On tree automata that certify termination of left-linear term rewriting systems

  • Authors:

  • Alfons Geser;Dieter Hofbauer;Johannes Waldmann;Hans Zantema

  • Affiliations:

  • Hochschule für Technik, Wirtschaft und Kultur (FH) Leipzig, Fachbereich EIT, Postfach 301166, D-04251 Leipzig, Germany;Mühlengasse 16, D-34125 Kassel, Germany;Hochschule für Technik, Wirtschaft und Kultur (FH) Leipzig, Fachbereich IMN, Postfach 301166, D-04251 Leipzig, Germany;Faculteit Wiskunde en Informatica, Technische Universiteit Eindhoven, Postbus 513, 5600 MB Eindhoven, The Netherlands

  • Venue:
  • Information and Computation
  • Year:
  • 2007

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Abstract

We present a new method for automatically proving termination of left-linear term rewriting systems on a given regular language of terms. It is a generalization of the match bound method for string rewriting. To prove that a term rewriting system terminates we first construct an enriched system over a new signature that simulates the original derivations. The enriched system is an infinite system over an infinite signature, but it is locally terminating: every restriction of the enriched system to a finite signature is terminating. We then construct iteratively a finite tree automaton that accepts the enriched given regular language and is closed under rewriting modulo the enriched system. If this procedure stops, then the enriched system is compact: every enriched derivation involves only a finite signature. Therefore, the original system terminates. We present two methods to construct the enrichment: roof heights for left-linear systems, and match heights for linear systems. For linear systems, the method is strengthened further by a forward closure construction. Using these methods, we give examples for automated termination proofs that cannot be obtained by standard methods.