solving for argument filterings (2006)

by M Codish, P Schneider-Kamp, V Lagoon, R Thiemann, J SAT Giesl
Venue:In: Proc. 13th International Conference on Logic for Programming, Artificial Intelligence and Reasoning, LNAI
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SAT Solving for Termination Analysis with Polynomial Interpretations

by Carsten Fuhs, Jürgen Giesl, Aart Middeldorp, Peter Schneider-Kamp, René Thiemann, Harald Zankl , 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 ..."
Abstract - Cited by 54 (22 self) - Add to MetaCart
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.
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The termination competition

by Claude Marché, Hans Zantema - In Proc. RTA ’07, LNCS 4533 , 2007
"... Abstract. Since 2004, a Termination Competition is organized every year. This competition boosted a lot the development of automatic termination tools, but also the design of new techniques for proving termination. We present the background, results, and conclusions of the three first editions, and ..."
Abstract - Cited by 23 (1 self) - Add to MetaCart
Abstract. Since 2004, a Termination Competition is organized every year. This competition boosted a lot the development of automatic termination tools, but also the design of new techniques for proving termination. We present the background, results, and conclusions of the three first editions, and discuss perspectives and challenges for the future. 1 Motivation and
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Proving Termination using Recursive Path Orders and SAT solving

by Peter Schneider-Kamp, René Thiemann, Elena Annov, Michael Codish, Jürgen Giesl - IN PROC. FROCOS ’07, LNAI 4720 , 2007
"... We introduce a propositional encoding of the recursive path order with status (RPO). RPO is a combination of a multiset path order and a lexicographic path order which considers permutations of the arguments in the lexicographic comparison. Our encoding allows us to apply SAT solvers in order to d ..."
Abstract - Cited by 17 (8 self) - Add to MetaCart
We introduce a propositional encoding of the recursive path order with status (RPO). RPO is a combination of a multiset path order and a lexicographic path order which considers permutations of the arguments in the lexicographic comparison. Our encoding allows us to apply SAT solvers in order to determine whether a given term rewrite system is RPO-terminating. Furthermore, to apply RPO within the dependency pair framework, we combined our novel encoding for RPO with an existing encoding for argument filters. We implemented our contributions in the termination prover AProVE. Our experiments show that due to our encoding, combining termination provers with SAT solvers improves the performance of RPO-implementations by orders of magnitude.
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Satisfying KBO Constraints

by Harald Zankl, Aart Middeldorp - In Proc. RTA ’07, LNCS 4533 , 2007
"... Abstract. This paper presents two new approaches to prove termination of rewrite systems with the Knuth-Bendix 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 ..."
Abstract - Cited by 9 (3 self) - Add to MetaCart
Abstract. This paper presents two new approaches to prove termination of rewrite systems with the Knuth-Bendix 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 Knuth-Bendix order orients the rules of the encoded rewrite system from left to right. 1
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SAT Solving for Termination Proofs with Recursive Path Orders and Dependency Pairs

by Michael Codish, Jürgen Giesl, Peter Schneider-Kamp, René Thiemann
"... 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 ..."
Abstract - Cited by 8 (2 self) - Add to MetaCart
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 inter-related 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 (so-called 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.

KBO Orientability

by Harald Zankl, Nao Hirokawa, Aart Middeldorp - J AUTOM REASONING , 2009
"... This article presents three new approaches to prove termination of rewrite systems with the Knuth-Bendix order efficiently. The constraints for the weight function and for the precedence are encoded in (pseudo-)propositional logic or linear arithmetic and the resulting formula is tested for satisfia ..."
Abstract - Cited by 8 (3 self) - Add to MetaCart
This article presents three new approaches to prove termination of rewrite systems with the Knuth-Bendix order efficiently. The constraints for the weight function and for the precedence are encoded in (pseudo-)propositional logic or linear arithmetic and the resulting formula is tested for satisfiability using dedicated solvers. Any satisfying assignment represents a weight function and a precedence such that the induced Knuth-Bendix order orients the rules of the encoded rewrite system from left to right. This means that in contrast to the dedicated methods our approach does not directly solve the problem but transforms it to equivalent formulations for which sophisticated back-ends exist. In order to make all approaches complete we present a method to compute upper bounds on the weights. Furthermore, our encodings take dependency pairs into account to increase the applicability of the order.

Predictive labeling with dependency pairs using SAT

by Adam Koprowski, Aart Middeldorp - in: Proc. 21st CADE, LNAI 4603, 2007
"... Abstract. This paper combines predictive labeling with dependency pairs and reports on its implementation. Our starting point is the method of proving termination of rewrite systems using semantic labeling with infinite models in combination with lexicographic path orders. We replace semantic labeli ..."
Abstract - Cited by 7 (4 self) - Add to MetaCart
Abstract. This paper combines predictive labeling with dependency pairs and reports on its implementation. Our starting point is the method of proving termination of rewrite systems using semantic labeling with infinite models in combination with lexicographic path orders. We replace semantic labeling with predictive labeling to weaken the quasi-model constraints and we combine it with dependency pairs (usable rules and argument filtering) to increase the power of the method. Encoding the resulting search problem as a propositional satisfiability problem and calling a state-of-the-art SAT solver yields a powerful technique for proving termination automatically. 1
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Unifying the Knuth-Bendix, recursive path and polynomial orders

by Akihisa Yamada, Toshiki Sakabe, Keiichirou Kusakari - In Proc. PPDP ’13 , 2013
"... 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 ma ..."
Abstract - Cited by 4 (2 self) - Add to MetaCart
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.
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A SAT-based implementation for RPO termination

by Elena Annov, Michael Codish, René Thiemann, Peter Schneider-Kamp, Jürgen Giesl - LOUISIANA TECHNICAL UNIVERSITY , 2000
"... This paper introduces a propositional encoding of the recursive path order (RPO) on terms which is a combination of a multiset path order and a lexicographic path order which considers permutations of the arguments in the lexicographic comparison. The proposed encoding allows us to use SAT solvers ..."
Abstract - Cited by 4 (2 self) - Add to MetaCart
This paper introduces a propositional encoding of the recursive path order (RPO) on terms which is a combination of a multiset path order and a lexicographic path order which considers permutations of the arguments in the lexicographic comparison. The proposed encoding allows us to use SAT solvers in order to determine whether a given term rewrite system is RPO terminating. An implementation is described.
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Lazy Abstraction for Size-Change Termination

by Michael Codish, Carsten Fuhs, Jürgen Giesl, Peter Schneider-Kamp , 2010
"... Size-change termination is a widely used means of proving termination where source programs are first abstracted to size-change graphs which are then analyzed to determine if they satisfy the sizechange termination property. Here, the choice of the abstraction is crucial to the success of the meth ..."
Abstract - Cited by 3 (0 self) - Add to MetaCart
Size-change termination is a widely used means of proving termination where source programs are first abstracted to size-change graphs which are then analyzed to determine if they satisfy the sizechange termination property. Here, the choice of the abstraction is crucial to the success of the method, and it is an open problem how to choose an abstraction such that no critical loss of precision occurs. This paper shows how to couple the search for a suitable abstraction and the test for size-change termination via an encoding to a single SAT instance. In this way, the problem of choosing the right abstraction is solved en passant by a SAT solver. We show that for the setting of term rewriting, the integration of this approach into the dependency pair framework works smoothly and gives rise to a new class of size-change reduction pairs. We implemented size-change reduction pairs in the termination prover AProVE and evaluated their usefulness in extensive experiments.
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