A representation of L-domains by information systems

Lecture Notes in Computer Science

2020

Self Cite

Based on the investigation of the proof system of a disjunctive propositional logic introduced by Yi-xiang Chen and Achim Jung, this paper establishes a purely syntactic representation of algebraic L-domains. The central tools used here are logical states which build a bridge between the logical proof systems and algebraic L-domains. A notion of consequence relation is also made to determine Scott-continuous functions between algebraic L-domains. More precisely, a category of certain proof systems with consequence relations is shown to be equivalent to that of algebraic L-domains with Scott-continuous functions.As an application, a subclass of disjunctive propositional logic is found to provide a logical representation for Scott domains. This allows gaining deep insight into the subject of capturing domains in terms of logic.

Section: Proposition 31 ([4]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Disjunctive Propositional Logic and Scott Domains

Lecture Notes in Computer Science

2020

Self Cite

“…In Larsen and Winskel (1984), Larsen and Winskel showed how to use the concrete representation of information systems to advantage in solving recursive domain equations. Since then, many scholars have established several kinds of information systems for the representations of various domains (Hoofman 1993;Huang et al 2015;Spreen et al 2008;Vickers 2004;Wang and Li 2021;Wu et al 2016).…”

Section: Introductionmentioning

confidence: 99%

Consistent disjunctive sequent calculi and Scott domains

Math. Struct. Comp. Sci.

2021

Self Cite

Based 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.

“…And soon after, Larsen and Winskel (1984) proved that there is an equivalence between categories of such information systems with approximable mappings as morphisms and algebraic bounded complete domains with Scott-continuous functions as morphisms. As to representations of various domains, many scholars have devised several kinds of information systems and similar structures (Hoofman (1993); Huang et al (2015); Spreen et al (2008); Vickers (1993); Wu et al (2016)).…”

Section: Introductionmentioning

confidence: 99%

A representation of proper BC domains based on conjunctive sequent calculi

Math. Struct. Comp. Sci.

2019

Self Cite

We 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.