On the Complexity of the Small Term Reachability Problem for Terminating Term Rewriting Systems

Authors Franz Baader , Jürgen Giesl

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Author Details

Franz Baader
  • Theoretical Computer Science, TU Dresden, Germany
  • SCADS.AI Dresden/Leipzig, Germany
Jürgen Giesl
  • RWTH Aachen University, Aachen, Germany

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Franz Baader and Jürgen Giesl. On the Complexity of the Small Term Reachability Problem for Terminating Term Rewriting Systems. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


Motivated by an application where we try to make proofs for Description Logic inferences smaller by rewriting, we consider the following decision problem, which we call the small term reachability problem: given a term rewriting system R, a term s, and a natural number n, decide whether there is a term t of size ≤ n reachable from s using the rules of R. We investigate the complexity of this problem depending on how termination of R can be established. We show that the problem is NP-complete for length-reducing term rewriting systems. Its complexity increases to N2ExpTime-complete (NExpTime-complete) if termination is proved using a (linear) polynomial order and to PSpace-complete for systems whose termination can be shown using a restricted class of Knuth-Bendix orders. Confluence reduces the complexity to P for the length-reducing case, but has no effect on the worst-case complexity in the other two cases.

Subject Classification

ACM Subject Classification
  • Theory of computation → Equational logic and rewriting
  • Theory of computation → Automated reasoning
  • Theory of computation → Complexity theory and logic
  • Rewriting
  • Termination
  • Confluence
  • Creating small terms
  • Derivational complexity
  • Description Logics
  • Proof rewriting


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