Abstract

In this short paper we give a criterion of modal definability of a first-order universal Horn sentence with exactly one positive atom in terms of its graph. As a consequence we obtain that every modal logic axiomatized by a single modal Horn formula (i.e. of the form K+ φ where φ is a modal Horn formula) is Kripke complete. Modal definability of first-order formulas has been intensively studied in modal logic, and even applied to automatic reasoning [9]. On the one hand, it has a nice Goldblatt-Thomason characterization [4], on the other hand, the problem “decide whether a first-order formula is modally definable ” is in general undecidable [2]. But the cause of this undecidability is in the undecidability of first-order logic, so when we restrict attention to a fragment with decidable implication, we are likely to obtain an algorithmic criterion for modal definability, as in this paper. Also this research is motivated by scrutinizing Theorem 5.9 of [3] saying that if L1 and L2 are Kripke complete and Horn axiomatizable unimodal logics, then L1 × L2 = [L1, L2] and studying whether Horn axiomatizability implies Kripke completeness. We give the positive answer to the last question for the case of a single universal Horn sentence with exactly one positive atom, but in general this problem seems to be open. Consider the classical first-order language LfΛ in the signature consisting of only binary predicates Rλ indexed by a finite set Λ. An atom is a formula of the form xiRλxj, where xi and xj are object variables and λ ∈ Λ. Universal Horn sentences (in short, Horn formulas) are closed (i.e. without free variables) formulas of the form ∀x1...∀xn(ψ → φ), where ψ is a conjunction of atoms and φ is an atom. Allowing ∨ in ψ as in [3] is equivalent to considering conjunctions of such formulas. Universal Horn sentences can be represented by tuples of the form D = (WD, (RDλ: λ ∈ Λ), α, β, λ0), where W D = {x1,..., xn} is a finite set, R