Thomas Ehrhard. Projecting sequential algorithms on the strongly stable functions. To appear in Annals of Pure and Applied Logic vol. 77 No. 3.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the sequential algorithms model includes algorithms which compute functionals which are sensitive to non functional aspects of the behaviour of their arguments. The bidomains model also contains non sequential functions; while the strongly stable model, in the light of a recent result by Ehrhard [Ehr], can be seen as the extensional collapse of the sequential algorithms model. In short, all these models are unsatisfactory because they contain junk . On the other side of the coin, we have Milner s result that an order extensional model is fully abstract i all its compact elements are de ....

Thomas Ehrhard. Projecting sequential algorithms on the strongly stable functions. To appear in Annals of Pure and Applied Logic vol. 77 No. 3.

....solution to our full abstraction problem, the sequential algorithms form a bridge between the highly intensional world which is being mapped by games semantics (with its close connections with syntax) and the various abstract notions of extensional sequential functional. For example, Ehrhard [13] has shown that the collapse of the sequential algorithms model of PCF without errors is isomorphic to one in a category of hypocerence spaces and strongly stable functions, which can also be characterised more generally as the sequentially realizable functionals [27] 1.3 Contribution The ....

....as a structure preserving functor which identi es two innocent strategies if and only if they are intrinsically equivalent [21] The extensional collapse of the error free games and sequential algorithms models of SPCF is a sound model of PCF, although it is not fully abstract. Ehrhard [13] has characterised the collapse of the sequential algorithms model in terms of the strongly stable functions on hypercoherence spaces. Theorem 5.25 (Ehrhard) The extensional collapse of the (error free) sequential algorithms model of PCF is isomorphic to the strongly stable model of PCF. By ....

T. Ehrhard. Projecting sequential algorithms on strongly stable functions. Annals of Pure and Applied Logic, 77, 1996.

....tous les DIC r eflexifs. On en d eduit un crit ere d isomorphisme pour ces mod eles, ainsi qu une preuve s emantique de l existence de 2 0 mod eles fortement stables non fortement isomorphes. La notion de forte stabilit e a et e introduite par Bucciarelli et Ehrhard dans [10] On renvoie a [11, 12, 14] pour les liens existant entre la notion de forte stabilit e et la s equentialit e. Le cadre le plus g en eral dans lequel la forte stabilit e a et e d efinie est celui des DI domaines avec coh erence (DIC) voir [11] On peut construire des mod eles du lambda calcul pur dans ce cadre g en eral ....

T. Ehrhard, Projecting sequential algorithms on strongly stable functions, Annals of Pure and Applied Logic, Vol 77, no. 3, 1996, P. 201-244.

.... shown to have an extensional model in the category of sequential algorithms by Cartwright, Curien and Felleisen [8] More recently, there has been renewed interest in sequentially realisable functionals [19] which can be e ectively presented in various forms, such as strongly stable functionals [13] Although this suggests that the sequentially realisable functionals are signi cant from a semantic perspective, it does not answer the question: what is the nature of the programming constructs associated with them However, the most natural alternative to PCF as a language embodying sequential ....

T. Ehrhard. Projecting sequential algorithms on strongly stable functions. Annals of Pure and Applied Logic, 77, 1996.

....some time specifically to the third (and much the least well known) of the above classes of computable functionals. This class is especially interesting, as it embodies an intuitively sequential notion 3 of computability that is stronger than PCF. Its discovery is in essence due to Ehrhard [Ehr96], who showed that the functionals intensionally computable by sequential algorithms coincide with the strongly stable functions in a certain domain theoretic model. Since then, the implicit notion of computability has been brought more clearly into focus via a realizability model discovered ....

T. Ehrhard. Projecting sequential algorithms on strongly stable functions. Ann. Pure Appl. Logic, 77, 1996.

....sense computable, but they are not implementable even in full ML. Both of the above notions have been widely known since the early work of Scott [10] and Plotkin [9] The third class of computable functions the SR functions 2 has emerged more recently from work of Bucciarelli and Ehrhard [1, 2], van Oosten [7] and the present author [4] The SR functions form a larger class than the PCF computable functions, but they are nonetheless all expressible in an impure sequential language such as ML. They admit a wide variety of mathematical characterizations, and have many pleasing theoretical ....

T. Ehrhard. Projecting sequential algorithms on strongly stable functions. Annals of Pure and Applied Logic, 77:201--244, 1996.

....of dI domains with coherence, the following very satisfactory result on strong stability may be obtained. Theorem 4.5.5. The category whose objects are dI domains with coherence and whose morphisms are strongly stable functions is cartesian closed. This category gives rise to a model of pcf. In [ Ehrhard, 1994b ] Ehrhard shows that any strongly stable function which arises from the model is the extensional component of a sequential algorithm. More precisely, a cartesian closed category is constructed whose objects are triples h E; X; i. In such a triple, E is a sequential structure, X is a ....

T. Ehrhard. Projecting sequential algorithms on strongly stable functions. Journal of Pure and Applied Logic, To appear, 1994.

....the sequential algorithms model includes algorithms which compute functionals which are sensitive to non functional aspects of the behaviour of their arguments. The bidomains model also contains non sequential functions; while the strongly stable model, in the light of a recent result by Ehrhard [10], can be seen as the extensional collapse of the sequential algorithms model. In short, all these models are unsatisfactory because they contain junk . On the other side of the coin, we have Milner s result that an order extensional model is fully abstract iff all its compact elements are ....

