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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.
A fully syntactic ACRPO
 Information and Computation
, 1999
"... . We present the first fully syntactic (i.e., noninterpretationbased) ACcompatible recursive path ordering (RPO). It is simple, and hence easy to implement, and its behaviour is intuitive as in the standard RPO. The ordering is ACtotal, and defined uniformly for both ground and nonground ter ..."
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Cited by 21 (4 self)
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. We present the first fully syntactic (i.e., noninterpretationbased) ACcompatible recursive path ordering (RPO). It is simple, and hence easy to implement, and its behaviour is intuitive as in the standard RPO. The ordering is ACtotal, and defined uniformly for both ground and nonground terms, as well as for partial precedences. More importantly, it is the first one that can deal incrementally with partial precedences, an aspect that is essential, together with its intuitive behaviour, for interactive applications like KnuthBendix completion. 1
Termination of AssociativeCommutative Rewriting by Dependency Pairs
 9th International Conference on Rewriting Techniques and Applications, volume 1379 of Lecture
, 1998
"... A new criterion for termination of rewriting has been described by Arts and Giesl in 1997. We show how this criterion can be generalized to rewriting modulo associativity and commutativity. We also show how one can build weak ACcompatible reduction orderings which may be used in this criterion. ..."
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Cited by 13 (1 self)
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A new criterion for termination of rewriting has been described by Arts and Giesl in 1997. We show how this criterion can be generalized to rewriting modulo associativity and commutativity. We also show how one can build weak ACcompatible reduction orderings which may be used in this criterion.
A total ACcompatible ordering based on RPO
 Theoretical Computer Science
, 1995
"... We define a simplification ordering on terms which is ACcompatible and total on nonAC equivalent ground terms, without any restrictions on the signature like the number of ACsymbols or free symbols. Unlike previous work by Narendran and Rusinowitch [NR91], our ACRPO ordering is not based on poly ..."
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Cited by 12 (7 self)
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We define a simplification ordering on terms which is ACcompatible and total on nonAC equivalent ground terms, without any restrictions on the signature like the number of ACsymbols or free symbols. Unlike previous work by Narendran and Rusinowitch [NR91], our ACRPO ordering is not based on polynomial interpretations, but on a simple extension of the wellknown RPO ordering (with a total (arbitrary) precedence on the function symbols). This solves an open question posed e.g. by Bachmair [Bac92]. A second difference is that this ordering is also defined on terms with variables, which makes it applicable in practice for complete theorem proving strategies with builtin ACunification and for orienting nonground rewrite systems. The ordering is defined in a simple way by means of rewrite rules, and can be easily implemented, since its main component is RPO. 1 Introduction Automated termination proofs are wellknown to be crucial for using rewritinglike methods in theorem proving an...
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.
Semantic Unification for Convergent Systems
, 1994
"... Equation solving is the process of nding a substitution of terms for variables that makes two terms equal in a given theory, while semantic uni cation is the process that generates a basis set of such unifying substitutions. A simpler variant of the problem is semantic matching, where the substituti ..."
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Cited by 4 (2 self)
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Equation solving is the process of nding a substitution of terms for variables that makes two terms equal in a given theory, while semantic uni cation is the process that generates a basis set of such unifying substitutions. A simpler variant of the problem is semantic matching, where the substitution is made in only one of the terms. Semantic uni cation and matching constitute an important component of theorem proving and programming language interpreters. In this thesis we formulate a uni cation procedure based on a system of transformation rules that looks at goals in a lazy, topdown fashion, and prove its soundness and completeness for equational theories described by convergent rewrite systems ( nite sets of equations that compute unique output values when applied from lefttoright to input values). We consider di erent variants of the system of transformation rules. We describe syntactic restrictions on the equations under which simpler sets of transformation rules are su cient for generating a complete set of semantic matchings. We show that our rstorder uni cation procedure, with slight modi cations, can be used to solve the satis ability problem in combinatory logic together with a convergent set of algebraic axioms, resulting in a complete higherorder uni cation procedure for the given algebra. We also provide transformation rules to handle sit
An ACCompatible KnuthBendix Order
"... We introduce a family of ACcompatible KnuthBendix simplification orders which are ACtotal on ground terms. Our orders preserve attractive features of the original KnuthBendix orders such as existence of a polynomialtime algorithm for comparing terms; computationally e#cient approximations, for ..."
