zbMATH — the first resource for mathematics

\(p\)-adic valued logical calculi in simulations of the slime mould behaviour. (English) Zbl 1398.03113
Summary: In this paper we consider possibilities for applying \(p\)-adic valued logic BL to the task of designing an unconventional computer based on the medium of slime mould (order Physarales, class Myxomecetes, subclass Myxogastromycetidae), the giant amoebozoa that looks for attractants and reaches them by means of propagating complex networks. If it is assumed that at any time step \(t\) of propagation the slime mould can discover and reach not more than \(p-1\) attractants, then this behaviour can be coded in terms of \(p\)-adic numbers. As a result, this behaviour implements some \(p\)-adic valued arithmetic circuits and can verify \(p\)-adic valued logical propositions.
03B50 Many-valued logic
92B05 General biology and biomathematics
Full Text: DOI
[1] Adamatzky, A. (2010). Physarum machines: Computers from slime mould. Singapore: World Scientific.
[2] Adamatzky, A., Erokhin, V., Grube, M., Schubert, T., & Schumann, A. (2012). Physarum chip project: Growing computers from slime mould. International Journal of Unconventional Computing,8, 319-323.
[3] Baaz, M., Fermüller, C., & Zach, R. (1993). Systematic construction of natural deduction systems for many-valued logics. In 23rd IEEE International Symposium on Multiple-Valued Logic (ISMVL 1993), May 24-27, 1993, Sacramento, California, USA, proceedings (pp. 208-213). Washington, DC: IEEE Computer Society.
[4] Ilić-Stepić, A., & Ognjanović, Z. (2014). Logics for reasoning about processes of thinking with information coded by p-adic numbers. Studia Logica,103, 145-174.
[5] Ilić-Stepić, A., Ognjanović, Z., & Ikodinović, N. (2014). Conditional p-adic probability logic. International Journal of Approximate Reasoning,55, 1843-1865.
[6] Ilić-Stepić, A., Ognjanović, Z., Ikodinović, N., & Perović, A. (2012). A p-adic probability logic. Mathematical Logic Quarterly,58, 263-280.
[7] Khrennikov, A. (1990). Mathematical methods of the non-Archimedean physics. Russian Mathematical Surveys,45(4), 87-125. · Zbl 0722.46040
[8] Khrennikov, A. (1994). p-Adic valued distributions in mathematical physics. Dordrecht: Kluwer Academic. · Zbl 0833.46061
[9] Khrennikov, A. (1995). p-Adic probability interpretation of Bell’s inequality. Physics Letters A,200, 219-223. · Zbl 1020.81534
[10] Khrennikov, A. (1997). Non-Archimedean analysis: Quantum paradoxes, dynamical systems and biological models. Dordrecht: Kluwer Academic. · Zbl 0920.11087
[11] Khrennikov, A. (1998). Human subconscious as a p-adic dynamical system. Journal of Theoretical Biology,193, 179-196.
[12] Khrennikov, A. (2000). p-Adic discrete dynamical systems and collective behaviour of information states in cognitive models. Discrete Dynamics in Nature and Society,5, 59-69. · Zbl 1229.37128
[13] Khrennikov, A. (2001). Interpretations of probability and their p-adic extensions. Theory of Probability & Its Applications,46, 256-273. · Zbl 1012.81005
[14] Khrennikov, A. (2008). Toward theory of p-adic valued probabilities. Studies in Logic, Grammar and Rhetoric,14(27), 137-154.
[15] Khrennikov, A., Yamada, S., & van Rooij, A. (1999). The measure-theoretical approach to p-adic probability theory. Annales mathématiques Blaise Pascal,6(1), 21-32. · Zbl 0941.60010
[16] Koblitz, N. (1984). p-Adic numbers, p-adic analysis and zeta functions (2nd ed.). Berlin: Springer. · Zbl 0364.12015
[17] Milošević, M. (2010). A propositional p-adic probability logic. Publications de l’Institut Mathématique (nouvelle série),87(101), 75-83.
[18] Nakagaki, T., Iima, M., Ueda, T., Nishiura, Y., Saigusa, T., Tero, A., & Showalter, K. (2007). Minimum-risk path finding by an adaptive amoeba network. Physical Review Letters,99, 68-104.
[19] Nakagaki, T., Yamada, H., & Toth, A. (2000). Maze-solving by an amoeboid organism. Nature,407, 470.
[20] Nakagaki, T., Yamada, H., & Toth, A. (2001). Path finding by tube morphogenesis in an amoeboid organism. Biophysical Chemistry,92, 47-52.
[21] Saigusa, T., Tero, A., Nakagaki, T., & Kuramoto, Y. (2008). Amoebae anticipate periodic events. Physical Review Letters,100, 018101.
[22] Schumann, A. (2007a). Non-Archimedean valued predicate logic. Bulletin of the Section of Logic,36, 67-78. · Zbl 1286.03094
[23] Schumann, A. (2007b). p-Adic multiple-validity and p-adic valued logical calculi. Journal of Multiple-Valued Logic and Soft Computing,13, 29-60. · Zbl 1129.03007
[24] Schumann, A. (2008a). Non-Archimedean fuzzy and probability logic. Journal of Applied Non-Classical Logics,18, 29-48. · Zbl 1187.03024
[25] Schumann, A. (2008b). Non-Archimedean valued and p-adic valued fuzzy cellular automata. Journal of Cellular Automata,3, 337-354. · Zbl 1158.68026
[26] Schumann, A. (2008c). Non-well-founded probabilities on streams. In D. Dubois, et al. (Eds.), Soft methods for handling variability and imprecision (pp. 59-65). New York, NY: Springer.
[27] Schumann, A. (2010). Non-Archimedean valued extension of logic LП and p-adic valued extension of logic BL. Journal of Uncertain Systems,4, 99-115.
[28] Schumann, A. (2014a). p-Adic valued fuzzyness and experiments with Physarum polycephalum. In J. Li, D. Zhang, M. Li, & Y. Lei (Eds.), 2014 11th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD) (pp. 466-472). Washington, DC: IEEE Computer Society.
[29] Schumann, A. (2014b). Reflexive games and non-Archimedean probabilities. p-Adic Numbers. Ultrametric Analysis and Applications,6, 66-79. · Zbl 1312.91025
[30] Schumann, A., & Pancerz, K. (2013). Towards an object-oriented programming language for Physarum polycephalum computing. In M. Szczuka, L. Czaja, & M. Kacprzak (Eds.), Proceedings of the Workshop on Concurrency, Specification and Programming (CS &P ’2013) (pp. 389-397). Bialystok: Bialystok University of Technology.
[31] Schumann, A., & Pancerz, K. (2014). Towards an object-oriented programming language for Physarum polycephalum computing: A petri net model approach. Fundamenta Informaticae,133, 271-285.
[32] Shirakawa, T., Gunji, Y.-P., & Miyake, Y. (2011). An associative learning experiment using the plasmodium of Physarum polycephalum. Nano Communication Networks,2, 99-105.
[33] Tero, A., Nakagaki, T., Toyabe, K., Yumiki, K., & Kobayashi, R. (2010). A method inspired by physarum for solving the Steiner problem. International Journal of Unconventional Computing,6, 109-123.
[34] Watanabe, S., Tero, A., Takamatsu, A., & Nakagaki, T. (2011). Traffic optimisation in railroad networks using an algorithm mimicking an amoeba-like organism, Physarum plasmodium. Biosystems,105, 225-232.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.