T-supermodularity of Choquet integrals

SISY 2011 9th International Symposium on Intelligent Systems and Informatics Subotica, Serbia September 8–10, 2011 PROCEEDINGS Committees Honorary Chairs WILLIAM A. GRUVER, IEEE Division X Director MIOMIR VUKOBRATOVIĆ, Institut Mihajlo Pupin, Beograd, Serbia Founding Honorary Chair IMRE J. RUDAS, Óbuda University, Budapest, Hungary Honorary Committee LÁSZLÓ T. KÓCZY, Széchenyi István University, Győr, Hungary ÉVA PATAKI, Subotica Tech, Serbia International Advisory Board KAORU HIROTA, Tokyo Institute of Technology, Japan MUDER JENG, National Taiwan Ocean University, Taiwan T. T. LEE, National Taiwan Univ., Taiwan OUSSAMA KHATIB, Stanford University, USA EMIL M. PETRIU, University of Ottawa, Canada HIDEYUKI TAKAGI, Kyushu University, Japan General Chairs Endre Pap, University of Novi Sad, Serbia János Fodor, Óbuda University, Budapest, Hungary Technical Program Committee Bernard de Baets, Genth, Belgium Péter Baranyi, BME, Hungary György Bárdossy, Hungarian Ac. of Sci. Barnabás Bede, Óbuda University Balázs Benyó, BME, Hungary Ivana Berkovic, Technical Faculty Mihajlo Pupin, Zrenjanin Róbert Fullér, ELTE, Hungary Michel Grabisch, Paris, France László Horváth, Óbuda University Zsolt Csaba Johanyák, Kecskemét College, Hungary Aleksandar Jovanovic, Belgrade, Serbia Jozef Kelemen, Silisian University Erich Peter Klement, Linz, Austria Levente Kovács, BME, Hungary Radko Mesiar, Bratislava, Slovakia Gyula Mester, Subotica Tech, Serbia, and Univ. of Szeged, Hungary Zora Konjovic, Novi Sad, Serbia Miloš Rackovic, Novi Sad, Serbia Dragica Radoslav, Technical Faculty Mihajlo Pupin, Zrenjanin Dušan Surla, Novi Sad, Serbia József K. Tar, Óbuda University, Hungary Dušan Teodorovic, Belgrade, Serbia József Tick, Óbuda University, Hungary Domonkos Tikk, BME, Hungary Szilveszter Pletl, Subotica Tech, Serbia, and Univ. of Szeged, Hungary Organizing Committee Chair Márta Takács, Óbuda University, Hungary Organizing Committee Attila L. Bencsik, Óbuda University Gizella Csikós-Pajor, Subotica Tech József Gáti, Óbuda University Orsolya Hölvényi, Óbuda University Gyula Kártyás, Óbuda University Ilona Reha, Óbuda University Ivana Štajner-Papuga, Univ. of Novi Sad Anita Szabó, Subotica Tech Lívia Szedmina, Subotica Tech Secretary General Anikó Szakál, Óbuda University, Budapest, Hungary [email protected] Table of Contents On the Discreet Charm of Robot Programming Jozef Kelemen An Approach for Intelligent Mobile Robot Motion Planning and Trajectory Tracking in Structured Static Environments Marko Susic, Aleksandar Cosic, Aleksandar Ribic, Dusko Katic Modeling and Simulation of Quad-rotor Dynamics and Spatial Navigation A. Rodic, G. Mester 2-DOF Control Solutions for BLDC-m Drives Alexandra-Iulia Stînean, Stefan Preitl, Radu-Emil Precup, Claudia-Adina Dragos, MirceaBogdan Radac RFPT-based Decentralized Adaptive Control of Partially, Roughly Modeled, Coupled Dynamic Systems Teréz A. Várkonyi, József K. Tar, János F. 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However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. SISY 2011 • 2011 IEEE 9th International Symposium on Intelligent Systems and Informatics • September 8-10, 2011, Subotica, Serbia � -supermodularity of Choquet integrals Martin Kalina Maddalena Manzi Biljana Mihailović Department of Mathematics Faculty of Civil Engineering Slovak University of Technology, Radlinského 11, 813 68 Bratislava, Slovakia Email: [email protected] Deams “Bruno de Finetti” University of Trieste Piazzale Europa, 1 34127 Trieste, Italy Email: [email protected] Faculty of Technical Sciences University of Novi Sad Trg Dositeja Obradovića 6, 21000 Novi Sad, Serbia Email: [email protected] Abstract—This paper is based on the investigation of a new property for Choquet integral: � -supermodularity. With