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Conditional \(p\)-adic probability logic. (English) Zbl 1433.03062
Summary: In this paper we present the proof-theoretical approach to \(p\)-adic valued conditional probabilistic logics. We introduce two such logics denoted by \(\mathrm{CPL}_{\mathbf{Z}_{\mathbf{p}}}\) and \(\mathrm{CPL}_{\mathbf{Q}_{\mathbf{p}}}^{\mathrm{fin}}\). Each of these logics extends classical propositional logic with a list of binary (conditional probability) operators. Formulas are interpreted in Kripke-like models that are based on \(p\)-adic probability spaces. Axiomatic systems with infinitary rules of inference are given and proved to be sound and strongly complete. The decidability of the satisfiability problem for each logic is proved.

MSC:
03B48 Probability and inductive logic
03B25 Decidability of theories and sets of sentences
68T27 Logic in artificial intelligence
Software:
TETRAD
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[1] Neapolitan, R. E., Probabilistic reasoning in expert systems, (1990), Wiley New York
[2] Pearl, J., Probabilistic reasoning in intelligent systems, (1988), Morgan Kaufmann San Francisco
[3] Spirtes, P.; Glymour, C.; Scheines, V., Causation, prediction, and search, (1993), Springer New York · Zbl 0806.62001
[4] Fagin, R.; Halpern, J. Y.; Moses, Y.; Vardi, M. Y., Reasoning about knowledge, (2003), MIT Press Cambridge · Zbl 1060.03008
[5] Thrun, S.; Burgard, W.; Fox, D., Probabilistic robotics, (2005), MIT Press Cambridge · Zbl 1081.68703
[6] Kersting, K.; Raedt, L. D., Bayesian logic programming: theory and tool, (Getoor, L.; Taskar, B., Introduction to Statistical Relational Learning, (2007), MIT Press Cambridge)
[7] Bacchus, F., Lp, a logic for representing and reasoning with statistical knowledge, Comput. Intell., 6, 209-231, (1990)
[8] Fagin, R.; Halpern, J.; Megiddo, N., A logic for reasoning about probabilities, Inf. Comput., 87, 1-2, 78-128, (1990) · Zbl 0811.03014
[9] Lehmann, D., Generalized qualitative probability: savage revised, (Proceedings of 12th Conference on Uncertainty in Artificial Intelligence (UAI-96), (1996)), 381-388
[10] Ognjanović, Z.; Rašković, M.; Marković, Z., Probability logics, Zb. Rad. Log. Comput. Sci., 12, 20, 35-111, (2009) · Zbl 1224.03005
[11] Rašković, M.; Djordjević, R., Probability quantifiers and operators, (1996), VESTA Belgrade · Zbl 0933.03044
[12] Pap, E., Handbook of measure theory, (2002), Elsevier · Zbl 0998.28001
[13] Ikodinović, N.; Rašković, M.; Ognjanović, Z.; Marković, Z., Measure logic, (LNAI, vol. 4724, (2007)), 128-138 · Zbl 1148.68485
[14] Rašković, M.; Ognjanović, Z.; Marković, Z., A logic with approximate conditional probabilities that can model default reasoning, Int. J. Approx. Reason., 49, 52-66, (2008) · Zbl 1184.68520
[15] Coletti, G.; Scozzafava, R., Probabilistic logic in a coherent setting, (2002), Kluwer Dordrecht · Zbl 1005.60007
[16] Biazzo, V.; Gilio, A.; Lukasiewicz, T.; Sanfilippo, G., Probabilistic logic under coherence: complexity and algorithms, Ann. Math. Artif. Intell., 45, 1-2, 35-81, (2005) · Zbl 1083.03027
[17] Gilio, A., Generalizing inference rules in a coherence-based probabilistic default reasoning, Int. J. Approx. Reason., 53, 413-434, (2012) · Zbl 1242.68330
[18] Flaminio, T.; Montagna, F., A logical and algebraic approach to conditional probability, Arch. Math. Log., 44, 4, 245-262, (2005) · Zbl 1064.03016
[19] Gilio, A.; Sanfilippo, G., Quasi conjunction, quasi disjunction, t-norms and t-conorms: probabilistic aspects, Inf. Sci., 245, 146-167, (2013) · Zbl 1320.68188
[20] Khrennikov, A. Y., Interpretations of probability, (2009), Walter de Gruyter Berlin, Germany · Zbl 1369.81014
[21] Khrennikov, A. Y., p-adic valued distributions in mathematical physics, (1994), Kluwer Academic Publishers Dordrecht · Zbl 0833.46061
[22] Keynes, J. M., A treatise on probability, (1921), MacMillan and Co. London · JFM 48.0615.08
[23] Marković, Z.; Rašković, M.; Ognjanović, Z., A probabilistic extension of intuitionistic logic, Math. Log. Q., 49, 415-424, (2003) · Zbl 1022.03011
[24] Marković, Z.; Rašković, M.; Ognjanović, Z., Completeness theorem for a logic with imprecise and conditional probabilities, Publ. Inst. Math., 78, 35-49, (2005) · Zbl 1144.03019
[25] Ognjanović, Z.; Rašković, M., Some probability logics with new types of probability operators, J. Log. Comput., 9, 2, 181-195, (1999) · Zbl 0941.03022
[26] Rašković, M., Classical logic with some probability operators, Publ. Inst. Math., 53, 1-3, (1993) · Zbl 0799.03018
