Conference Paper
Satisfiability of Systems of Ordinal Notations with the Subterm Property is Decidable.
DOI: 10.1007/3540542337_155 Conference: Automata, Languages and Programming, 18th International Colloquium, ICALP91, Madrid, Spain, July 812, 1991, Proceedings
Source: DBLP
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 "Algorithms for, and complexity of, orientability problem for various versions of the recursive path orders were considered in [5] [12] [13]. The problems of solving ordering constraints for lexicographic , recursive path orders and for KBO are NPcomplete, see [1] [7] [9] [10] [16] [17]. However, to check if 1 orients l ! "
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ABSTRACT: We consider two decision problems related to the KnuthBendix 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 polynomialtime algorithm for orientability builds upon an algorithm for solving systems of homogeneous linear inequalities over integers. The polynomialtime 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 Pcomplete. Also we show that if a system is orientable using a realvalued instance of KBO, then it is also orientable using an integervalued instance of KBO.Information and Computation 11/2001; 183(2). DOI:10.1016/S08905401(03)00021X · 0.60 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 wellknown 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). "
Article: Constraint Solving for Term Orderings Compatible with Abelian Semigroups, Monoids and Groups
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ABSTRACT: It is crucial for the performance of ordered resolution or paramodulationbased 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 Ecompatible, 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 wellfounded Ecompatible ordering for these E, the constraint satisfiability problem is NPhard even for conjunctions of inequations.Constraints 10/2001; DOI:10.1023/B:CONS.0000036021.31386.cc · 1.15 Impact Factor 
 "The KnuthBendix order is used in the stateoftheart 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 KnuthBendix ordering constraints was proved only recently in [Korovin and Voronkov 2000]. "
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ABSTRACT: We show that the problem of solving KnuthBendix ordering constraints is NPcomplete, as a corollary we show that the existential firstorder theory of any term algebra with the KnuthBendix ordering is NPcomplete.ACM Transactions on Computational Logic 12/2000; 6(2). DOI:10.1145/1055686.1055692 · 0.64 Impact Factor
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