S. Vickers. Information systems for continuous posets. Theoretical Computer Science, 114:201-229, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the opens of X. For abstract valuations coming from non empty compact subsets, these two classes of subbasic opens are familiar: K 2 2U iff K U , and K 2 U iff K U 6= Furthermore, for every X, the two modalities 2 and satisfy the usual properties known from the Vietoris power locale [7,2]: Proposition 3.1 2 and are Scott continuous. 2; 2X = X = PVX (U [ V ) U [ V 2(U V ) 2U 2V U [ 2(U [ V ) U [ 2V 2U (U V ) 2U V These properties immediately follow from the defining properties of abstract valuations. 3.2 Sobriety of PVX Theorem 3.2 For every ....

....s X = s Y ffi f holds. 4.3 The Multiplication of the Monad Now, we look for a family of maps mX : PV (PVX) PVX. In the following, the index X is often omitted. Given fi in PV (PVX) and an open U of X, we have to define mfi(U) as an element of A. From the analogy with the Vietoris power locale [7], we expect that the inverse image function m Gamma satisfies the equations m Gamma (2U) 22U and m Gamma (U) U . Hence, mfi(U) 1 iff fi is in m Gamma (2U) 22U iff fi(2U) 1. Since for a in A, a = 1 implies a 6= 0, 2U is a subset of U . By monotonicity of fi, fi(2U) 1 ....

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S. Vickers. Information systems for continuous posets. Theoretical Computer Science, 116(2):201--229, 1993.

....satisfy reflexivity and this makes it possible to have different logical systems on the left and right of the turnstile. Direct descriptions of powerlocales in terms of presentational schemes have been carried through in a number of contexts: for algebraic dcpos in [Plo83] continuous dcpos in [Vic93], completions of quasimetric spaces in [Vic97] and for strongly algebraic (SFP) domains in [Abr91] The second author presented part of this work to the logic group in Padova and wants to thank the audience for their interesting comments. Steve Vickers gave also detailed comments that helped in ....

S. Vickers. Information systems for continuous posets. Theoretical Computer Science, 114:201--229, 1993.

....those results which, in non categorical form, first appeared in [12] We should also mention the work of Guitart and Riguet, 5] who have proved a constructive, but non ord enriched, version of Theorem 17 using methods quite different from ours. Vickers too has proved a version of our Theorem 17 [18]. The results here are fundamentally indebted to Eilenberg and Mac Lane, for without the language of categories Proposition 11, which separates our contributions from those of Raney, is difficult to state succinctly and, more importantly, difficult to even conjecture. Finally we thank the referee ....

....X A such that R Delta = R = Delta R. Thus we have (9y) aRy x) if and only if aRx if and only if (9b) a bRx) Since rel is a 2 category, in fact an ord category, so is kar(rel) transformations being containments of arrows. This observation is central to our considerations. Vickers [18] uses the term infosys for an object of kar(rel) and we adopt this name. We refer to the arrows as modules. For Vickers they are the lower approximable semimappings. If X and A are orders then a module is an (order )ideal as studied in [3] In the present context the conditions on R : X A ....

S. Vickers. Information systems for continuous posets. Theoretical Computer Science B, 1993.

....des Saarlandes, Postfach 151150 D 66041 Saarbrucken, Germany e mail: heckmann cs.uni sb.de Abstract The probabilistic power domain construction of Jones and Plotkin [6, 7] is defined by a construction on dcpo s. We present alternative definitions in terms of information systems a la Vickers [12], and in terms of locales. On continuous domains, all three definitions coincide. 1 Introduction To model probabilistic and randomized algorithms in the semantic framework of dcpo s and Scott continuous functions, Jones and Plotkin introduce in [6, 7] the probabilistic power domain construction ....

....the information systems of [9] for the class of bounded complete algebraic domains (Scott Domains) The information systems of [3, 4] present bounded complete continuous domains, but this is still not sufficient, since PD does not preserve bounded completeness as shown in [6, Section 4. 5] In [12], Vickers introduced a kind of information systems (infosyses) suitable to cover all continuous domains. In the paper at hand, we show that PD can be described in terms of these infosyses. This was already conjectured by Vickers at the end of his paper. We also looked for a localic description of ....

