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A Guide to LP, The Larch Prover
, 1991
"... This guide provides an introduction to LP (the Larch Prover), Release 2.2. It describes how LP can be used to axiomatize theories in a subset of multisorted firstorder logic and to provide assistance in proving theorems. It also contains a tutorial overview of the equational termrewriting technolo ..."
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Cited by 129 (6 self)
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This guide provides an introduction to LP (the Larch Prover), Release 2.2. It describes how LP can be used to axiomatize theories in a subset of multisorted firstorder logic and to provide assistance in proving theorems. It also contains a tutorial overview of the equational termrewriting technology that provides, along with induction rules and other usersupplied nonequational rules of inference, part of LP's inference engine.
Normalised Rewriting and Normalised Completion
, 1994
"... We introduce normalised rewriting, a new rewrite relation. It generalises former notions of rewriting modulo E, dropping some conditions on E. For example, E can now be the theory of identity, idempotency, the theory of Abelian groups, the theory of commutative rings. We give a new completion algor ..."
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Cited by 19 (2 self)
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We introduce normalised rewriting, a new rewrite relation. It generalises former notions of rewriting modulo E, dropping some conditions on E. For example, E can now be the theory of identity, idempotency, the theory of Abelian groups, the theory of commutative rings. We give a new completion algorithm for normalised rewriting. It contains as an instance the usual AC completion algorithm, but also the wellknown Buchberger's algorithm for computing standard bases of polynomial ideals. We investigate the particular case of completion of ground equations, In this case we prove by a uniform method that completion modulo E terminates, for some interesting E. As a consequence, we obtain the decidability of the word problem for some classes of equational theories. We give implementation results which shows the efficiency of normalised completion with respect to completion modulo AC. 1 Introduction Equational axioms are very common in most sciences, including computer science. Equations can ...
Modular & Incremental Proofs of ACTermination
 Journal of Symbolic Computation
, 2002
"... Recently, the framework of rewriting modules was proposed and provided modular and incremental termination criteria. In this paper, we extend these results to the important case of Associative and Commutative rewriting by means of ACdependency pairs. ..."
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Cited by 15 (3 self)
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Recently, the framework of rewriting modules was proposed and provided modular and incremental termination criteria. In this paper, we extend these results to the important case of Associative and Commutative rewriting by means of ACdependency pairs.
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.
Termination and Completion modulo Associativity, Commutativity and Identity
 Theoretical Computer Science
, 1992
"... Rewriting with associativity, commutativity and identity has been an open problem for a long time. In 1989, Baird, Peterson and Wilkerson introduced the notion of constrained rewriting, to avoid the problem of nontermination inherent to the use of identities. We build up on this idea in two ways: b ..."
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Cited by 11 (3 self)
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Rewriting with associativity, commutativity and identity has been an open problem for a long time. In 1989, Baird, Peterson and Wilkerson introduced the notion of constrained rewriting, to avoid the problem of nontermination inherent to the use of identities. We build up on this idea in two ways: by giving a complete set of rules for completion modulo these axioms; by showing how to build appropriate orderings for proving termination of constrained rewriting modulo associativity, commutativity and identity. 1 Introduction Equations are ubiquitous in mathematics and the sciences. Among the most common equations are associativity, commutativity and identity (existence of a neutral element). Rewriting is an efficient way of reasoning with equations, introduced by Knuth and Bendix [12]. When rewriting, equations are used in one direction chosen once and for all. Unfortunately, orientation alone is not a complete inference rule: given a set of equational axioms E, there may be equal terms...
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.
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 ...
A Guide to LP, The Larch Prover
, 1991
"... This guide provides an introduction to LP (the Larch Prover), Release 2.2. It describes how LP can be used to axiomatize theories in a subset of multisorted firstorder logic and to provide assistance in proving theorems. It also contains a tutorial overview of the equational termrewriting techno ..."
Abstract
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This guide provides an introduction to LP (the Larch Prover), Release 2.2. It describes how LP can be used to axiomatize theories in a subset of multisorted firstorder logic and to provide assistance in proving theorems. It also contains a tutorial overview of the equational termrewriting technology that provides, along with induction rules and other usersupplied nonequational rules of inference, part of LP's inference engine. v Contents 1 Introduction 1 2 The proof life cycle 2 3 Getting started 4 3.1 Typesetting conventions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 3.2 Online help : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 3.3 Entering commands : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 3.4 Naming and displaying objects : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 3.5 Recording sessions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 3.6 Se...
Termination of AssociativeCommutative Rewriting using Dependency Pairs Criteria
, 2002
"... In 1997, Arts and Giesl proposed new criteria for proving termination of rewriting, based on the socalled dependency pairs. We show how these criteria can be generalized to rewriting modulo associativity and commutativity. ..."
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In 1997, Arts and Giesl proposed new criteria for proving termination of rewriting, based on the socalled dependency pairs. We show how these criteria can be generalized to rewriting modulo associativity and commutativity.