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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 i ..."
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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...
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.
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.
Weakly Distributive Domains ⋆
"... Abstract. In our previous work [17] we have shown that for anyωalgebraic meetcpo D, if all higherorder stable function spaces built from D areωalgebraic, then D is finitary. This accomplishes the first of a possible, twostep process in solving the problem raised in [1, 2]: whether the category ..."
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Abstract. In our previous work [17] we have shown that for anyωalgebraic meetcpo D, if all higherorder stable function spaces built from D areωalgebraic, then D is finitary. This accomplishes the first of a possible, twostep process in solving the problem raised in [1, 2]: whether the category of stable bifinite domains of AmadioDrosteGöbel [1, 6] is the largest cartesian closed full subcategory within the category ofωalgebraic meetcpos with stable functions. This paper presents results on the second step, which is to show that for any ωalgebraic meetcpo D satisfying axioms M and I to be contained in a cartesian closed full subcategory usingωalgebraic meetcpos with stable functions, it must not violate MI ∞. We introduce a new class of domains called weakly distributive domains and show that for these domains to be in a cartesian closed category usingωalgebraic meetcpos, property MI ∞ must not be violated. We further demonstrate that principally distributive domains (those for which each principle ideal is distributive) form a proper subclass of weakly distributive domains, and Birkhoff’s M3 and N5 [5] are weakly distributive (but nondistributive). We introduce also the notion of meetgenerators in constructing stable functions and show that if anωalgebraic meetcpo D contains an infinite number of meetgenerators, then [D→D] fails I. However, the original problem of Amadio and Curien remains open. 1
On an Open Problem of Amadio and Curien: the Finite Antichain Condition 1 Abstract
"... More than a dozen years ago, Amadio [1] (see Amadio and Curien [2] as well) raised the question of whether the category of stable bifinite domains of AmadioDroste [1,6,7] is the largest cartesian closed full subcategory of the category of ωalgebraic meetcpos with stable functions. A solution to ..."
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More than a dozen years ago, Amadio [1] (see Amadio and Curien [2] as well) raised the question of whether the category of stable bifinite domains of AmadioDroste [1,6,7] is the largest cartesian closed full subcategory of the category of ωalgebraic meetcpos with stable functions. A solution to this problem has two major steps: (1) Show that for any ωalgebraic meetcpo D, if all higherorder stable function spaces built from D are ωalgebraic, then D is finitary (i.e., it satisfies the socalled axiom I); (2) Show that for any ωalgebraic meetcpo D, if D violates MI ∞ , then [D → D] violates either M or I. We solve the first part of the problem in this paper, i.e., for any ωalgebraic meetcpo D, if the stable function space [D → D] satisfies M, then D is finitary. Our notion of (mub, meet)closed set, which is introduced for step 1, will also be used for treating some example cases in step 2. 1
Stable Power Domains
, 1998
"... In the category of stable dcpo's, free constructions w.r.t. algebraic theories exist. From this, we obtain various stable power domain constructions. After handling their properties in general, we concentrate on the stable Plotkin power construction. For continuous ground domains, it is explici ..."
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In the category of stable dcpo's, free constructions w.r.t. algebraic theories exist. From this, we obtain various stable power domain constructions. After handling their properties in general, we concentrate on the stable Plotkin power construction. For continuous ground domains, it is explicitly described in terms of saturated compact sets. In case of algebraic ground domains, this description is isomorphic to Buneman's lossless power domains.
PreCoherence Spaces with Approximation Structure: A Model for Intuitionistic Linear Logic Which is Not a Model of Classical Linear Logic
"... By using additional structure inherent in coherence spaces a new model for intuitionistic linear logic is constructed which is not a model for classical linear logic. The new class of spaces contains also the empty space, whence it yields a logical model, not only a typetheoretic one. 1 ..."
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By using additional structure inherent in coherence spaces a new model for intuitionistic linear logic is constructed which is not a model for classical linear logic. The new class of spaces contains also the empty space, whence it yields a logical model, not only a typetheoretic one. 1
Under consideration for publication in Math. Struct. in Comp. Science The Largest Cartesian Closed Category of Domains, Considered Constructively
, 2002
"... A conjecture of Smyth 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 effectively strongly algebraic cpo, where it was not made precise what is meant by an effectively algebraic and an effectively stron ..."
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A conjecture of Smyth 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 effectively strongly algebraic cpo, where it was not made precise what is meant by an effectively algebraic and an effectively strongly algebraic cpo. Notions of an effectively strongly algebraic cpo and an effective SFP domain are introduced and shown to be (effectively) equivalent. Moreover, the conjecture is shown to hold if instead of being effectively algebraic, [D → D] is only required to be ωalgebraic and D is forced to have a completeness test, that is a procedure which decides for any two finite sets X and Y of compact cpo elements whether X is a complete set of upper bounds of Y. As a consequence, the category of effective SFP objects and continuous maps turns out to be the largest Cartesian closed full subcategory of the category of ωalgebraic cpo’s that have a completeness test. It is then studied whether such a result also holds in a constructive framework, where one considers categories with constructive domains as objects, that is, domains consisting only of the constructive (computable) elements of an indexed ωalgebraic cpo, and computable maps as morphisms. This is indeed the case: the category of constructive SFP domains is the largest constructively Cartesian closed weakly indexed effectively full subcategory of the category of constructive domains that have a completeness test and satisfy a further effectivity requirement.
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 effectively strongly algebraic cpo, where it was left open what exactly is meant by an effectively algebraic and an effec ..."
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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 effectively strongly algebraic cpo, where it was left open what exactly is meant by an effectively algebraic and an effectively strongly algebraic cpo. First, notions of an effectively strongly algebraic cpo and an effective 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.