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Abstract congruence closure
 Journal of Automated Reasoning
"... Abstract. We describe the concept of an abstract congruence closure and provide equational inference rules for its construction. The length of any maximal derivation using these inference rules for constructing an abstract congruence closure is at most quadratic in the input size. The framework is u ..."
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Cited by 36 (3 self)
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Abstract. We describe the concept of an abstract congruence closure and provide equational inference rules for its construction. The length of any maximal derivation using these inference rules for constructing an abstract congruence closure is at most quadratic in the input size. The framework is used to describe the logical aspects of some wellknown algorithms for congruence closure. It is also used to obtain an efficient implementation of congruence closure. We present experimental results that illustrate the relative differences in performance of the different algorithms. The notion is extended to handle associative and commutative function symbols, thus providing the concept of an associativecommutative congruence closure. Congruence closure (modulo associativity and commutativity) can be used to construct ground convergent rewrite systems corresponding to a set of ground equations (containing AC symbols). Key words: term rewriting, congruence closure, associativecommutative theories. 1.
Abstract congruence closure and specializations
 17th International Conference on Automated Deduction, volume 1831 of Lecture Notes in Artificial Intelligence
, 2000
"... We use the uniform framework of abstract congruence closure to study the congruence closure algorithms described by Nelson and Oppen [9], Downey, Sethi and Tarjan [7] and Shostak [11]. The descriptions thus obtained abstracts from certain implementation details while still allowing for comparison be ..."
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Cited by 22 (4 self)
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We use the uniform framework of abstract congruence closure to study the congruence closure algorithms described by Nelson and Oppen [9], Downey, Sethi and Tarjan [7] and Shostak [11]. The descriptions thus obtained abstracts from certain implementation details while still allowing for comparison between these different algorithms. Experimental results are presented to illustrate the relative efficiency and explain differences in performance of these three algorithms. The transition rules for computation of abstract congruence closure are obtained from rules for standard completion enhanced with an extension rule that enlarges a given signature by new constants.
Abstract canonical inference
 ACM Trans. on Computational Logic
, 2007
"... An abstract framework of canonical inference is used to explore how different proof orderings induce different variants of saturation and completeness. Notions like completion, paramodulation, saturation, redundancy elimination, and rewritesystem reduction are connected to proof orderings. Fairness ..."
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Cited by 17 (10 self)
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An abstract framework of canonical inference is used to explore how different proof orderings induce different variants of saturation and completeness. Notions like completion, paramodulation, saturation, redundancy elimination, and rewritesystem reduction are connected to proof orderings. Fairness of deductive mechanisms is defined in terms of proof orderings, distinguishing between (ordinary) “fairness, ” which yields completeness, and “uniform fairness, ” which yields saturation.
Decision Problems in Ordered Rewriting
 In 13th IEEE Symposium on Logic in Computer Science (LICS
, 1997
"... A term rewrite system (TRS) terminates iff its rules are contained in a reduction ordering ?. In order to deal with any set of equations, including inherently nonterminating ones (like commutativity), TRS have been generalised to ordered TRS (E; ?), where equations of E are applied in whatever dir ..."
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Cited by 16 (7 self)
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A term rewrite system (TRS) terminates iff its rules are contained in a reduction ordering ?. In order to deal with any set of equations, including inherently nonterminating ones (like commutativity), TRS have been generalised to ordered TRS (E; ?), where equations of E are applied in whatever direction agrees with ?. The confluence of terminating TRS is wellknown to be decidable, but for ordered TRS the decidability of confluence has been open. Here we show that the confluence of ordered TRS is decidable if ordering constraints for ? can be solved in an adequate way, which holds in particular for the class of LPO orderings. For sets E of constrained equations, confluence is shown to be undecidable. Finally, ground reducibility is proved undecidable for ordered TRS. 1 Introduction Term rewrite systems (TRS) have been applied to many problems in symbolic computation, automated theorem proving, program synthesis and verification, and logic programming among others. Two fundamental pr...
Buchberger's algorithm: A constraintbased completion procedure
, 1994
"... We present an extended completion procedure with builtin theories defined by a collection of associativity and commutativity axioms and additional ground equations, and reformulate Buchberger's algorithm for constructing Gröbner bases for polynomial ideals in this formalism. The presentation of com ..."
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Cited by 15 (2 self)
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We present an extended completion procedure with builtin theories defined by a collection of associativity and commutativity axioms and additional ground equations, and reformulate Buchberger's algorithm for constructing Gröbner bases for polynomial ideals in this formalism. The presentation of completion is at an abstract level, by transition rules, with a suitable notion of fairness used to characterize a wide class of correct completion procedures, among them Buchberger's original algorithm for polynomial rings over a field.
Abstract saturationbased inference
 IN PROCEEDINGS OF THE 18TH ANNUAL SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE
, 2003
"... Solving goals—like deciding word problems or resolving constraints—is much easier in some theory presentations than in others. What have been called “completion processes”, in particular in the study of equational logic, involve finding appropriate presentations of a given theory to solve easily a g ..."
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Cited by 12 (5 self)
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Solving goals—like deciding word problems or resolving constraints—is much easier in some theory presentations than in others. What have been called “completion processes”, in particular in the study of equational logic, involve finding appropriate presentations of a given theory to solve easily a given class of problems. We provide a general prooftheoretic setting within which completionlike processes can be modelled and studied. This framework centers around wellfounded orderings of proofs. It allows for abstract definitions and very general characterizations of saturation processes and redundancy criteria.
Paramodulation with NonMonotonic Orderings
 In 14th IEEE Symposium on Logic in Computer Science (LICS
, 1999
"... All current completeness results for ordered paramodulation require the term ordering Ø to be wellfounded, monotonic and total(izable) on ground terms. Here we introduce a new proof technique where the only properties required for Ø are wellfoundedness and the subterm property 1 . The technique ..."
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Cited by 9 (7 self)
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All current completeness results for ordered paramodulation require the term ordering Ø to be wellfounded, monotonic and total(izable) on ground terms. Here we introduce a new proof technique where the only properties required for Ø are wellfoundedness and the subterm property 1 . The technique is a relatively simple and elegant application of some fundamental results on the termination and confluence of ground term rewrite systems (TRS). By a careful further analysis of our technique, we obtain the first KnuthBendix completion procedure that finds a convergent TRS for a given set of equations E and a (possibly nontotalizable) reduction ordering Ø whenever it exists 2 . Note that being a reduction ordering is the minimal possible requirement on Ø, since a TRS terminates if, and only if, it is contained in a reduction ordering. Keywords: term rewriting, automated deduction. 1 Introduction Deduction with equality is fundamental in mathematics, logics and many applications of ...