Automata for reduction properties solving
Journal of Symbolic Computation
Term rewriting and all that
Right-Linear Finite Path Overlapping Term Rewriting Systems Effectively Preserve Recognizability
RTA '00 Proceedings of the 11th International Conference on Rewriting Techniques and Applications
Ground reducibility is EXPTIME-complete
Information and Computation
Characterizing Confluence by Rewrite Closure and Right Ground Term Rewrite Systems
Applicable Algebra in Engineering, Communication and Computing
On the Normalization and Unique Normalization Properties of Term Rewrite Systems
CADE-21 Proceedings of the 21st international conference on Automated Deduction: Automated Deduction
Non-linear Rewrite Closure and Weak Normalization
LICS '09 Proceedings of the 2009 24th Annual IEEE Symposium on Logic In Computer Science
Termination of rewriting with right-flat rules
RTA'07 Proceedings of the 18th international conference on Term rewriting and applications
Confluence of shallow right-linear rewrite systems
CSL'05 Proceedings of the 19th international conference on Computer Science Logic
Decidability of termination for semi-constructor TRSs, left-linear shallow TRSs and related systems
RTA'06 Proceedings of the 17th international conference on Term Rewriting and Applications
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A rewrite closure is an extension of a term rewrite system with new rules, usually deduced by transitivity. Rewrite closures have the nice property that all rewrite derivations can be transformed into derivations of a simple form. This property has been useful for proving decidability results in term rewriting. Unfortunately, when the term rewrite system is not linear, the construction of a rewrite closure is quite challenging. In this paper, we construct a rewrite closure for term rewrite systems that satisfy two properties: the right-hand side term in each rewrite rule contains no repeated variable (right-linear) and contains no variable occurring at depth greater than one (right-shallow). The left-hand side term is unrestricted, and in particular, it may be non-linear. As a consequence of the rewrite closure construction, we are able to prove decidability of the weak normalization problem for right-linear right-shallow term rewrite systems. Proving this result also requires tree automata theory. We use the fact that right-shallow right-linear term rewrite systems are regularity preserving. Moreover, their set of normal forms can be represented with a tree automaton with disequality constraints, and emptiness of this kind of automata, as well as its generalization to reduction automata, is decidable. A preliminary version of this work was presented at LICS 2009 (Creus 2009).