Proving termination with multiset orderings

  • Nachum Dershowitz
  • Zohar Manna
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 71)


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 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 multisets 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, Stanford Artificial Intelligence Laboratory, Stanford, California.


Production System Natural Number Termination Function Small Element Order Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1979

Authors and Affiliations

  • Nachum Dershowitz
    • 1
  • Zohar Manna
    • 1
  1. 1.Stanford University and Weizmann InstituteUSA

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