Bi-rewriting, a term rewriting technique for monotonic order relations

Abstract

We propose an extension of rewriting techniques to derive inclusion relations $a \subseteq b$ between terms built from monotonic operators. Instead of using only a rewriting relation $\xrightarrow{ \supseteq }$ and rewriting a to b, we use another rewriting relation $\xrightarrow{ \supseteq }$ as well and seek a common expression c such that $a{\xrightarrow{ \subseteq }^*}c$ and $b{\xrightarrow{ \supseteq }^*}c$. Each component of the bi-rewriting system $\left\langle {\xrightarrow{ \subseteq },\xrightarrow{ \supseteq }} \right\rangle$ is allowed to be a subset of the corresponding inclusion ⊆ or ⊇. In order to assure the decidability and completeness of the proof procedure we study the commutativity of $\xrightarrow{ \subseteq }$ and $\xrightarrow{ \supseteq }$. We also extend the existing techniques of rewriting modulo equalities to bi-rewriting modulo a set of inclusions. We present the canonical bi-rewriting system corresponding to the theory of non-distributive lattices.

This work has been partially supported by the project TESEU (TIC 91-430) funded by the CICYT