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Reduction Order


A strict order > on the set of terms of a term rewriting system is called a reduction order if

1. The set of terms is well ordered with respect to >, that is, all its nonempty subsets contain their least elements,

2. This order is compatible with functions (operations) of the system, i.e.,

 t_i>t_i^'=>f(t_1,...,t_i,...,t_n)>f(t_1,...,t_i^',...,t_n),

and

3. For any substitution theta (cf. unification), s>t=>stheta>ttheta.

If x>y holds for every rewriting rule x->y, then this term rewriting system is finitely terminating.


See also

Church-Rosser Property, Confluent, Critical Pair, Knuth-Bendix Completion Algorithm, Term Rewriting System

This entry contributed by Alex Sakharov (author's link)

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References

Baader, F. and Nipkow, T. Term Rewriting and All That. Cambridge, England: Cambridge University Press, 1999.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1037, 2002.

Referenced on Wolfram|Alpha

Reduction Order

Cite this as:

Sakharov, Alex. "Reduction Order." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ReductionOrder.html

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