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S. Vickers. Information systems for continuous posets. Theoretical Computer Science, 116(2):201--229, 1993.

....domains as simply as possible in terms of densely ordered sets of tokens. This has been done in [Smy77] using R structures (an R structure is a transitively ordered set of tokens such that the predecessors of each token from a directed set) The idea has recently been taken up by Vickers [Vic93], who works with an even simpler notion of Infosys (or densely ordered set) as well as with R structures. As far as concerns the topics of the present paper, Smy77] addresses the effective solution of domain equations via sequences of embeddings of R structures (but does not have an explicit ....

....of Infosys (or densely ordered set) as well as with R structures. As far as concerns the topics of the present paper, Smy77] addresses the effective solution of domain equations via sequences of embeddings of R structures (but does not have an explicit cpo of R structures) the applications in [Vic93], on the other hand, have little bearing on the present paper. 5.3 Information systems for Boolean algebras Consider the category of classical propositional logics, CPL, whose objects are given by tuples (A; 0; 1) where is a pre order, and are binary operations, is a unary ....

S. J. Vickers. Information systems for continuous posets. Theoretical Computer Science, 114:201--229, 1993.

....Power Domains, Information Systems, and Locales [Hec94c] The probabilistic power domain construction PD of Jones and Plotkin [Jon90, JP89] is defined by a construction on dcpo s. In the summarized paper, we present alternative definitions in terms of information systems a la Vickers [Vic93], and in terms of locales. On continuous domains, all three definitions coincide. The subsections of this section correspond to the sections of the summarized paper. 6.1 Introduction In [Jon90, JP89] it is shown that PD preserves ( continuity of dcpo s. On the other hand, it does not preserve ....

....systems of [Sco82] for the class of bounded complete algebraic domains (Scott Domains) The information systems of [Hoo90, Hoo92] present bounded complete continuous domains, but this is still not sufficient, since PD does not preserve bounded completeness as shown in [Jon90, Section 4. 5] In [Vic93], Vickers introduced a kind of information systems (infosyses) suitable to cover all continuous domains. In the summarized paper, we show that PD can be described in terms of these infosyses. This was already conjectured by Vickers. We also looked for a localic description of the power ....

[Article contains additional citation context not shown here]

S. Vickers. Information systems for continuous posets. Theoretical Computer Science, 116(2):201--229, 1993.

....subset can be viewed as a proposition about, or a property of, a program. A comprehensive analysis of the underlying logic for the cartesian closed category of the so called bi finite domains was developed by Abramsky [1] Various other categories of domains have also been studied in logical form [44, 119, 72]. Several cartesian closed categories of algebraic domains, including the so called Scott domains, have been employed in the semantics of computation. They are used to obtain a non trivial model of the untyped calculus [5] based on a domain isomorphic to its own function space [106, 107] They ....

S. J. Vickers. Information systems for continuous posets. Theoretical Computer Science, 114:201--229, 1993.

.... we have W[0, Fr [0,q) qQ ) 0,q) # q q [0,q ) The symbolic generator [0,q) corresponds to q R x , and the relations to the property of being rounded (the direction of the relation) and upper ( This is the rounded ideal completion of the continuous information system ([17], where it is called an infosys ) whose tokens are the extended positive rationals ordered by (with q) in other words, in the infosys sense is numerical . Hence it is a continuous dcpo (i.e. its frame is constructively completely distributive) Classically, it is the space whose ....

....the module law: for X xy M y d iff there is some e for which B d (x) B e (y) M. We thus see that left X modules are equivalent to subsets of XQ satisfying (i) Then (iii) holds iff inf x M x = 0, and (ii) is equivalent to 4.5 (iv) This presentation leads to a useful embedding. Recall [17] that a continuous information system is a set D (of tokens) with a transitive, interpolative order (so o = Then its rounded ideal completion RIdl(D) is a continuous dcpo. Now the order just defined on XQ is transitive and interpolative. Definition 4.9 The ball domain Ball(X) is the ....

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S.J. Vickers, Information systems for continuous posets, Theoretical Computer Science 114 (1993) 201--229.

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S. Vickers. Information systems for continuous posets. Theoretical Computer Science, 114:201-229, 1993.

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