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30
ContextSensitive Rewriting Strategies
, 1997
"... Contextsensitive rewriting is a simple restriction of rewriting which is formalized by imposing fixed restrictions on replacements. Such a restriction is given on a purely syntactic basis: it is (explicitly or automatically) specified on the arguments of symbols of the signature and inductively ..."
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Cited by 45 (32 self)
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Contextsensitive rewriting is a simple restriction of rewriting which is formalized by imposing fixed restrictions on replacements. Such a restriction is given on a purely syntactic basis: it is (explicitly or automatically) specified on the arguments of symbols of the signature and inductively extended to arbitrary positions of terms built from those symbols. Termination is not only preserved but usually improved and several methods have been developed to formally prove it. In this paper, we investigate the definition, properties, and use of contextsensitive rewriting strategies, i.e., particular, fixed sequences of contextsensitive rewriting steps. We study how to define them in order to obtain efficient computations and to ensure that contextsensitive computations terminate whenever possible. We give conditions enabling the use of these strategies for rootnormalization, normalization, and infinitary normalization. We show that this theory is suitable for formalizing ...
Approximating dependency graphs using tree automata techniques
 In Proc. IJCAR 2001, LNAI 2083
, 2001
"... Abstract. The dependency pair method of Arts and Giesl is the most powerful technique for proving termination of term rewrite systems automatically. We show that the method can be improved by using tree automata techniques to obtain better approximations of the dependency graph. This graph determine ..."
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Cited by 16 (5 self)
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Abstract. The dependency pair method of Arts and Giesl is the most powerful technique for proving termination of term rewrite systems automatically. We show that the method can be improved by using tree automata techniques to obtain better approximations of the dependency graph. This graph determines the ordering constraints that need to be solved in order to conclude termination. We further show that by using our approximations the dependency pair method provides a decision procedure for termination of rightground rewrite systems. 1
Closure of HedgeAutomata Languages by Hedge Rewriting
, 2008
"... We consider rewriting systems for unranked ordered terms, i.e. trees where the number of successors of a node is not determined by its label, and is not a priori bounded. The rewriting systems are defined such that variables in the rewrite rules can be substituted by hedges (sequences of terms) ins ..."
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Cited by 10 (2 self)
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We consider rewriting systems for unranked ordered terms, i.e. trees where the number of successors of a node is not determined by its label, and is not a priori bounded. The rewriting systems are defined such that variables in the rewrite rules can be substituted by hedges (sequences of terms) instead of just terms. Consequently, this notion of rewriting subsumes both standard term rewriting and word rewriting. We investigate some preservation properties for two classes of languages of unranked ordered terms under this generalization of term rewriting. The considered classes include languages of hedge automata (HA) and some extension (called CFHA) with contextfree languages in transitions, instead of regular languages. In particular, we show that the set of unranked terms reachable from a given HA language, using a so called inverse contextfree rewrite system, is a HA language. The proof, based on a HA completion procedure, reuses and combines known techniques with nontrivial adaptations. Moreover, we prove, with different techniques, that the closure of CFHA languages with respect to restricted contextfree rewrite systems, the symmetric case of the above rewrite systems, is a CFHA language. As a consequence, the problems of ground reachability and regular hedge model checking are decidable in both cases. We give several counter examples showing that we cannot relax the restrictions.
Layered transducing term rewriting system and its recognizability preserving property
 In: Proc. 13th RTA Conf., Copenhagen (Denmark
, 2000
"... A term rewriting system which effectively preserves recognizability (EPRTRS) has good mathematical properties. In this paper, a new subclass of TRSs, layered transducing TRSs (LTTRSs) is defined and its recognizability preserving property is discussed. The class of LTTRSs contains some EPRTRSs, ..."
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Cited by 10 (0 self)
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A term rewriting system which effectively preserves recognizability (EPRTRS) has good mathematical properties. In this paper, a new subclass of TRSs, layered transducing TRSs (LTTRSs) is defined and its recognizability preserving property is discussed. The class of LTTRSs contains some EPRTRSs, e.g., {f(x) → f(g(x))} which do not belong to any of the known decidable subclasses of EPRTRSs. Bottomup linear tree transducer, which is a wellknown computation model in the tree language theory, is a special case of LTTRS. We present a sufficient condition for an LTTRS to be an EPRTRS. Also reachability and joinability are shown to be decidable for LTTRSs. 1
The Confluence Problem for Flat TRSs
 in &quot;Proceedings of the 8th International Conference on Artificial Intelligence and Symbolic Computation (AISC’06
, 2006
"... Abstract. We prove that the properties of reachability, joinability and confluence are undecidable for flat TRSs. Here, a TRS is flat if the heights of the left and righthand sides of each rewrite rule are at most one. Key words: Term rewriting system, Decision problem, Confluence, Flat. 1 ..."
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Cited by 7 (2 self)
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Abstract. We prove that the properties of reachability, joinability and confluence are undecidable for flat TRSs. Here, a TRS is flat if the heights of the left and righthand sides of each rewrite rule are at most one. Key words: Term rewriting system, Decision problem, Confluence, Flat. 1
Decidability of termination for semiconstructor TRSs, leftlinear shallow TRSs and related systems
 Proc. of 17th Int’l Conference on Rewriting Techniqes and Applications, Seattle, (RTA2006), LNCS 4098
, 2006
"... We consider several classes of term rewriting systems and prove that termination is decidable for these classes. By showing the cycling property of infinite dependency chains, we prove that termination is decidable for semiconstructor case, which is a superclass of rightground TRSs. By analyzing ar ..."
