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16
A Completion Procedure for Computing a Canonical Basis for a kSubalgebra
 IN COMPUTERS AND MATHEMATICS
, 1989
"... A completion procedure for computing a canonical basis for a ksubalgebra is proposed. Using this canonical basis, the membership problem for a ksubalgebra can be solved. The approach follows Buchberger's approach for computing a Gröbner basis for a polynomial ideal and is based on rewriting con ..."
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Cited by 31 (0 self)
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A completion procedure for computing a canonical basis for a ksubalgebra is proposed. Using this canonical basis, the membership problem for a ksubalgebra can be solved. The approach follows Buchberger's approach for computing a Gröbner basis for a polynomial ideal and is based on rewriting concepts. A canonical basis produced by the completion procedure shares many properties of a Grobner basis such as reducing an element of a ksubalgebra to 0 and generating unique normal forms for the equivalence classes generated by a ksubalgebra. In contrast to Shannon and Sweedler's approach using tag variables, this approach is direct. One of the limitations of the approach however is that the procedure may not terminate for some term orderings thus giving an infinite canonical basis. The procedure is illustrated using examples.
Computing Gröbner bases in monoid and group rings
 Proc. ISSAC'93
, 1995
"... Following Buchberger's approach to computing a Gröbner basis of a polynomial ideal in polynomial rings, a completion procedure for finitely generated right ideals in Z[H] is given, where H is an ordered monoid presented by a finite, convergent semiThue system (\Sigma; T). Taking a finite set F ` Z[ ..."
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Cited by 21 (8 self)
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Following Buchberger's approach to computing a Gröbner basis of a polynomial ideal in polynomial rings, a completion procedure for finitely generated right ideals in Z[H] is given, where H is an ordered monoid presented by a finite, convergent semiThue system (\Sigma; T). Taking a finite set F ` Z[H] we get a (possibly infinite) basis of the right ideal generated by F, such that using this basis we have unique normal forms for all p 2 Z[H] (especially the normal form is 0 in case p is an element of the right ideal generated by F). As the ordering and multiplication on H need not be compatible, reduction has to be defined carefully in order to make it Noetherian. Further we no longer have p \Delta x! p 0 for p 2 Z[H]; x 2 H. Similar to Buchberger's spolynomials, confluence criteria are developed and a completion procedure is given. In case T = ; or (\Sigma; T) is a convergent, 2monadic presentation of a group providing inverses of length 1 for the generators or (\Sigma; T) is a convergent presentation of a commutative monoid, termination can be shown. So in this cases finitely generated right ideals admit finite Gröbner bases. The connection to the subgroup problem is discussed.
An Eunification algorithm for analyzing protocols that use modular exponentiation
, 2003
"... Modular multiplication and exponentiation are common operations in modern cryptography. Uni cation problems with respect to some equational theories that these operations satisfy are investigated. Two dierent but related equational theories are analyzed. A uni cation algorithm is given for one of ..."
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Cited by 20 (0 self)
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Modular multiplication and exponentiation are common operations in modern cryptography. Uni cation problems with respect to some equational theories that these operations satisfy are investigated. Two dierent but related equational theories are analyzed. A uni cation algorithm is given for one of the theories which relies on solving syzygies over multivariate integral polynomials with noncommuting indeterminates. For the other theory, in which the distributivity property of exponentiation over multiplication is assumed, the uni ability problem is shown to be undecidable by adapting a construction developed by one of the authors to reduce Hilbert's 10th problem to the solvability problem for linear equations over semirings. A new algorithm for computing strong Grobner bases of right ideals over the polynomial semiring Z<X 1 ; : : : ; Xn> is proposed; unlike earlier algorithms proposed by Baader as well as by Madlener and Reinert which work only for right admissible term orderings with the boundedness property, this algorithm works for any right admissible term ordering. The algorithms for some of these uni cation problems are expected to be integrated into Research supported in part by the NSF grant nos. CCR0098114 and CDA9503064, the ONR grant no. N000140110429, and a grant from the Computer Science Research Institute at Sandia National Labs.
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 ...
Ideal Membership in Polynomial Rings over the Integers
 J. Amer. Math. Soc
"... Abstract. We present a new approach to the ideal membership problem for polynomial rings over the integers: given polynomials f0, f1,..., fn ∈ Z[X], where X = (X1,..., XN) is an Ntuple of indeterminates, are there g1,..., gn ∈ Z[X] such that f0 = g1f1 + · · · + gnfn? We show that the degree of th ..."
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Cited by 16 (2 self)
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Abstract. We present a new approach to the ideal membership problem for polynomial rings over the integers: given polynomials f0, f1,..., fn ∈ Z[X], where X = (X1,..., XN) is an Ntuple of indeterminates, are there g1,..., gn ∈ Z[X] such that f0 = g1f1 + · · · + gnfn? We show that the degree of the polynomials g1,..., gn can be bounded by (2d) 2O(N2) (h + 1) where d is the maximum total degree and h the maximum height of the coefficients of f0,..., fn. Some related questions, primarily concerning linear equations in R[X], where R is the ring of integers of a number field, are also treated.
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...
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.
Superposition Theorem Proving for Abelian Groups Represented as Integer Modules
 Theoretical Computer Science
, 1996
"... We define a superposition calculus specialized for abelian groups represented as integer modules, and show its refutational completeness. This allows to substantially reduce the number of inferences compared to a standard superposition prover which applies the axioms directly. Specifically, equation ..."
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Cited by 13 (4 self)
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We define a superposition calculus specialized for abelian groups represented as integer modules, and show its refutational completeness. This allows to substantially reduce the number of inferences compared to a standard superposition prover which applies the axioms directly. Specifically, equational literals are simplified, so that only the maximal term of the sums is on the lefthand side. Only certain minimal superpositions need to be considered; other superpositions which a standard prover would consider become redundant. This not only reduces the number of inferences, but also reduces the size of the ACunification problems which are generated. That is, ACunification is not necessary at the top of a term, only below some nonACsymbol. Further, we consider situations where the axioms give rise to variable overlaps and develop techniques to avoid these explosive cases where possible. 1 Introduction Historically, starting from plain resolution, more and more problematic axioms ha...
Solving Linear Equations Over Polynomial Semirings
 RUTGER UNIVERSITY (NJ
"... We consider the problem of solving linear equations over various semirings. In particular, solving of linear equations over polynomial rings with the additional restriction that the solutions must have only nonnegative coefficients is shown to be undecidable. Applications to undecidability proofs o ..."
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Cited by 10 (4 self)
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We consider the problem of solving linear equations over various semirings. In particular, solving of linear equations over polynomial rings with the additional restriction that the solutions must have only nonnegative coefficients is shown to be undecidable. Applications to undecidability proofs of several unification problems are illustrated, one of which, unification modulo one associativecommutative function and one endomorphism, has been a longstanding open problem. The problem of solving multiset constraints is also shown to be undecidable.
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...