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Show CHAPTER 4 TERM REWRITING SYSTEMS AND UNIFICATION This chapter is dedicated to an extension of the narrowing method to some classes of nonterminating, but confluent term rewriting systems. As discussed in the last chapter, the development of unification algorithms for equational theories is usually done in the context of term rewriting systems derived from equational theories. We are here interested in exploring classes of term rewriting systems for which narrowing yields a complete unification algorithm. The class of term rewriting systems under consideration in this chapter is called closed I inear term rewriting systems. The linearity condition says that no variable can appear more than once in the lefthand side of a rewrite rule. This is also called the !eft-linear property in literature. The closure property is a special case of the confluence property. The importance of this investigation stems from the fact that closed linear term rewriting systems are actually equational programs written in the equational language described in [42], except for a few minor differences. There exist term rewriting systems, however, that are closed linear but for which narrowing fails to generate complete sets of unifiers for some E-unifiable terms. A further condition that is required for a completeness proof, and which we impose here, is called nonrepetition. Intuitively, if a term rewriting system is nonterminating, there are nonterminating reduction sequences, in which some rewrite rule{s) must be used for reductions infinitely many times. For example, suppose that f(t 1, ... ,tn) - > g(s1, ... ,sm) is a rewrite rule that is used infinitely many times in a nonterminating reduction sequence. We then have an |