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Cited by 6 (3 self)
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We consider several classes of term rewriting systems and prove that termination is decidable for these classes. By showing the cycling property of infinite dependency chains, we prove that termination is decidable for semiconstructor case, which is a superclass of rightground TRSs. By analyzing argument propagation cycles in the dependency graph, we show that termination is also decidable for leftlinear shallow TRSs. Moreover we extend these by combining these two techniques. 1
Innermost Reachability and Context Sensitive Reachability Properties are Decidable for Linear RightShallow Term Rewriting Systems
 Proc. 19th International Conference on Rewriting Techniques and Applications (RTA’08 ), Voronkov, A.(ed.), Hagenberg
, 2008
"... Abstract. A reachability problem is a problem used to decide whether s is reachable to t by R or not for a given two terms s, t and a term rewriting system R. Since it is known that this problem is undecidable, effort has been devoted to finding subclasses of term rewriting systems in which the reac ..."
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Cited by 5 (2 self)
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Abstract. A reachability problem is a problem used to decide whether s is reachable to t by R or not for a given two terms s, t and a term rewriting system R. Since it is known that this problem is undecidable, effort has been devoted to finding subclasses of term rewriting systems in which the reachability is decidable. However few works on decidability exist for innermost reduction strategy or contextsensitive rewriting. In this paper, we show that innermost reachability and contextsensitive reachability are decidable for linear rightshallow term rewriting systems. Our approach is based on the tree automata technique that is commonly used for analysis of reachability and its related properties. 1
Reduction strategies for leftlinear term rewriting systems
 Processes, Terms and Cycles: Steps on the Road to Infinity:Essays Dedicated to Jan Willem Klop on the Occasion of His 60th Birthday, volume 3838 of Lecture Notes in Computer Science
, 2005
"... Abstract. Huet and Lévy (1979) showed that needed reduction is a normalizing strategy for orthogonal (i.e., leftlinear and nonoverlapping) term rewriting systems. In order to obtain a decidable needed reduction strategy, they proposed the notion of strongly sequential approximation. Extending the ..."
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Cited by 4 (0 self)
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Abstract. Huet and Lévy (1979) showed that needed reduction is a normalizing strategy for orthogonal (i.e., leftlinear and nonoverlapping) term rewriting systems. In order to obtain a decidable needed reduction strategy, they proposed the notion of strongly sequential approximation. Extending their seminal work, several better decidable approximations of leftlinear term rewriting systems, for example, NV approximation, shallow approximation, growing approximation, etc., have been investigated in the literature. In all of these works, orthogonality is required to guarantee approximated decidable needed reductions are actually normalizing strategies. This paper extends these decidable normalizing strategies to leftlinear overlapping term rewriting systems. The key idea is the balanced weak ChurchRosser property. We prove that approximated external reduction is a computable normalizing strategy for the class of leftlinear term rewriting systems in which every critical pair can be joined with root balanced reductions. This class includes all weakly orthogonal leftnormal systems, for example, combinatory logic CL with the overlapping rules pred · (succ · x) → x and succ · (pred · x) → x, for which leftmostoutermost reduction is a computable normalizing strategy. 1
Proving termination of rewrite systems using bounds
 In Proc. 18th RTA, volume 4533 of LNCS
, 2007
"... Abstract. The use of automata techniques to prove the termination of string rewrite systems and leftlinear term rewrite systems is advocated by Geser et al. in a recent sequence of papers. We extend their work to nonleftlinear rewrite systems. The key to this extension is the introduction of so ..."
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Cited by 3 (1 self)
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Abstract. The use of automata techniques to prove the termination of string rewrite systems and leftlinear term rewrite systems is advocated by Geser et al. in a recent sequence of papers. We extend their work to nonleftlinear rewrite systems. The key to this extension is the introduction of socalled raise rules and the use of tree automata that are not quite deterministic. Furthermore, we present negative solutions to two open problems related to string rewrite systems. 1
Closure of Tree Automata Languages under Innermost Rewriting
, 2008
"... Preservation of regularity by a term rewriting system (TRS) states that the set of reachable terms from a tree automata (TA) language (aka regular term set) is also a TA language. It is an important and useful property, and there have been many works on identifying classes of TRS ensuring it; unfort ..."
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Cited by 2 (0 self)
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Preservation of regularity by a term rewriting system (TRS) states that the set of reachable terms from a tree automata (TA) language (aka regular term set) is also a TA language. It is an important and useful property, and there have been many works on identifying classes of TRS ensuring it; unfortunately, regularity is not preserved for restricted classes of TRS like shallow TRS. Nevertheless, this property has not been studied for important strategies of rewriting like the innermost strategy – which corresponds to the call by value computation of programming languages. We prove that the set of innermostreachable terms from a TA language by a shallow TRS is not necessarily regular, but it can be recognized by a TA with equality and disequality constraints between brothers. As a consequence we conclude decidability of regularity of the reachable set of terms from a TA language by innermost rewriting and shallow TRS. This result is in contrast with plain (not necessarily innermost) rewriting for which we prove undecidability. We also show that, like for plain rewriting, innermost rewriting with linear and rightshallow TRS preserves regularity.