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Logics for reasoning about processes of thinking with information coded by $$p$$-adic numbers. (English) Zbl 1382.03045
Summary: In this paper we present two types of logics (denoted $$L^D_{Q_p}$$ and $$L^{\mathrm{thinking}}_{Z_p}$$) where certain $$p$$-adic functions are associated to propositional formulas. Logics of the former type are $$p$$-adic valued probability logics. In each of these logics we use probability formulas $$K_{r,\rho}\alpha$$ and $$D_\rho\alpha,\beta$$ which enable us to make sentences of the form “the probability of $$\alpha$$ belongs to the $$p$$-adic ball with the center $$r$$ and the radius $$\rho$$”, and “the $$p$$-adic distance between the probabilities of $$\alpha$$ and $$\beta$$ is less than or equal to $$\rho$$”, respectively. Logics of the later type formalize processes of thinking where information are coded by $$p$$-adic numbers. We use the same operators as above, but in this formalism $$K_r$$,$$_\rho\alpha$$ means “the p-adic code of the information $$\alpha$$ belongs to the $$p$$-adic ball with the center $$r$$ and the radius $$\rho$$”, while $$D_\rho\alpha$$, $$\beta$$ means “the $$p$$-adic distance between codes of $$\alpha$$ and $$\beta$$ are less than or equal to $$\rho''$$. The corresponding strongly complete axiom systems are presented and decidability of the satisfiability problem for each logic is proved.

##### MSC:
 03B48 Probability and inductive logic 03B25 Decidability of theories and sets of sentences 03B42 Logics of knowledge and belief (including belief change) 68T27 Logic in artificial intelligence
##### Keywords:
$$p$$-adic; probability logic; coding information
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##### References:
  Albeverio, S., and A.Yu. Khrennikov, Representation of the Weyl group in spaces of square integrable functions with respect to p-adic valued Gaussian distributions, Journal of Physics A 29:5515-5527, 1996. · Zbl 0903.46073  Albeverio, S., R. Cianci, and A.Yu. Khrennikov, A representation of quantum field Hamiltonians in a p-adic Hilbert space, Theoretical and Mathematical Physics 112(3):355-374, 1997. · Zbl 0954.03024  Albeverio, S.; Khrennikov, A.Yu.; Kloeden, P.E., Memory retrieval as p-adic dynamical system., BioSystems, 49, 105-115, (1999)  Aref’eva, Ya. I., B. Dragivch, P. H. Frampton, and I. V. Volovich, The vave function of the Universe and p-adic gravity, International Journal of Modern Physics A 6(24):4341-4358, 1991. · Zbl 0733.53039  Bachman, G., Introduction to p-adic Numbers and Valuation Theory, Polytechnic institute of Brooklyn, Mathematics Department, Brooklyn, New-York.  Baker, A. J., An Introduction to p-adic Numbers and p-adic Analysis, Department of Mathematics, University of Glasgow, Glasgow, Scotland.  De Grande-De Kimpe, N., and A.Yu. Khrennikov, Non-Archimedean Laplace transform, The Bulletin of the Belgian Mathematical Society 3:225-237, 1996. · Zbl 0845.46047  Fagin, R., J. Halpern, and N. Megiddo, A logic for reasoning about probabilities, Information and Computation 87(1-2):78-128, 1990. · Zbl 0811.03014  Gouvea, F. Q., p-adic Numbers: An Introduction, 2nd ed., Springer, ISBN 3-540-62911-4, 2000. · Zbl 0786.11001  Ilić Stepić, A., A logic for reasoning about qualitative probability, Publications de l’institute mathematique, Nouvelle Série, tome, vol. 87, 2010, pp. 97-108. · Zbl 0954.03024  Ilić Stepić, A., Z. Ognjanović, N. Ikodinović, and A. Perović, A $$p$$-adic probability logic, Mathematical Logic Quarterly 58(4-5):263-280, 2012. · Zbl 1251.03027  Keynes, J. M., Treatise on Probability, MacMillan and Co., London, 1921. · JFM 48.0615.08  Khrennikov, A., Human subconscious as a p-adic dynamical system, Journal of Theoretical Biology, 193, 179-196, (1998)  Khrennikov, A., Theory of p-adic valued probabilities, studies in logic, Grammar and Rhetoric, 14, 35-71, (2008)  Khrennikov, A., Theory of p-adic valued probabilities, studies in logic, Grammar and Rhetoric, 14, 137-154, (2008)  Khrennikov, A.Yu., Mathematical methods of the non-Archimedean physics, Uspekhi Matematicheskikh Nauk, 45, 79-110, (1990)  Khrennikov, A.Yu., p-adic Valued Distributions in Mathematical Physics, Kluwer Academic Publishers, Dordrecht, 1994. · Zbl 0833.46061  Khrennikov, A.Yu., P-adic probability interpretation of bell’s inequality, Physics Letters A, 200, 219-223, (1995) · Zbl 1020.81534  Khrennikov, A.Yu., Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Acad. Publ., Dordrecht, The Netherlands, 1997. · Zbl 0920.11087  Khrennikov, A.Yu., P-adic discrete dynamical systems and collective behaviour of information states in cognitive models., Discrete Dynamics in Nature and Society, 5, 59-69, (2000) · Zbl 1229.37128  Khrennikov, A.Yu., Interpretations of Probability, Walter de Gruyter, Berlin, Germany, 2008. · Zbl 0998.81508  Koblitz, N., p-adic Numbers, p-adic Analysis and Zeta-Functions, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo. · Zbl 0364.12015  Koc, C. K., A Tutorial on p-adic Arithmetic, Electrical and Computer Engineering, Oregon State University, Corvallis, Oregon 97331. · Zbl 0941.03022  Marković, Z., Z. Ognjanović, and M. Rašković, A probabilistic extension of intuitionistic logic, Mathematical Logic Quarterly 49(4):415-424, 2003. · Zbl 1022.03011  Nilsson, N., Probabilistic logic, Artificial Intelligence, 28, 71-87, (1986) · Zbl 0589.03007  Ognjanović, Z., and N. Ikodinović, A logic with higher order conditional probabilities, Publications de l’institute Mathematique, Nouvelle série, tome, vol. 82, 2007, pp. 141-154.  Ognjanović, Z.; Rašković, M., Some probability logic with new types of probability operators., Journal of Logic Computation, 9, 181-195, (1999) · Zbl 0941.03022  Ognjanović, Z.; Rašković, M., Some first-order probability logics, Theoretical Computer Science, 247, 191-212, (2000) · Zbl 0954.03024  Ognjanović, Z.; Perović, A.; Rašković, M., Logic with the qualitative probability operator., Logic Journal of IGPL, 16, 105-120, (2008) · Zbl 1138.03024  Ognjanović, Z., M. Rašković, and Z. Marković, Probability logics, Zbornik Radova, Subseries Logic in Computer Science 12(20):35-111, 2009. · Zbl 1224.03005  Rašković, M., Z. Marković, and Z.Ognjanović, A logic with approximate conditional probabilities that can model default reasoning, International Journal of Approximate Reasoning 49:52-66, 2008. · Zbl 1184.68520  Robert, A. M., A Course in p-adic Analysis, Springer, New York, ISBN 0-387-98669-3, 2000. · Zbl 0947.11035  Vladimirov, V.S.; Volovich, I.V.; Zelenov, E.I., The spectral theory in the p-adic quantum mechanics., Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, 54, 275-302, (1990) · Zbl 0709.22010
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