Multi-sorted logic, models and logical geometry

Abstract

Let $\Theta$ be a variety of algebras, $(H, \Psi, f)$ be a model, where $H$ is an algebra from $\Theta$, $\Psi$ is a set of relation symbols $\varphi$, $f$ is an interpretation of all $\varphi$ in $H$. Let $X^0$ be an infinite set of variables, $\Gamma$ be a collection of all finite subsets in $X^0$ (collection of sorts), $\widetilde\Phi$ be the multi-sorted algebra of formulas. These data define a knowledge base $KB(H,\Psi, f)$. In the paper the notion of isomorphism of knowledge bases is considered. We give sufficient conditions which provide isomorphism of knowledge bases. We also study the problem of necessary and sufficient conditions for isomorphism of two knowledge bases.