this definition we want to extend the property of supermodularity for Choquet integral with respect to a supermodular fuzzy measure, in particular by using Frank �-norms. Keywords: Triangular norms, fuzzy measures, fuzzy sets, Choquet integral, � -supermodularity. I. I NTRODUCTION In this article we introduce and study � -supermodularity for Choquet integral. This notion is connected with �-norms, that we recall in the first section. Preliminary classes of fuzzy measures (such as subadditive, submodular, symmetric) and fuzzy sets are respectively given in Section III ([8], [16], [20]) and in Section IV. � -supermodularity for Choquet integral is defined in Section V and with this concept we prove in Proposition 1 that Choquet integral is � -supermodular with respect to the minimum �norm in the family of all measurable functions � : � → [0, 1] on a finite set �. A special case of � -supermodularity is considered in Corollary 1. In conclusion we provide some examples. II. T RIANGULAR NORMS A triangular norm (briefly �-norm) � is defined to be a two-place function � : [0, 1] × [0, 1] → [0, 1] fulfilling the following properties: 1) � (1, �) = �, for each � ∈ [0, 1]; 2) � (�, �1 ) ≤ � (�, �2 ), for all �, �1 , �2 ∈ [0, 1], if �1 ≤ �2 ; 3) � (�, �) = � (�, �), for all �, � ∈ [0, 1]; 4) � (�, � (�, �)) = � (� (�, �), �), for all �, �, � ∈ [0, 1]. Note that a �-norm defines an abelian monoid on [0, 1] with unit 1 and annihilator 0 and where the semigroup operation is order-preserving and commutative. Given a �-norm � , the two-place function � : [0, 1] × [0, 1] → [0, 1], defined by �(�, �) = 1 − � (1 − �, 1 − �) 978-1-4577-1974-5/11/$26.00 ©2011 IEEE is called a �-conorm (or the dual of � ). Obviously � fulfills monotonicity, commutativity, associativity and �(�, 0) = � �������� ���������. Here we deal with the Frank family of �-norms �� , � ∈ [0, ∞]. For each � ∈ [0, ∞], the Frank t-norms are defined by the formulas ∙ the minimum t-norm, �� (�, �) := min{�, �}, ∙ the product t-norm, �� (�, �) := � ⋅ �, ∙ the Łukasiewicz t-norm �� (�, �) := max{0, � + � − 1}, ∙ if � ∈ (0, ∞) ∖ {1}, ] [ (�� − 1)(�� − 1) . �� (�, �) := log� 1 + �−1 The basic t-conorms (dual to four basic t-norms) are: ∙ the maximum t-conorm, �� (�, �) := max{�, �}, ∙ the probabilistic sum, �� (�, �) := � + � − � ⋅ �, ∙ the Łukasiewicz t-conorm �� (�, �) := min{1, � + �}, ∙ if � ∈ (0, ∞) ∖ {1}, ] [ (�1−� − 1)(�1−� − 1) �� (�, �) := 1 − log� 1 + . �−1 The family {�� ∣� ∈ [0, ∞]} appeared first in Frank’s [9] investigation of the functional equation � + � = � (�, �) + �(�, �), ∀�, � ∈ [0, 1], (1) where � is a triangular norm and � is an associative function on the unit square. Note that the only strict solutions of (1) are just �-norms �� for � ∈ [0, ∞] (where �0 = �� and �∞ = �� ) and the corresponding �� are just the dual �-conorms, i.e., �� (�, �) = 1−�� (1−�, 1−�). In particular Frank [9] showed that the �-norms �� , 0 ≤ � ≤ ∞, form a single family in the sense that �� , �� and �� are the limits of �� corresponding to their subscripts. III. F UZZY MEASURES For applications, several distinguished classes of fuzzy measures are important. We list some of them in the next definitions for the sake of selfcontainedness, though these well known properties can be found, e. g., in [8], [16], [20]. Definition 1: Let � = {1, 2, . . . , �}, � ∈ � be a fixed set of criteria. A mapping � : 2� → [0, 1] is called a fuzzy – 71 – M. Kalina et al. • T-supermodularity of Choquet Integrals measure whenever �(∅) = 0, �(�) = 1 and for all � ⊆ � ⊆ �, it holds �(�) ≤ �(�). Distinguished classes of fuzzy measures are determined by their respective properties. Definition 2: A fuzzy measure � on � is called: 1) additive if ∀�, � ∈ 2� , �(� ∪ �) ≤ �(�) + �(�), �∩� = ∅, �(�∪�) ≥ �(�)+�(�), 4) submodular whenever ∀�, � ∈ 