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The Largest Cartesian Closed Category of Stable Domains
 Theoretical Computer Science
"... This paper shows that Axiom d and Axiom I are important when one works within the realm of Scottdomains. In particular, it has been shown that (i) if [D ! s D] has a countable basis, then D must be finitary, for any Scottdomain D; (ii) if [D ! s D] is bounded complete, then D must be distributive, ..."
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This paper shows that Axiom d and Axiom I are important when one works within the realm of Scottdomains. In particular, it has been shown that (i) if [D ! s D] has a countable basis, then D must be finitary, for any Scottdomain D; (ii) if [D ! s D] is bounded complete, then D must be distributive, for any finitary Scottdomain D. Therefore, the category of dIdomains is the largest cartesian closed category within omegaalgebraic, bounded complete domains, with the exponential being the stable function space. 1 Introduction Among Scott's many insights which shaped the whole area of domain theory, one is that the partial ordering of a domain should be interpreted as the ordering about information. "Thus," wrote Scott [16], "x v y means that
Factorisation Systems on Domains
, 1996
"... We present a cartesian closed category of continuous domains containing the classical examples of Scottdomains with continuous functions and Berry's dIdomains with stable functions as full cartesian closed subcategories. Furthermore, the category is closed with respect to bilimits and there is an ..."
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We present a cartesian closed category of continuous domains containing the classical examples of Scottdomains with continuous functions and Berry's dIdomains with stable functions as full cartesian closed subcategories. Furthermore, the category is closed with respect to bilimits and there is an algebraic and a generalised topological description of its morphisms. 1 Introduction There are two kinds of morphism that are studied in classical domain theory, Scottcontinuous ones and stable ones. Berry introduced stable maps to model sequentiality in the calculus [Ber78]. A stable map, in addition to being continuous (i.e. preserving directed suprema), also preserves bounded binary infima. So, for first order the stable functions are a subset of the continuous ones, but at higher order types the continuous and the stable function space become incomparable. Stability captures sequentiality to some extend, e.g. POR is not a stable function, yet the stable model of PCF fails to be fully...
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. 0 Introduction In this section, we provide a common framework for the summarized papers. In Subsection 0.1, Moggi's approach to specify denotational semantics by means of strong monads is introduced. In Subsection 0.2, we specialize this approach to languages with a binary choice construct. Strong monads can be obtained in at least two ways: as free constructions w.r.t. algebraic theories (Subsection 0.3), and by using second order functions (Subsection 0.4). Finally, formal definitions of those concepts which are used in all...
The Largest Cartesian Closed Category of Domains, Considered Constructively
, 2000
"... A conjecture of Smyth [10] is discussed which says that if D and [D # D] are effectively algebraic directedcomplete partial orders with least element (cpo's), then D is an e#ectively strongly algebraic cpo, where it was left open what exactly is meant by an effectively algebraic and an e#ectively ..."
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A conjecture of Smyth [10] is discussed which says that if D and [D # D] are effectively algebraic directedcomplete partial orders with least element (cpo's), then D is an e#ectively strongly algebraic cpo, where it was left open what exactly is meant by an effectively algebraic and an e#ectively strongly algebraic cpo. First, notions of an e#ectively strongly algebraic cpo and an e#ective SFP object are introduced. The effective SFP objects are just the constructive (computable) objects in the effectively given category [9] of indexed # algebraic cpo's. Theorem Every effective SFP object is an effectively strongly algebraic cpo, and vice versa. Moreover, this equivalence holds effectively. This shows that the given notion of an effective SFP object is stable. In e#ectivity considerations of # algebraic cpo's it is usual to require that the partial order be decidable on the compact elements. Here, we use a stronger assumption. Theorem If D is an indexed #algebraic cpo that has a comp...
Maximality and Totality of Stable Functions in the Category of Stable Bifinite Domains ∗
"... This paper studies maximality and totality of stable functions in the category of stable bifinite domains. We present three main results: (1) every maximumpreserving function is a maximal element in the stable function spaces; (2) a maximal stable function f: D → E is maximumpreserving if D is maxi ..."
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This paper studies maximality and totality of stable functions in the category of stable bifinite domains. We present three main results: (1) every maximumpreserving function is a maximal element in the stable function spaces; (2) a maximal stable function f: D → E is maximumpreserving if D is maximumseparable and E is completely separable; and (3) a stable bifinite domain D is maximumseparable if and only if for any locally distributive stable bifinite domain E, each maximal stable function f: D → E is maximumpreserving.