Proving termination with multiset orderings

Nachum Dershowitz, Zohar Manna

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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 ~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, Stanford Artificial Intelligence Laboratory, Stanford, California.

Original languageEnglish
Title of host publicationAutomata, Languages and Programming - 6th Colloquium
EditorsHermann A. Maurer
PublisherSpringer Verlag
Number of pages15
ISBN (Print)9783540095101
StatePublished - 1979
Externally publishedYes
Event6th International Colloquium on Automata, Languages and Programming, ICALP 1979 - Graz, Austria
Duration: 16 Jul 197920 Jul 1979

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume71 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference6th International Colloquium on Automata, Languages and Programming, ICALP 1979


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