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.

Increasing interpretations

“…A careful inspection of the proofs in Section 5 reveals that our results extend to polynomial interpretations with rational [8,22], and real coefficients [22] as well as negative constants [14]. Furthermore it generalizes to the matrix method [5] since there the interpretation of a dependency pair amounts to a natural number. Consequently the dependency graph is labeled in exactly the same fashion.…”

“…Other approaches allow less restrictive kinds of polynomials including max [7]. Furthermore matrix [5], quasi-periodic [29], and arctic [18] interpretations do also extend the termination proving power significantly. All these extensions share the property that the rewrite rules under consideration must weakly decrease and at least one rule has to decrease strictly.…”

“…Hence it is not so astonishing that the problematic SCC can be handled by matrix interpretations [5] of dimension two. Actually all of the tools (dedicated to proving termination) participating in the TRS category of the 2007 and 2008 editions of the international termination competition can handle this system.…”

Proving Termination by Dependency Pairs and Inductive Theorem Proving

“…10 For example, by Theorem 7 one could switch to termination methods like [28] exploiting types. 11 One cannot assume that DP problems (P, R) always have a specific form, e.g., that P includes A → A whenever R includes a → a. The reason is that a DP problem (P, R) can result from arbitrary DP processors that were applied before.…”

“…SORT(add(x, xs)) → SORT(del(max(add(x, xs)), add(x, xs))) (11) SORT(add(x, xs)) → DEL(max(add(x, xs)), add(x, xs))…”