader s mind. The concept of gaining predicates enables easy proofs of completeness of such reduction strategies. Restricting the set of admissible patterns of a program can lead to considerably simpler reduction strategies. In Berry s example ([3]) h(True, False, x) True h(False, x, True) True h(x, True, False) True all three arguments in the term h(t 1 , t 2 , t 3 ) have to be reduced in parallel (alternately) 7 , since the term s semantics may not be even if any one argument denotes . Hence, one may only permit sequential ....

G. Berry. Stable models of typed -calculi. In Proceedings of the 5th ICALP Conference, LNCS 62, pages 73--89, Udine, Italy, 1978.

....order: f 6 s g ( 8x; y 2 D: x 6 y ) f(x) f(y) g(x) Directed suprema, as well as bounded infima, with respect to the Berry order are taken pointwise. The cartesian closed categories DCPO and DCPO are defined analogously. The theory of meet cpo s and stable functions is due to Berry [7], who used them to study the semantics of sequential computations. 1.2.5 Domain equations Let D be any one of the pointed categories DCPO , CPO , DCPO , CPO , DCPO bc , or CPO bc . The objects of D are called domains. One of the main features of these categories of domains ....

G. Berry. Stable models of typed -calculi. In Proceedings of the 5th International Colloquium on Automata, Languages and Programming, Springer LNCS 62, pages 72--89, 1978.

....and f v s g iff for all x v y, f(x) f(y)ug(x) This is the first step in an extensive development of Stable Domain Theory in which stable functions under the stable ordering take the place which continuous functions play in standard Domain Theory. Stable Domain theory was introduced by Berry [Berry, 1978, Berry, 1979] Some more recent references are [Girard, 1986, Coquand et al. 1987, Taylor, 1990, Ehrhard, 1993] Berry s motivation in introducing stable functions was actually to try to capture the notion of sequentially computable function at higher types. For the theory of sequential ....

G. Berry. Stable models of typed -calculi. In Proceedings of the 5th International Colloquium on Automata, Languages and Programming, volume 62 of Lecture Notes in Computer Science, pages 72--89. Springer Verlag, 1978.

....are exhibited as axioms and generic properties are established from them. The stability theorem presented here (also called fundamental theorem) is the most significant contribution of this approach. It inherits the conceptual clarity of its domain theoretic ancestor, Berry stability theorem [Be], and the genericity of its own rewriting theoretic principles. Informally, the theorem states that there exists from every term M a cone of minimal computations to the set of head normal forms. ARS. The framework of Axiomatic Rewriting Systems (ARSs) encompasses many existing Rewriting Systems ....

....Proof See appendix. 5 A categorical approach We have just established a very useful rewriting theorem but we would like to go one step further and connect our work with mainstream denotational semantics. Our motto is that the fundamental theorem is really a stability theorem in the spirit of Berry [Be]. To understand this, assume that a functor [ Gamma] C] Gamma S interprets the terms of the ARS (G; in a small category S. Assume now that the functor verifies a series of conditions corresponding in fact to open stability of Section 2. We prove (see theorem 5.4) that the restriction of ....

[Article contains additional citation context not shown here]

G. Berry, "Stable models of typed -calculi", in C. Bohm, G. Ausiello, eds, Third colloquium on automata, languages and programming (Udine, 1978). Lecture Notes in Computer Science 62, Springer, Berlin, 1978.

....and f v s g iff for all x v y, f(x) f(y) u g(x) This is the first step in an extensive development of Stable Domain Theory in which stable functions under the stable ordering take the place which continuous functions play in standard Domain Theory. Stable Domain theory was introduced by Berry [Berry, 1978, Berry, 1979] Some more recent references are [Girard, 1986, Coquand et al. 1987, Taylor, 1990, Ehrhard, 1993] Berry s motivation in introducing stable functions was actually to try to capture the notion of sequentially computable function at higher types. For the theory of sequential ....

G. Berry. Stable models of typed -calculi. In Proceedings of the 5th International Colloquium on Automata, Languages and Programming, volume 62 of Lecture Notes in Computer Science, pages 72--89. Springer Verlag, 1978.