2� , Definition 5: Let � and � be fuzzy sets. The standard intersection of � and �, � ∩ �, is defined by �(� ∪ �) + �(� ∩ �) ≤ �(�) + �(�), ∀�, � ∈ 2 , �¯(�) := 1 − � (�), �(� ∪ �) + �(� ∩ �) ≥ �(�) + �(�), 6) symmetric if for any subsets �, �, ∣�∣ = ∣�∣ implies �(�) = �(�). The conjugate or dual of a fuzzy measure � is a fuzzy measure � defined by �(�) := �(�) − �(�� ), ∀� ∈ �. Similar to the way operations on ordinary sets are treated, we can generalize the standard union and the standard intersection for an arbitrary class of fuzzy sets: if {�� ∣� ∈ �} is a class of fuzzy sets, where � is an arbitrary index set, then ∪�∈� �� is the fuzzy set having membership function sup�∈� �� (�), � ∈ �, and ∩�∈� �� is the fuzzy set having membership function inf �∈� �� (�), � ∈ �. Definition 6: Let � be a fuzzy set. The standard complement of � , �¯ , is defined by the membership function 5) supermodular whenever � ∀� ∈ �. (� ∧ �)(�) = � (�) ∧ �(�), 3) superadditive (supermeasure) whenever ∀�, � ∈ 2� , (� ∨ �)(�) = � (�) ∨ �(�), �(� ∪ �) = �(�) + �(�), 2) subadditive (submeasure) whenever ∀�, � ∈ 2� , Definition 4: Let � and � be fuzzy sets. The standard union of � and �, � ∪ �, is defined by ∀� ∈ �. Two or more of the three basic operations can also be combined. For example, the difference � − � of fuzzy sets ¯ so that we have the � and � can be expressed as � ∩ �, following membership function (� − �)(�) := max[0, � (�) − �(�)] for all � ∈ �. � ⊆ �. For other kinds of measures, like belief, plausibility, possibility and necessity, there is a deep description in [19]. Observe that if a fuzzy measure is both submodular and supermodular, it is modular and thus a probability measure on �. Evidently, each supermodular fuzzy measure is also superadditive and similarly, each submodular fuzzy measure is subadditive. IV. F UZZY SETS Following [21], [23], we recall the definition of a fuzzy set. Let � be a nonempty set, a fuzzy subset � of � is defined by a characteristic function � : � → [0, 1] which associates with each � in � its “grade of membership”, � (�) in � . To distinguish between the characteristic function of a nonfuzzy set and the characteristic function of a fuzzy set, the latter will be referred to as a membership function. A standard fuzzy set is called normalized if A. Operations on Fuzzy Sets Operations on fuzzy sets are performed using triangular norms. Being an associative operation, a t-norm T may be extended to an arbitrary finite number of elements, then we � �� . denote it by ��=1 The extension of the operations intersection, union and complementation in ordinary set theory to fuzzy sets was always done pointwise: one considered two two-place functions � : [0, 1] × [0, 1] → [0, 1], � : [0, 1] × [0, 1] → [0, 1] and one-place function � : [0, 1] → [0, 1] and extended them in the usual way: if � , � are two membership functions of fuzzy sets � and �, then � (�, �)(�) =� (� (�), �(�)), �(�, �)(�) =�(� (�), �(�)), � (� )(�) =� (� (�)). sup � (�) = 1 �∈� Since any crisp set � can be defined by its characteristic function �� : � → {0, 1}, it is a special standard fuzzy set. Let ℱ(�) denote the family of all membership functions of all fuzzy sets on a finite set � ∕= ∅. For �, � ∈ ℱ(�), � ≤ � means that � (�) ≤ �(�) ∀� ∈ �. Then (� ∨ �)(�) = max{� (�), �(�)} and (� ∧ �)(�) = min{� (�), �(�)}. Definition 3: If � (�) ≤ �(�) for any � ∈ �, we say that the fuzzy set � is included in the fuzzy set � and we write � ⊂ �. If � ⊂ � and � ⊂ � , we say that � and � are equal, which we write as � = �. (2) (3) (4) In his first paper Zadeh [22] suggested to use � (�, �) = �� (�, �) = min(�, �) for intersection, �(�, �) = �� (�, �) = max(�, �) for union and � (�) = 1 − � for complementation. Alsina et al. [1] and Prade [17] suggested to use a �-norm for intersection and its �-conorm