Matrix Interpretations for Proving Termination of Term Rewriting

• Published in: Journal of automated reasoning Vol. 40; no. 2-3; pp. 195 - 220
• Main Authors: Endrullis, Jörg, Waldmann, Johannes, Zantema, Hans
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01-03-2008; Springer Nature B.V
• Subjects: Artificial Intelligence; Boolean algebra; Computer Science; Mathematical analysis; Mathematical Logic and Formal Languages; Mathematical Logic and Foundations; Solvers; State of the art; Studies; Symbolic and Algebraic Manipulation; Symbols; Transformations; Transforms; Vectors (mathematics); Term rewriting; Termination; Matrix interpretations; Satisfiability
• ISSN: 0168-7433; 1573-0670
• 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.