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35
SAT Solving for Termination Analysis with Polynomial Interpretations
, 2007
"... Polynomial interpretations are one of the most popular techniques for automated termination analysis and the search for such interpretations is a main bottleneck in most termination provers. We show that one can obtain speedups in orders of magnitude by encoding this task as a SAT problem and by a ..."
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Cited by 52 (23 self)
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Polynomial interpretations are one of the most popular techniques for automated termination analysis and the search for such interpretations is a main bottleneck in most termination provers. We show that one can obtain speedups in orders of magnitude by encoding this task as a SAT problem and by applying modern SAT solvers.
A SATBased Approach to Size Change Termination with Global Ranking Functions
"... Abstract. We describe a new approach to proving termination with size change graphs. This is the first decision procedure for size change termination (SCT) which makes direct use of global ranking functions. It handles a welldefined and significant subset of SCT instances, designed to be amenable t ..."
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Cited by 20 (8 self)
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Abstract. We describe a new approach to proving termination with size change graphs. This is the first decision procedure for size change termination (SCT) which makes direct use of global ranking functions. It handles a welldefined and significant subset of SCT instances, designed to be amenable to a SATbased solution. We have implemented the approach using a stateoftheart Boolean satisfaction solver. Experimentation indicates that the approach is a viable alternative to the complete SCT decision procedure based on closure computation and local ranking functions. Our approach has the extra benefit of producing an explicit witness to prove termination in the form of a global ranking function. 1
Sat solving for argument filterings
 In Logic for Programming, Artificial Intelligence and Reasoning (LPAR
, 2006
"... Abstract. This paper introduces a propositional encoding for lexicographic path orders in connection with dependency pairs. This facilitates the application of SAT solvers for termination analysis of term rewrite systems based on the dependency pair method. We address two main interrelated issues a ..."
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Cited by 18 (10 self)
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Abstract. This paper introduces a propositional encoding for lexicographic path orders in connection with dependency pairs. This facilitates the application of SAT solvers for termination analysis of term rewrite systems based on the dependency pair method. We address two main interrelated issues and encode them as satisfiability problems of propositional formulas that can be efficiently handled by SAT solving: (1) the combined search for a lexicographic path order together with an argument filtering to orient a set of inequalities; and (2) how the choice of the argument filtering influences the set of inequalities that have to be oriented. We have implemented our contributions in the termination prover AProVE. Extensive experiments show that by our encoding and the application of SAT solvers one obtains speedups in orders of magnitude as well as increased termination proving power. 1
Proving Termination of ContextSensitive Rewriting with MUTERM
, 2007
"... Contextsensitive rewriting (CSR) is a restriction of rewriting which forbids reductions on selected arguments of functions. Proving termination of CSR is an interesting problem with several applications in the fields of term rewriting and programming languages. Several methods have been developed f ..."
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Cited by 17 (15 self)
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Contextsensitive rewriting (CSR) is a restriction of rewriting which forbids reductions on selected arguments of functions. Proving termination of CSR is an interesting problem with several applications in the fields of term rewriting and programming languages. Several methods have been developed for proving termination of CSR. The new version of MUTERM which we present here implements all currently known techniques. Furthermore, we show how to combine them to furnish MUTERM with an expert which is able to automatically perform the termination proofs. Finally, we provide a first experimental evaluation of the tool.
Constraints for argument filterings
 In Proc. of the 8th International Workshop on Termination
, 2006
"... Abstract. The dependency pair method is a powerful method for automatically proving termination of rewrite systems. When used with traditional simplification orders like LPO and KBO, argument filterings play a key role. In this paper we propose an encoding of argument filterings in propositional log ..."
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Cited by 11 (5 self)
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Abstract. The dependency pair method is a powerful method for automatically proving termination of rewrite systems. When used with traditional simplification orders like LPO and KBO, argument filterings play a key role. In this paper we propose an encoding of argument filterings in propositional logic. By incorporating propositional encodings of simplification orders, the search for suitable argument filterings is turned into a satisfiability problem. Preliminary experimental results show that our logicbased approach is significantly faster than existing implementations. 1
Satisfiability of Nonlinear (Ir)rational Arithmetic
 16th International Conference on Logic for Programming, Artificial Intelligence and Reasoning, LPAR’10
, 2010
"... Abstract. We present a novel way for reasoning about (possibly ir)rational quantifierfree nonlinear arithmetic by a reduction to SAT/SMT. The approach is incomplete and dedicated to satisfiable instances only but is able to produce models for satisfiable problems quickly. These characteristics s ..."
