22,558 research outputs found
Identification of control targets in Boolean molecular network models via computational algebra
Motivation: Many problems in biomedicine and other areas of the life sciences
can be characterized as control problems, with the goal of finding strategies
to change a disease or otherwise undesirable state of a biological system into
another, more desirable, state through an intervention, such as a drug or other
therapeutic treatment. The identification of such strategies is typically based
on a mathematical model of the process to be altered through targeted control
inputs. This paper focuses on processes at the molecular level that determine
the state of an individual cell, involving signaling or gene regulation. The
mathematical model type considered is that of Boolean networks. The potential
control targets can be represented by a set of nodes and edges that can be
manipulated to produce a desired effect on the system. Experimentally, node
manipulation requires technology to completely repress or fully activate a
particular gene product while edge manipulations only require a drug that
inactivates the interaction between two gene products. Results: This paper
presents a method for the identification of potential intervention targets in
Boolean molecular network models using algebraic techniques. The approach
exploits an algebraic representation of Boolean networks to encode the control
candidates in the network wiring diagram as the solutions of a system of
polynomials equations, and then uses computational algebra techniques to find
such controllers. The control methods in this paper are validated through the
identification of combinatorial interventions in the signaling pathways of
previously reported control targets in two well studied systems, a p53-mdm2
network and a blood T cell lymphocyte granular leukemia survival signaling
network.Comment: 12 pages, 4 figures, 2 table
Polynomial Identity Testing and the Ideal Proof System: PIT is in NP if and only if IPS can be p-simulated by a Cook-Reckhow proof system
The Ideal Proof System (IPS) of Grochow & Pitassi (FOCS 2014, J. ACM, 2018)
is an algebraic proof system that uses algebraic circuits to refute the
solvability of unsatisfiable systems of polynomial equations. One potential
drawback of IPS is that verifying an IPS proof is only known to be doable using
Polynomial Identity Testing (PIT), which is solvable by a randomized algorithm,
but whose derandomization, even into NSUBEXP, is equivalent to strong lower
bounds. However, the circuits that are used in IPS proofs are not arbitrary,
and it is conceivable that one could get around general PIT by leveraging some
structure in these circuits. This proposal may be even more tempting when IPS
is used as a proof system for Boolean Unsatisfiability, where the equations
themselves have additional structure.
Our main result is that, on the contrary, one cannot get around PIT as above:
we show that IPS, even as a proof system for Boolean Unsatisfiability, can be
p-simulated by a deterministically verifiable (Cook-Reckhow) proof system if
and only if PIT is in NP. We use our main result to propose a potentially new
approach to derandomizing PIT into NP
A general-purpose approach to computer-aided dynamic analysis of a flexible helicopter
A general purpose mathematical formulation is described for dynamic analysis of a helicopter consisting of flexible and/or rigid bodies that undergo large translations and rotations. Rigid body and elastic sets of generalized coordinates are used. The rigid body coordinates define the location and the orientation of a body coordinate frame (global frame) with respect to an inertial frame. The elastic coordinates are introduced using a finite element approach in order to model flexible components. The compatibility conditions between two adjacent elements in a flexible body are imposed using a Boolean matrix, whereas the compatibility conditions between two adjacent bodies are imposed using the Lagrange multiplier approach. Since the form of the constraint equations depends upon the type of kinematic joint and involves only the generalized coordinates of the two participating elements, then a library of constraint elements can be developed to impose the kinematic constraint in an automated fashion. For the body constraints, the Lagrange multipliers yield the reaction forces and torques of the bodies at the joints. The virtual work approach is used to derive the equations of motion, which are a system of differential and algebraic equations that are highly nonlinear. The formulation presented is general and is compared with hard-wired formulations commonly used in helicopter analysis
Об одном подходе к решению логических комбинаторных задач
