Kripke Completeness of Strictly Positive Modal Logics over Meet-semilattices with Operators

Our concern is the completeness problem for strongly positive (SP) theories, that is, sets of implications between SP-terms built from propositional variables, conjunction and modal diamond operators. Originated in logic, algebra and computer science, SP-theories have two natural semantics: meet-semilattices with monotone operators providing Birkhoff-style calculi, and first-order relational structures (aka Kripke frames) often used as the intended structures in applications. Here we lay foundations of a completeness theory that aims to answer the question whether the two semantics define the same consequence relations for a given SP-theory. In this paper, we investigate connections between various consequence relations for the fragment of propositional multi-modal logic that comprises implications σ → τ , where σ and τ are strictly positive modal formulas [8] constructed from propositional variables using conjunction ∧, unary diamond operators ✸i, and the constant ‘truth’ ⊤. We call such formulas SP-implications and we call sets of SP-implications SP-theories . §1. Background. Consequence relations between SP-theories and SP-implications have been studied in knowledge representation, universal algebra, and modal provability logic. 1.1. Description logic EL. In knowledge representation, ontologies are used to define vocabularies for domains of interest together with logical relationships between the vocabulary terms [4, 54, 5]. The description logic EL [6, 3] is a widely used ontology language, in which such relationships are given by means of (notational variants of) SP-implications. A typical example of an EL ontology is SNOMED CT [65] that provides a standardised medical vocabulary for the healthcare systems of more than twenty countries. SNOMED CT consists of about 300,000 SP-implications covering most aspects of medicine and healthcare. For example, the SP-implication Viral pneumonia → ✸causative agentVirus ∧✸finding siteLung says that viral pneumonia is caused by a virus and found in lungs. EL is the logical underpinning of the profile OWL2EL of the Web Ontology Language OWL2 [58] designed by W3C for writing up ontologies. Under the EL semantics, SP-implications are interpreted in relational structures known as Kripke