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On the recursive decomposition ordering with lexicographical status and other related orderings
 Journal of Automated Reasoning
, 1990
"... This paper studies three orderings, useful in theorem proving, especially for proving termination of term rewriting systems: the recursive decomposition ordering with status, the recursive path ordering with status and the closure ordering. It proves the transitivity of the recursive path ordering ..."
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This paper studies three orderings, useful in theorem proving, especially for proving termination of term rewriting systems: the recursive decomposition ordering with status, the recursive path ordering with status and the closure ordering. It proves the transitivity of the recursive path ordering, the strict inclusion of the recursive path ordering in the recursive decomposition ordering, the totality of the recursive path ordering { therefore of the recursive decomposition ordering {, the strict inclusion of the recursive decomposition ordering in the closure ordering and the stability of the closure ordering by instanciation. 1
Knowledge Representation and Classical Logic
, 2007
"... Mathematical logicians had developed the art of formalizing declarative knowledge long before the advent of the computer age. But they were interested primarily in formalizing mathematics. Because of the important role of nonmathematical knowledge in AI, their emphasis was too narrow from the perspe ..."
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Mathematical logicians had developed the art of formalizing declarative knowledge long before the advent of the computer age. But they were interested primarily in formalizing mathematics. Because of the important role of nonmathematical knowledge in AI, their emphasis was too narrow from the perspective of knowledge representation, their formal languages were not sufficiently expressive. On the other hand, most logicians were not concerned about the possibility of automated reasoning; from the perspective of knowledge representation, they were often too generous in the choice of syntactic constructs. In spite of these differences, classical mathematical logic has exerted significant influence on knowledge representation research, and it is appropriate to begin this handbook with a discussion of the relationship between these fields. The language of classical logic that is most widely used in the theory of knowledge representation is the language of firstorder (predicate) formulas. These are the formulas that John McCarthy proposed to use for representing declarative knowledge in his advice taker paper [176], and Alan Robinson proposed to prove automatically using resolution [236]. Propositional logic is, of course, the most important subset of firstorder logic; recent
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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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.
Combinations of Simplifying Conditional Term Rewriting Systems
 In Proceedings of the 3rd International Workshop on Conditional Term Rewriting Systems
, 1992
"... . A conditional term rewriting system (CTRS) is called simplifying if there exists a simplification ordering ? on terms such that the lefthand side of any rewrite rule is greater than the righthand side and the terms occurring in the conditions of that rule. If a simplifying join CTRS consists of ..."
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. A conditional term rewriting system (CTRS) is called simplifying if there exists a simplification ordering ? on terms such that the lefthand side of any rewrite rule is greater than the righthand side and the terms occurring in the conditions of that rule. If a simplifying join CTRS consists of finitely many rules, it is terminating and the applicability of a rewrite rule is decidable by recursively reducing the terms in the conditions. Consider two finite CTRSs R1 and R2 which may share constructors (symbols that do not occur at the root position of the lefthand side of any rewrite rule) but no other function symbols. It will be shown that the combined CTRS R = R1[R2 is simplifying if and only if R1 and R2 are simplifying. Moreover, confluence is a modular property of finite simplifying join CTRSs. 1 Introduction During the past decade, term rewriting has gained an enormous importance in fields of computer science concerned with symbolic manipulation. Among others, it may be vie...
Recursively Defined (Quasi) Orders on Terms
, 1997
"... We study the problems involved in the recursive definition of (quasi) orders on terms, focussing on the question of establishing welldefinedness, and the properties required for partial and quasiorders: irreflexivity and transitivity, and reflexivity and transitivity, respectively. These propertie ..."
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We study the problems involved in the recursive definition of (quasi) orders on terms, focussing on the question of establishing welldefinedness, and the properties required for partial and quasiorders: irreflexivity and transitivity, and reflexivity and transitivity, respectively. These properties are in general difficult to establish and this has in many cases come down in the literature as folklore results. Here we present a general scheme that allows us to show that relations are welldefined and represent partial or quasiorders. Known path orders as semantic, recursive and lexicographic path order as well as KnuthBendix order fit into the scheme. Additionally we will also discuss how to obtain other properties commonly found in term orders (amongst which wellfoundedness) as an integrated feature of the scheme. Keywords: Path Orders, term rewriting, semantic path order , recursive path order , lexicographic path order, KnuthBendix order . Contents 1 Introduction 2 2 Prelimi...