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20
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 43 (30 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 15 (4 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
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 9 (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 "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
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 7 (1 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.
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 4 (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 4 (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
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.
Decidable CallbyNeed Computations in Term Rewriting
, 2004
"... The theorem of Huet and Lévy stating that for orthogonal rewrite systems (i) every reducible term contains a needed redex and (ii) repeated contraction of needed redexes results in a normal form if the term under consideration has a normal form, forms the basis of all results on optimal normalizing ..."
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Cited by 2 (2 self)
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The theorem of Huet and Lévy stating that for orthogonal rewrite systems (i) every reducible term contains a needed redex and (ii) repeated contraction of needed redexes results in a normal form if the term under consideration has a normal form, forms the basis of all results on optimal normalizing strategies for orthogonal rewrite systems. However, needed redexes are not computable in general. In the paper we show how the use of approximations and elementary tree automata techniques allows one to obtain decidable conditions in a simple and elegant way. Surprisingly, by avoiding complicated concepts like index and sequentiality we are able to cover much larger classes of rewrite systems. We also study modularity aspects of the classes in our hierarchy. It turns out that none of the classes is preserved under signature extension. By imposing various conditions we recover the preservation under signature extension. By imposing some more conditions we are able to strengthen the signature extension results to modularity for disjoint and constructorsharing combinations.
Levels of Undecidability in Rewriting
, 2011
"... Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This classification gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarc ..."
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Cited by 2 (1 self)
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Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This classification gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas, and continuing into the analytic hierarchy, where quantification over function variables is allowed. In this paper we give an overview of how the main properties of first order term rewriting systems are classified in these hierarchies. We consider properties related to normalization (strong normalization, weak normalization and dependency problems) and properties related to confluence (confluence, local confluence and the unique normal form property). For all of these we distinguish between the single term version and the uniform version. Where appropriate, we also distinguish between ground and open terms. Most uniform properties are Π 0 2complete. The particular problem of local confluence turns out to be Π 0 2complete for ground terms, but only Σ 0 1complete (and thereby recursively enumerable) for open terms. The most surprising result concerns dependency pair problems without minimality flag: we prove this problem to be Π 1 1complete, hence not in the arithmetical hierarchy, but properly in the analytic hierarchy. Some of our results are new or have appeared in our earlier publications [35, 7]. Others are based on folklore constructions, and are included for completeness as their precise classifications have hardly been noticed previously.