Weak Convergence and Uniform Normalization in Infinitary Rewriting
نویسنده
چکیده
We study infinitary term rewriting systems containing finitely many rules. For these, we show that if a weakly convergent reduction is not strongly convergent, it contains a term that reduces to itself in one step (but the step itself need not be part of the reduction). Using this result, we prove the starkly surprising result that for any orthogonal system with finitely many rules, the system is weakly normalizing under weak convergence iff it is strongly normalizing under weak convergence iff it is weakly normalizing under strong convergence iff it is strongly normalizing under strong convergence. As further corollaries, we derive a number of new results for weakly convergent rewriting: Systems with finitely many rules enjoy unique normal forms, and acyclic orthogonal systems are confluent. Our results suggest that it may be possible to recover some of the positive results for strongly convergent rewriting in the setting of weak convergence, if systems with finitely many rules are considered. Finally, we give a number of counterexamples showing failure of most of the results when infinite sets of rules are allowed.
منابع مشابه
Infinitary Normalization
In infinitary orthogonal first-order term rewriting the properties confluence (CR), Uniqueness of Normal forms (UN), Parallel Moves Lemma (PML) have been generalized to their infinitary versions CR∞, UN∞, PML∞, and so on. Several relations between these properties have been established in the literature. Generalization of the termination properties, Strong Normalization (SN) and Weak Normalizat...
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