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Domains for Computation in Mathematics, Physics and Exact Real Arithmetic
 Bulletin of Symbolic Logic
, 1997
"... We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability dist ..."
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We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chao...
Information Systems for Continuous Posets
, 1993
"... The method of information systems is extended from algebraic posets to continuous posets by taking a set of tokens with an ordering that is transitive and interpolative but not necessarily reflexive. This develops results of Raney on completely distributive lattices and of Hoofman on continuous S ..."
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Cited by 15 (3 self)
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The method of information systems is extended from algebraic posets to continuous posets by taking a set of tokens with an ordering that is transitive and interpolative but not necessarily reflexive. This develops results of Raney on completely distributive lattices and of Hoofman on continuous Scott domains, and also generalizes Smyth's "Rstructures". Various constructions on continuous posets have neat descriptions in terms of these continuous information systems; here we describe HoffmannLawson duality (which could not be done easily with Rstructures) and Vietoris power locales. 2 We also use the method to give a partial answer to a question of Johnstone's: in the context of continuous posets, Vietoris algebras are the same as localic semilattices.
Probabilistic Power Domains, Information Systems, and Locales
 Mathematical Foundations of Programming Semantics VIII, volume 802 of Lecture Notes in Computer Science
, 1994
"... 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 ..."
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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 PD . It forms a computational monad in the sense of [8] in the category of dcpo's and continuous functions and various of its subcategories of `domains'. Every probabilistic powerdomain PDX is equipped with a family of binary operations + p indexed by a real number p between 0 and 1 such that A+ p B denotes the result of choosing A with probability p and B with probability 1 \Gamma p. Other applications of PD were found in [1]. The probabilistic powerdomain of the upper power space [10] of a second ...
ICategories as a framework for solving domain equations
, 1993
"... An abstract notion of category of information systems or Icategory is introduced as a generalisation of Scott's wellknown category of information systems. As in the theory of partial orders, Icategories can be complete or !algebraic, and it is shown that !algebraic Icategories can be obt ..."
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Cited by 7 (1 self)
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An abstract notion of category of information systems or Icategory is introduced as a generalisation of Scott's wellknown category of information systems. As in the theory of partial orders, Icategories can be complete or !algebraic, and it is shown that !algebraic Icategories can be obtained from a certain completion of countable Icategories. The proposed axioms for a complete Icategory introduce a global partial order on the morphisms of the category, making them a cpo. An initial algebra theorem for a class of functors continuous on the cpo of morphisms is proved, thus giving canonical solution of domain equations; an effective version of these results for !algebraic Icategories is also provided. Some basic examples of Icategories representing the categories of sets, Boolean algebras, Scott domains and continuous Scott domains are constructed. 1 Introduction A distinctive feature of information systems representing Scott domains, as expressed in [Sco82, LW84], is that th...
Information Systems Revisited: The General Continuous Case∗
"... In this paper a new notion of continuous information system is introduced. It is shown that the information systems of this kind generate exactly the continuous domains. The new information systems are of the same logicoriented style as the information systems first introduced by Scott in 1982: the ..."
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In this paper a new notion of continuous information system is introduced. It is shown that the information systems of this kind generate exactly the continuous domains. The new information systems are of the same logicoriented style as the information systems first introduced by Scott in 1982: they consist of a set of tokens, a consistency predicate and an entailment relation satisfying a set of natural axioms. It is shown that continuous information systems are closely related to abstract bases. bases and/or continuous domains are equivalent, it follows that the category of continuous information systems is also equivalent to that of continuous domains. In applications mostly subclasses of continuous domains are considered. The domains have e.g. to be pointed, algebraic, boundedcomplete or FS. Conditions are presented that when fulfilled by an continuous information system force the generated domain to belong to the required subclass. In most cases the requirements are not only sufficient but also necessary. 1
Semantics of Binary Choice Constructs
"... This paper is a summary of the following six publications: (1) Stable Power Domains [Hec94d] (2) Product Operations in Strong Monads [Hec93b] (3) Power Domains Supporting Recursion and Failure [Hec92] (4) Lower Bag Domains [Hec94a] (5) Probabilistic Domains [Hec94b] (6) Probabilistic Power Domains, ..."
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This paper is a summary of the following six publications: (1) Stable Power Domains [Hec94d] (2) Product Operations in Strong Monads [Hec93b] (3) Power Domains Supporting Recursion and Failure [Hec92] (4) Lower Bag Domains [Hec94a] (5) Probabilistic Domains [Hec94b] (6) Probabilistic Power Domains, Information Systems, and Locales [Hec94c] After a general introduction in Section 0, the main results of these six publications are summarized in Sections 1 through 6.
Continuous DCPOs as complete information systems
, 2002
"... Continuous DCPOs are shown to be precisely the complete objects (in the sense of the completions of Brummer, Giuli and Herrlich, [2], [1]) in a certain category of generalized information systems (related to those of Vickers, [9]), with naturally de ned morphisms and embeddings. ..."
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Continuous DCPOs are shown to be precisely the complete objects (in the sense of the completions of Brummer, Giuli and Herrlich, [2], [1]) in a certain category of generalized information systems (related to those of Vickers, [9]), with naturally de ned morphisms and embeddings.
semifunCtors
, 1990
"... Girard categories (GC's) were defined in [14] as categorical models for linear logic. It was shown that the Kleisli category of a GC is Cartesian closed. ..."
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Girard categories (GC's) were defined in [14] as categorical models for linear logic. It was shown that the Kleisli category of a GC is Cartesian closed.
Logic of Fixed Points and Scott Topology∗
"... Abstract. Domains with Scott topologies are frequently defined as sets of fixed points of retractions or, more generally, arbitrary Scott continuous transformations of domains. For the case of retractions, the logical approach to this technique was developed by Hoofman under the name of continuous ..."
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Abstract. Domains with Scott topologies are frequently defined as sets of fixed points of retractions or, more generally, arbitrary Scott continuous transformations of domains. For the case of retractions, the logical approach to this technique was developed by Hoofman under the name of continuous information systems. We present this approach for general Scott continuous transformations, together with its underlying intuition, a number of applications to closure operations and finitary retractions in the algebraic case, and a spectrum of outstanding open problems. 1
Systems
"... A logicoriented representation of Ldomains in the style of Scott’s information sys ..."
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A logicoriented representation of Ldomains in the style of Scott’s information sys