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Theorem, any nonterminating system must be self-embedding in the sense that it allows for the derivation of some term from a simpler one; thus termination is guaranteed jf every rule in the system as a reduction in some simplification ordering.Most 01 the orderings that have been used for proving tennination are indeed simplication orderings ; using this notion often allows for much easier proofs. A particularly useful class of simplification orderings, the 'recursive path orderings', is defined . Examples of the use of simplification orderings in termination proofs are given.

A common tool for proving the termination of programs is the well-founded set , a set ordered in such a way as to admit no infinite descending sequences. The basic approach is to find a termination function that maps the values of the program variables into some well-founded set, such that the value of the termination function is repeatedly reduced throughout the computation. All too often, the termination functions required are difficult to find and are of a complexity out of proportion to the program under consideration. Multisets ( bags ) over a given well-founded set S are sets that admit multiple occurrences of elements taken from S . The given ordering on S induces an ordering on the finite multisets over S . This multiset ordering is shown to be well-founded. The multiset ordering enables the use of relatively simple and intuitive termination functions in otherwise difficult termination proofs. In particular, the multiset ordering is used to prove the termination of production systems , programs defined in terms of sets of rewriting rules.

A common tool for proving the termination of programs is the well-founded set, a set ordered in such a way as to admit no infinite descending sequences. The basic approach is to find a termination functio~ that maps the values of the program variables into some well-founded set, such that the value of the termination function is continually reduced throughout the computation. All too often, the termination functions required are difficult to find and are of a complexity out of proportion to the program under consideration. However, by providing more sophisticated well-founded sets, the corresponding termination functions can be simplified.Given a well-founded set S, we consider ~Itisets over S, "sets" that admit multiple occurrences of elements taken from S. We define an ordering on all finite multisets over S that is induced by the given ordering on S. This multiset ordering is shown to be well-founded. The value of the multiset ordering is that it permits the use of relatively simple and intuitive termination functions in otherwise difficult termination proofs. In particular, we apply the multiset ordering to prove the termination of production systems, programs defined in terms of sets of rewriting rules.An extended version of this paper appeared as Memo AIM-310,