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Termination of Term Rewriting Using Dependency Pairs
 Comput. Sci
, 2000
"... We present techniques to prove termination and innermost termination of term rewriting systems automatically. In contrast to previous approaches, we do not compare left and righthand sides of rewrite rules, but introduce the notion of dependency pairs to compare lefthand sides with special subter ..."
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Cited by 208 (46 self)
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We present techniques to prove termination and innermost termination of term rewriting systems automatically. In contrast to previous approaches, we do not compare left and righthand sides of rewrite rules, but introduce the notion of dependency pairs to compare lefthand sides with special subterms of the righthand sides. This results in a technique which allows to apply existing methods for automated termination proofs to term rewriting systems where they failed up to now. In particular, there are numerous term rewriting systems where a direct termination proof with simplification orderings is not possible, but in combination with our technique, wellknown simplification orderings (such as the recursive path ordering, polynomial orderings, or the KnuthBendix ordering) can now be used to prove termination automatically. Unlike previous methods, our technique for proving innermost termination automatically can also be applied to prove innermost termination of term rewriting systems that are not terminating. Moreover, as innermost termination implies termination for certain classes of term rewriting systems, this technique can also be used for termination proofs of such systems.
Argument Filtering Transformation
 In Proc. 1st PPDP, LNCS 1702
, 1999
"... To simplify the task of proving termination of term rewriting systems, several elimination methods, such as the dummy elimination, the distribution elimination, the general dummy elimination and the improved general dummy elimination, have been proposed. In this paper, we show that the argument lter ..."
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Cited by 39 (2 self)
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To simplify the task of proving termination of term rewriting systems, several elimination methods, such as the dummy elimination, the distribution elimination, the general dummy elimination and the improved general dummy elimination, have been proposed. In this paper, we show that the argument ltering method combining with the dependency pair technique is essential in all the above elimination methods. We present remarkable simple proofs for the soundness of these elimination methods based on this observation. Moreover, we propose a new elimination method, called the argument ltering transformation, which is not only more powerful than all the other elimination methods but also especially useful to make clear the essential relation hidden behind these methods.
Termination of Term Rewriting
, 2000
"... Contents 1 Introduction 2 2 Semantical methods 3 2.1 Wellfounded monotone algebras . . . . . . . . . . . . . . . . . . . . 3 2.2 Polynomial interpretations . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Polynomial interpretations modulo AC . . . . . . . . . . . . . . . . . 13 2.4 Lexicograp ..."
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Cited by 26 (6 self)
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Contents 1 Introduction 2 2 Semantical methods 3 2.1 Wellfounded monotone algebras . . . . . . . . . . . . . . . . . . . . 3 2.2 Polynomial interpretations . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Polynomial interpretations modulo AC . . . . . . . . . . . . . . . . . 13 2.4 Lexicographic combinations . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 A hierarchy of termination 17 3.1 Simple termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Total termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3 The hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4 Syntactical methods 25 4.1 Recursive path order . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2 Justi cation of recursive path order . . . . . . . . . . . . . . . . . . . 30 4.3 Extensions of recursive path order . . . . . . . . . . . . . . . . . . . 36 4.
Simple Termination of Rewrite Systems
 Theoretical Computer Science
, 1997
"... In this paper we investigate the concept of simple termination. A term rewriting system is called simply terminating if its termination can be proved by means of a simplification order. The basic ingredient of a simplification order is the subterm property, but in the literature two different defini ..."
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Cited by 16 (2 self)
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In this paper we investigate the concept of simple termination. A term rewriting system is called simply terminating if its termination can be proved by means of a simplification order. The basic ingredient of a simplification order is the subterm property, but in the literature two different definitions are given: one based on (strict) partial orders and another one based on preorders (or quasiorders). We argue that there is no reason to choose the second one, while the first one has certain advantages. Simplification orders are known to be wellfounded orders on terms over a finite signature. This important result no longer holds if we consider infinite signatures. Nevertheless, wellknown simplification orders like the recursive path order are also wellfounded on terms over infinite signatures, provided the underlying precedence is wellfounded. We propose a new definition of simplification order, which coincides with the old one (based on partial orders) in case of finite signatu...
Eliminating Dummy Elimination
 IN PROCEEDINGS OF THE 17TH INTERNATIONAL CONFERENCE ON AUTOMATED DEDUCTION, CADE17
, 2000
"... This paper is concerned with methods that automatically prove termination of term rewrite systems. The aim of dummy elimination, a method to prove termination introduced by Ferreira and Zantema, is to transform a given rewrite system into a rewrite system whose termination is easier to prove. We ..."
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Cited by 5 (3 self)
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This paper is concerned with methods that automatically prove termination of term rewrite systems. The aim of dummy elimination, a method to prove termination introduced by Ferreira and Zantema, is to transform a given rewrite system into a rewrite system whose termination is easier to prove. We show that dummy elimination is subsumed by the more recent dependency pair method of Arts and Giesl. More precisely, if dummy elimination succeeds in transforming a rewrite system into a socalled simply terminating rewrite system then termination of the given rewrite system can be directly proved by the dependency pair technique. Even stronger, using dummy elimination as a preprocessing step to the dependency pair technique does not have any advantages either. We show that to a large extent these results also hold for the argument ltering transformation of Kusakari et al.
Termination, ACTermination and Dependency Pairs of Term Rewriting Systems
 Ph.D. thesis, JAIST
, 2000
"... Copyright c ○ 2000 by Keiichirou KUSAKARI Recently, Arts and Giesl introduced the notion of dependency pairs, which gives effective methods for proving termination of term rewriting systems (TRSs). In this thesis, we extend the notion of dependency pairs to ACTRSs, and introduce new methods for eff ..."
