Scott Continuous Functions Research Articles

A note on information systems for continuous semi-lattices

Wang et al. recently provided a representation for continuous semi-lattices by SL-information systems and raised a question in the conclusion section of [15]. That is, whether a modification of the notion of an approximable mapping can be made so that an equivalence to the category of continuous semi-lattices is obtained. This category has Scott-continuous functions that respect non-empty finite meets as morphisms. In this note, we solve this problem by introducing the concept of SL-approximable mappings between SL-information systems. As an application, we propose the BC-approximable mappings between BC-information systems and establish a categorical equivalence between BC-information systems with BC-approximable mappings and bounded-complete domains with Scott-continuous functions that preserve non-empty infima.

Robustness, Scott continuity, and computability

AbstractRobustness is a property of system analyses, namely monotonic maps from the complete lattice of subsets of a (system’s state) space to the two-point lattice. The definition of robustness requires the space to be a metric space. Robust analyses cannot discriminate between a subset of the metric space and its closure; therefore, one can restrict to the complete lattice of closed subsets. When the metric space is compact, the complete lattice of closed subsets ordered by reverse inclusion is $\omega$ -continuous, and robust analyses are exactly the Scott-continuous maps. Thus, one can also ask whether a robust analysis is computable (with respect to a countable base). The main result of this paper establishes a relation between robustness and Scott continuity when the metric space is not compact. The key idea is to replace the metric space with a compact Hausdorff space, and relate robustness and Scott continuity by an adjunction between the complete lattice of closed subsets of the metric space and the $\omega$ -continuous lattice of closed subsets of the compact Hausdorff space. We demonstrate the applicability of this result with several examples involving Banach spaces.

Representations of Domains via CF-approximation Spaces

Representations of domains mean in a general way representing a domain as a suitable family endowed with set-inclusion order of some mathematical structures. In this paper, representations of domains via CF-approximation spaces are considered. Concepts of CF-approximation spaces and CF-closed sets are introduced. It is proved that the family of CF-closed sets in a CF-approximation space endowed with set-inclusion order is a continuous domain and that every continuous domain is isomorphic to the family of CF-closed sets of some CF-approximation space endowed with set-inclusion order. The concept of CF-approximable relations is introduced using a categorical approach, which later facilitates the proof that the category of CF-approximation spaces and CF-approximable relations is equivalent to that of continuous domains and Scott continuous maps.

Continuous R-valuations

We introduce continuous $R$-valuations on directed-complete posets (dcpos, for short), as a generalization of continuous valuations in domain theory, by extending values of continuous valuations from reals to so-called Abelian d-rags $R$. Like the valuation monad $\mathbf{V}$ introduced by Jones and Plotkin, we show that the construction of continuous $R$-valuations extends to a strong monad $\mathbf{V}^R$ on the category of dcpos and Scott-continuous maps. Additionally, and as in recent work by the two authors and C. Th\'eron, and by the second author, B. Lindenhovius, M. Mislove and V. Zamdzhiev, we show that we can extract a commutative monad $\mathbf{V}^R_m$ out of it, whose elements we call minimal $R$-valuations. We also show that continuous $R$-valuations have close connections to measures when $R$ is taken to be $\mathbf{I}\mathbb{R}^\star_+$, the interval domain of the extended nonnegative reals: (1) On every coherent topological space, every non-zero, bounded $\tau$-smooth measure $\mu$ (defined on the Borel $\sigma$-algebra), canonically determines a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation; and (2) such a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation is the most precise (in a certain sense) continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation that approximates $\mu$, when the support of $\mu$ is a compact Hausdorff subspace of a second-countable stably compact topological space. This in particular applies to Lebesgue measure on the unit interval. As a result, the Lebesgue measure can be identified as a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation. Additionally, we show that the latter is minimal.

Bounded complete domains and their logical form

We establish a framework of approximable disjunctive sequent calculus, which is sound and complete with respect to approximable FD-algebras. We show that the category of approximable FD-algebras with approximable mappings is equivalent to that of bounded complete domains with Scott continuous functions. This extends Abramsky's logical representation of Scott domains as domain prelocales to a continuous setting. We also consider some domain constructions applied to approximable FD-algebras and show how to construct the approximable FD-algebras we need for defining the semantics of programming languages. According to a substructure relation, we define a pointed ω-chain complete class of approximable FD-algebras, on which the domain constructions are made continuous and then the initial solutions to recursive domain equations are transformed into the fixed-points of such continuous functions.

A representation of L-domain by formal concept analysis

In this paper, we represent L-domains by means of formal concept analysis. Based on the attributive continuous formal contexts, we propose the notions of LDF-contexts and \({\mathcal {F}}\)-morphisms and show that they provide concrete representations of L-domains and Scott-continuous functions between them, respectively. Moreover, the category of LDF-contexts with \(\mathcal {F}\)-morphisms is proven to be equivalent to that of L-domains with Scott-continuous functions as morphisms.

