Abstract
In this chapter we present some of the mathematical tools used in AI, mostly related to decision making. One of the main tools are general aggregation functions (operators) with some special important cases as triangular norms and copulas. We present briefly some of their important applications. Important extension of the classical field of real numbers is related to the so called pseudo-operations: pseudo-addition and pseudo-multiplication. Here we present some of the important cases. Then corresponding pseudo additive measures and corresponding integrals are introduced, which make the base for the so called pseudo-analysis. The usefulness of the pseudo-analysis, which is a approach to nonlinearity, uncertainty and optimization, is illustrated with a short overview of some important applications. Very important tools in modeling decision making with overlaped events are general fuzzy measures and corresponding integrals as Choquet integral. We present a fuzzy integral approach to Cumulative Prospect theory. Further, we present many different generalizations of the Choquet integral.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Adams, D.R.: Choquet integrals in potential theory. Publ. Mat. 42(1), 3–66 (1998)
Aumann, R.J., Shapley, L.S.: Values of Non-Atomic Games. Princeton University Press, Princton (1974)
Baccelli, F., Cohen, G., Olsder, G.J., Quadra, J.P.T: Synchronization and Linearity: An Algebra for Discrete Event Systems. Wiley, New York (1992)
Benvenuti, P., Mesiar, R., Vivona D.: Monotone set functions-based integrals. In: Pap, E. (eds.) Handbook of Measure Theory, vol. II, pp. 1329–1379 . Elsevier, Amsterdam (2002)
Bernard, C., Ghossoub, M.: Static portfolio choice under cumulative prospect theory. Math. Financ. Econ. 2, 277–306 (2010)
Burgin, M.S.: Nonclassical models of the natural numbers. Uspekhi Mat. Nauk 32, 209–210 (1977). (in Russian)
Burgin, M., Meissner, G.: \(1 + 1 = 3:\) Synergy arithmetics in economics. Appl. Math. 8, 133–134 (2017)
Czachor, M.: A Loophole of all “Loophole-Free” Bell-type theorems. Found. Sci. 25, 971–985 (2020). https://doi.org/10.1007/s10699-020-09666-0
Czachor, M.: Non-Newtonian mathematics instead of non-Newtonian physics: dark matter and dark energy from a mismatch of arithmetics. Found. Sci. 26, 75–95 (2021). https://doi.org/10.1007/s10699-020-09687-9(0123456789(),-volV)(0123456789().,-volV)
Carbajal-Hernandez, J.J., Sanchez-Fernandez, L.P., Carrasco-Ochoa, J.A.: Assessment and prediction of air quality using fuzzy logic and autoregressive models. Atmos. Environ. 60, 3–50 (2012)
Choquet, G.: Theory of capacities. Ann. lnst. Fourier 5, 131–295 (1953)
Deisenroth, M.P., Faisal, A.A., Ong, C.S.: Mathematics for Machine Learning. Cambridge University Press, Cambridge (2020)
Delić, M., Nedović, E., Pap, E.: Extended power-based aggregation of distance functions and application in image segmentation. Inf. Sci. 494, 155–173 (2019)
Denneberg, D.: Non-additive Measure and Integral. Kluwer, Dordrecht (1994)
Do, Y., Thiele, C.: \(L^p\) theory for outer measures and two themes of Lennart Carlson united. Bull. Amer. Math. Soc. 52(2), 249–296 (2015)
Dubois, D., Pap, E., Prade, H.: Hybrid probabilistic-possibilistic mixtures and utility functions. In: Fodor, J., de Baets, B., Perny, P. (eds. ) Preferences and Decisions Under Incomplete Knowledge, volume 51 of Studies in Fuzziness and Soft Computing, pp. 51–73. Springer, Berlin (2000)
Even, Y., Lehrer, E.: Decomposition-integral: unifying Choquet and concave integrals. Econom. Theory 56, 33–58 (2014)
Gal, G.: On a Choquet-Stieltjes type integral on intervals. Math. Slovaca 69, 801–814 (2019)
Gilboa, I., Schmeidler, D.: Additive Representations of non-additive measures and the Choquet integral. Ann. Oper. Res. 52, 43–65 (1994)
Grabisch, M.: Set Functions. Games and Capacities in Decision Making. Springer, Cham (2016)
Grabisch, M., Marichal, J.L., Mesiar, R., Pap, E.: Aggregation Functions. Encyclopedia of Mathematics and Its Applications, vol. 127, Cambridge University Press, Cambridge (2009)
Grossman, M., Katz, R.: Non-Newtonian Calculus. Lee Press, Pigeon Cove (1972)
