Larsen, K., Winskel, G.: Using information systems to solve recursive domain equations. Information and Computation, Vol. 91(2):232--258, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Scott continuous functions form a category which is equivalent to the category of information systems together with approximable maps. Note that here the Scott continuous functions are those set functions which preserve filtered colimits (i.e. directed suprema) For details see both [Sco82] and [WL83]. Of course, it is by definition that a Scott domain has a least element. We now extend Scott s results to structures which are just like Scott domains but which do not necessarily possess a least element; we shall call these Scott predomains. The literature describes many different kinds of ....

....) 3. X A XrY B Y implies X rY . We refer to a morphism r: A B as a preapproximable map; the identity on A is just A and composition is the usual composition of relations. Remark 7.2. 7 This definition is clearly very similar to that of an information system as given in [Sco82] and [WL83]. In Scott s original paper, the token sets contain a distinguished element Delta which plays the role of a least element in the corresponding domain. However, if this requirement is removed, the resulting information systems still represent Scott domains as is noted in [WL83] the (consistent) ....

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G. Winskel and K.G. Larsen. Using information systems to solve recursive domain equations effectively. Technical Report 51, University of Cambridge Computer Laboratory, 1983. 165

....points from which the others can be derived as joins. Moreover, Scott continuous maps between Scott domains can be described as relations (approximable mappings) involving tokens. The information systems are crucial when one comes to solving domain equations (Scott [20] and Larsen and Winskel [14]) essentially because a colimit of a chain of domains (obtained by iterating the constructor in the domain equation) is constructed by taking the set theoretic union of the corresponding information systems. It is well known that algebraic posets (otherwise known as algebraic dcpos) can be ....

K. Larsen and G. Winskel, Using information systems to solve recursive domain equations effectively, in G. Kahn, D.B. MacQueen and G. Plotkin, eds, Semantics of Data Types (Lecture Notes in Computer Science 173, Springer-Verlag, Berlin, 1984) 109-129.

.... calculus it is not always the case that it will give rise to a square model, but it is often the case (see Section 5. 1) Roughly speaking prime webs are to (binary) prime algebraic Scott domains what Scott s information systems [33] are to Scott domains in the continuous semantics (cf. also [20]) and prime event structures to DI domains in the stable semantics (see for ex. 10] but, categorically speaking, the situation is less neat here (remark 3 in section 3.2) Interest of the present class. The rst interest of the class lies in the simplicity of the interpretations of terms in ....

K.G. Larsen and G. Winskel, Using information systems to solve recursive domain equations, In Lecture Notes in Computer Science, vol. 173 (Semantics of data types), p. 109-130, Spriger-Verlag, 1984.

....for its interpretation. The whole discussion leads to the following domain equation D = P ] D D] where P ] is the upper powerdomain functor, in the category of prime algebraic lattices. It is well known that each prime algebraic lattice can be described by an information system ([42]) and also by means of intersection types ( 22] Really we have developed in previous sections a system of intersection and union types; we will use this system now to build a model, which actually is the initial solution of our domain equation. Because of rules ( and ( I) the set of ....

K.G.Larsen, G.Winskell, "Using Information Systems to Solve Recursive Domain Equations Effectively", LNCS 173, Springer-Verlag, Berlin 1984, 109-130.

....exactly by those types that cannot be non trivially equated to any type whose outermost operator is Phi. We can now face the problem of representing the initial solution of equation (5) Indeed, applying to the case of EATS the technique to solve domain equations using Information Systems (see [26]) we know that it suffices, for each domain D n in the direct limit D 1 = lim D n (that gives the solution of (5) to put into a single bag the types that represent its compacts, and then to take the space of filters. Now D 0 is the one element domain, hence no basic type constant but the ....

K. Larsen, G. Winskel, "Using Information Systems to Solve Recursive Domain Equations Effectively", LNCS 173, Springer-Verlag, Berlin, 1984, 109-130. Alessi, Dezani and de' Liguoro/A Convex Powerdomain over Lattices: its Logic and -Calculus 57

....it should be clear that all domains, parametrizations, and continuous functors considered are in some fairly obvious sense e ective. This could be made precise by either introducing numberings for the compact elements of the domains [3, 5, 6] or by representing domains by information systems [9, 16, 20]. In [3] and [20] e ective versions of the density theorem and the continuous choice principle for functionals of nite types are proved. In the following we assume that all domains and parametrizations considered are e ective. For a domain D we let D e denote the set of elements x 2 D such ....