Рассматривается известная логическая задача, авторство которой приписывается А. Эйнштейну. Предлагается, алгебраический подход к ее решению, который сводится к последовательному решению системы уравнений с булевыми переменными.Розглядається відома логічна задача, авторство якої приписують А. Ейнштейну. Пропонується алгебраїчний підхід до її розв’язання, який зводиться до послідовного розв’язання системи лінійних рівнянь з булевими змінними.The paper deals with the well-known logical problem which statement is attributed to A. Einstein. An algebraic approach to its solution is suggested which reduces to successive solving the system of equations in boolean variables
New developments in the theory of Groebner bases and applications to formal verification
We present foundational work on standard bases over rings and on Boolean
Groebner bases in the framework of Boolean functions. The research was
motivated by our collaboration with electrical engineers and computer
scientists on problems arising from formal verification of digital circuits. In
fact, algebraic modelling of formal verification problems is developed on the
word-level as well as on the bit-level. The word-level model leads to Groebner
basis in the polynomial ring over Z/2n while the bit-level model leads to
Boolean Groebner bases. In addition to the theoretical foundations of both
approaches, the algorithms have been implemented. Using these implementations
we show that special data structures and the exploitation of symmetries make
Groebner bases competitive to state-of-the-art tools from formal verification
but having the advantage of being systematic and more flexible.Comment: 44 pages, 8 figures, submitted to the Special Issue of the Journal of
Pure and Applied Algebr
Aspects of Algebraic Quantum Theory: a Tribute to Hans Primas
This paper outlines the common ground between the motivations lying behind
Hans Primas' algebraic approach to quantum phenomena and those lying behind
David Bohm's approach which led to his notion of implicate/explicate order.
This connection has been made possible by the recent application of orthogonal
Clifford algebraic techniques to the de Broglie-Bohm approach for relativistic
systems with spin.Comment: 18 pages. No figure
On the Complexity of Solving Quadratic Boolean Systems
A fundamental problem in computer science is to find all the common zeroes of
quadratic polynomials in unknowns over . The
cryptanalysis of several modern ciphers reduces to this problem. Up to now, the
best complexity bound was reached by an exhaustive search in
operations. We give an algorithm that reduces the problem to a combination of
exhaustive search and sparse linear algebra. This algorithm has several
variants depending on the method used for the linear algebra step. Under
precise algebraic assumptions on the input system, we show that the
deterministic variant of our algorithm has complexity bounded by
when , while a probabilistic variant of the Las Vegas type
has expected complexity . Experiments on random systems show
that the algebraic assumptions are satisfied with probability very close to~1.
We also give a rough estimate for the actual threshold between our method and
exhaustive search, which is as low as~200, and thus very relevant for
cryptographic applications.Comment: 25 page
A discussion on the origin of quantum probabilities
We study the origin of quantum probabilities as arising from non-boolean
propositional-operational structures. We apply the method developed by Cox to
non distributive lattices and develop an alternative formulation of
non-Kolmogorvian probability measures for quantum mechanics. By generalizing
the method presented in previous works, we outline a general framework for the
deduction of probabilities in general propositional structures represented by
lattices (including the non-distributive case).Comment: Improved versio
ADAM: Analysis of Discrete Models of Biological Systems Using Computer Algebra
Background: Many biological systems are modeled qualitatively with discrete
models, such as probabilistic Boolean networks, logical models, Petri nets, and
agent-based models, with the goal to gain a better understanding of the system.
The computational complexity to analyze the complete dynamics of these models
grows exponentially in the number of variables, which impedes working with
complex models. Although there exist sophisticated algorithms to determine the
dynamics of discrete models, their implementations usually require
labor-intensive formatting of the model formulation, and they are oftentimes
not accessible to users without programming skills. Efficient analysis methods
are needed that are accessible to modelers and easy to use. Method: By
converting discrete models into algebraic models, tools from computational
algebra can be used to analyze their dynamics. 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.
Results: A method for efficiently identifying attractors, and 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. 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, i.e., while the number of nodes in a biological network may
be quite large, each node is affected only by a small number of other nodes,
and robustness, i.e., small number of attractors
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