The Behaviour of Affine Boolean Sequential Networks

Milligan, D. K.; Wilson, M.

doi:10.1080/09540099308915693

Cited by 18 publications

(13 citation statements)

References 7 publications

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“…An affine transformation of a vector space V is a map T : V → V given by T (v) = L(v) + b for some linear map L : V → V and some fixed b ∈ V . In this section, we review some results and terminology from linear algebra and discuss previous work of Elspas [14] and Milligan and Wilson [20] concerning the dynamics of affine transformations of F 2 -vector spaces.…”

Section: Dynamics Of Affine Transformationsmentioning

confidence: 99%

“…In Section 4, we give a formula for |Per r (T )| for any positive integer r when T is the SDS map of an SIAECA with the identity update order. To prove our theorems, we borrow many techniques from Elspas [14] and Milligan and Wilson [20] (which we discuss in Section 3). In [14] and [20], the authors determine the dynamics of affine maps from the elementary divisors of certain linear maps.…”

Section: Introductionmentioning

confidence: 99%

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Flexible toggles and symmetric invertible asynchronous elementary cellular automata

Defant

2018

Discrete Mathematics

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A sequential dynamical system (SDS) consists of a graph G with vertices v1, . . . , vn, a state set A, a collection of "vertex functions" {fv i } n i=1 , and a permutation π ∈ Sn that specifies how to compose these functions to yield the SDS map [G, {fv i } n i=1 , π] : A n → A n . In this paper, we study symmetric invertible SDS defined over the cycle graph Cn using the set of states F2. These are, in other words, asynchronous elementary cellular automata (ECA) defined using ECA rules 150 and 105. Each of these SDS defines a group action on the set F n 2 of n-bit binary vectors. Because the SDS maps are products of involutions, this relates to generalized toggle groups, which Striker recently introduced. In this paper, we further generalize the notion of a generalized toggle group to that of a flexible toggle group; the SDS maps we consider are examples of Coxeter elements of flexible toggle groups.Our main result is the complete classification of the dynamics of symmetric invertible SDS defined over cycle graphs using the set of states F2 and the identity update order π = 123 · · · n. More precisely, if T denotes the SDS map of such an SDS, then we obtain an explicit formula for |Perr(T )|, the number of periodic points of T of period r, for every positive integer r. It turns out that if we fix r and vary n and T , then |Perr(T )| only takes at most three nonzero values.

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Section: Dynamics Of Affine Transformationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Flexible toggles and symmetric invertible asynchronous elementary cellular automata

Defant

2018

Discrete Mathematics

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“…[93] and [92] present applications of the Boolean case in control theory. Furthermore, the affine case was studied by [73]. An interesting contribution was made by Paul Cull ([26]), who extended the considerations to nonlinear functions, and showed how to reduce them to the linear case.…”

Section: Currently Available Techniques For Dynamics Forecastmentioning

confidence: 99%

Boolean monomial control systems

Delgado-Eckert

2009

Mathematical and Computer Modelling of Dynamical Systems

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“…In general, a synchronous Boolean network contains n nodes (x 1 , x 2 , x 3 …, x n ), with each node updating itself with each other in a synchronous manner (Milligan and Wilson, 1993;Farrow et al, 2004). Each node has only 1 Boolean value: 1 (ON) or 0 (OFF) at 1 moment.…”

Section: Synchronous Boolean Network and Basic Definitionsmentioning

confidence: 99%

An efficient algorithm for computing fixed length attractors based on bounded model checking in synchronous Boolean networks with biochemical applications

Li¹,

Yang²,

Zheng³

et al. 2015

Genet. Mol. Res.

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ABSTRACT. Genetic regulatory networks are the key to understanding biochemical systems. One condition of the genetic regulatory network under different living environments can be modeled as a synchronous Boolean network. The attractors of these Boolean networks will help biologists to identify determinant and stable factors. Existing methods identify attractors based on a random initial state or the entire state simultaneously. They cannot identify the fixed length attractors directly. The complexity of including time increases exponentially with respect to the attractor number and length of attractors. This study used the bounded model checking to quickly locate fixed length attractors. Based on the SAT solver, we propose a new algorithm for efficiently computing the fixed length attractors, which is more suitable for large Boolean networks and numerous attractors' networks. After comparison An algorithm for computing fixed length attractors using the tool BooleNet, empirical experiments involving biochemical systems demonstrated the feasibility and efficiency of our approach.

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The Behaviour of Affine Boolean Sequential Networks

Cited by 18 publications

References 7 publications

Flexible toggles and symmetric invertible asynchronous elementary cellular automata

Flexible toggles and symmetric invertible asynchronous elementary cellular automata

Boolean monomial control systems

An efficient algorithm for computing fixed length attractors based on bounded model checking in synchronous Boolean networks with biochemical applications

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