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15
Normalized Rewriting: an alternative to Rewriting modulo a Set of Equations
, 1996
"... this paper is to make the similarity between KnuthBendix completion and the Buchberger algorithm explicit, by describing a general algorithm called Snormalized completion where S is a parameter, such that both algorithms are Normalized Rewriting: an alternative to Rewriting modulo a Set of Equatio ..."
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Cited by 27 (0 self)
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this paper is to make the similarity between KnuthBendix completion and the Buchberger algorithm explicit, by describing a general algorithm called Snormalized completion where S is a parameter, such that both algorithms are Normalized Rewriting: an alternative to Rewriting modulo a Set of Equations 3 instances of this general algorithm for a particular choice of S. This has been achieved in two steps.
An algebraic approach for the unsatisfiability of nonlinear constraints
 In Computer Science Logic (CSL), volume 3634 of LNCS
"... Abstract. We describe a simple algebraic semidecision procedure for detecting unsatisfiability of a (quantifierfree) conjunction of nonlinear equalities and inequalities. The procedure consists of Gröbner basis computation plus extension rules that introduce new definitions, and hence it can be de ..."
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Cited by 17 (2 self)
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Abstract. We describe a simple algebraic semidecision procedure for detecting unsatisfiability of a (quantifierfree) conjunction of nonlinear equalities and inequalities. The procedure consists of Gröbner basis computation plus extension rules that introduce new definitions, and hence it can be described as a criticalpair completionbased logical procedure. This procedure is shown to be sound and refutationally complete. When projected onto the linear case, our procedure reduces to the Simplex method for solving linear constraints. If only finitely many new definitions are introduced, then the procedure is also terminating. Such terminating, but potentially incomplete, procedures are used in “incompletenesstolerant ” applications. 1
String rewriting and Gröbner bases  a general approach to monoid and group rings
 Proceedings of the Workshop on Symbolic Rewriting Techniques, Monte Verita
, 1995
"... The concept of algebraic simplification is of great importance for the field of symbolic computation in computer algebra. In this paper we review some fundamental concepts concerning reduction rings in the spirit of Buchberger. The most important properties of reduction rings are presented. The tech ..."
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Cited by 15 (5 self)
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The concept of algebraic simplification is of great importance for the field of symbolic computation in computer algebra. In this paper we review some fundamental concepts concerning reduction rings in the spirit of Buchberger. The most important properties of reduction rings are presented. The techniques for presenting monoids or groups by string rewriting systems are used to define several types of reduction in monoid and group rings. Grobner bases in this setting arise naturally as generalizations of the corresponding known notions in the commutative and some noncommutative cases. Several results on the connection of the word problem and the congruence problem are proven. The concepts of saturation and completion are introduced for monoid rings having a finite convergent presentation by a semiThue system. For certain presentations, including free groups and contextfree groups, the existence of finite Grobner bases for finitely generated right ideals is shown and a procedure to com...
Rewrite Systems for Natural, Integral, and Rational Arithmetic
, 1997
"... We give algebraic presentations of the sets of natural numbers, integers, and rational numbers by convergent rewrite systems which moreover allow efficient computations of arithmetical expressions. We then use such systems in the general normalised completion algorithm, in order to compute Gröbner b ..."
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Cited by 12 (1 self)
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We give algebraic presentations of the sets of natural numbers, integers, and rational numbers by convergent rewrite systems which moreover allow efficient computations of arithmetical expressions. We then use such systems in the general normalised completion algorithm, in order to compute Gröbner bases of polynomial ideals over Q.
Combining Algebra and Universal Algebra in FirstOrder Theorem Proving: The Case of Commutative Rings
 In Proc. 10th Workshop on Specification of Abstract Data Types
, 1995
"... . We present a general approach for integrating certain mathematical structures in firstorder equational theorem provers. More specifically, we consider theorem proving problems specified by sets of firstorder clauses that contain the axioms of a commutative ring with a unit element. Associativeco ..."
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Cited by 7 (4 self)
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. We present a general approach for integrating certain mathematical structures in firstorder equational theorem provers. More specifically, we consider theorem proving problems specified by sets of firstorder clauses that contain the axioms of a commutative ring with a unit element. Associativecommutative superposition forms the deductive core of our method, while a convergent rewrite system for commutative rings provides a starting point for more specialized inferences tailored to the given class of formulas. We adopt ideas from the Grobner basis method to show that many inferences of the superposition calculus are redundant. This result is obtained by the judicious application of the simplification techniques afforded by convergent rewriting and by a process called symmetrization that embeds inferences between single clauses and ring axioms. 1 Introduction 1.1 Motivation Specifications of programs include both symbols with their usual mathematical meaning as well as additional f...
