Matrix Interpretations for Proving Termination of Term Rewriting

Abstract

Abstract. 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 non-total well-founded ordering. Function symbols are interpreted by linear mappings represented by matrices. This method allows 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. Our implementation performs well on the Termination Problem Data Base: better than 5 out of 6 tools that participated in the 2005 termination competition in the category of term rewriting.

Models for Logics and Conditional Constraints in Automated Proofs of Termination

“…We also have In the following section we discuss an interesting application of convex polytopic domains to improve the well-known matrix interpretations [5,2].…”

“…As an interesting specialization of this general idea, Section 4 introduces convex matrix interpretations as a new, twofold extension of the framework introduced by Endrullis et al [5] for TRSs, where rather than using vectors x of natural numbers (or non-negative numbers, as in [2]), we use convex sets satisfying a matrix inequality Ax ≥ b. Section 5 discusses existing approaches to deal with the obtained numeric conditional constraints: Farkas' Lemma and results from Algebraic Geometry.…”

Termination by Quasi-periodic Interpretations

“…As the main property of weakly monotone algebras we recall the following theorem from [4]. The approach for proving SN(R) now is trying to prove SN(DP(R) top /R) by finding a suitable weakly monotone algebra such that according to Theorem 2 rules from DP(R) may be removed.…”

Exact DFA Identification Using SAT Solvers

“…The power of these solvers can be used in other problems by translating these problems into SAT instances, and subsequently running a SAT solver on these translated problems. This approach is very competitive for several problems, see, e.g., [9,10,11]. We adopt this approach for DFA identification.…”