Counterexamples to Termination for the Direct Sum of Term Rewriting Systems

The direct sum of two term rewriting systems is the union of systems having disjoint sets of function symbols. It is shown that the direct sum of two term rewriting systems is not terminating, even if these systems are both terminating. Keyword: Term rewriting system, termination Introduction A term rewriting system R is a set of rewriting rules M → N , where M and N are terms [1, 3, 5]. The direct sum system R1 ⊕ R2 is defined as the union of two term rewriting systems with disjoint function symbols [8]. It was proved [8] that for any term rewriting systems R1 and R2, R1 ⊕R2 is confluent iff R1 and R2 are confluent. By replacing confluent with terminating in the above proposition, the analogous conjecture for the terminating property has the form: R1 ⊕R2 is terminating iff R1 and R2 are terminating. However, the answer to this conjecture is negative against our expectation. We show the counterexamples to this conjecture and its modifications.