A unified ordering for termination proving

Abstract

We introduce a reduction order called the weighted path order (WPO) that subsumes many existing reduction orders. WPO compares weights of terms as in the Knuth-Bendix order (KBO), while WPO allows weights to be computed by a wide class of interpretations. We investigate summations, polynomials and maximums for such interpretations. We show that KBO is a restricted case of WPO induced by summations, the polynomial order (POLO) is subsumed by WPO induced by polynomials, and the lexicographic path order (LPO) is a restricted case of WPO induced by maximums. By combining these interpretations, we obtain an instance of WPO that unifies KBO, LPO and POLO. In order to fit WPO in the modern dependency pair framework, we further provide a reduction pair based on WPO and partial statuses. As a reduction pair, WPO also subsumes matrix interpretations. We finally present SMT encodings of our techniques, and demonstrate the significance of our work through experiments.

Reducing Relative Termination to Dependency Pair Problems

“…We implemented the proposed method in the termination tool NaTT 3 and showed its significance through experiments. Using results of this paper and [29], NaTT can prove relative termination of R quot w.r.t. B rand , and relative termination of R test ∪ R f g w.r.t.…”

“…• the weighted path order with partial status [29], and • (2-or 3-dimensional) matrix interpretations [6].…”

The Termination and Complexity Competition

“…NaTT [51] combines the dependency pair framework with the weighted path order [52]. MU-TERM [1] is particularly suitable for TRSs with modified reduction relations (like innermost, context-sensitive, equational, or conditional rewriting).…”

Nagoya Termination Tool

“…• the Knuth-Bendix order (KBO) [15] and its variants including KBO with status [20], the generalized KBO [19] and the transfinite KBO [18,22], • the recursive path order [3] and the lexicographic path order (LPO) [14], -polynomial interpretations (POLO) [1,17] and its variants, including certain forms 5 of POLO with negative constants [11] and max-POLO [6], -the matrix interpretation method [4,13], and the weighted path order (WPO) [25,26].…”

“…Proving termination of term rewrite systems (TRSs) has been an active field of research. In this paper, we describe the Nagoya Termination Tool (NaTT), a termination prover for TRS, which is available at http://www.trs.cm.is.nagoya-u.ac.jp/NaTT/ NaTT is powerful and fast; its power comes from the novel implementation of the weighted path order (WPO) [25,26] that subsumes most of the existing reduction pairs, and its efficiency comes from the strong cooperation with stateof-the-art satisfiability modulo theory (SMT) solvers. In principle, any solver that complies with the SMT-LIB Standard 3 version 2.0 can be incorporated as a back-end into NaTT.…”