# Proving termination with multiset orderings

## Abstract

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.

## Keywords

Production System Natural Number Termination Function Small Element Order Type## Preview

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## References

- Floyd, R. W. [1967],
*Assigning meanings to programs*, Proc. Symp. in Applied Mathematics, vol. 19 (J. T. Schwartz, ed.), American Mathematical Society, Providence, RI, pp. 19–32.Google Scholar - Gentzen, G. [1938],
*New version of the consistency proof for elementary number theory*, The collected papers of Gerhart Gentzen (M. E. Szabo, ed.), North Holland, Amsterdam (1969), pp. 252–286.Google Scholar - Gorn, S. [Sept. 1965],
*Explicit definitions and linguistic dominoes*, Proc. Conf. on Systems and Computer Science, London, Ontario, pp. 77–115.Google Scholar - Iturriaga, R. [May 1967],
*Contributions to mechanical mathematics*, Ph.D. thesis, Carnegie-Mellon Univ., Pittsburgh, PA.Google Scholar - Knuth, D. E. and P. B. Bendix [1969],
*Simple word problems in universal algebras*, Computational Problems in Universal Algebras (J. Leech, ed.), Pergamon Press, Oxford, pp. 263–297.Google Scholar - Lankford, D. S. [May 1975],
*Canonical algebraic simplification in computational logic*, Memo ATP-25, Automatic Theorem Proving Project, Univ. of Texas, Austin, TX.Google Scholar - Lipton, R. J. and L. Snyder [Aug 1977],
*On the halting of tree replacement systems*, Proc. Conf. on Theoretical Computer Science, Waterloo, Ontario, pp. 43–46.Google Scholar - Manna, Z. and S. Ness [Jan 1970],
*On the termination of Markov algorithms*, Proc. Third Hawaii Intl. Conf. on Systems Sciences, Honolulu, HI, pp. 789–792.Google Scholar - Plaisted, D. [July 1978],
*Well-founded orderings for proving the termination of rewrite rules*, Memo R-78-932, Dept. of Computer Science, Univ. of Illinois, Urbana, IL.Google Scholar - Plaisted, D. [Oct. 1978],
*A recursively defined ordering for proving termination of term rewriting systems*, Memo R-78-943, Dept. of Computer Science, Univ. of Illinois, Urbana, IL.Google Scholar