D. S. Scott, "Some ordered sets in computer science", In: I. Rival (ed.), Ordered Sets, 1982, pp.677--718.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....topology of the models of the continuous, stable and strongly stable semantics is connected) 1 Introduction Many familiar models of the type free lambda calculus are constructed by order theoretic methods. Computational motivations and intuitions justified Scott s view of models (see [41] [42] as partially ordered sets and of functions as monotonic functions over these sets. After Scott, a large number of mathematical models for the lambda calculus, arising from syntax free constructions, have been introduced in various categories of domains (see [1] 45] 1 Email: ....

Scott D.S., Some ordered sets in computer science, In: Ordered sets (I. Rival ed.), Proc. of the NATO Advanced Study Institute (Banff, Canada), Reidel (1981), 677--718.

....is a very rich and complex structure (see e.g. 3, 10, 24, 25, 49] syntactical techniques are usually difficult to use in the study of lambda theories. Therefore, semantic methods have been extensively investigated. Computational motivations and intuitions justify Scott s view of models (see [44, 45]) as partially ordered sets (sets of observations or informations) and of computable functions as monotonic functions over these sets. After Scott, mathematical models of lambda calculus in various categories of domains (see [1, 48] were classified into semantics according to the nature of their ....

D.S. Scott, "Some ordered sets in computer science", In: Ordered sets (I. Rival ed.), Proc. of the NATO Advanced Study Institute (Banff, Canada), Reidel (1981), pp. 677-718

....structures we return to in Section 7.2. The basic idea behind Ginsberg s bilattices is to label sentences or rather our degrees of belief in sentences as more than simply true or false. Ginsberg mentions previous attempts in this direction. For instance, Dana Scott considers in [Sco82] situations where statements can be partially ordered by their truth and falsity let such a truth ordering be denoted by t . Sandewall introduces in [San85] ordering on statements based on the amount of information they contain 3 let such an information ordering be denoted by k . One can ....

....is needed to clarify these issues. Regarding the mathematical tools employed in the thesis, there is a place for improvement, too. For instance, lattices were mainly used, but in several cases a top element was added to the structure to make it a lattice some alternatives are provided in Scott [Sco82] Related mathematical structures include intersection structures, domains and information systems, all thoroughly discussed in [DP90] The logical framework itself also deserves more attention. It is claimed that the logic is exactly what was needed for instance, negation is not introduced ....

Dana S. Scott. Some ordered sets in computer science. In I. Rival, editor, Ordered Sets, pages 677--718. D. Reidel Publishing Company, 1982.

....recursive, then the function (2 ) x ly.f (x,y) f s (x) has to be recursive, i.e. the index of ly.f (x,y) depends uniformly effectively on x or, also, s exists as a recursive function. In familiar CCC s for denotational semantics, say, where morphisms are continuous functions (see Scott (1976, 1982), Plotkin (1978) 1) requires that a function is continuous iff it is so in each argument in the uniform sense of (2) Note also that the Fact above gives a characterization of acceptable G del numberings of PR, as defined in Rogers(1967) Consider first a function g: w PR and set L 1 ....

Scott, D. (1982) "Some ordered sets in Computer Science," in Ordered Sets (Rival Ed.), 22 Reidel.

.... x 2 8 = x by finding a solution for it over, say, the structure of the the complex numbers. In Eff one may find solutions to all relevant domain equations as follows. Consider a constructive version of Scott domains (essentially, the computable substructures of the effectively given domains in Scott[1982]) When taking continuous and computable maps as morphisms, they form a Cartesian Closed Category, which can be fully and faithfully embedded in Eff (when Eff is constructed out of Kleene s w ) In this way the limit constructions, needed to solve the equations, can be carried on within Eff. The ....

Scott, D. [1982] "Some ordered sets in Computer Science," in Ordered Sets (Rival Ed.), Reidel.

....structures we return to in Section 7.2. The basic idea behind Ginsberg s bilattices is to label sentences or rather our degrees of belief in sentences as more than simply true or false. Ginsberg mentions previous attempts in this direction. For instance, Dana Scott considers in [Sco82] situations where statements can be partially ordered 2 As said in [Gan96a] the two approaches were developed independently, and have not been merged, yet. 14 CHAPTER 2. BACKGROUND by their truth and falsity let such a truth ordering be denoted by t . Sandewall introduces in [San85] ....

....issues. Regarding the mathematical tools employed in the thesis, there is a place for improvement, too. For instance, lattices were mainly used, but in several cases a top element was added to the structure to make it a lattice some 124 CHAPTER 8. CONCLUSION alternatives are provided in Scott [Sco82]. Related mathematical structures include intersection structures, domains and information systems, all thoroughly discussed in [DP90] The logical framework itself also deserves more attention. It is claimed that the logic is exactly what was needed for instance, negation is not introduced ....