K. G. Larsen, G. Winskel. Using Information Systems to Solve Recursive Domain Equations Eectively. Proceedings of the Conference on Abstract Datatypes, Sophia-Antipolis, France, Lecture Notes in Computer Science, 173:109-129, Springer, 1984.

....on domains by an increasing union of webs. Not only do we get a presentation of the models which is now much more easy to handle, even in the case, but it is now almost trivial to deal with limit ordinal stages. This general method of solving recursive equations on domains is presented in [25] for Scott domains; there the webs are Scott s information systems [36] The most simple classes of webbed domains where one can solve recursive equations are Girard s coherent spaces [16] and Krivine s spaces of initial segments [23, 24] In the rst case the webs are of the shape (D; where ....

....2 ug for u 2 S(C C) v 2 S(C) It is straightforward to prove that Tr and A 0 are inverse order isomorphisms, and hence isomorphisms between the two domains. Hint: 7 points to check) Formally the proof of 8.2.1 does not di er from the case. For the semantics this is worked out in [25] in a slightly di erent formalism, in [24] for pcs s with trivial coherence; 16] is not really relevant for pcs s with trivial preorders since it deals with the more accurate class of stable functions. Remark. A corollary of Lemma 8.2.1 is that the problem of solving D = D D] in the ....

K.G Larsen and G. Winskel. Using information systems to solve recursive domain equations. In Lecture Notes in Computer Science, volume 173 (Semantics of data types), pages 109-130. Springer-Verlag, 1984.

....is the least typical and Example 1.3 the most typical as illustrations of what is usually done with information systems, in the way of constructing domains of computation. Several variations on the de nition of an information system are possible. For example, one may, like Larsen and Winskel in [LW84], dispense with (in Example 1.1 above we did not bother to specify ) A more signi cant technical simpli cation can be achieved by requiring, in e ect, that every compact element of a domain be represented by a token. We arrive at the (inhabited) propositional languages of Fourman and Grayson ....

....representing (species of) stable domains; see Section 7 below. These remarks may tend to suggest that the theory of information systems should be assimilated to that of Stone like dualities. However this overlooks the most distinctive feature of information systems, as expounded already in [Sco82, LW84] and in the context of the presentation of domains for sequentiality by concrete data structures in [BC81] We refer to the idea that the collection of all information systems (of a given variety) itself has an information ordering . The importance of this is that domain equations, of the form ....

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K. G. Larsen and G. Winskel. Using information systems to solve recursive domain equations eectively. In D. B. MacQueen G. Kahn and G. Plotkin, editors, Semantics of Data Types, pages 109-130, Berlin, 1984. Springer-Verlag. Lecture Notes in Computer Science Vol. 173.

....whether or not they have other nice domain theoretic properties. One of the desirable properties is the existence of a universal (or even saturated) domain [4] in a certain category. The other related property to have is a framework for solving domain equations by fixed point construction, as in [5, 8]. It is the purpose of the paper to establish these results for quasiprime generated information systems which represent quasi prime algebraic domains. One of the most useful results on universal domains is given in the work of Droste and Gobel [1] who introduced the Fraiss e J onsson theorem in ....

....the result of Droste and Gobel for showing the existence of a saturated (universal, homogeneous) quasi prime algebraic domain. Our main definition here is the notion of q embeddings for quasi prime algebraic domains. The appropriate notion of embeddings for Scott domains (call them s embeddings) [5, 2] and for dI domains (call them r embeddings r for rigid) 8] are well known . However, none of the these embeddings works for quasi prime algebraic domains, for the following reasons: ffl The s embeddings are too general: under this embedding the colimit of an chain of finite Scott domains ....

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Larsen, K. and Winskel, G. Using information systems to solve recursive domain equations effectively. In Lecture Notes in Computer Science 173, 1984.