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Cited by 4 (0 self)
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We introduce a family of ACcompatible KnuthBendix simplification orders which are ACtotal on ground terms. Our orders preserve attractive features of the original KnuthBendix orders such as existence of a polynomialtime algorithm for comparing terms; computationally e#cient approximations, for instance comparing weights of terms; and preference of light terms over heavy ones. This makes these orders especially suited for automated deduction where e#cient algorithms on orders are desirable.
A total ACcompatible ordering with RPO scheme
, 1997
"... . Like Kapur and Sivakumar in [KS97], we present an ACcompatible simplification ordering total on ground terms that follows the same scheme as the recursive path ordering (RPO). The first improvement with respect to their work is that our ordering has a simpler definition, and as a consequence we ca ..."
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Cited by 3 (1 self)
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. Like Kapur and Sivakumar in [KS97], we present an ACcompatible simplification ordering total on ground terms that follows the same scheme as the recursive path ordering (RPO). The first improvement with respect to their work is that our ordering has a simpler definition, and as a consequence we can obtain simpler proofs for the properties of the ordering and get a better understanding of the difficulties of finding this kind of orderings. But the main improvement is that, due to the simplicity of our definition, we can provide nontrivial extensions of the ordering to terms with variables, which is crucial in practice for proving termination of term rewriting systems modulo AC and for theorem proving strategies with builtin ACunification. 1 Introduction Rewritebased methods with builtin associativity and commutativity (AC) properties for some of the operators are wellknown to be crucial in theorem proving and programming. Therefore a lot of work has been done on the development...
AssociativeCommutative Reduction Orderings via HeadPreserving Interpretations
, 1995
"... We introduce a generic definition of reduction orderings on term algebras containing associativecommutative (hereafter denoted AC) operators. These orderings are compatible with the AC theory hence makes them suitable for use in deduction systems where AC operators are builtin. Furthermore, they ..."
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Cited by 2 (0 self)
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We introduce a generic definition of reduction orderings on term algebras containing associativecommutative (hereafter denoted AC) operators. These orderings are compatible with the AC theory hence makes them suitable for use in deduction systems where AC operators are builtin. Furthermore, they have the nice property of being total on AC classes of ground terms, a required property for example to avoid failure in ACcompletion, or to insure completeness of ordered strategies in firstorder theorem proving with builtin AC operators. We show that the two definitions already known of such total and ACcompatible orderings [24, 25] are actually instances of our definition. Finally, we find new such orderings which have more properties, first an ordering based on an integer polynomial interpretation, answering positively to a question left open by Narendran and Rusinowitch, and second an ordering which allow to orient the distributivity axiom in the usual way, answering positively to a ...
Combination of Compatible Reduction Orderings that are Total on Ground Terms (Extended Abstract)
 In 12th Ann. IEEE Symp. on Logic in Computer Science
, 1997
"... Franz Baader LuFg Theoretical Computer Science, RWTH Aachen Ahornstraße 55, 52074 Aachen, Germany email: baader@informatik.rwthaachen.de 1 Introduction Reduction orderings that are total on ground terms play an important role in many areas of automated deduction. For example, unfailing completio ..."
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Franz Baader LuFg Theoretical Computer Science, RWTH Aachen Ahornstraße 55, 52074 Aachen, Germany email: baader@informatik.rwthaachen.de 1 Introduction Reduction orderings that are total on ground terms play an important role in many areas of automated deduction. For example, unfailing completion [4]a variant of KnuthBendix completion that avoids failure due to incomparable critical pairspresupposes such an ordering. In addition, using a reduction ordering that is total on ground terms, one can show that any finite set of ground equations has a decidable word problem [13, 20]. It is very easy to obtain such orderings. Indeed, many of the standard methods for constructing reduction orderings yield orderings that are total on ground terms: both KnuthBendix orderings [12] and lexicographic path orderings [10] are total on ground terms if they are based on a total precedence ordering on the set of function symbols. Things become more complex if one is interested in reduction or...