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Cited by 10 (1 self)
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Abstract. We present a novel way for reasoning about (possibly ir)rational quantifierfree nonlinear arithmetic by a reduction to SAT/SMT. The approach is incomplete and dedicated to satisfiable instances only but is able to produce models for satisfiable problems quickly. These characteristics suffice for applications such as termination analysis of rewrite systems. Our prototype implementation, called MiniSmt, is made freely available. Extensive experiments show that it outperforms current SMT solvers especially on rational and irrational domains.
Satisfying KBO Constraints
 In Proc. RTA ’07, LNCS 4533
, 2007
"... Abstract. This paper presents two new approaches to prove termination of rewrite systems with the KnuthBendix order efficiently. The constraints for the weight function and for the precedence are encoded in (pseudo)propositional logic and the resulting formula is tested for satisfiability. Any sat ..."
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Cited by 9 (3 self)
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Abstract. This paper presents two new approaches to prove termination of rewrite systems with the KnuthBendix order efficiently. The constraints for the weight function and for the precedence are encoded in (pseudo)propositional logic and the resulting formula is tested for satisfiability. Any satisfying assignment represents a weight function and a precedence such that the induced KnuthBendix order orients the rules of the encoded rewrite system from left to right. 1
Synthesizing shortest linear straightline programs over GF(2) using SAT
 In Proc. SAT ’10, volume 6175 of LNCS
, 2010
"... Abstract. Nontrivial linear straightline programs over the Galois field of two elements occur frequently in applications such as encryption or highperformance computing. Finding the shortest linear straightline program for a given set of linear forms is known to be MaxSNPcomplete, i.e., there i ..."
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Cited by 8 (1 self)
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Abstract. Nontrivial linear straightline programs over the Galois field of two elements occur frequently in applications such as encryption or highperformance computing. Finding the shortest linear straightline program for a given set of linear forms is known to be MaxSNPcomplete, i.e., there is no ǫapproximation for the problem unless P = NP. This paper presents a nonapproximative approach for finding the shortest linear straightline program. In other words, we show how to search for a circuit of XOR gates with the minimal number of such gates. The approach is based on a reduction of the associated decision problem (“Is there a program of length k?”) to satisfiability of propositional logic. Using modern SAT solvers, optimal solutions to interesting problem instances can be obtained. 1
SAT Solving for Termination Proofs with Recursive Path Orders and Dependency Pairs
"... This paper introduces a propositional encoding for recursive path orders (RPO), in connection with dependency pairs. Hence, we capture in a uniform setting all common instances of RPO, i.e., lexicographic path orders (LPO), multiset path orders (MPO), and lexicographic path orders with status (LPO ..."
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Cited by 8 (2 self)
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This paper introduces a propositional encoding for recursive path orders (RPO), in connection with dependency pairs. Hence, we capture in a uniform setting all common instances of RPO, i.e., lexicographic path orders (LPO), multiset path orders (MPO), and lexicographic path orders with status (LPOS). This facilitates the application of SAT solvers for termination analysis of term rewrite systems (TRSs). We address four main interrelated issues and show how to encode them as satisfiability problems of propositional formulas that can be efficiently handled by SAT solving: (A) the lexicographic comparison w.r.t. a permutation of the arguments; (B) the multiset extension of a base order; (C) the combined search for a path order together with an argument filter to orient a set of inequalities; and (D) how the choice of the argument filter influences the set of inequalities that have to be oriented (socalled usable rules). We have implemented our contributions in the termination prover AProVE. Extensive experiments show that by our encoding and the application of SAT solvers one obtains speedups in orders of magnitude as well as increased termination proving power.
Decreasing diagrams and relative termination
 In IJCAR. LNCS
"... Abstract. In this paper we use the decreasing diagrams technique to show that a leftlinear term rewrite system R is confluent if all its critical pairs are joinable and the critical pair steps are relatively terminating with respect to R. We further show how to encode the rulelabeling heuristic fo ..."
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Cited by 7 (1 self)
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Abstract. In this paper we use the decreasing diagrams technique to show that a leftlinear term rewrite system R is confluent if all its critical pairs are joinable and the critical pair steps are relatively terminating with respect to R. We further show how to encode the rulelabeling heuristic for decreasing diagrams as a satisfiability problem. Experimental data for both methods are presented. 1