.... calculus it is not always the case that it will give rise to a square model, but it is often the case 3 (see Section 5. 1) Roughly speaking prime webs are to (binary) prime algebraic Scott domains what Scott s information systems [32] are to Scott domains in the continuous semantics (cf. also [20]) and prime event structures to DI domains in the stable semantics (see for ex. 10] but, categorically speaking, the situation is less neat here (remark 2 in section 3.2) Interest of the present class. The rst interest of the class lies in the simplicity of the interpretations of terms in its ....

K.G. Larsen and G. Winskel, Using information systems to solve recursive domain equations, In Lecture Notes in Computer Science, vol. 173 (Semantics of data types), p. 109-130, Spriger-Verlag, 1984.

.... S ffl j (1; any) any; 0) S ffl ; S ( 1; any) finite; 0) S ( finite; any) a configuration structure is axiomatisable by sequents of the form (finite; 1) iff it is closed under nonempty bounded intersections and directed unions; this is essentially due to Larsen and Winskel [10] as axiomatisations of the form (finite; 1) correspond to Scott information systems. The entries with ( s indicate that only the implication from right to left holds; a counterexample is provided by the collection of all co finite proper subsets of the natural numbers. A minor, but useful, ....

K.G. Larsen & G. Winskel (1991): Using information systems to solve recursive domain equations. Information and Computation 91(2), pp. 232--258.

....systems. It is straightforward to show the resulting structure forms a category that is closed under lifting, products, sums, function space and the upper and lower powerdomains, all defined in a manner exactly parallel to the definitions for information systems. Furthermore, the techniques in [LW84] for constructing a large cpo of information systems and solving recursive domain equations via a least fixed point calculation, carry over exactly into this richer setting. 4.2 Constraints over an information system We now show how to generate certain kinds of constraint systems from an ....

Kim G. Larsen and Glynn Winskel. Using information systems to solve recursive domain equations effectively. In G. Kahn, D.B.MacQueen, and G.Plotkin, editors, Semantics of Data Types, International Symposium, Sophia Antipolis, LNCS 173. Springer Verlag, 1984.

....to provide a new kind of object, along with related operations on them. In this setting, the constructor and accessor operations are replaced with primitive constraints. In this chapter, we formalize the constraint system as a variant of Dana Scott s information system (see [31] and more recently [42]) We will begin by informally describing the information system, using this as an intuitive basis for the formal description of a constraint system. After this, we shall describe the various constraint systems found as datatypes in Jo. Finally, we shall provide a many sorted constraint system ....

....not contained in any other element. It is not the purpose of this exposition to elaborate on this topic further, because we are proceeding to discuss a very similar structure (the constraint system) in more detail. Interested readers are referred to the papers by Scott [31] and Winskel and Larsen [42]. 2.1.2 Constraint Systems The constraint system is similar to the information system, except that we shift our emphasis away from computing elements of a domain towards computing the solution of constraint net12 works. Remember that a constraint network is a graph of variables connected by ....

G. Winskel and K. G. Larsen. Using information systems to solve recursive domain equations effectively. Technical Report 51, University of Cambridge Computer Laboratory, 198?

....attention to the fundamental questions Kreisel originally was interested in. Obviously this work owes much to other sources. In particular I have made use of work by Scott [Sco82] whose notion of an information system is taken as a basis to introduce domains) Roscoe [Ros87] Larsen and Winskel [LW84] and Berger [Ber93] The paper is organized as follows. Section 1 treats information systems, and in section 2 it is shown that the partial orders defined by them are exactly the (Scott) domains with countable basis. Section 3 gives a characterization of the continuous functions between domains, ....

K.G. Larsen and G. Winskel. Using information systems to solve recursive domain equations effectively. In Proceedings of the Conference on Abstract Datatypes, Sophia--Antipolis, France, pages 109--129, Berlin, Heidelberg, New York, 1984. Springer. Lecture Notes in Computer Science, Vol. 173.

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K. G. Larsen and G. Winskel. Using information systems to solve recursive domain equations e#ectively. In Proc. Semantics of Data Types, 1984, LNCS 173.

....T 1 , T k may contain the T j s. We shall write # T . T as an abbreviation for the k tuple with j component j T . T, and confuse a closed expression for a path order with the path order itself. Simultaneous recursive equations for path orders can be solved using information systems [21, 12]. Here, it will be convenient to give a concrete, inductive characterisation based on a language of paths : p, q : P abs p . 16) Above, P ranges over finite sets of paths. We use P q as notation for pairs in the function space ( P) Q. The language is complemented by formation ....