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Cited by 5 (0 self)
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Copyright c ○ 2000 by Keiichirou KUSAKARI Recently, Arts and Giesl introduced the notion of dependency pairs, which gives effective methods for proving termination of term rewriting systems (TRSs). In this thesis, we extend the notion of dependency pairs to ACTRSs, and introduce new methods for effectively proving ACtermination. Since it is impossible to directly apply the notion of dependency pairs to ACTRSs, we introduce the head parts in terms and show an analogy between the root positions in infinite reduction sequences by TRSs and the head positions in those by ACTRSs. Indeed, this analogy is essential for the extension of dependency pairs to ACTRSs. Based on this analogy, we define ACdependency pairs. To simplify the task of proving termination and ACtermination, several elimination transformations such as the dummy elimination, the distribution elimination, the general dummy elimination and the improved general dummy elimination, have been proposed. In this thesis, we show that the argument filtering method combined with the ACdependency pair technique is essential in all the elimination transformations above. We present remarkable simple proofs for the soundness of these elimination transformations based on this observation. Moreover, we propose a new elimination transformation, called the argument filtering transformation, which is not only more powerful than all the other elimination transformations but also especially useful to make clear an essential relationship among them.
Reducing righthand sides for termination
 In Processes, Terms and Cycles, LNCS 3838:173–197
, 2005
"... Abstract. We propose two transformations on term rewrite systems (TRSs) based on reducing right hand sides: one related to the transformation order and a variant of dummy elimination. Under mild conditions we prove that the transformed system is terminating if and only if the original one is termina ..."
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Cited by 5 (1 self)
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Abstract. We propose two transformations on term rewrite systems (TRSs) based on reducing right hand sides: one related to the transformation order and a variant of dummy elimination. Under mild conditions we prove that the transformed system is terminating if and only if the original one is terminating. Both transformations are very easy to implement, and make it much easier to prove termination of some TRSs automatically. 1
Relative Undecidability in Term Rewriting
, 1996
"... . For two hierarchies of properties of term rewriting systems related to confluence and termination, respectively, we prove relative undecidability : for implications X ) Y in the hierarchies the property X is undecidable for term rewriting systems satisfying Y . 1 Introduction In this paper we c ..."
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Cited by 3 (3 self)
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. For two hierarchies of properties of term rewriting systems related to confluence and termination, respectively, we prove relative undecidability : for implications X ) Y in the hierarchies the property X is undecidable for term rewriting systems satisfying Y . 1 Introduction In this paper we consider finite term rewriting systems (TRSs) over finite signatures. For these systems termination and confluence are desired properties that are sometimes very hard to prove. Classical results ([7, 9]) state that they are undecidable: no decision procedure exists getting an arbitrary finite TRS as input and giving as output whether the TRS is terminating (confluent) or not. In this paper we don't consider only termination and confluence, but also a number of related properties. For termination they are linearly ordered by implication: PT ) !T ) TT ) ST ) NSE ) SN ) NL ) AC The acronyms stand for polynomial termination (PT), !termination (!T), total termination (TT), simple termination (ST)...
Relative Undecidability in the Termination Hierarchy of Single Rewrite Rules
 Proceedings of the Colloquium on Trees in Algebra and Programming, Lecture Notes in Computer Science
, 1997
"... . For a hierarchy of properties of term rewriting systems, related to termination, we prove relative undecidability even in the case of single rewrite rules: for implications X ) Y in the hierarchy the property X is undecidable for rewrite rules satisfying Y . 1 Introduction A fundamental problem ..."
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Cited by 3 (2 self)
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. For a hierarchy of properties of term rewriting systems, related to termination, we prove relative undecidability even in the case of single rewrite rules: for implications X ) Y in the hierarchy the property X is undecidable for rewrite rules satisfying Y . 1 Introduction A fundamental problem in the theory of term rewriting is the detection of termination: for a fixed system of rewrite rules, determine whether there are infinite rewrite sequences. Besides termination a number of related properties are of interest, linearly ordered by implication: polynomial termination ) !termination ) total termination ) simple termination ) nonselfembeddingness ) termination ) nonloopingness ) acyclicity We call this the termination hierarchy. Apart from polynomial termination, all properties in the termination hierarchy are known to be undecidable ([11, 15, 13, 18, 8, 9]). In [9] we showed the stronger result of relative undecidability : for all implications X ) Y in the termination hier...
OmegaTermination is Undecidable for Totally Terminating Term Rewriting Systems
 Journal of Symbolic Computation
, 1996
"... We give a complete proof of the fact that the following problem is undecidable: Given: A term rewriting system, where the termination of its rewrite relation is provable by a total reduction order on ground terms, Wanted: Does there exist a strictly monotonic interpretation in the positive integer ..."
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Cited by 3 (2 self)
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We give a complete proof of the fact that the following problem is undecidable: Given: A term rewriting system, where the termination of its rewrite relation is provable by a total reduction order on ground terms, Wanted: Does there exist a strictly monotonic interpretation in the positive integers that proves termination? Keywords: term rewriting, termination, total termination, omegatermination, termination hierarchy, termination type 1 Introduction Termination of a term rewriting system (TRS), i.e. the nonexistence of an infinite rewrite reduction, is one of the key notions in term rewriting. It is the basis of a number of decision algorithms for properties undecidable in the general case. It is known that termination is undecidable for TRSs, even for the restricted case of TRS with only unary function symbols [7], and for the case of onerule TRSs [2]. The following true hierarchy of TRSs is called the termination hierarchy [12]. termination ) nonselfembedding ) simple te...