for union of fuzzy sets. V. C HOQUET I NTEGRAL Now we are focusing our attention to the integral with respect to nonadditive fuzzy measures, known as the Choquet integral, which is useful in many fields such as mathematical economics and multicriteria decision making. – 72 – SISY 2011 • 2011 IEEE 9th International Symposium on Intelligent Systems and Informatics • September 8-10, 2011, Subotica, Serbia A. Choquet Integrals for Nonnegative Functions Let ℱ(�) denote the family of all membership functions of all fuzzy sets on a finite set � ∕= ∅, � a �-algebra of subsets of �, i.e. � = 2� and � : � → [0, 1] a monotone measure, such that (�, �, �) is a monotone measure space. Definition 7: The Choquet integral of a nonnegative measurable function � with respect to a∫ monotone measure � on measurable set �, denoted by (�) � � ��, is defined by the formula ∫ ∫ 1 (�) � �� = (�) �(�� ∩ �) ��, � 0 where ∫ �� = {�∣� (�) ≥ �} for � ∈ ∫[0, 1]. When � = �, (�) � � �� is usually written as (�) � ��. Since � : � → [0, 1] in Definition 7, where � is an arbitrary universal set, is measurable, we know that �� = {�∣� (�) ≥ �} ∈ � for � ∈ [0, 1] and, therefore, �� ∩ � ∈ �. So, �(�� ∩ �) is well defined for all � ∈ [0, 1]. Furthermore, �� is a class of sets that are nonincreasing with respect to � and so are sets in �� ∩ �. Since monotone measure � is a nondecreasing set function, we know that �(�� ∩ �) is a nondecreasing function of � and, therefore, the above Riemann integral makes sense. Thus, the Choquet integral of a nonnegative measurable function with respect to a monotone measure on a measurable set is well defined. B. � -supermodularity for Choquet integrals Now we define � -supermodularity for Choquet integrals. Definition 8: A Choquet integral Ch� (� ) : ℱ(�) → [0, 1] defined for all � ∈ ℱ(�) by ∫ Ch� (� ) = (�) � ��, is � -supermodular if ∀�, � ∈ ℱ(�) and Frank �-norms it satisfies the following relation: Ch� (� (�, �)) + Ch� (�(�, �)) ≥ Ch� (� ) + Ch� (�). (5) If only the equality holds, we say that Choquet integral is � -modular. When we use the minimum �-norm and its dual �-conorm, we have ∧-supermodular Choquet integral, i. e. standard supermodularity. So, in this case we say simply that Choquet integral is supermodular and for supermodular Choquet integrals the following proposition holds (see [14]). Proposition 1: Consider � = {�1 , ⋅ ⋅ ⋅ , �� }. Let � be a fuzzy measure which is supermodular on 2� . Then the mapping Ch� (� ) : ℱ(�) → [0, 1] defined for all � ∈ ℱ(�) by ∫ Ch� (� ) = (�) � ��. is supermodular on ℱ(�) and so the following relation holds: Ch� (� ∨ �) + Ch� (� ∧ �) ≥ Ch� (� ) + Ch� (�). (6) Following [8] and [16], we see in the following proposition that a Choquet integral is also superadditive if and only if the fuzzy measure � is supermodular. Proposition 2: Let Ch� : ℱ(�) → [0, 1] be a Choquet integral based on a fuzzy measure � on � = {1, . . . , �}. Then we have that Ch� is superadditive if and only if the fuzzy measure � is supermodular: Ch� (� + �) ≥ Ch� (� ) + Ch� (�), for all � and � ∈ ℱ(�). Consider � -supermodularity of Choquet integral with respect to Frank �-norms, then the following proposition holds: Corollary 1: Consider � = {�1 , ⋅ ⋅ ⋅ , �� } and two membership functions � and �, such that � ∧ � = 0. Let � be a supermodular fuzzy measure on 2� . Then a mapping Ch� : ℱ(�) → [0, 1] defined for all � ∈ ℱ(�) by ∫ Ch� (� ) = (�) � ��, is � -supermodular on ℱ(�), where � is a Frank �-norm. So the following relation holds: Ch� (� (�, �)) + Ch� (�(�, �)) ≥ Ch� (� ) + Ch� (�). (7) Example 1: Consider � = {�1 , �2 } and a supermodular measure �, such that �(�1 ) = �1 with 0 < �1 < 1 and �(�2 ) = 0. In the following table we consider the membership functions � and �, such that � ∧ � = 0 and � = �� . � � �1 0 0.8 �2 0.7 0 So we have ∫ ∫1 1) (C) � �� = 0 �({� ∈ � : � (�) ≥ �})�� = 0; ∫ ∫1 2) (C) ��� = 0 �({� ∈ � : �(�) ≥ �})�� = ∫ 0.8 �({�1 })�� = 0.8�1 . 