Confluent Term Rewriting Systems with Membership Conditions

1. I n t r o d u c t i o n Many term rewriting systems and their modifications are considered in logic, automated theorem proving, and programming language [2,3,4,6,8,9]. A fundamental property of term rewriting systems is the confluence property. A few sufficient criteria for the confluence are well known [2,3,4,5,8,9]. However, if a term rewriting system is nonterminating and nonlinear, we know few criteria for the confluence of the system [7,10]. In this paper, we study the confluence of membership-conditional term rewriting systems that are nonterminating and nonlinear. In a membership-conditional term rewriting system, the rewriting rule can have membership conditions. We explain this concept with an example. We first consider a classical term rewriting system R that is nonterminating and nonlinear: { /(x,x) ~ 0 n f(g(x), x) ~ 1 2 ~ g(2) The diagram in Figure 1 illustrates that R is not confluent: