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28
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 rewritin ..."
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Cited by 30 (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.
An Efficient Incremental Algorithm for Solving Systems of Linear Diophantine Equations
, 1994
"... In this paper, we describe an algorithm for solving systems of linear Diophantine equations based on a generalization of an algorithm for solving one equation due to Fortenbacher [3]. It can solve a system as a whole, or be used incrementally when the system is a sequential accumulation of several s ..."
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Cited by 29 (0 self)
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In this paper, we describe an algorithm for solving systems of linear Diophantine equations based on a generalization of an algorithm for solving one equation due to Fortenbacher [3]. It can solve a system as a whole, or be used incrementally when the system is a sequential accumulation of several subsystems. The proof of termination of the algorithm is difficult, whereas the proofs of completeness and correctness are straightforward generalizations of Fortenbacher's proof.
33 Basic Test Problems: A Practical Evaluation of Some Paramodulation Strategies
, 1996
"... Introduction Many researchers who study the theoretical aspects of inference systems believe that if inference rule A is complete and more restrictive than inference rule B, then the use of A will lead more quickly to proofs than will the use of B. The literature contains statements of the sort &qu ..."
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Cited by 24 (5 self)
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Introduction Many researchers who study the theoretical aspects of inference systems believe that if inference rule A is complete and more restrictive than inference rule B, then the use of A will lead more quickly to proofs than will the use of B. The literature contains statements of the sort "our rule is complete and it heavily prunes the search space; therefore it is efficient". 2 These positions are highly questionable and indicate that the authors have little or no experience with the practical use of automated inference systems. Restrictive rules (1) can block short, easytofind proofs, (2) can block proofs involving simple clauses, the type of clause on which many practical searches focus, (3) can require weakening of redundancy control such as subsumption and demodulation, and (4) can require the use of complex checks in deciding whether such rules should be applied. The only way to determ
Adventures in AssociativeCommutative Unification
 Journal of Symbolic Computation
, 1989
"... We have discovered an efficient algorithm for matching and unification in associativecommutative (AC) equational theories. In most cases of AC unification our method obviates the need for solving diophantine equations, and thus avoids one of the bottlenecks of other associativecommutative unificat ..."
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Cited by 22 (0 self)
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We have discovered an efficient algorithm for matching and unification in associativecommutative (AC) equational theories. In most cases of AC unification our method obviates the need for solving diophantine equations, and thus avoids one of the bottlenecks of other associativecommutative unification techniques. The algorithm efficiently utilizes powerful constraints to eliminate much of the search involved in generating valid substitutions. Moreover, it is able to generate solutions lazily, enabling its use in an SLDresolutionbased environment like Prolog. We have found the method to run much faster and use less space than other associativecommutative unification procedures on many commonly encountered AC problems. 1 Introduction Associativecommutative (AC) equational theories surface in a number of computer science applications, including term rewriting, automatic theorem proving, software verification, and database retrieval. As a simple example, consider trying to find a sub...
Solving Systems of Linear Diophantine Equations: An Algebraic Approach
 IN PROC. 16TH MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE, WARSAW, LNCS 520
, 1991
"... We describe through an algebraic and geometrical study, a new method for solving systems of linear diophantine equations. This approach yields an algorithm which is intrinsically parallel. In addition to the algorithm, we give a geometrical interpretation of the satisfiability of an homogeneous syst ..."
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Cited by 18 (1 self)
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We describe through an algebraic and geometrical study, a new method for solving systems of linear diophantine equations. This approach yields an algorithm which is intrinsically parallel. In addition to the algorithm, we give a geometrical interpretation of the satisfiability of an homogeneous system, as well as upper bounds on height and length of all minimal solutions of such a system. We also show how our results apply to inhomogeneous systems yielding necessary conditions for satisfiability and upper bounds on the minimal solutions.
DoubleExponential Complexity of Computing a Complete Set of ACUnifiers
 In Proceedings 7th IEEE Symposium on Logic in Computer Science
"... A new algorithm for computing a complete set of unifiers for two terms involving associativecommutative function symbols is presented. The algorithm is based on a nondeterministic algorithm given by the authors in 1986 to show the NPcompleteness of associativecommutative unifiability. The algori ..."
