"Algorithms for, and complexity of, orientability problem for various versions of the recursive path orders were considered in   . The problems of solving ordering constraints for lexicographic , recursive path orders and for KBO are NP-complete, see      . However, to check if 1 orients l ! "
[Show abstract][Hide abstract] ABSTRACT: We consider two decision problems related to the Knuth-Bendix order (KBO). The first problem is orientability: given a system of rewrite rules R, does there exist an instance of KBO which orients every ground instance of every rewrite rule in R. The second problem is whether a given instance of KBO orients every ground instance of a given rewrite rule. This problem can also be reformulated as the problem of solving a single ordering constraint for the KBO. We prove that both problems can be solved in polynomial time. The polynomial-time algorithm for orientability builds upon an algorithm for solving systems of homogeneous linear inequalities over integers. The polynomial-time algorithm for solving a single ordering constraint does not need to solve systems of linear inequalities and can be run in time O(n 2 ). We show that the orientability problem is P-complete. Also we show that if a system is orientable using a realvalued instance of KBO, then it is also orientable using an integer-valued instance of KBO.
Information and Computation 11/2001; 183(2). DOI:10.1016/S0890-5401(03)00021-X · 0.83 Impact Factor
"The latter semantics is in some cases easier to check, and is used in applications like the computation of saturated sets of ordering constrained clauses that can be used for deduction with other clauses containing arbitrary new (e.g., Skolem) symbols, but it is less restrictive and hence less powerful for refutational theorem proving. The satissability problem for ordering constraints was rst shown decidable for the well-known recursive path orderings (RPO) introduced by N. Dershowitz (Dershowitz, 1982), for xed signatures (Comon, 1990; Jouannaud and Okada, 1991) and extended ones (Nieuwenhuis and Rubio, 1995; Nieuwenhuis, 1993). NP algorithms ((xed and extended signatures) were given in (Nieuwenhuis , 1993; Narendran et al., 1999). "
[Show abstract][Hide abstract] ABSTRACT: It is crucial for the performance of ordered resolution or paramodulation-based deduction systems that they incorporate specialized techniques to work efficiently with standard algebraic theories E. Essential ingredients for this purpose are term orderings that are E-compatible, for the given E, and algorithms deciding constraint satisfiability for such orderings. Here we introduce a uniform technique providing the first such algorithms for some orderings for abelian semigroups, abelian monoids and abelian groups, which we believe will lead to reasonably efficient techniques for practice. Our algorithms are in NP, and hence optimal, since in addition we show that, for any well-founded E-compatible ordering for these E, the constraint satisfiability problem is NP-hard even for conjunctions of inequations.
"The Knuth-Bendix order is used in the state-of-the-art theorem provers, for example, E [Schulz 1999], SPASS [Weidenbach, Afshordel, Brahm, Cohrs, Engel, Keen, Theobalt and Topic 1999], Vampire [Riazanov and Voronkov 1999], and Waldmeister [Hillenbrand, Buch, Vogt and Löchner 1997]. There is extensive literature on solving recursive path ordering constraints (e.g., [Comon 1990, Jouannaud and Okada 1991, Nieuwenhuis 1993, Narendran, Rusinowitch and Verma 1999]). The decidability of Knuth-Bendix ordering constraints was proved only recently in [Korovin and Voronkov 2000]. "
[Show abstract][Hide abstract] ABSTRACT: We show that the problem of solving Knuth-Bendix ordering constraints is NP-complete, as a corollary we show that the existential first-order theory of any term algebra with the Knuth-Bendix ordering is NP-complete.
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