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A Survey on Intersections of Maximal Partial Clones of Boolean Partial Functions Couceiro, Miguel ; in Multiple-Valued Logic (ISMVL), 2012 42nd IEEE International Symposium on (2012) We survey known results and present some new ones about intersections of maximal partial clones on a 2-element set. Detailed reference viewed: 52 (0 UL)An ordinal approach to risk measurement ; Couceiro, Miguel in Mathematical and Statistical Methods for Actuarial Sciences and Finance (2012) In this short note, we aim at a qualitative framework for modeling multivariate risk. To this extent, we consider completely distributive lattices as underlying universes, and make use of lattice ... [more ▼] In this short note, we aim at a qualitative framework for modeling multivariate risk. To this extent, we consider completely distributive lattices as underlying universes, and make use of lattice functions to formalize the notion of risk measure. Several properties of risk measures are translated into this general setting, and used to provide axiomatic characterizations. Moreover, a notion of quantile of a lattice-valued random variable is proposed, which is shown to retain several desirable properties of its real-valued counterpart. [less ▲] Detailed reference viewed: 66 (1 UL)Axiomatizations of quasi-Lovász extensions of pseudo-Boolean functions Couceiro, Miguel ; Marichal, Jean-Luc in Aequationes Mathematicae (2011), 82(3), 213-231 We introduce the concept of quasi-Lov\'asz extension as being a mapping $f\colon I^n\to\R$ defined on a nonempty real interval $I$ containing the origin and which can be factorized as $f(x_1,\ldots,x_n)=L ... [more ▼] We introduce the concept of quasi-Lov\'asz extension as being a mapping $f\colon I^n\to\R$ defined on a nonempty real interval $I$ containing the origin and which can be factorized as $f(x_1,\ldots,x_n)=L(\varphi(x_1),\ldots,\varphi(x_n))$, where $L$ is the Lov\'asz extension of a pseudo-Boolean function $\psi\colon\{0,1\}^n\to\R$ (i.e., the function $L\colon\R^n\to\R$ whose restriction to each simplex of the standard triangulation of $[0,1]^n$ is the unique affine function which agrees with $\psi$ at the vertices of this simplex) and $\varphi\colon I\to\R$ is a nondecreasing function vanishing at the origin. These functions appear naturally within the scope of decision making under uncertainty since they subsume overall preference functionals associated with discrete Choquet integrals whose variables are transformed by a given utility function. To axiomatize the class of quasi-Lov\'asz extensions, we propose generalizations of properties used to characterize the Lov\'asz extensions, including a comonotonic version of modularity and a natural relaxation of homogeneity. A variant of the latter property enables us to axiomatize also the class of symmetric quasi-Lov\'asz extensions, which are compositions of symmetric Lov\'asz extensions with $1$-place nondecreasing odd functions. [less ▲] Detailed reference viewed: 86 (2 UL)Axiomatizations of Lovász extensions of pseudo-Boolean functions Couceiro, Miguel ; Marichal, Jean-Luc in Fuzzy Sets & Systems (2011), 181(1), 28-38 Three important properties in aggregation theory are investigated, namely horizontal min-additivity, horizontal max-additivity, and comonotonic additivity, which are defined by certain relaxations of the ... [more ▼] Three important properties in aggregation theory are investigated, namely horizontal min-additivity, horizontal max-additivity, and comonotonic additivity, which are defined by certain relaxations of the Cauchy functional equation in several variables. We show that these properties are equivalent and we completely describe the functions characterized by them. By adding some regularity conditions, these functions coincide with the Lovász extensions vanishing at the origin, which subsume the discrete Choquet integrals. We also propose a simultaneous generalization of horizontal min-additivity and horizontal max-additivity, called horizontal median-additivity, and we describe the corresponding function class. Additional conditions then reduce this class to that of symmetric Lovász extensions, which includes the discrete symmetric Choquet integrals. [less ▲] Detailed reference viewed: 93 (2 UL)Agents in the shadow (cooperative games with non-cooperating players) Waldhauser, Tamás ; Couceiro, Miguel ; Marichal, Jean-Luc Scientific Conference (2011, July 19) The aim of this talk is to report on recent investigations about lattice derivatives of Boolean and pseudo-Boolean functions and their interpretations in game theory. The partial lattice derivatives of a ... [more ▼] The aim of this talk is to report on recent investigations about lattice derivatives of Boolean and pseudo-Boolean functions and their interpretations in game theory. The partial lattice derivatives of a (pseudo-) Boolean function are analogues of the classical partial derivatives, with the difference operation replaced by the minimum or the maximum operation. We focus on commutation properties of these lattice differential operators and relate them to local monotonicity properties. The least and greatest functions that can be obtained from a given function