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Cited by 17 (0 self)
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A new algorithm for computing a complete set of unifiers for two terms involving associativecommutative function symbols is presented. The algorithm is based on a nondeterministic algorithm given by the authors in 1986 to show the NPcompleteness of associativecommutative unifiability. The algorithm is easy to understand, its termination can be easily established. More importantly, its complexity can be easily analyzed and is shown to be doubly exponential in the size of the input terms. The analysis also shows that there is a doubleexponential upper bound on the size of a complete set of unifiers of two input terms. Since there is a family of simple associativecommutative unification problems which have complete sets of unifiers whose size is doubly exponential, the algorithm is optimal in its order of complexity in this sense. This is the first associativecommutative unification algorithm whose complexity has been completely analyzed. The approach can also be used to show a singl...
Canonical Bases: Relations with Standard Bases, Finiteness Conditions and Application to Tame Automorphisms
, 1994
"... Canonical bases for ksubalgeras of k[x 1 ; : : : ; xn ] are analogs of standard bases for ideals. They form a set of generators, which allows to answer the membership problem by a reduction process. Unfortunately, they may be infinite even for finitely generated subalgeras. We redefine canonical ..."
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Cited by 14 (1 self)
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Canonical bases for ksubalgeras of k[x 1 ; : : : ; xn ] are analogs of standard bases for ideals. They form a set of generators, which allows to answer the membership problem by a reduction process. Unfortunately, they may be infinite even for finitely generated subalgeras. We redefine canonical bases, and for that we recall some properties of monoids, kalgebras of monoids and "binomial" ideals, which play an essential role in our presentation and the implementation we made in the IBM computer algebra system Scratchpad II. We complete the already known relations between standard bases and canonical bases by generalizing the notion of standard bases for ideals of any ksubalgebra admitting a finite canonical basis. We also have a way of finding a set of generators of the ideal of relations between elements of a canonical basis, which is a standard basis for some ordering. We then turn to finiteness conditions, and investigate the case of integrally closed subalgebras. We sho...
On the Hardness of the Shortest Vector Problem
, 1998
"... An ndimensional lattice is the set of all integral linear combinations of n linearly independent vectors in R^m. One of the most studied algorithmic problems on lattices is the shortest vector problem (SVP): given a lattice, find the shortest nonzero vector in it. We prove that the shortest vector ..."
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Cited by 12 (1 self)
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An ndimensional lattice is the set of all integral linear combinations of n linearly independent vectors in R^m. One of the most studied algorithmic problems on lattices is the shortest vector problem (SVP): given a lattice, find the shortest nonzero vector in it. We prove that the shortest vector problem is NPhard (for randomized reductions) to approximate within some constant factor greater than 1 in any norm l_p (p>1). In particular, we prove the NPhardness of approximating SVP in the Euclidean norm within any factor less than sqrt 2. The same NPhardness results hold for deterministic nonuniform reductions. A deterministic uniform reduction is also given under a reasonable number theoretic conjecture concerning the distribution of smooth numbers. In proving the NPhardness of SVP we develop a number of technical tools that might be of independent interest. In particular, a lattice packing is constructed with the property that the number of unit spheres contained in an ndimensional ball of radius greater then 1 + (sqrt 2) grows exponentially in n, a new constructive version of Sauer's lemma(a combinatorial result somehow related to the notion of VCdimension) is presented, considerably simplifying all previously known constructions.
Complete Solving of Linear Diophantine Equations and Inequations without Adding Variables
 PROC. OF 1ST INTERNATIONAL CONFERENCE ON PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING
, 1995
"... In this paper, we present an algorithm for solving directly linear Diophantine systems of both equations and inequations. Here directly means without adding slack variables for encoding inequalities as equalities. This algorithm is an extension of the algorithm due to Contejean and Devie [9] for so ..."
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Cited by 11 (1 self)
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In this paper, we present an algorithm for solving directly linear Diophantine systems of both equations and inequations. Here directly means without adding slack variables for encoding inequalities as equalities. This algorithm is an extension of the algorithm due to Contejean and Devie [9] for solving linear Diophantine systems of equations, which is itself a generalization of the algorithm of Fortenbacher [6] for solving a single linear Diophantine equation. All the nice properties of the algorithm of Contejean and Devie are still satisfied by the new algorithm: it is complete, i.e. provides a (finite) description of the set of solutions, it can be implemented with a bounded stack, and it admits an incremental version. All of these characteristics enable its easy integration in the CLP paradigm.