[27] Rašković, M.; Ognjanović, Z.; Marković, Z., A logic with conditional probabilities, (9th European Conference JELIA’04 Logics in Artificial Intelligence, Lecture Notes in Artificial Intelligence, (2004)), 226-238 · Zbl 1111.68688
[28] Milošević, M.; Ognjanović, Z., A first-order conditional probability logic with iterations, Publ. Inst. Math., 93, 107, 19-27, (2013) · Zbl 1313.03008
[29] Ilić-Stepić, A., A logic for reasoning about qualitative probability, Publ. Inst. Math., 87, 97-108, (2010) · Zbl 1289.03006
[30] Ilić-Stepić, A.; Ognjanović, Z.; Ikodinović, N.; Perović, A., A p-adic probability logic, Math. Log. Q., 58, 4-5, 263-280, (2012) · Zbl 1251.03027
[31] Khrennikov, A. Y., p-adic probability interpretation of Bell’s inequality, Phys. Lett. A, 200, 3-4, 219-223, (1995) · Zbl 0910.60001
[32] Dragovich, B.; Dragovich, A., p-adic modeling of the genome and the genetic code, Comput. J., 53, 4, 432-442, (2010)
[33] Khrennikov, A. Y., p-adic discrete dynamical systems and collective behaviour of information states in cognitive models, Discrete Dyn. Nat. Soc., 5, 59-69, (2000) · Zbl 1229.37128
[34] Khrennikov, A. Y., Human subconscious as p-adic dynamical system, J. Theor. Biol., 193, 179-196, (1998)
[35] Albeverio, S.; Khrennikov, A. Y.; Kloeden, P., Memory retrieval as p-adic dynamical system, Biosystems, 49, 105-115, (1999)
[36] Bachman, G., Introduction to p-adic numbers and valuation theory, (1964), Academic Press
[37] Schikof, W., Ultrametric calculus, an introduction to p-adic analysis, (1984), Cambridge University Press
[38] Koc, C. K., A tutorial on p-adic arithmetic, (2002), Oregon State University Corvallis, Oregon
[39] Adams, E. W., The logic of conditionals, (1975), Reidel Dordrecht
[40] Kraus, S.; Lehmann, D.; Magidor, M., Nonmonotonic reasoning, preferential models and cumulative logics, Artif. Intell., 44, 167-207, (1990) · Zbl 0782.03012
[41] Friedman, N.; Halpern, J., Plausibility measure and default reasoning, J. ACM, 48, 4, 648-685, (2001) · Zbl 1127.68438
[42] Schumann, A., p-adic multiple-validity and p-adic valued logical calculi, J. Mult.-Valued Log. Soft Comput., 13, 1-2, 29-60, (2007) · Zbl 1129.03007
[43] Ikodinović, N.; Ognjanović, Z., A logic with coherent conditional probabilities, (Proceedings of the 8th European Conference Symbolic and Quantitative Approaches to Reasoning with Uncertainty ECSQARU 2005, Lecture Notes in Computer Science, vol. 3571, (2005)), 726-736 · Zbl 1122.03015
[44] Cassels, J. W.S., Local fields, (1986), Cambridge University Press London · Zbl 0595.12006
[45] Cassels, J. W.S.; Fronlich, A., Algebraic number theory, (1967), Academic Press London
[46] Hasse, H., Number theory, (1980), Springer Berlin
[47] Arefeva, Y. I.; Dragivch, B.; Frampton, P.; Volovich, I., The wave function of the universe and p-adic gravity, Int. J. Mod. Phys. A, 6, 24, 4341-4358, (1991) · Zbl 0733.53039
[48] Frampton, P.; Okada, Y., A p-adic string n-point function, Phys. Lett. B, 60, 484-486, (1988)
[49] Freund, P.; Witten, E., Adelic string amplitudes, Phys. Lett. B, 199, 191-195, (1987)
[50] Vladimirov, V., On the freund-Witten adelic formula for veneziano amplitudes, Lett. Math. Phys., 27, 123-131, (1993) · Zbl 0779.11034
[51] Volovich, I., p-adic string, Class. Quantum Gravity, 4, 83-87, (1978)
[52] De Grande-De Kimpe, N.; Khrennikov, A. Y., Non-Archimedean Laplace transform, Bull. Belg. Math. Soc. Simon Stevin, 3, 225-237, (1996) · Zbl 0845.46047
[53] Khrennikov, A. Y., Mathematical methods of the non-Archimedean physics, Usp. Mat. Nauk, 45, 4, 79-110, (1990)
[54] Vladimirov, V.; Volovich, I.; Zelenov, E., The spectral theory in the p-adic quantum mechanics, Izv. Akad. Nauk SSSR, Ser. Mat., 54, 2, 275-302, (1990) · Zbl 0709.22010
[55] Albeverio, S.; Khrennikov, A. Y., Representation of the Weyl group in spaces of square integrable functions with respect to p-adic valued Gaussian distributions, J. Phys. A, 29, 5515-5527, (1996) · Zbl 0903.46073
[56] Albeverio, S.; Cianci, R.; Khrennikov, A. Y., A representation of quantum field Hamiltonians in a p-adic Hilbert space, Theor. Math. Phys., 112, 3, 355-374, (1997)
[57] Khrennikov, A. Y., Non-Archimedean analysis: quantum paradoxes, dynamical systems and biological models, (1997), Kluwer Academic Publishers Dordrecht · Zbl 0920.11087
[58] Gouvea, F. Q., p-adic numbers: an introduction, (2000), Springer Berlin
[59] Robert, A. M., A course in p-adic analysis, (2000), Springer Berlin · Zbl 0947.11035
[60] Monna, A.; Springer, T., Integration non-archimedienne, Indag. Math., 25, 634-653, (1963) · Zbl 0147.11803
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