T. Ehrhard, Projecting sequential algorithms on the strongly stable functions, submitted for publication, 1993.

....the sequential algorithms model includes algorithms which compute functionals which are sensitive to non functional aspects of the behaviour of their arguments. The bidomains model also contains non sequential functions; while the strongly stable model, in the light of a recent result by Ehrhard [Ehr], can be seen as the extensional collapse of the sequential algorithms model. In short, all these models are unsatisfactory because they contain junk . On the other side of the coin, we have Milner s result that an order extensional model is fully abstract iff all its compact elements are ....

Thomas Ehrhard. Projecting sequential algorithms on the strongly stable functions. To appear in Annals of Pure and Applied Logic vol. 77 No. 3.

....give rise to exactly the PCF computable functionals, others give rise to a quite different and larger class of sequentially computable functionals. This class is closely related to the strongly stable functionals of Bucciarelli and Ehrhard [8] It was already implicit in the work of Ehrhard [10] that these functionals were in some sense computable by sequential algorithms, but the realizability models have shed new light on this class of functionals and have provided fresh evidence that it really is a mathematically natural class of higher type computable functions. This seems to me to ....

....Ehrhard [7] ffl The extensional collapse of the Berry Curien sequential algorithms model [5] ffl The category of presheaves over the monoid of sequential endofunctions on N N . ffl The modified realizability model over B. The equivalences between these constructions are due to Ehrhard [10, 11], van Oosten [24] and the author. The diversity of these descriptions contribute to the impression that the class of SR functionals is somehow a mathematically natural structure. Moreover, the above descriptions can be used to show many good theoretical properties of E(B) Not surprisingly, the ....

T. Ehrhard. Projecting sequential algorithms on strongly stable functions. Ann. Pure Appl. Logic, 77, 1996.

.... of this paper is, that a natural generalization of function realizability to partial functions from IN to IN (yielding a total combinatory algebra) gives a type structure of higher type functionals which coincides with the relevant part of Ehrhard and Bucciarelli s dI domains with coherence ([3, 4, 2]) Research supported by the Dutch National Research Foundation NWO This could be interesting for a number of reasons. First, it provides another handle on Ehrhard s work, which is complicated and rather heavily loaded with definitions; however, the fact that dI domains with coherence have a ....

....definitions; however, the fact that dI domains with coherence have a completely independent generation process (which process is well known in logic) seems to me to enhance their naturalness as a mathematical structure. Of course, the result in this paper calls for comparison with the result in [3], viz. that dI domains with coherence are the extensional collapse of another domain theoretic structure, sequential structures and sequential algorithms. My result is essentially different in that it relates the dI domains with coherence to something which is defined independently of any domain ....

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T. Ehrhard, Projecting sequential algorithms on strongly stable functions, Annals of Pure and Applied Logic 77(1996), pp.201-244

....the sequential algorithms model includes algorithms which compute ifunctionalsj which are sensitive to non functional aspects of the behaviour of their arguments. The bidomains model also contains non sequential functions; while the strongly stable model, in the light of a recent result by Ehrhard [14], can be seen as the iextensional collapsej of the sequential algorithms model. In short, all these models are unsatisfactory because they contain ijunkj. On the other side of the coin, we have Milner s result that an order extensional model is fully abstract ioe all its compact elements are ....

T. Ehrhard, Projecting sequential algorithms on the strongly stable functions, Annals of Pure and Applied Logic, 1993.

....P of P together with a morphism p : P A such that for any Q 2 P and any morphism f : Q A, there exists a (not necessarily unique) morphism f 0 : Q P such that p ffi f 0 = f . We shall say that f 0 is a lifting of f along p. A very similar lifting condition played an essential role in [Ehr96]. Saying that (P; p) is a rigid P unfolding of A means furthermore that Id P is the only morphism g : P P such that p ffi g = p. By proposition 5, if (P 0 ; p 0 ) is another rigid P unfolding of A, there is exactly one morphism f : P P 0 such that p 0 ffi f = p, and f is an ....

Thomas Ehrhard. Projecting sequential algorithms on strongly stable functions. Annals of Pure and Applied Logic, 77:201--244, 1996.

....link. There are also other investigations [11] along the same lines which were already lead in the intuitionistic case. Specifically, Ehrhard s extensional collapse of sequential algorithms, which Lamarche [10] and Curien showed could be thought of as games model, to strongly stable functions in [7, 8]. To track down the relation between the present attempt and those former investigations should serve well the general purpose of bridging both kinds of models. Let us recall that the category we described in [4] is a simple adaptation of AJM games category [2, 3] which models classical MELL for ....

Thomas Ehrhard. Projecting sequential algorithms on strongly stable functions. Annals of Pure and Applied Logic, 77(3), February 1996.

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Thomas Ehrhard. Projecting sequential algorithms on the strongly stable functions. To appear in Annals of Pure and Applied Logic vol. 77 No. 3.

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T. Ehrhard. Projecting sequential algorithms on strongly stable functions. Annals of Pure and Applied Logic, 77:201-244, 1996.

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