Grbić, T., Medić, S., Perović, A., Mihailović, B., Novković, N., Duraković, N.: A premium principle based on the \(g\)-integral. Stoch. Anal. Appl. 35(3), 465–477 (2017)
Greco, S., Mesiar, R., Rindone, F.: Discrete bipolar universal integral. Fuzzy Sets Syst. 252, 55–65 (2014)
Joe, H.: Dependence Modeling with Copulas. Monographs on Statistics and Applied Probability, vol. 134. CRC Press, Boca Raton (2015)
Hadžić, O., Pap, E.: Fixed Point Theory in Probabilistic Metric Spaces. Mathematics and Its Applications, vol. 536. Kluwer Academic Publishers, Dordrecht (2001)
Kaluszka, M., Krzeszowiec, M.: Pricing insurance contracts under cumulative prospect theory. Insur. Math. Econ. 50, 159–166 (2012)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer, Amsterdam (2000)
Klement, E.P., Mesiar, R., Pap, E.: Integration with respect to decomposable measures based on a conditionally distributive semiring on the unit interval. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 8, 701–717 (2000)
Klement, E.P., Mesiar, R., Pap, E.: A universal integral as common frame for Choquet and Sugeno integral. IEEE Trans. Fuzzy Syst. 18(1), 178–187 (2010)
Klement, E.P., Li, J., Mesiar, R., Pap, E.: Integrals based on monotone set functions. Fuzzy Sets Syst. 281, 88–102 (2015)
Kolokoltsov, V.N.: Nonexpansive maps and option pricing theory. Kybernetika 34(6), 713–724 (1998)
Kuczma, M., Gilányi, A. (ed.): An Introduction to the Theory of Functional Equations and Inequalities, Cauchy’s Equation and Jensens Inequality, 2nd edn. Birkhäuser, Basel (2009)
Lehrer, E.: A new integral for capacities. Econom. Theory 39, 157–176 (2009)
Lehrer, E., Teper, R.: The concave integral over large spaces. Fuzzy Sets Syst. 159, 2130–2144 (2008)
Litvinov, G.L., Maslov, V.P. (eds): Idempotent Mathematics and Mathematical Physics. Contemporary Mathematics, vol. 377. American Mathematical Society, Providence (2003)
Mao, T., Yang, F.: Characterizations of risk aversion in cumulative prospect theory. Mathematics and Financial Economics. https://doi.org/10.1007/s11579-018-0229-0
Maslov, V.P.: A new superposition principle for optimization problems. Uspekhi Mat Nauk 42(3), 39–48 (1987)
Maslov, V.P., Samborskii, S.N. (eds.): Idempotent Analysis. Advances in Soviet Mathematics, vol. 13. American Mathematical Society, Providence (1992)
Mesiar, R.: Choquet-like integral. J. Math. Anal. Appl. 194, 477–488 (1995)
Mesiar, R., Li, J., Pap, E.: Pseudo-concave integrals. In: NLMUA 2011, Advances in Intelligent Systems and Computing, vol. 100, pp. 43–49. Springer, Berlin (2011)
Mesiar, R., Li, J., Pap, E.: The Choquet integral as Lebesgue integral and related inequalities. Kybernetika 46, 1098–1107 (2010)
Mesiar, R., Li, J., Pap, E.: Discrete pseudo-integrals. J. Approx. Reason. 54, 357–364 (2013)
Mesiar, R., Li, J., Pap, E.: Superdecomposition integral. Fuzzy Sets Syst. 259, 3–11 (2015)
Mesiar, R., Pap, E.: Idempotent integral as limit of \(g\)-integrals. Fuzzy Sets Syst. 102, 385–392 (1999)
Mesiar, R., Rybarik, J.: Pan-operations structure. Fuzzy Sets Syst. 74, 365–369 (1995)
Mihailović, B., Pap, E.: Asymmetric integral as a limit of generated Choquet integrals based on absolutely monotone real set functions. Fuzzy Sets Syst. 181, 39–49 (2011)
Mihailović, B., Pap, E., Štrboja, M., Simićevic A.: A unified approach to the monotone integral-based premium principles under the CPT theory. Fuzzy Sets Syst. 398, 78–97 (2020)
Murofushi, T., Sugeno, M.: A theory of fuzzy measures: representations, the Choquet integral, and null sets. J. Math. Anal. Appl. 159, 532–549 (1991)
Narukawa, Y., Murofushi, T.: The \(n\)-step Choquet integral on finite spaces. In: Proceedings of the 9th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, Annecy, pp. 539–543 (2002)
Nedović, Lj., Pap, E.: Aggregation of sequence of fuzzy measures. Iranian J. Fuzzy 17(2), 39–55 (2020)
Nelsen, R.B.: An Introduction to Copulas. Lecture Notes in Statistics. Springer, Berlin (1999)
Pap, E.: An integral generated by decomposable measure. Univ. Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 20(1), 135–144 (1990)