Information systems for continuous semi-lattices

We consider two problems on representation theory for domains by information systems. The first problem is how to represent continuous semi-lattices as information systems, and the second one is which subclass of the continuous information systems designed by Spreen et al. for continuous domains can be used to represent bounded complete domains. We solve the first problem by defining a notion of SL-information systems. This is a special case of the continuous information systems. We also discuss the morphism part of SL-information systems, which corresponds to Scott-continuous functions between continuous semi-lattices. Noting that each bounded complete domain is a special continuous semi-lattices, we provide a correct solution to the second problem by properly strengthening the conditions of SL-information systems.

Consistent disjunctive sequent calculi and Scott domains

AbstractBased on the framework of disjunctive propositional logic, we first provide a syntactic representation for Scott domains. Precisely, we establish a category of consistent disjunctive sequent calculi with consequence relations, and show it is equivalent to that of Scott domains with Scott-continuous functions. Furthermore, we illustrate the approach to solving recursive domain equations by introducing some standard domain constructions, such as lifting and sums. The subsystems relation on consistent finitary disjunctive sequent calculi makes these domain constructions continuous. Solutions to recursive domain equations are given by constructing the least fixed point of a continuous function.

Representation of bifinite domains by BF-closure spaces

Abstract In this paper, we propose the notion of BF-closure spaces as concrete representation of bifinite domains. We prove that every bifinite domain can be obtained as the set of F-closed sets of some BF-closure space under set inclusion. Furthermore, we obtain that the category of bifinite domains and Scott-continuous functions is equivalent to that of BF-closure spaces and F-morphisms.

Continuous L-domains in logical form

We introduce a framework of approximable disjunctive propositional logic, which is the logic that results from a disjunctive propositional logic by adding an additional connective. The Lindenbaum algebra of this logic is an approximable dD-algebra. We show that for any approximable dD-algebra, its approximable filters ordered by set inclusion form a continuous L-domain. Conversely, every continuous L-domain can be represented as an approximable dD-algebra. Moreover, we establish a categorical equivalence between the category of approximable dD-algebras with approximable dD-algebra morphisms and that of continuous L-domains with Scott-continuous functions. This extends Abramsky's Domain Theory in Logical Form to the world of continuous L-domains. As an application, we give an affirmative answer to an open problem of Chen and Jung.

Continuous Domains in Formal Concept Analysis*

Formal Concept Analysis (FCA) has been proven to be an effective method of restructuring complete lattices and various algebraic domains. In this paper, the notion of contractive mappings over formal contexts is proposed, which can be viewed as a generalization of interior operators on sets into the framework of FCA. Then, by considering subset-selections consistent with contractive mappings, the notions of attribute continuous formal contexts and continuous concepts are introduced. It is shown that the set of continuous concepts of an attribute continuous formal context forms a continuous domain, and every continuous domain can be restructured in this way. Moreover, the notion of F-morphisms is identified to produce a category equivalent to that of continuous domains with Scott continuous functions. The paper also investigates the representations of various subclasses of continuous domains including algebraic domains and stably continuous semilattices.

Thompson aggregators, Scott continuous Koopmans operators, and Least Fixed Point theory

We reconsider the theory of Thompson aggregators proposed by Marinacci and Montrucchio (Marinacci and Montrucchio, 2010). We prove the existence of a Least Fixed Point (LFP) solution to the Koopmans equation. It is a recursive utility function. Our proof turns on demonstrating the Koopmans operator is a Scott continuous function when its domain is an order bounded subset of a space of bounded functions defined on the commodity space. Kleene’s Fixed Point Theorem yields the construction of the LFP by an iterative procedure. We argue the LFP solution is the Koopmans equation’s principal solution. It is constructed by an iterative procedure requiring less information (according to an information ordering) than approximations for any other fixed point. Additional distinctions between the LFP and GFP (Greatest Fixed Point) are presented. A general selection criterion for multiple solutions for functional equations and recursive methods is proposed.

Compatibilities between continuous semilattices

We define compatibilities between continuous semilattices as Scott continuous functions from their pairwise cartesian products to $\{0,1\}$ that are zero preserving in each variable. It is shown that many specific kinds of mathematical objects can be regarded as compatibilities, among them monotonic predicates, Galois connections, completely distributive lattices, isotone mappings with images being chains, semilattice morphisms etc. Compatibility between compatibilities is also introduced, it is shown that conjugation of non-additive real-valued or lattice valued measures is its particular case.

