Journal of Automated Reasoning

, Volume 40, Issue 2, pp 195–220

Matrix Interpretations for Proving Termination of Term Rewriting


  • Jörg Endrullis
    • Department of Computer ScienceVrije Universiteit Amsterdam
  • Johannes Waldmann
    • Hochschule für Technik, Wirtschaft und Kultur (FH) Leipzig
    • Department of Computer ScienceTechnische Universiteit Eindhoven

DOI: 10.1007/s10817-007-9087-9

Cite this article as:
Endrullis, J., Waldmann, J. & Zantema, H. J Autom Reasoning (2008) 40: 195. doi:10.1007/s10817-007-9087-9


We present a new method for automatically proving termination of term rewriting. It is based on the well-known idea of interpretation of terms where every rewrite step causes a decrease, but instead of the usual natural numbers we use vectors of natural numbers, ordered by a particular nontotal well-founded ordering. Function symbols are interpreted by linear mappings represented by matrices. This method allows us to prove termination and relative termination. A modification of the latter, in which strict steps are only allowed at the top, turns out to be helpful in combination with the dependency pair transformation. By bounding the dimension and the matrix coefficients, the search problem becomes finite. Our implementation transforms it to a Boolean satisfiability problem (SAT), to be solved by a state-of-the-art SAT solver.


Term rewritingTerminationMatrix interpretationsSatisfiability

Copyright information

© Springer Science+Business Media B.V. 2007