Abstract
This paper presents a combinatorial study to characterise the dynamics of intersecting Boolean automata circuits and more specifically that of double Boolean automata circuits. Explicit formulae are given to count the number of periodic configurations and attractors of these networks and a conjecture proposes a comparison between the number of attractors of isolated circuits and that of double circuits. The aim of this study is to give intuition on the way circuits interact and how a circuits intersection modifies the “degrees of freedom” of the overall network.
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References
Adams, W., Shanks, D.: Strong primality tests that are not sufficient. Mathematics of Computation (American Mathematical Society) 39(159), 255–300 (1982)
Apostol, T.M.: Introduction to analytic number theory. Springer, Heidelberg (1976)
Aracena, J., Ben Lamine, S., Mermet, O., Cohen, O., Demongeot, J.: Mathematical modeling in genetic networks: relationships between the genetic expression and both chromosomic breakage and positive circuits. IEEE Transactions on Systems, Man, and Cybernetics 33, 825–834 (2003)
Demongeot, J., Noual, M., Sené, S.: Combinatorics of boolean automata circuits dynamics (2011) (submitted)
Elena, A.: Algorithme pour la simulation dynamique des réseaux de régulation génétique. Master’s thesis, University J. Fourier (2004)
Goles, E.: Comportement oscillatoire d’une famille d’automates cellulaires non uniformes. Ph.D. thesis, Université scientifique et médicale de Grenoble, France (1980), http://tel.archives-ouvertes.fr/tel-00293368/fr/
Hopfield, J.J.: Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences of the USA 79, 2554–2558 (1982)
Kauffman, S.A.: Metabolic stability and epigenesis in randomly constructed genetic nets. Journal of Theoretical Biology 22, 437–467 (1969)
McCulloch, W.S., Pitts, W.: A logical calculus of the ideas immanent in nervous activity. Bulletin of Mathematical Biophysics 5, 115–133 (1943)
Puri, Y., Ward, T.: Arithmetic and growth of periodic orbits. Journal of Integer Sequences 4(2) (2001)
Puri, Y., Ward, T.: A dynamical property unique to the Lucas sequence. The Fibonacci Quarterly. The Official Journal of the Fibonacci Association 39(5), 398–402 (2001)
Remy, É., Ruet, P., Thieffry, D.: Graphic requirements for multistability and attractive cycles in a Boolean dynamical framework. Advances in Applied Mathematics 41, 335–350 (2008)
Ribenboim, P.: The New Book of Prime Number Records. Springer, Heidelberg (1996)
Richard, A.: Positive circuits and maximal number of fixed points in discrete dynamical systems. Discrete Applied Mathematics 157, 3281–3288 (2009)
Riordan, J.: An Introduction to Combinatorial Analysis. Wiley, New York (1980)
Sloane, N.J.A.: The on-line encyclopedia of integer sequences, OEIS (2008)
Thomas, R.: Boolean formalisation of genetic control circuits. Journal of Theoretical Biology 42, 563–585 (1973)
Thomas, R.: On the relation between the logical structure of systems and their ability to generate multiple steady states or sustained oscillations. Springer Series in Synergetics, vol. 9, pp. 180–193 (1981)
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Noual, M. (2012). Dynamics of Circuits and Intersecting Circuits. In: Dediu, AH., Martín-Vide, C. (eds) Language and Automata Theory and Applications. LATA 2012. Lecture Notes in Computer Science, vol 7183. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28332-1_37
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DOI: https://doi.org/10.1007/978-3-642-28332-1_37
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-28331-4
Online ISBN: 978-3-642-28332-1
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