Buchberger Algorithm Research Articles

Incorporating Decision Procedures in Implicit Induction

In this paper we present an approach to integrating reasoning specialists into cover set induction based on constraint contextual rewriting. The approach has been successfully used to incorporate decision procedures into the SPIKE prover. Computer experiments on non-trivial verification problems illustrating the effectiveness of the proposed technique are given. The generality of the approach allows for the integration of computer algebra algorithms and techniques into induction theorem provers. To illustrate this, we discuss the integration of the Buchberger algorithm into our framework.

Gröbner Bases in Orders of Algebraic Number Fields

We prove that any orderOof any algebraic number field K is a reduction ring. Rather than showing the axioms for a reduction ring hold, we start from scratch by well-orderingO, defining a division algorithm, and demonstrating how to use it in a Buchberger algorithm which computes a Gröbner basis given a finite generating set for an ideal. It is shown that our theory of Gröbner bases is equivalent to the ideal membership problem and in fact, a total of eight characterizations are given for a Gröbner basis. Additional conclusions and questions for further investigation are revealed at the end of the paper.

Strong Gröbner bases and cyclic codes over a finite-chain ring.

Abstract We present a unified theory of Grobner bases over a principal ideal ring with a view to applications in Coding Theory, for example to the structure of codes over a finite-chain ring, e.g. a Galois ring. Our theory is effective and yields an extension of Buchberger's algorithm to principal ideal rings. In particular, given a set of generators for a cyclic code over a Galois ring, we can compute a ‘minimal strong’ Grobner basis for the code. We give some applications to cyclic codes.

A Machine-Checked Implementation of Buchberger's Algorithm

We present an implementation of Buchberger's algorithm that has been proved correct within the proof assistant Coq. The implementation contains the basic algorithm plus two standard optimizations.

Declarative Representation of Algorithms of Completion by Critical Pairs

A declarative representation of critical pairs for completion algorithms is described. The representation can be considered as a generic program that, after parametrization of some of its functions, realizes the Knuth-Bendix algorithm for solving the word problem in universal algebras or Buchberger algorithm for constructing Grobner bases of polynomial ideals. The properties of termination and correction are considered. A realization of the representation in the APS algebraic system is described.

Minimised Geometric Buchberger Algorithm for Integer Programming

Recently, various algebraic integer programming (IP) solvers have been proposed based on the theory of Grobner bases. The main difficulty of these solvers is the size of the Grobner bases generated. In algorithms proposed so far, large Grobner bases are generated by either introducing additional variables or by considering the generic IP problem IPA,C. Some improvements have been proposed such as Hosten and Sturmfels' method (GRIN) designed to avoid additional variables and Thomas' truncated Grobner basis method which computes the reduced Grobner basis for a specific IP problem IPA,C(b) (rather than its generalisation IPA,C). In this paper we propose a new algebraic algorithm for solving IP problems. The new algorithm, called Minimised Geometric Buchberger Algorithm, combines Hosten and Sturmfels' GRIN and Thomas' truncated Grobner basis method to compute the fundamental segments of an IP problem IPA,C directly in its original space and also the truncated Grobner basis for a specific IP problem IPA,C(b). We have carried out experiments to compare this algorithm with others such as the geometric Buchberger algorithm, the truncated geometric Buchberger algorithm and the algorithm in GRIN. These experiments show that the new algorithm offers significant performance improvement.

Totally Ordered Commutative Monoids

A totally ordered monoid—or tomonoid , for short—is a commutative semigroup with identity S equipped with a total order \les that is translation invariant, i.e. , that satisfies: \forall x, y, z∈ S, x\les y\;\Rightarrow \; x+z \les y+z. We call a tomonoid that is a quotient of some totally ordered free commutative monoid formally integral. Our most significant results concern characterizations of this condition by means of constructions in the lattice \Z n that are reminiscent of the geometric interpretation of the Buchberger algorithm that occurs in integer programming. In particular, we show that every two-generator tomonoid is formally integral. In addition, we give several (new) examples of tomonoids that are not formally integral, we present results on the structure of nil tomonoids and we show how a valuation-theoretic construction due to Hion reveals relationships between formally integral tomonoids and ordered commutative rings satisfying a condition introduced by Henriksen and Isbell.

