Abstract
This paper analyzes the possible limit set structures for the standard threshold block-sequential finite dynamical systems. As a special case of their work on Neural Networks (generalized threshold functions), Goles and Olivos (1981 [2]) showed that for the single block case (parallel update) one may only have fixed points and 2-cycles as ω-limit sets. Barrett et al (2006 [1]), but also Goles et al (1990 [3]) as a special case, proved that for the case with n blocks (sequential update) the only ω-limit sets are fixed points. This paper generalizes and unifies these results to standard threshold systems with block-sequential update schemes.
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References
Barrett, C.L., Hunt III, H.B., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: Complexity of reachability problems for finite discrete sequential dynamical systems. Journal of Computer and System Sciences 72, 1317–1345 (2006)
Goles, E., Olivos, J.: Comportement periodique des fonctions a seuil binaires et applications. Discrete Applied Mathematics 3, 93–105 (1981)
Goles, E., Martinez, S.: Neural and automata networks: Dynamical behaviour and applications. Kluwer Academic Publishers (1990)
Kuhlman, C., Mortveit, H.S., Murrugarra, D., Anil Kumar, V.S.: Bifurcations in boolean networks. Discrete Mathematics and Theoretical Computer Science (2011), accepted, peer-reviewed article for Automata, November 21-23, Santiago, Chile (2011)
Laubenbacher, R., Paraigis, B.: Limits of sequential dynamical systems (preprint)
Laubenbacher, R., Pareigis, B.: Equivalence relations on finite dynamical systems. Advances in Applied Mathematics 26, 237–251 (2001)
Mortveit, H.S., Reidys, C.M.: Discrete, sequential dynamical systems. Discrete Mathematics 226, 281–295 (2001)
Mortveit, H.S., Reidys, C.M.: An introduction to sequential dynamical systems. Springer, Universitext (2007)
Robert, F.: Discrete iterations. a metric study. Springer Series in Computational Mathematics, vol. (6). Springer (1986)
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Mortveit, H.S. (2012). Limit Cycle Structure for Block-Sequential Threshold Systems. In: Sirakoulis, G.C., Bandini, S. (eds) Cellular Automata. ACRI 2012. Lecture Notes in Computer Science, vol 7495. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33350-7_69
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DOI: https://doi.org/10.1007/978-3-642-33350-7_69
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33349-1
Online ISBN: 978-3-642-33350-7
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