An algebraic characterization of equivalent preferential models

Abstract

Preferential model is one of the important semantical structures in nonmonotonic logic. This paper aims to establish an isomorphism theorem for preferential models, which gives us a purely algebraic characterization of the equivalence of preferential models. To this end, we present the notions of local similarity and local simulation. Based on these notions, two operators Δ(·) and μ(·) over preferential models are introduced and explored respectively. Together with other two existent operators ρ(·) and Π_D(·), we introduce an operator ∂_D(·). Then the isomorphism theorem is obtained in terms of ∂_D(·), which asserts that for any two preferential models M₁ and M₂, they generate the same preferential inference if and only if ∂_D(M₁) and ∂_D(M₂) are isomorphic. Based on ∂_D(·), we also get an alternative model-theoretical characterization of the well-known postulate Weaken Disjunctive Rationality. Moreover, in the finite language framework, we show that Δ(μ(·)) is competent for the task of eliminating redundancy, and provide a representation result for k-relations.