0 Then we consider the functions �� (�, �) and �� (�, �) and their respective Choquet integrals: �� (�, �) �� (�, �) �1 0 0.8 �2 0 0.7 ∫ 1) (C) ∫ �� (�, �)�� = 0; 2) (C) �� (�, �)�� = ∫1 �({� ∈ � : (� + � − � ⋅ �)(�) ≥ �})�� = ∫ 0.8 ∫00.7 �({�})�� + 0.7 �({�1 })�� = 0.7 + 0.1�1 0 and we see 0.7 + 0.1�1 > 0.8�1 . So Choquet integral is � -supermodular on ℱ(�) with respect to the supermodular fuzzy measure � and � = �� . Consider also the following measure, which is supermodular (see [3] and [14]). Example 2: Let (2� , ⊆,′ , ∅, �) be the complemented bounded lattice with intersection and union as the lattice – 73 – M. Kalina et al. • T-supermodularity of Choquet Integrals operations, where � = {�1 , �2 , �3 }. Let � : 2� → [0, 1] be given for all � ∈ 2� by { 1 if �1 ∈ �, 4−∣�∣ �(�) = 0 otherwise. Now we consider the following membership functions � and �, such that � ∧ � = 0. So we have � � �1 0.1 0 �2 0 0.3 �3 0.5 0 ∫1 � �� = 0 �({� ∈ � : � (�) ≥ �})�� = �({�1 , �3 })�� = 0.05; 0 ∫ ∫1 2) (C) ��� = 0 �({� ∈ � : �(�) ≥ �})�� = ∫ 0.3 �({�2 })�� = 0. 0 Then we consider the functions �� (�, �) and �� (�, �) and their respective Choquet integrals: 1) (C) ∫ 0.1 �� (�, �) �� (�, �) ∫ �1 0 0.1 �2 0 0.3 �3 0 0.5 ∫ 1) (C) ∫ �� (�, �)�� = 0; 2) (C) �� (�, �)�� = ∫1 �({� ∈ � : (� + � − � ⋅ �)(�) ≥ �})�� = ∫00.1 ∫ 0.3 ∫ 0.5 �({�})��+ 0.1 �({�2 , �3 })��+ 0.3 �({�3 })�� = 0 0.1. Finally we can see 0.1 > 0.05 and so Choquet integral is � -supermodular on ℱ(�). [6] D. Butnariu and E. P. Klement. Triangular norm-based measures and their Markov kernel representation. J. Math. Anal. Appl., 162:111–143, 1991. [7] D. Butnariu and E. P. Klement. Triangular Norm-Based Measures and Games with Fuzzy Coalitions, volume 10 of Theory and Decision Library, Series C: Game Theory, Mathematical Programming and Operations Research. Kluwer, Dordrecht, 1993. [8] D. Denneberg. Non-additive measure and integral, volume 27 of Theory and Decision Library. Series B: Mathematical and Statistical Methods. Kluwer Academic Publishers Group, Dordrecht, 1994. [9] M. J. Frank. On the simoultaneous associativity of � (�, �) and � + � − � (�, �). Aequationes Math., 19:194–226, 1979. [10] E. P. Klement. Characterization of finite fuzzy measures using Markoffkernels. J. Math. Anal. Appl., 75:330–339, 1980. [11] E. P. Klement. Characterization of fuzzy measures constructed by means of triangular norms. J. Math. Anal. Appl., 86:345–358, 1982. [12] E. P. Klement. Construction of fuzzy �-algebras using triangular norms. J. Math. Anal. Appl., 85:543–565, 1982. [13] H. König. New facts around the Choquet integral. preprint n. 62, 2002. [14] M. Manzi. 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C ONCLUDING REMARKS In this paper we have analyzed � -supermodularity and showed under which conditions Choquet integral is � supermodular. In a next step we want to study Sugeno and Shilkret integrals and show under which conditions they are � -super- or submodular. ACKNOWLEDGMENT The first author was supported by the Science and Technology Assistance Agency under the contract No. APVV-007310 and by VEGA grant agency, grant numbers 1/0080/10 and 1/0297/11. The third author was supported by the project MNTRS 174009 and by Vojvodina Provincial Secretariat for Science and Technological Development. R EFERENCES [1] C. Alsina, E. Trillas, and L. Valverde. On non-distributive logical connectives for fuzzy sets theory. Busefal, 3:18–29, 1980. [2] G. Barbieri and H. Weber. A representation theorem and a Lyapunov theorem for �� -measures: The solution of two problems of Butnariu and Klement. J. Math. Anal. Appl., 244:408–424, 2000. [3] S. Bodjanova and M. Kalina. 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