Term Rewriting that Preserves Models in KR-Logic

Abstract

In human proofs of mathematical problems, such as proofs in a group theory, term rewriting is usually used. When we consider Herbrand semantics for the first-order logic with constraints ($\text {FOL}_\mathrm{c}$), correct representation of evaluable terms cannot be obtained due to lack of representation power of the logic. In place of $\text {FOL}_\mathrm{c}$ with Herbrand semantics, we use $\text {KRL}_\mathrm{c}$ (KR-Logic with built-in constraints). We propose a class of term rewriting rules, and prove that they preserve the sets of all models in KR-Logic. Representation and computation by the rewriting rules in KR-Logic is well established in the space of $\text {KRL}_\mathrm{c}$. This paper opens a new method of logical problem solving, with $\text {KRL}_\mathrm{c}$ being the representation space and $\text {ECLS}_\mathrm{N}$ being the computation space. This theory integrates logical inference and functional rewriting under the broader concept of equivalent transformation.