Restricting Tree Grammars with Term Rewriting

Authors Jan Bessai, Lukasz Czajka, Felix Laarmann, Jakob Rehof

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Author Details

Jan Bessai
  • TU Dortmund, Germany
Lukasz Czajka
  • TU Dortmund, Germany
Felix Laarmann
  • TU Dortmund, Germany
Jakob Rehof
  • TU Dortmund, Germany


We thank Christoph Stahl for creating the tikz figures. We also thank our reviewers for their insightful and useful comments which improved the final version of the paper.

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Jan Bessai, Lukasz Czajka, Felix Laarmann, and Jakob Rehof. Restricting Tree Grammars with Term Rewriting. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We investigate the problem of enumerating all terms generated by a tree-grammar which are also in normal form with respect to a set of directed equations (rewriting relation). To this end we show that deciding emptiness and finiteness of the resulting set is EXPTIME-complete. The emptiness result is inspired by a prior result by Comon and Jacquemard on ground reducibility. The finiteness result is based on modification of pumping arguments used by Comon and Jacquemard. We highlight practical applications and limitations. We provide and evaluate a prototype implementation. Limitations are somewhat surprising in that, while deciding emptiness and finiteness is EXPTIME-complete for linear and nonlinear rewrite relations, the linear case is practically feasible while the nonlinear case is infeasible, even for a trivially small example. The algorithms provided for the linear case also improve on prior practical results by Kallat et al.

Subject Classification

ACM Subject Classification
  • Theory of computation → Tree languages
  • Theory of computation → Automata extensions
  • Theory of computation → Equational logic and rewriting
  • tree automata
  • tree grammar
  • term rewriting
  • normalization
  • emptiness
  • finiteness


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