....judgements p P p # . Recall that P means P.#p # P # . p P p # . P : P q : Q q : P p : P # # #p : ###AP# p 1 : P pn : P p : T j [ # T . # abs p : j T . P P q #Q P##Q P # ####AP# P P # abs p Using information systems as in [12] yields the same representation, except for the tagging with abs in recursive types, done to help in the proof of adequacy in Sect. 4.1. So rather than the straight equality between a recursive type and its unfolding which we are used to from [12] we get an isomorphism abs : T j [ T . j ....

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K. G. Larsen and G. Winskel. Using information systems to solve recursive domain equations e#ectively. In Proc. Semantics of Data Types, 1984, LNCS 173.

....variables other than x, we need to make use of the strength map given by (16) see Proposition 3.5 below. t : Q # u : P t] Q # P (# P) t (33) Recursive type definitions. Simultaneous recursive equations for path orders can be solved using information systems [47, 30]. Here, it will be convenient to give a concrete, inductive characterisation based on a language of paths: abs p . 34) Above, P ranges over finite sets of paths. We use P q as notation for pairs in the function space P Q. The language is complemented by formation rules using 10 P ....

....p : T j [ # T . Figure 1: Paths judgements p : P, meaning that p is an element of P, displayed in Fig. 1 alongside rules defining the ordering on P using judgements p p # . We remind the reader that P P # means P.#p # P # . p p # . Using information systems as in [30] yields the same representation, except for the tagging with abs in recursive types, done to help in the proof of adequacy in Sect. 3.4.1. So rather than the straight equality between a recursive type and its unfolding which we are used to from [30] we get an isomorphism abs : T j [ T . ....

[Article contains additional citation context not shown here]

K. G. Larsen and G. Winskel. Using information systems to solve recursive domain equations e#ectively. In Proc. Semantics of Data Types, 1984, LNCS 173.

....#P # a a C # a a F C = P P F = P # P . We can solve such recursive equations for path orders by several techniques, ranging from the sophisticated method of [7] providing inductive and coinductive characterisations, to simple methods essentially based on inductive definitions. Paralleling [14], path orders under the order Q and (#p, p # P. p P p # p # ) form a (large) cpo with respect to which all the constructions we have just seen can be made Scott continuous. Solutions to equations like those above are then obtained as (simultaneous) least fixed points. 4. An ....

K. G. Larsen and G. Winskel. Using information systems to solve recursive domain equations effectively. LNCS 173, 1984.

....(in F) It is our choice of path for abstractions which narrows us to a linear process passing language, one where the input process can be run at most once to yield a single (computation) path. Fortunately the simple technique for solving recursive domain equations via information systems in [14] suces to solve such equations. A path order can be regarded as an information system in which every nite subset of is consistent and in which the entailment relation is given by the partial order of so fp g p i p p . Path orders under the order PE Q ( P Q (8p; p ....

G.Winskel and K.Larsen. Using information systems to solve recursive domain equations eectively. LNCS 173, 1984.

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Larsen, K., Winskel, G.: Using information systems to solve recursive domain equations. Information and Computation, Vol. 91(2):232--258, 1991.

No context found.

K. G. Larsen and G. Winskel. Using information systems to solve recursive domain equations e#ectively. In Proc. Semantics of Data Types, 1984, LNCS 173.

No context found.

K.G.Larsen, G.Winskell, "Using Information Systems to Solve Recursive Domain Equations Effectively", LNCS 173, Springer-Verlag, Berlin 1984, 109-130.

No context found.

K. G. Larsen and G. Winskel. Using information systems to solve recursive domain equations e#ectively. In Proc. Semantics of Data Types, 1984, LNCS 173.

No context found.

K.G. Larsen and G. Winskel, Using information systems to solve recursive domain equations eectively, in: Mathematical Foundations of Programming Language Semantics, Lecture Notes in Comp. Sci. 173(1984).

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K. Larsen, G. Winskel, "Using Information Systems to Solve Recursive Domain Equations Effectively", LNCS 173, Springer-Verlag, Berlin, 1984, 109-130.

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