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Natural termination
 Theoretical Computer Science
"... Abstract. We generalize the various path orderings and the conditions under which they work, and describe an implementation of this general ordering. We look at methods for proving termination of orthogonal systems and give a new solution to a problem of Zantema's. 1 ..."
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Cited by 83 (11 self)
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Abstract. We generalize the various path orderings and the conditions under which they work, and describe an implementation of this general ordering. We look at methods for proving termination of orthogonal systems and give a new solution to a problem of Zantema's. 1
Knowledge Representation and Classical Logic
"... 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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Cited by 10 (4 self)
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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
On the Relation Between Primitive Recursion, Schematization, and Divergence
 Proceeding 3rd Conference on Algebraic and Logic Programming, Volterra (Italy), volume 632 of Lecture Notes in Computer Science
, 1992
"... The paper presents a new schematization of infinite families of terms called the primal grammars, based on the notion of primitive recursive rewrite systems. This schematization is presented by a generating term and a canonical rewrite system. It is proved that the class of primal grammars covers co ..."
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Cited by 7 (2 self)
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The paper presents a new schematization of infinite families of terms called the primal grammars, based on the notion of primitive recursive rewrite systems. This schematization is presented by a generating term and a canonical rewrite system. It is proved that the class of primal grammars covers completely the class of crossed rewrite systems. This proof contains a construction of a primal grammar from a crossed rewrite system.
NonLooping String Rewriting
, 1996
"... Reductions des mots de la forme t ! + R utv, appelees boucles, sont la raison la plus frequente des reductions innies. Regardees comme un modele de comptation, reductions innies sont ingratuites pour quelle raison leur decouverte statique est importante. Ils existent des systemes de reecriture de ..."
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Cited by 7 (0 self)
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Reductions des mots de la forme t ! + R utv, appelees boucles, sont la raison la plus frequente des reductions innies. Regardees comme un modele de comptation, reductions innies sont ingratuites pour quelle raison leur decouverte statique est importante. Ils existent des systemes de reecriture des mots qui admettent des reductions innies quand meme ils ne admettent pas des boucles. Leur nontermination est particulierement dicile a decouvrer. Nous presentons des conditions necessaires pour l'existence de boucles et par cette maniere etablissons une methode a reconna^tre le cas dicile. Nous presontons en detail quatre criteria relevants: (i) l'existence de boucles est characteris e par l'existence de fermetures en avant boucles; (ii) elimination des poupees, une methode de transformation qui preserve le nontermination, preserve aussi l'existence de boucles; (iii) l'introduction des poupees, une methode de transformation assistant l'elimination des ...
Well behaved derivations in onerule semiThue systems
, 1995
"... This paper is a contribution to the investigation of the problem, does a given onerule semiThue system have an infinite derivation? It is an open question as to whether an algorithm exists for this problem, whose complement is sometimes called the uniform termination problem for onerule semiThue ..."
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Cited by 6 (1 self)
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This paper is a contribution to the investigation of the problem, does a given onerule semiThue system have an infinite derivation? It is an open question as to whether an algorithm exists for this problem, whose complement is sometimes called the uniform termination problem for onerule semiThue systems. It is well known that the corresponding problem for semiThue systems with finite sets of rules is undecidable. The importance of the general problem of termination in various types of computing systems is acknowledged by all. The more specific area of uniform termination of various rewriting systems is also important, as seen in the abundant literature, with many interesting results obtained from several points of view. In this category there are three references that are most relevant to this paper: [1] points out the importance of the uniform termination problem for certain practical computing problems in the area of theorem proving; [2] and [7] give good accounts of various termination problems; [7] has a particularly useful description of the peculiarities of termination problems for onerule semiThue systems among termination problems in general. The first stimulus for my own research in this area was the conference presentation of [7]. Two suitable general references on semiThue systems are [3] and [6]. There are only a few papers directed specifically at the unform termination problem for onerule semiThue systems. The first such seems to be Winfried Kurth's Ph.D. dissertation [4] (see also [5]), which contains an interesting broad introduction to the problem, as well as some farreaching preliminary results. One of these is a
Chain Properties of Rule Closures
 Formal Aspects of Computing
, 1990
"... This article presents an introduction to the generalization of the crossed rule approach to the detection of KnuthBendix completion procedure divergence. It introduces the closure chains, which are special rule closures constructed by means of particular substitution operations and operators, as a ..."
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Cited by 5 (2 self)
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This article presents an introduction to the generalization of the crossed rule approach to the detection of KnuthBendix completion procedure divergence. It introduces the closure chains, which are special rule closures constructed by means of particular substitution operations and operators, as a suitable formalism for a progress in this direction. Supporting substitution algebra is developed first, followed by considerations concerning rule closures in general, and concluded by investigation of closure chain properties.
Vademecum of Divergent Term Rewriting Systems
 CRIN, Research Report
, 1990
"... This paper presents two structural patterns to detect divergence of the completion procedure, followed by a detailed overview of dioeerent examples of divergent rewrite systems. Further it introduces five different empirical methods to avoid divergence, applicable during a session with a rewrite rul ..."
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Cited by 4 (1 self)
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This paper presents two structural patterns to detect divergence of the completion procedure, followed by a detailed overview of dioeerent examples of divergent rewrite systems. Further it introduces five different empirical methods to avoid divergence, applicable during a session with a rewrite rule laboratory.
Deciding Innermost Loops
 PROC. RTA '08
, 2008
"... We present the first method to disprove innermost termination of term rewrite systems automatically. To this end, we first develop a suitable notion of an innermost loop. Second, we show how to detect innermost loops: One can start with any technique amenable to find loops. Then our novel procedur ..."
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Cited by 4 (4 self)
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We present the first method to disprove innermost termination of term rewrite systems automatically. To this end, we first develop a suitable notion of an innermost loop. Second, we show how to detect innermost loops: One can start with any technique amenable to find loops. Then our novel procedure can be applied to decide whether a given loop is an innermost loop. We implemented and successfully evaluated our method in the termination prover AProVE.