Closed-form displacement and coupler curve analysis of planar multi-loop mechanisms using Gröbner bases

The displacement analysis problem for planar and spatial mechanisms can be written as a system of algebraic equations in particular as a system of multi-variate polynomial equations. Elimination theory based on resultants and polynomial continuation are some of the methods which have been used to solve this problem. This paper explores an alternate approach, based on Gröbner bases, to solve the displacement analysis problem for planar mechanisms. It is shown that the reduced set of generators obtained using Buchberger's algorithm for Gröbner bases not only yields the input–output polynomial for the mechanism, but also provides comprehensive information on the number of closures and the relationships between various links of the mechanism. Numerical examples illustrating the applicability of Gröbner bases to displacement analysis of 10- and 12-link mechanisms, and determination of coupler curve equation for an 8-link mechanism are presented. It is seen that even though the Gröbner bases method is versatile enough to handle finitely solvable as well as over-constrained systems of equations, it can run into computational problems due to rapidly growing numerical coefficients and/or the set of generators. The examples presented show how these difficulties can be overcome by artificially decoupling complex mechanisms to help facilitate their closed-form analysis.

A Fast Algorithm for Gröbner Basis Conversion and its Applications

The Gröbner walk method converts a Gröbner basis by partitioning the computation of the basis into several smaller computations following a path in the Gröbner fan of the ideal generated by the system of equations. The method works with ideals of zero-dimension as well as positive dimension. Typically, the target point of the walking path lies on the intersection of very many cones, which ends up with initial forms of a considerable number of terms. Therefore, it is crucial to the performance of the conversion to change the target point since we have to compute a Gröbner basis with respect to the elimination term order of such large initial forms. In contrast to heuristic methods found in the literature, in this paper the author presents a deterministic method to vary the target point in order to ensure the generality of the position, i.e. we always have just a few terms in the initial forms. It turns out that this theoretical result brings a dramatic speed-up in practice. We have implemented the Gröbner walk method together with the deterministic method for varying the target point in the kernel of Mathematica. Our experiments show the superlative performance of our improved Gröbner walk method in comparison with other known methods. Our best performance is 3 × 104times faster than the direct computation of the reduced Gröbner basis with respect to pure lexicographic term order (using the Buchberger algorithm and the sugar cube strategy). We also discuss the complexity of the conversion algorithm and prove a degree bound for polynomials in the target Gröbner basis. In the second part of the paper, we present some applications of the conversion method for implicitization and geometric reasoning. We compare the efficiency of the improved Gröbner walk method with other methods for elimination such as multivariate resultant methods.

M RC - Data Structures and Procedures for Computing in Monoid and Group Rings

Grobner bases and Buchberger's algorithm have been generalized to monoid and group rings. In this paper we present the data structures and procedures we used to implement in a first step the concept of prefix reduction in the system MRC 1.0 together with some examples.

Stability and margin of stability tests for multidimensional filters

In this paper we give simplified proofs of the stability and the margin of stability criteria of Zeheb and Walach for n-dimensional recursive digital filters. We base our proofs on the method of Lagrange multipliers. Both the stability and margin of stability questions are converted into the problem of locating the zeros of a system of polynomial equations. These systems can then be solved using Buchberger's algorithm for computing the Groebner basis of a polynomial ideal.

Gröbner Bases, Invariant Theory and Equivariant Dynamics

This paper is about algorithmic invariant theory as it is required within equivariant dynamical systems. The question of generic bifurcation equations (arbitrary equivariant polynomial vector) requires the knowledge of fundamental invariants and equivariants. We discuss computations which are related to this for finite groups and semi-simple Lie groups. We consider questions such as the completeness of invariants and equivariants. Efficient computations are gained by the Hilbert series driven Buchberger algorithm because computation of elimination ideals is heavily required. Applications such as orbit space reduction are presented.

A new efficient algorithm for computing Gröbner bases ( F4)

This paper introduces a new efficient algorithm for computing Gröbner bases. To avoid as much intermediate computation as possible, the algorithm computes successive truncated Gröbner bases and it replaces the classical polynomial reduction found in the Buchberger algorithm by the simultaneous reduction of several polynomials. This powerful reduction mechanism is achieved by means of a symbolic precomputation and by extensive use of sparse linear algebra methods. Current techniques in linear algebra used in Computer Algebra are reviewed together with other methods coming from the numerical field. Some previously untractable problems (Cyclic 9) are presented as well as an empirical comparison of a first implementation of this algorithm with other well known programs. This comparison pays careful attention to methodology issues. All the benchmarks and CPU times used in this paper are frequently updated and available on a Web page. Even though the new algorithm does not improve the worst case complexity it is several times faster than previous implementations both for integers and modulo p computations.

Symbolic Analysis of Robot Base Parameter Set Using Grobner-Basis

We analyzed base parameters for closed-loop robots using robot symbolic analysis based on the completion procedure in polynomial ideal theory. The robot dynamics regressor is represented as a matrix of multivariate polynomials and reduced to normal form based on Buchberger's algorithm by constructing reduced Grobner basis from kinematic constrained equations. The linear independence of the reduced regressor's column vectors is studied by Gauss-Jordan elimination. Original dynamic parameters are regrouped and some eliminated, depending on results. This omits the need to solve kinematic constrained equations explicitly, deriving all base parameters systematically in theory. An example is shown using robot symbolic analysis system: ROSAM 11.