.... authors building on Winskel s work on safe nets [28] have done in [12, 13] for PT nets (see [7, 9] for related approaches) In Winskel s work which in turn builds on the previous work [15] the denotation of a safe net is a coherent finitary prime algebraic Scott domain [25] or dI domain [1]. Winskel shows that there exists a coreflection a particularly nice form of adjunction between the category Dom of (coherent) finitary prime algebraic domains and the category Safe of safe Petri nets. This coreflection factorizes through the chain of coreflections Safe Occ PES Dom U [ F ....

....(T Theta ; 4; #) where 4 and # are the restrictions to the set of transitions of Theta of, respectively, the flow ordering and the conflict relation implicitly defined by Theta. Finitary prime algebraic domains or dI domains introduced by G. Berry while studying sequentiality of functions [1] are particular Scott s domains which are distributive and in which each finite element is preceded only by a finite number of elements of the domain. Here we are interested in their coherent version, i.e. in the version in which the underlying partial order is pairwise complete. Definition ....

G. Berry. Stable Models of Typed -calculi. In Proceedings ICALP `78 , LNCS, n. 62, pp. 72--89, 1978.

....construction stays within CONT as was shown in [Jon90] but again it is unknown if it stays within FS domains. Hence we may have to accept the fact that there is no single category of domains which meets all needs. In his attempts to capture sequentiality in a mathematical model, Gerard Berry [Ber78, Ber79] developed an alternative domain theory based on the notion of stability, seemingly incompatible with Scott continuity (see [Gun92, Section 5.2] for an exposition) Through the very recent work of Fran cois Lamarche [Lam] and Mathias Kegelmann [Keg95] we now see that both Scott continuity ....

.... to reach Milner s model [JS93] The technical advantages and disadvantages to each construction are described succinctly in [JS93] The second approach, domain theoretic, comes from Berry s notion of stable functions, a subset of the continuous functions on dI domains that does not include por [Ber78] The stable model is quite interesting, but not fully abstract; in fact, even though it seems closer at first order type to the fully abstract model, the inequational theory is incomparable to that of Plotkin s model [JM91] Bucciarelli and Ehrhard [BE91, Ehr93] refined the model with stronger ....

G. Berry. Stable models of typed -calculi. In Proceedings of the 5th International Colloquium on Automata, Languages and Programming, volume 62 of Lecture Notes in Computer Science, pages 72--89. Springer Verlag, 1978.

....space respect the above intuition, although it was motivated from the study of sequentiality. This paper studies Berry s category of dI domains with stable functions, which is a relatively intricate, yet elegant framework for semantics of programming languages. This category was first shown in [2] to be cartesian closed and to provide a model for the typed calculus. Since then, stable domains have found many applications, such as in linear logic [10,22] concurrency [19] polymorphism [4] and sequential computations [5] A dI domain is a bounded complete, algebraic cpo which is ....

....theory. The first kind establishes certain properties, such as cartesian closure, existence of an universal object, etc. of a particular category of domains, demonstrating that these domains can serve as a suitable semantic space. Results on Scott domains [11] SFP domains [15] and dI domains [2] can be categorized as the first kind. The second kind of results has a different emphasis. Rather than establishing the suitability of a certain class of domains for semantic modeling, these results show that categories with certain properties just do not exist beyond a limit. The work of Smyth ....

[Article contains additional citation context not shown here]

G. Berry, Stable models of typed -calculi, in: Lecture Notes in Computer Science 62 (1978) 72--89.

No context found.

G. Berry, "Stable models of typed -calculi", in C. Bohm, G. Ausiello, eds, Third colloquium on automata, languages and programming (Udine, 1978). Lecture Notes in Computer Science 62, Springer, Berlin, 1978.

No context found.

G. Berry. Stable models of typed -calculi. In Proc. 5 th Coll. on Automata, Languages and Programming, number 62 in Lecture Notes in Computer Science, pages 72--89. Springer-Verlag, July 1978.

No context found.

G. Berry. Stable models of typed -calculi. In Proceedings of the 5th International Colloquium on Automata, Languages and Programming, volume 62 of Lecture Notes in Computer Science, pages 72--89. Springer Verlag, 1978.

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