Multi-sorted logic, models and logical geometry

TL;DR

This paper explores the conditions under which two multi-sorted logical knowledge bases are isomorphic, providing both sufficient and necessary criteria within the framework of algebraic models and logical geometry.

Contribution

It introduces a formal approach to determine isomorphism of knowledge bases in multi-sorted logic using algebraic and geometric methods, extending prior work in logical model theory.

Findings

Provided sufficient conditions for knowledge base isomorphism.

Analyzed necessary and sufficient conditions for knowledge base equivalence.

Developed a framework connecting algebraic models with 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.