Application of computer-generated finite-difference equations to decision and inverse problems in elasticity

The use of appropriate finite-difference equations in static isotropic elasticity problems, based on experimental measurements of an elastic quantity at a small number of equidistant points and computer algebra elimination techniques such as the famous Buchberger algorithm for Gröbner bases, is proposed for the inverse elasticity problem concerning the decision about the existence of a singularity (concentrated force, hole, inclusion, crack, etc.) and, further, the determination of the related characteristic loading and geometrical quantities. Both elementary problems and non-trivial problems concerning a concentrated force, a crack, a circular hole and the torsion of an elliptical bar illustrate the present computational approach. Further possibilities are also suggested in brief.

A Parallel Implementation of Buchberger's Algorithm overZpforp ≤ 31991

Gröbner bases of ideals of polynomials are known to have many applications. They have been applied to problems in commutative algebra, statistics, graph theory, robotics and differential equations. Their use as a research tool, however, is limited by their computational complexity. These two facts have inspired numerous attempts to parallelize Buchberger's algorithm to compute them.In this paper, we describe a parallel implementation developed on the Cray T3D using the extensions to C provided by ac. The program is based on the publicly available package Macaulay which computes Gröbner bases of homogeneous ideals overZpfor primesp≤ 31991. The efficiency is nearly 100% on up to 16 processors for moderately sized problems. Above 16 processors, the efficiency drops.

An Algebraic Computational Approach to the Identifiability of Fourier Models

Computer algebra and in particular Gröbner bases are powerful tools in experimental design (Pistone and Wynn, 1996,Biometrika83, 653–666). This paper applies this algebraic methodology to the identifiability of Fourier models. The choice of the class of trigonometric models forces one to deal with complex entities and algebraic irrational numbers. By means of standard techniques we have implemented a version of the Buchberger algorithm that computes Gröbner bases over the complex rational numbers and other simple algebraic extensions of the rational numbers. Some examples are fully carried out.

Global identifiability of linear compartmental models--a computer algebra algorithm.

A priori global identifiability deals with the uniqueness of the solution for the unknown parameters of a model and is, thus, a prerequisite for parameter estimation of biological dynamic models. Global identifiability is however difficult to test, since it requires solving a system of algebraic nonlinear equations which increases both in nonlinearity degree and number of terms and unknowns with increasing model order. In this paper, a computer algebra tool, GLOBI (GLOBal Identifiability) is presented, which combines the topological transfer function method with the Buchberger algorithm, to test global identifiability of linear compartmental models. GLOBI allows for the automatic testing of a priori global identifiability of general structure compartmental models from general multi input-multi output experiments. Examples of usage of GLOBI to analyze a priori global identifiability of some complex biological compartmental models are provided.

Parallelism in MuPAD

S INGULAR is a computer algebra system designed to be a power-house for polynomial computations with special features for algebraic geometry and singularity theory.The development of S INGULAR started in 1986, motivated by the question of the existence of complete intersection singularities which are not quasi-homogeneous, but whose Poincare-complex is exact. The existence of such singularities could be shown in 1987 with an implementation of Mora's algorithm on an Atari in Modula-2 [11]. In 1990, S INGULAR was ported to Unix, and a first public-domain distribution followed. The code of S INGULAR was rewritten in C/C++. An interpreted user-level programming language and user-written libraries were added in 1993. With the release of version 1.0 in May 1997, many important features were added, including:• Polynomial factorization• Factorizing Buchberger algorithm• FGLM algorithm• Primary decomposition of ideals• MP links for general and efficient storage and inter-process communicationsThe goal of the poster is to discuss and demonstrate these new features of S INGULAR where, in combination with the posters on MP and MuPAD, special emphasize will be paid to S INGULAR 's MP interface and its resulting possibilities and problems.

A Variant of the Buchberger Algorithm for Integer Programming

In this paper we modify Buchberger's S-pair reduction algorithm for computing a Grobner basis of a toric ideal so as to apply it to an integer program (IP) in inequality form with fixed right-hand sides and fixed upper bounds on the variables. We formulate the algorithm in the original space and interpret the reduction steps geometrically. In fact, three variants of this algorithm are presented, and we give elementary proofs for their correctness. A relationship among these (exact) algorithms, iterative improvement heuristics, and the Kernighan--Lin procedure is established. Computational results are also presented.