On fuzzy monotone convergence [formula omitted]-cotopological spaces

In this paper, we generalize the concept of a monotone convergence space (also called a d-space) to the setting of a Q-cotopological space, where Q is a commutative and integral quantale. We establish a D-completion for every stratified Q-cotopological space, which is a category reflection of the category SQ-CTop of stratified Q-cotopological spaces onto the full subcategory SQ-DCTop of monotone convergence Q-cotopological spaces. By introducing the notion of a tapered set, a direct characterization of the completion is obtained: the D-completion of each stratified Q-cotopological space X consists exactly of those tapered closed sets in X. We show that the D-completion can be applied to obtain a universal fuzzy directed completion of a Q-ordered set by endowing it with the Scott cotopology, taking the D-completion, and then passing to the specialization Q-order. Consequently, the category Q-DOrd of fuzzy directed complete Q-ordered sets and Scott continuous functions is reflective in the category Q-Ordσ of Q-ordered sets and Scott continuous functions.

A representation of continuous lattices based on closure spaces

In this paper, we establish the link between continuous lattices and closure spaces. By generalizing the notion of algebraic closure space to continuous closure space, we show that continuous lattices can be represented by continuous closure spaces, just as algebraic lattices can be represented by algebraic closure spaces. We also introduce the notion of approximable mappings between continuous closure spaces, which produces the category equivalent to that of continuous lattices with Scott-continuous functions. These results demonstrate the capability of closure spaces in representing continuous domains.

The Order-Sobrification Monad

We investigate the so-called order-sobrification monad proposed by Ho et al. (Log Methods Comput Sci 14:1–19, 2018) for solving the Ho–Zhao problem, and show that this monad is commutative. We also show that the Eilenberg–Moore algebras of the order-sobrification monad over dcpo’s are precisely the strongly complete dcpo’s and the algebra homomorphisms are those Scott-continuous functions preserving suprema of irreducible subsets. As a corollary, we show that this monad gives rise to the free strongly complete dcpo construction over the category of posets and Scott-continuous functions. A question related to this monad is left open alongside our discussion, an affirmative answer to which might lead to a uniform way of constructing non-sober complete lattices.

A logic for Lawson compact algebraic L-domains

In this paper, we build a logic which is named N-sequent calculus. Based on this logic, we provide two kinds of logical representations of Lawson compact algebraic L-domains: one in terms of logical algebras and the other in terms of logical syntax.The first representation takes the corresponding logical algebras as research objects. The use of prime filters achieves the connection between our logic and Lawson compact algebraic L-domains. This approach is inspired by Abramsky's SFP domain logic and the disjunctive propositional logic on algebraic L-domains introduced by Yixiang Chen and Achim Jung. However, there are essential differences between them at the morphisms part. For the second representation, we directly adopt N-sequent calculi themselves as objects instead of the logical algebras. Then we establish the category of N-sequent calculi with consequence relations equivalent to that of Lawson compact algebraic L-domains with Scott continuous maps. This demonstrates the capability of the syntax of the logic in representing domains.

A representation of proper BC domains based on conjunctive sequent calculi

AbstractWe build a logical system named a conjunctive sequent calculus which is a conjunctive fragment of the classical propositional sequent calculus in the sense of proof theory. We prove that a special class of formulae of a consistent conjunctive sequent calculus forms a bounded complete continuous domain without greatest element (for short, a proper BC domain), and each proper BC domain can be obtained in this way. More generally, we present conjunctive consequence relations as morphisms between consistent conjunctive sequent calculi and build a category which is equivalent to that of proper BC domains with Scott-continuous functions. A logical characterization of purely syntactic form for proper BC domains is obtained.

Consistent-linear FS-domains

In this paper, we consider bounded-complete domains with maps which preserve existing suprema, and introduce consistent-linear FS-domains. The main conclusions include: 1. The category with consistent-linear FS-domains as objects and Scott continuous functions as morphisms is a cartesian closed category. 2. A bounded-complete domain is completely distributive iff it is a distributive consistent-linear FS-domain. 3. A pointed dcpo L is continuous iff the consistent Hoare powerdomain HC(L) over L is continuous iff HC(L) is a distributive consistent-linear FS-domain.

A representation of continuous domains via relationally approximable concepts in a generalized framework of formal concept analysis

In this paper, in order to realize a representation of continuous domains, the notions of relationally consistent F-augmented contexts and relationally approximable concepts are introduced, which provides a generalized framework of formal concept analysis. We also introduce the notion of F-approximable mappings which serves as the morphism between relationally consistent F-augmented contexts. The main result is that the category of relationally consistent F-augmented contexts is equivalent to that of continuous domains with Scott continuous maps being morphisms. This provides a new approach to concretely representing continuous domains and demonstrates the efficiency of formal concept analysis in representing some important partially ordered structures.