Termination transformation by tree lifting ordering

Abstract

An extension of a modular termination result for term rewriting systems (TRSs, for short) by A. Middeldorp (1989) is presented. We intended to obtain this by adapting the dummy elimination transformation by M. C. F. Ferreira and H. Zantema (1995) under the presence of a non-collapsing non-duplicating terminating TRS whose function symbols are all to be eliminated. We propose a tree lifting ordering induced from a reduction order and a set G of function symbols, and use this ordering to transform a TRS R into R'; termination of R' implies that of RUS for any non-collapsing non-duplicating terminating TRS S whose function symbols are contained in G, provided that for any l → r in R (1) the root symbol of r is in G whenever that of l is in G; and (2) no variable appears directly below a symbol from G in l when G contains a constant. Because of conditions (1) and (2), our technique covers only a part of the dummy elimination technique; however, even when S is empty, there are cases that our technique has an advantage over the dummy elimination technique.

This work is partially supported by Grants from Ministry of Education, Science and Culture of Japan, #09245212 and #07680347.