DBases for Polynomial Ideals over Commutative Noetherian Rings
 8TH INTL. CONF. ON REWRITING TECHNIQUES AND APPLICATIONS
, 1997
"... We present a completionlike procedure for constructing Dbases for polynomial ideals over commutative Noetherian rings with unit. The procedure is described at an abstract level, by transition rules. Its termination is proved under certain assumptions about the strategy that controls the applica ..."
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Cited by 6 (4 self)
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We present a completionlike procedure for constructing Dbases for polynomial ideals over commutative Noetherian rings with unit. The procedure is described at an abstract level, by transition rules. Its termination is proved under certain assumptions about the strategy that controls the application of the transition rules. Correctness is established by proof simplication techniques.
SOLVING LINEAR BOUNDARY VALUE PROBLEMS VIA NONCOMMUTATIVE GRÖBNER BASES
"... A new approach for symbolically solving linear boundary value problems is presented. Rather than using generalpurpose tools for obtaining parametrized solutions of the underlying ODE and fitting them against the specified boundary conditions (which may be quite expensive), the problem is interpre ..."
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Cited by 5 (3 self)
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A new approach for symbolically solving linear boundary value problems is presented. Rather than using generalpurpose tools for obtaining parametrized solutions of the underlying ODE and fitting them against the specified boundary conditions (which may be quite expensive), the problem is interpreted as an operator inversion problem in a suitable Banach space setting. Using the concept of the oblique MoorePenrose inverse, it is possible to transform the inversion problem into a system of operator equations that can be attacked by virtue of noncommutative Gröbner bases. The resulting operator solution can be represented as an integral operator having the classical Green’s function as its kernel. Although, at this stage of research, we cannot yet give an algorithmic formulation of the method and its domain of admissible inputs, we do believe that it has promising perspectives of automation and generalization; some of these perspectives are discussed.
Superfluous Spolynomials in StrategyIndependent Gröbner Bases
 SYMBOLIC AND NUMERIC ALGORITHMS FOR SCIENTIFIC COMPUTING
, 2009
"... Using the machinery of proof orders originally introduced by Bachmair and Dershowitz in the context of canonical equational proofs, we give an abstract, strategyindependent presentation of Gröbner basis procedures and prove the correctness of two classical criteria for recognising superfluous Spol ..."
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Cited by 4 (1 self)
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Using the machinery of proof orders originally introduced by Bachmair and Dershowitz in the context of canonical equational proofs, we give an abstract, strategyindependent presentation of Gröbner basis procedures and prove the correctness of two classical criteria for recognising superfluous Spolynomials, Buchberger’s criteria 1 and 2, w.r.t. arbitrary fair and correct basis construction strategies. To do so, we develop a general method for proving the strategyindependent correctness of superfluous Spolynomial criteria which seems to be quite powerful. We also derive a new superfluous Spolynomial criterion which is a generalization of Buchberger1 for Gröbner basis procedures implementing a special form of eager simplification and is proved to be correct strategyindependently.
On Locally Minimal Nullstellensatz Proofs
 SATISFIABILITY MODULO THEORIES
, 2009
"... Hilbert’s weak Nullstellensatz guarantees the existence of algebraic proof objects certifying the unsatisfiability of systems of polynomial equations not satisfiable over any algebraically closed field. Such proof objects take the form of ideal membership identities and can be found algorithmically ..."
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Hilbert’s weak Nullstellensatz guarantees the existence of algebraic proof objects certifying the unsatisfiability of systems of polynomial equations not satisfiable over any algebraically closed field. Such proof objects take the form of ideal membership identities and can be found algorithmically using Gröbner bases and cofactorbased linear algebra techniques. However, these proof objects may contain redundant information: a proper subset of the equational assumptions used in these proofs may be sufficient to derive the unsatisfiability of the original polynomial system. For using Nullstellensatz techniques in SMTbased decision methods, a minimal proof object is often desired. With this in mind, we introduce a notion of locally minimal Nullstellensatz proofs and give idealtheoretic algorithms for their construction.
Cancellative Abelian Monoids in Refutational Theorem Proving
 PHD THESIS, INSTITUT FÜR INFORMATIK, UNIVERSITÄT DES SAARLANDES
, 1997
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