A Signature Based Border Basis Algorithm

Journal of Symbolic Computation, 2009

This work presents a new framework for Gröbner basis computations with Boolean polynomials. Boolean polynomials can be modelled in a rather simple way, with both coefficients and degree per variable lying in {0, 1}. The ring of Boolean polynomials is, however, not a polynomial ring, but rather the quotient ring of the polynomial ring over the field with two elements modulo the field equations x 2 = x for each variable x. Therefore, the usual polynomial data structures seem not to be appropriate for fast Gröbner basis computations. We introduce a specialised data structure for Boolean polynomials based on zero-suppressed binary decision diagrams (ZDDs), which is capable of handling these polynomials more efficiently with respect to memory consumption and also computational speed. Furthermore, we concentrate on high-level algorithmic aspects, taking into account the new data structures as well as structural properties of Boolean polynomials. For example, a new useless-pair criterion for Gröbner basis computations in Boolean rings is introduced. One of the motivations for our work is the growing importance of formal hardware and software verification based on Boolean expressions, which suffer -besides from the complexity of the problems -from the lack of an adequate treatment of arithmetic components. We are convinced that algebraic methods are more suited and we believe that our preliminary implementation shows that Gröbner bases on specific data structures can be capable to handle problems of industrial size.

Arxiv preprint arXiv:1101.3589, 2011

This paper describes a Buchberger-style algorithm to compute a Gröbner basis of a polynomial ideal, allowing for a selection strategy based on "signatures". We explain how three recent algorithms can be viewed as different strategies for the new algorithm, and how other selection strategies can be formulated. We describe a fourth as an example. We analyze the strategies both theoretically and empirically, leading to some surprising results.

Journal of Algebra and Its Applications, 2019

In this paper, we consider the problem of computing all possible order ideals and also sets connected to 1, and the corresponding border bases, for the vanishing ideal of a given finite set of points. In this context, two different approaches are discussed: based on the Buchberger–Möller Algorithm [H. M. Möller and B. Buchberger, The construction of multivariate polynomials with preassigned zeros, EUROCAM ’82 Conf., Computer Algebra, Marseille/France 1982, Lect. Notes Comput. Sci. 144, (1982), pp. 24–31], we first propose a new algorithm to compute all possible order ideals and the corresponding border bases for an ideal of points. The second approach involves adapting the Farr–Gao Algorithm [J. B. Farr and S. Gao, Computing Gröbner bases for vanishing ideals of finite sets of points, in 16th Int. Symp. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC-16, Las Vegas, NV, USA (Springer, Berlin, 2006), pp. 118–127] for finding all sets connected to 1, as well as ...

HAL (Le Centre pour la Communication Scientifique Directe), 2022

Proceedings of the 1993 international symposium on Symbolic and algebraic computation - ISSAC '93, 1993

Journal of Pure and Applied Algebra, 2005

This paper presents characterizations of border bases of zero-dimensional polynomial ideals that are analogous to the known characterizations of Gröbner bases. Based on a Border Division Algorithm, a variant of the usual Division Algorithm, we characterize border bases as border prebases with one of the following equivalent properties: special generation, generation of the border form ideal, confluence of the corresponding rewrite relation, reduction of S-polynomials to zero, and lifting of syzygies. The last characterization relies on a detailed study of the relative position of the border terms and their syzygy module. In particular, a border prebasis is a border basis if and only if all fundamental syzygies of neighboring border terms lift; these liftings are easy to compute.

Computing Research Repository, 2011

We demonstrate a method to parallelize the computation of a Gr\"obner basis for a homogenous ideal in a multigraded polynomial ring. Our method uses anti-chains in the lattice $\mathbb N^k$ to separate mutually independent S-polynomials for reduction.

2007

In the computation of Groebner bases using Buchberger's Algorithm, a key issue for improving the efficiency is to create techniques to help us avoid as many unnecessary pairs of polynomials from the non-computed set of pairs as possible. A good solution would be to avoid those pairs that can be easily ignored without computing their S-polynomials, and hence to process only on the set of pairs of generators of the module generated by syzygies. This paper details an improvment of Buchberger's Algorithm for computing Groebner bases by defining the module of solutions of a homogeneous linear equation with polynomial coefficients (called the syzygy module). As a consequence, we use these syzygy modules to give another equivalent condition for a set to be a Groebner basis for an ideal. As a result we demonastrate that this new condition can significantly improve the Buchberger's Algorithm to compute Groebner bases.

2017

Recent developments in formal datapath verification make efficient use of symbolic computer algebra algorithms for formal verification. The circuit is modeled as a set of polynomials over Boolean (or pseudo-Boolean) rings, and Gröbner basis (GB) reductions are performed over these polynomials to derive a canonical representation. GB reductions of Boolean polynomials tend to cause intermediate expression swell (term explosion problem) – often making the approach infeasible in a practical setting. To overcome these problems, this paper describes a logic synthesis analogue of GB reductions over Boolean polynomials, using the unate cube set algebra over characteristic sets. By representing Boolean polynomials as characteristic sets using Zero-suppressed BDDs (ZBDDs), implicit algorithms can be efficiently designed for GB-reduction on digital circuits. We show that imposition of circuit-topology based monomial orders on ZBDDs enables an implicit implementation of polynomial division, can...

Central European Journal of Mathematics

In this short note, we extend Faugére’s F4-algorithm for computing Gröbner bases to polynomial rings with coefficients in an Euclidean ring. Instead of successively reducing single S-polynomials as in Buchberger’s algorithm, the F4-algorithm is based on the simultaneous reduction of several polynomials.