Properties of Rewriting Systems
Observe that in both of the above rewriting systems, it is possible to get terms rewritten to a "simplest" term, where this term cannot be modified any further from the rules in the rewriting system. Terms which cannot be written any further are called normal forms. The potential existence or uniqueness of normal forms can be used to classify and describe certain rewriting systems. There are rewriting systems which do not have normal forms: a trivial example is the rewriting system on two terms a and b with a → b, b → a.
The property exhibited above where terms can be rewritten regardless of the choice of rewriting rule to obtain the same normal form is known as confluence. The property of confluence is linked with the property of having unique normal forms.
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