Continuous L-domains in logical form

Abstract

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.

Introduction

In Abramsky's landmark paper [1], a suitable programming logic for denotational semantics was presented axiomatically which developed domain theory in a logical point of view. Under this logical framework, the collection of all prime filters of the Lindenbaum algebra can be proved to form a Scott domain with respect to set inclusion, and every Scott domain can be obtained by this way up to isomorphism. Soon after, he generalized this observation for Scott domains to the more encompassing class of SFP-domains [2]. These works provided a systematic approach to moving back and forth between the denotational and logical models of some computation, mutually determining each other.

Abramsky's domain theory in logic form works well for some algebraic domains, the compact elements can be found to play a key role in this program. Following this way, Chen and Jung developed a framework of disjunctive propositional logic in which theories correspond to algebraic L-domains [7]. In [18], the authors introduced a logical language for describing Lawson compact algebraic L-domains. Continuous domain theory is a more active field and has a more widespread importance. This is a motivation to generalize the technique of Abramsky's domain logics and use it to show results of continuous domains.

Although the distance between algebraic and continuous domains is only a short step away, it is a hard task of extending Abramsky's domain logics from algebraic to continuous domains [13]. Jung, Kegelmann and Moshier defined a continuous sequent calculus for stably compact spaces and showed that much of Abramsky's work can be extended to this multi-lingual sequent calculus [12], [14]. The paper [17] also used the idea of multi-lingual sequent calculi to present a logical representation for bounded complete domains. It is worth noting that these multi-lingual sequent calculi did not define the interpretation $〚\cdot 〛$ of formulae. One explanation could be that the identity rule is absent. So there are still some key questions to be overcome.

There are many other frameworks of generalizing Abramsky's domain logics [4], [5], [15], [16], but the discussions on the continuous domains are involved little. In [7], Chen and Jung proposed an open problem: whether the disjunctive propositional logic presented by them for algebraic L-domains can be extended in some way such that all continuous L-domains are covered. In this paper, we provide a method to answer this question in the affirmative, hoping it can help the researchers to transfer Abramsky's domain theory in logic form to the continuous world based on those existing domain logics for algebraic domains.

To interpret the logical significance of the Plotkin powerdomain construction apparently, Abramsky enhanced the language of SFP-domains logic with two operators □ and ⋄. Introducing some unary connectives is a commonly used method in extending the language of a logic. For instance, the modal logic is the logic that expanded the classical logic by adding the necessity connective □ and the possibility connective ⋄. In [10], $\mathrm{H}\acute{\mathrm{a}}\mathrm{jek}$ expanded the language of the basic logic BL to that of ${\mathrm{BL}}_{△}$ by the connective △. It is well known that every continuous L-domain is a retract of an algebraic L-domain. In this paper, we want to describe the disjunctive propositional logic together with an extra connective playing the similar effects to a retraction. Moreover, the notion of stable open sets links algebraic L-domains and the disjunctive propositional logic. Hence we probably need to define the counterparts of stable open sets in continuous L-domains.

Chen has shown that the notion of stable dD-semilattices is a representation of algebraic L-domains [6]. However, the Lindenbaum algebra of a disjunctive propositional logic is a dD-semilattice, perhaps not a stable dD-semilattice. Section 3.1 of this paper revisits the syntactic framework of a disjunctive propositional logic, and bring a stable conditionality into a disjunctive propositional logic such that its Lindenbaum algebra is a stable dD-semilattice.

As a first step towards designing a logic for continuous L-domains, a concomitant disjunctive propositional logic is proposed in Section 3.2, by introducing a unary connective ∇ in the language of a stable disjunctive propositional logic. Then based on a stable disjunctive propositional logic and its concomitant, we define an approximable disjunctive propositional logic. In Section 3.3, we define the interpretation function of formulae and show that an approximable disjunctive propositional logic is complete with respect to its Lindenbaum algebra.

In Section 4.1, we discuss the relationship between the approximable disjunctive propositional logic and continuous L-domains. From the perspective of approximable dD-algebras, we show that every continuous L-domain has a logical presentation with the help of approximable filters and stable open sets. This extends Chen and Jung's work at the object level. On the side of morphisms, we extend their work in a different way. The morphisms between algebraic L-domains they discussed are stable functions. However, Scott continuous functions are the common choice in domain theory. In Section 4.2, we introduce the notion of approximable dD-morphisms and show that approximable dD-morphisms determine Scott continuous functions, and vice versa.