Deterministic tree pushdown automata and monadic tree rewriting systems
Journal of Computer and System Sciences
Handbook of theoretical computer science (vol. B)
Bottom-up tree pushdown automata: classification and connection with rewrite systems
Theoretical Computer Science
Linear generalized semi-monadic rewrite systems effectively preserve recognizability
Theoretical Computer Science
On the Confluence of Linear Shallow Term Rewrite Systems
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
On the Confluence of Linear Shallow Term Rewrite Systems
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
Right-Linear Finite Path Overlapping Term Rewriting Systems Effectively Preserve Recognizability
RTA '00 Proceedings of the 11th International Conference on Rewriting Techniques and Applications
Reachability and confluence are undecidable for flat term rewriting systems
Information Processing Letters
Termination of rewrite systems with shallow right-linear, collapsing, and right-ground rules
CADE' 20 Proceedings of the 20th international conference on Automated Deduction
RTA '08 Proceedings of the 19th international conference on Rewriting Techniques and Applications
Closure of Tree Automata Languages under Innermost Rewriting
Electronic Notes in Theoretical Computer Science (ENTCS)
Proceedings of the forty-second ACM symposium on Theory of computing
Journal of the ACM (JACM)
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Reachability and joinability are central properties of term rewriting. Unfortunately they are undecidable in general, and even for some restricted classes of term rewrite systems, like shallow term rewrite systems (where variables are only allowed to occur at depth 0 or 1 in the terms of the rules). Innermost rewriting is one of the most studied and used strategies for rewriting, since it corresponds to the "call by value" computation of programming languages. Henceforth, it is meaningful to study whether reachability and joinability are indeed decidable for a significant class of term rewrite systems with the use of the innermost strategy. In this paper we show that reachability and joinability are decidable for shallow term rewrite systems assuming that the innermost strategy is used. All of these results are obtained via the definition of the concept of weak normal form, and a construction of a finite representation of all weak normal forms reachable from every constant. For the particular left-linear shallow case and assuming that the maximum arity of the signature is a constant, these results are obtained with polynomial time complexity.