Dana S. Scott. Some ordered sets in computer science. In I. Rival, editor, Ordered Sets, pages 677--718. D. Reidel Publishing Company, 1982.

....than simply true or false is not a new one. There is an extensive literature discussing the ramifications of choosing the truth value assigned to a given statement from a continuum of possibilities instead of simply the two point set ft; fg. Typical examples are a suggestion of Scott s in 1982 [47] and one of Sandewall s in 1985 [46] 1 Scott notices that we can partially order statements by their truth or falsity, and looks at this as corresponding to an assignment to these statements of truth values chosen from some set L that is partially ordered by some relation t (the reason for the ....

D. S. Scott. Some ordered sets in computer science. In I. Rival, editor, Ordered Sets, pages 677--718. D. Reidel Publishing Company, Boston, 1982.

....p is the projection and q is the embedding. We abbreviate by writing hp; qi : E pe Gamma D. In this section we look at the relationship between normal substructures of pre orders and pe pairs from the point of view of approximable relations. We thereby generalize the theory exposited in [23] to the category of algebraic dcpo s. These results will be used to derive a universal domain technique for the Plotkin orders. Let A and B be pre orders. Write A B if there is an A 0 B such that A = A 0 . 3 As far as the author knows, this terminology was first used in [27] ....

....in the sense that every coherent algebraic dcpo is a retract of T . Since T is itself algebraic and coherent this provided a universal domain for a class of algebraic dcpo s that included the algebraic lattices but also contained certain desirable dcpo s without tops. In [22] 24] and [23], a third universal domain U is discussed. Although U is harder to understand than P or T it has the advantage of having every bounded complete algebraic dcpo as a projection (not just as a retract) There are instances in which a retraction universal domain does not have all of the ....

[Article contains additional citation context not shown here]

Scott, D. S., Some ordered sets in computer science. In: Ordered Sets, edited by I. Rival. D. Reidel Publishing Company, 1981, pp. 677--718.

....For example, it was desirable to have a class of domains which included such structures as the partial functions from natural numbers to natural numbers which under their usual ordering do not form a lattice. Such theories were proposed by Plotkin [Plo78] Berry [Ber78] and also Scott [Sco81, Sco82a, Sco82b]. Journal of Pure and Applied Algebra. vol. 63 (1990) pp. 49 66. An extended outline of this paper appeared in: Logic in Computer Science, edited by Y. Gurevich, pages 309 319, IEEE Computer Society Press, July 1988. y Department of Computer and Information Sciences, Philadelphia, PA 19104 ....

....universal domains so constructed are characterized by a property very similar to what model theories call countable saturation [CK73] We prove that a model with this property is unique up to isomorphism. We can apply this result to show that Scott s universal domain for the consistent completes [Sco81, Sco82a, Sco82b] is not saturated. The paper is divided into six sections which we overview briefly. Section two provides some definitions and establishes notation. A few basic propositions are also remarked. The third section discusses the coherence condition on the topology of a domain. We show how this ....

[Article contains additional citation context not shown here]

D. S. Scott. Some ordered sets in computer science. In I. Rival, editor, Ordered Sets, pages 677--718, D. Reidel, 1981.

....and only if, p id, p ffi p = p, and the image of p is a domain with respect to the restriction of the order on D. It is known that the partial order of finitary projections (with respect to the extensional ordering) is a domain, that we shall write Fp, and that this domain is embedded in [D D] [Sco81]. We obtain an embedding projection pair ( Phi 0 ; Psi 0 ) from Fp into D, from the composition of this embedding projection from Fp into [D D] with ( Phi; Psi) We now take for the interpretation of the set U the image of Phi 0 , which we again call U . This should cause no real confusion. ....

D. S. Scott. Some ordered sets in computer science. In I. Rival, editor, Ordered Sets, pages 677--718, D. Reidel, 1981.

....Lemma 28 The following operators are representable over U: ffi , Theta, Omega ; Phi; Delta) Delta) Delta) This means that we have solutions over the bounded complete domains for a quite substantial class of recursive equations. More discussion of U may be found in [Sco81] Sco82a] and [Sco82b] 42 Carl A. Gunter and Dana S. Scott 7.4 Representing operators on bifinite domains. The convex powerdomain ( Delta) cannot be representable over U because it does not preserve bounded completeness. We construct a domain over which this operator can be represented ....

D. S. Scott. Some ordered sets in computer science. In I. Rival, editor, Ordered Sets, pages 677--718. D. Reidel, 1981.

No context found.

D. S. Scott, "Some ordered sets in computer science", In: I. Rival (ed.), Ordered Sets, 1982, pp.677--718.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC