168 research outputs found
Reducing the size and number of linear programs in a dynamic Gr\"obner basis algorithm
The dynamic algorithm to compute a Gr\"obner basis is nearly twenty years
old, yet it seems to have arrived stillborn; aside from two initial
publications, there have been no published followups. One reason for this may
be that, at first glance, the added overhead seems to outweigh the benefit; the
algorithm must solve many linear programs with many linear constraints. This
paper describes two methods of reducing the cost substantially, answering the
problem effectively.Comment: 11 figures, of which half are algorithms; submitted to journal for
refereeing, December 201
Modifying Faug\`ere's F5 Algorithm to ensure termination
The structure of the F5 algorithm to compute Gr\"obner bases makes it very
efficient. However, while it is believed to terminate for so-called regular
sequences, it is not clear whether it terminates for all inputs. This paper has
two major parts. In the first part, we describe in detail the difficulties
related to a proof of termination. In the second part, we explore three
variants that ensure termination. Two of these have appeared previously only in
dissertations, and ensure termination by checking for a Gr\"obner basis using
traditional criteria. The third variant, F5+, identifies a degree bound using a
distinction between "necessary" and "redundant" critical pairs that follows
from the analysis in the first part. Experimental evidence suggests this third
approach is the most efficient of the three.Comment: 19 pages, 1 tabl
Computing the canonical representation of constructible sets
Constructible sets are needed in many algorithms of Computer Algebra, particularly in the GröbnerCover and other algorithms for parametric polynomial systems. In this paper we review the canonical form ofconstructible sets and give algorithms for computing it.Peer ReviewedPostprint (author's final draft
Highly Automated Formal Verification of Arithmetic Circuits
This dissertation investigates the problems of two distinctive formal verification techniques for verifying large scale multiplier circuits and proposes two approaches to overcome some of these problems. The first technique is equivalence checking based on recurrence relations, while the second one is the symbolic computation technique which is based on the theory of Gröbner bases. This investigation demonstrates that approaches based on symbolic computation have better scalability and more robustness than state-of-the-art equivalence checking techniques for verification of arithmetic circuits. According to this conclusion, the thesis leverages the symbolic computation technique to verify floating-point designs. It proposes a new algebraic equivalence checking, in contrast to classical combinational equivalence checking, the proposed technique is capable of checking the equivalence of two circuits which have different architectures of arithmetic units as well as control logic parts, e.g., floating-point multipliers
Solving a binary puzzle
A Binary puzzle is a Sudoku-like puzzle with values in each cell taken from the set (Formula presented.). Let (Formula presented.) be an even integer, a solved binary puzzle is an (Formula presented.) binary array that satisfies the following conditions: (1) no three consecutive ones and no three consecutive zeros in each row and each column; (2) the number of ones and zeros must be equal in each row and in each column; (3) there can be no repeated row and no repeated column. This paper proposes three approaches to solve the puzzle. The first method is based on a complete backtrack-based search algorithm. The idea is to propagate and fill an unsolved binary puzzle according to the three constraints, followed by a random guess if the puzzle remains unsolved. The second method of solving a binary puzzle is by representing it as an instance of a Boolean satisfiability problem which allows the solution for a binary puzzle to be obtained using SAT solvers. The third approach is based on expressing a binary puzzle as a system of polynomial equations over the binary field (Formula presented.). The set of solutions for the equation system implies the solutions for the binary puzzle and it is obtained by computing a Gröbner basis of the ideal generated by the polynomials. We experimentally compare the three approaches with binary puzzles of various sizes and different numbers of empty cells using a computer algebra system
A polyhedral approach to computing border bases
Border bases can be considered to be the natural extension of Gr\"obner bases
that have several advantages. Unfortunately, to date the classical border basis
algorithm relies on (degree-compatible) term orderings and implicitly on
reduced Gr\"obner bases. We adapt the classical border basis algorithm to allow
for calculating border bases for arbitrary degree-compatible order ideals,
which is \emph{independent} from term orderings. Moreover, the algorithm also
supports calculating degree-compatible order ideals with \emph{preference} on
contained elements, even though finding a preferred order ideal is NP-hard.
Effectively we retain degree-compatibility only to successively extend our
computation degree-by-degree. The adaptation is based on our polyhedral
characterization: order ideals that support a border basis correspond
one-to-one to integral points of the order ideal polytope. This establishes a
crucial connection between the ideal and the combinatorial structure of the
associated factor spaces
ADAM: Analysis of Discrete Models of Biological Systems Using Computer Algebra
Abstract Background
Many biological systems are modeled qualitatively with discrete models, such as probabilistic Boolean networks, logical models, Petri nets, and agent-based models, to gain a better understanding of them. The computational complexity to analyze the complete dynamics of these models grows exponentially in the number of variables, which impedes working with complex models. There exist software tools to analyze discrete models, but they either lack the algorithmic functionality to analyze complex models deterministically or they are inaccessible to many users as they require understanding the underlying algorithm and implementation, do not have a graphical user interface, or are hard to install. Efficient analysis methods that are accessible to modelers and easy to use are needed. Results
We propose a method for efficiently identifying attractors and introduce the web-based tool Analysis of Dynamic Algebraic Models (ADAM), which provides this and other analysis methods for discrete models. ADAM converts several discrete model types automatically into polynomial dynamical systems and analyzes their dynamics using tools from computer algebra. Specifically, we propose a method to identify attractors of a discrete model that is equivalent to solving a system of polynomial equations, a long-studied problem in computer algebra. Based on extensive experimentation with both discrete models arising in systems biology and randomly generated networks, we found that the algebraic algorithms presented in this manuscript are fast for systems with the structure maintained by most biological systems, namely sparseness and robustness. For a large set of published complex discrete models, ADAM identified the attractors in less than one second. Conclusions
Discrete modeling techniques are a useful tool for analyzing complex biological systems and there is a need in the biological community for accessible efficient analysis tools. ADAM provides analysis methods based on mathematical algorithms as a web-based tool for several different input formats, and it makes analysis of complex models accessible to a larger community, as it is platform independent as a web-service and does not require understanding of the underlying mathematics
Satisfiability Modulo Finite Fields
We study satisfiability modulo the theory of finite fields and
give a decision procedure for this theory. We implement our procedure
for prime fields inside the cvc5 SMT solver. Using this theory, we con-
struct SMT queries that encode translation validation for various zero
knowledge proof compilers applied to Boolean computations. We evalu-
ate our procedure on these benchmarks. Our experiments show that our
implementation is superior to previous approaches (which encode field
arithmetic using integers or bit-vectors)
Thomas Decomposition of Algebraic and Differential Systems
In this paper we consider disjoint decomposition of algebraic and non-linear
partial differential systems of equations and inequations into so-called simple
subsystems. We exploit Thomas decomposition ideas and develop them into a new
algorithm. For algebraic systems simplicity means triangularity, squarefreeness
and non-vanishing initials. For differential systems the algorithm provides not
only algebraic simplicity but also involutivity. The algorithm has been
implemented in Maple
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