Unifying the Knuth-Bendix, recursive path and polynomial orders

Abstract

We introduce a simplification order called the weighted path order (WPO). WPO compares weights of terms as in the Knuth-Bendix order (KBO), while WPO allows weights to be computed by an arbitrary interpretation which is weakly monotone and weakly simple. 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. We also present SMT encodings of our orders, as well as incorporating them in the dependency pair framework.

A unified ordering for termination proving

“…The revised version of WPO further subsumes POLO and matrix interpretations as a reduction pair. We also conclude that WPO does not subsume RPOLO by Example 18, which was left open in [28]. The experimental results in Sections 7.2 and 7.3 show a significant improvement due to the new definition of WPO as a reduction pair.…”

“…In the preliminary version of this paper [28], we simply applied argument filtering to obtain the reduction pair π WPO , ≻ π WPO from WPO. In this paper, we fully revise this approach and directly define a reduction pair by incorporating partial statuses [26] into WPO.…”

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].…”

“…For constant symbols, interpretations of the shape (1) are used. Since the operator max is not usually supported by SMT solvers, these interpretations are encoded as quantifierfree formulas using the technique presented in [25].…”

“…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.…”