Pap, E.: \(g\)-calculus. Univ. u Novom Sadu, Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 23(1), 145–150 (1993)
Pap, E.: Null-additive set functions. Mathematics and Its Applications, vol. 337. Kluwer Academic Publishers, Dordrecht (1995)
Pap, E. (ed.): Handbook of Measure Theory: I, II. Elsevier, Amsterdam (2002)
Pap, E.: Pseudo-additive measures and their applications. In: Pap, E. (ed.) Handbook of Measure Theory, vol. II, pp. 1403–1465. Elsevier, North-Holland (2002)
Pap, E.: A generalization of the utility theory using a hybrid idempotent-probabilistic measure. In: Litvinov, G.L., Maslov, V.P. (eds.) Proceedings of the Conference on Idempotent Mathematics and Mathematical Physics, Contemporary Mathematics, vol. 377, pp. 261–274. American Mathematical Society, Providence (2005)
Pap, E.: Generalized real analysis and its application. Int. J. Approx. Reason. 47, 368–386 (2008)
Pap, E. (ed.): Intelligent Systems: Models and Applications. Topics in Intelligent Engineering and Informatics, vol. 3. Springer, Berlin (2013)
Pap, E.: Sublinear Means. In: 36th Linz Seminar on Fuzzy Sets Theory: Functional Equations and Inequalities, Linz, 2–6 February 2, pp. 75–79 (2016)
Pap, E.: Three types of generalized Choquet integral. Bull. Un. Mat. Ital. 13(4), 545–553 (2020)
Pap, E., Štajner, I.: Generalized pseudo-convolution in the theory of probabilistic metric spaces, information, fuzzy numbers, system theory. Fuzzy Sets Syst. 102, 393–415 (1999)
Puhalskii, A.: Large Deviations and Idempotent Probability. CRC Press, Boca Raton (2001)
Quiggin, J.: Generalized Expected Utility Theory. The Rank-Dependent Model. Kluwer Academic Publishers, Boston (1993)
Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57, 517–587 (1989)
Shilkret, N.: Maxitive measure and integration. Indag. Math. 33, 109–116 (1971)
Sugeno, M.: Theory of fuzzy integrals and its applications. Ph.D. Thesis, Tokyo Institute of Technology (1974)
Sowlat, M.H., Gharibi, H., Yunesian, M., Mahmoudi, M.T., Lotfi, S.: A novel, fuzzy-based air quality index (FAQI) for air quality assessment. Atmos. Environ. 45, 2050–2059 (2011)
Sugeno, M., Murofushi, T.: Pseudo-additive measures and integrals. J. Math. Anal. Appl. 122, 197–222 (1987)
Tan, X., Han, L., Zhang, X., Zhou, W., Li, W., Qian, Y.: A review of current quality indexes and improvements under the multi-contaminant air pollution exposure. J. Environ. Manag. 279 (2021). https://doi.org/10.1016/j.jenvaman.2020.111681
Tversky, A., Kahneman, D.: Advances in prospect theory. Cumulative representation of uncertainty. J. Risk Uncertain. 5, 297–323 (1992)
Wang, S.S., Young, V.R., Panjer, H.H.: Axiomatic characterization of insurance prices. Insur.: Math. Econ. 21, 173–183 (1997)
Wakker, P.P.: Prospect Theory: For Risk and Ambiguity. Cambridge University Press, New York (2010)
Wakker, P., Tversky, A.: An axiomatization of cumulative prospect theory. J. Risk Uncertain. 7, 147–175 (1993)
Wang, Z., Klir, G.J.: Generalized Measure Theory. Springer, Boston (2009)
Yang, Q.: The pan-integral on the fuzzy measure space. Fuzzy Math. 3, 107–114 (1985). (in Chinese)
Yaari, M.E.: The dual theory of choice under risk. Econometrica 55, 95–115 (1987)
Young, V.R.: Premium principles. Encyclopedia of Actuarial Science. Wiley, New York (2006)
Zhang, D., Pap, E.: Fubini theorem and generalized Minkowski inequality for the pseudo-integral. J. Approx. Reason. 122, 9–23 (2020)
Zhang, D., Mesiar, R., Pap, E.: Pseudo-integral and generalized Choquet integral. Fuzzy Sets Syst. https://doi.org/10.1016/j.fss.2020.12.005
Acknowledgements
This research was supported by Science Fund of the Republic of Serbia, \(\#\)Grant No. 6524105, AI-ATLAS.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Pap, E. (2021). Mathematical Foundation of Artificial Intelligence. In: Pap, E. (eds) Artificial Intelligence: Theory and Applications. Studies in Computational Intelligence, vol 973. Springer, Cham. https://doi.org/10.1007/978-3-030-72711-6_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-72711-6_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-72710-9
Online ISBN: 978-3-030-72711-6
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)