A logical approach to stable domains

Authors:
Yi-Xiang Chen;Achim Jung
Affiliations:
The School of Mathematics, Physics and Informatics, Shanghai Normal University, Shanghai 200234, PR China;The School of Computer Science, The University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK
Venue:
Theoretical Computer Science
Year:
2006

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Abstract

Building on earlier work by Guo-Qiang Zhang on disjunctive information systems, and by Thomas Ehrhard, Pasquale Malacaria, and the first author on stable Stone duality, we develop a framework of disjunctive propositional logic in which theories correspond to algebraic L-domains. Disjunctions in the logic can be indexed by arbitrary sets (as in geometric logic) but must be provably disjoint. This raises several technical issues which have to be addressed before clean notions of axiom system and theory can be defined. We show soundness and completeness of the proof system with respect to distributive disjunctive semilattices, and prove that every such semilattice arises as the Lindenbaum algebra of a disjunctive theory. Via stable Stone duality, we show how to use disjunctive propositional logic for a logical description of algebraic L-domains.