f, by forming lattice derivatives with respect to all variables, are called the lower and the upper shadows of f, respectively. It turns out that the lower and upper shadows of f coincide with the alpha- and beta-effectivity functions of the cooperative game corresponding to f. Thus a function f has a unique shadow if and only if the two effectivity functions coincide, which is equivalent to the fact that certain two-player zero-sum games associated with (the cooperative game corresponding to) f are strictly determined. We formulate a conjecture about Boolean functions (i.e., simple games) with a unique shadow, and present proofs in some special cases as well as results of computer experiments that support our conjecture. [less ▲] Detailed reference viewed: 108 (0 UL)An algorithm for producing median formulas for Boolean functions Couceiro, Miguel ; Lehtonen, Erkko ; Marichal, Jean-Luc et al in Proc. of the Reed Muller 2011 Workshop (2011, July) We review various normal form representations of Boolean functions and outline a comparative study between them, which shows that the median normal form system provides representations that are more ... [more ▼] We review various normal form representations of Boolean functions and outline a comparative study between them, which shows that the median normal form system provides representations that are more efficient than the classical DNF, CNF and Reed–Muller (polynomial) normal form representations. We present an algorithm for producing median normal form representations of Boolean functions. [less ▲] Detailed reference viewed: 146 (3 UL)Axiomatizations of signed discrete Choquet integrals ; Couceiro, Miguel ; et al in International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems (2011), 19(2), 193-199 We study the so-called signed discrete Choquet integral (also called non-monotonic discrete Choquet integral) regarded as the Lovász extension of a pseudo-Boolean function which vanishes at the origin. We ... [more ▼] We study the so-called signed discrete Choquet integral (also called non-monotonic discrete Choquet integral) regarded as the Lovász extension of a pseudo-Boolean function which vanishes at the origin. We present axiomatizations of this generalized Choquet integral, given in terms of certain functional equations, as well as by necessary and sufficient conditions which reveal desirable properties in aggregation theory. [less ▲] Detailed reference viewed: 87 (2 UL)Associative polynomial functions over bounded distributive lattices Couceiro, Miguel ; Marichal, Jean-Luc in Order : A Journal on the Theory of Ordered Sets and its Applications (2011), 28(1), 1-8 The associativity property, usually defined for binary functions, can be generalized to functions of a given fixed arity n>=1 as well as to functions of multiple arities. In this paper, we investigate ... [more ▼] The associativity property, usually defined for binary functions, can be generalized to functions of a given fixed arity n>=1 as well as to functions of multiple arities. In this paper, we investigate these two generalizations in the case of polynomial functions over bounded distributive lattices and present explicit descriptions of the corresponding associative functions. We also show that, in this case, both generalizations of associativity are essentially the same. [less ▲] Detailed reference viewed: 92 (4 UL)Self-commuting lattice polynomial functions on chains Couceiro, Miguel ; Lehtonen, Erkko in Aequationes Mathematicae (2011), 81(3), 263-278 We provide sufficient conditions for a lattice polynomial function to be self-commuting. We explicitly describe self-commuting polynomial functions on chains. Detailed reference viewed: 64 (0 UL)A survey on the arity gap Couceiro, Miguel ; Lehtonen, Erkko ; Waldhauser, Tamás in 41st IEEE International Symposium on Multiple-Valued Logic (ISMVL 2011) (2011) The arity gap of a function of several variables is defined as the minimum decrease in the number of essential variables when essential variables are identified. We present a brief survey on the research ... [more ▼] The arity gap of a function of several variables is defined as the minimum decrease in the number of essential variables when essential variables are identified. We present a brief survey on the research done on the arity gap, from the first studies of this notion up to recent developments. [less ▲] Detailed reference viewed: 27 (0 UL)On equational definability of function classes Couceiro, Miguel ; Lehtonen, Erkko ; Waldhauser, Tamás in 41st IEEE International Symposium on Multiple-Valued Logic (ISMVL 2011) (2011) We propose a notion of functional equation for functions of fixed arity, which is based on a pair of clones. We present necessary conditions for a class of functions to be definable by such equations, and ... [more ▼] We propose a notion of functional equation for functions of fixed arity, which is based on a pair of clones. We present necessary conditions for a class of functions to be definable by such equations, and show that for certain choices of clones these conditions are also sufficient. [less ▲] Detailed reference viewed: 73 (0 UL)Axiomatizations of the discrete Choquet integral and extensions Couceiro, Miguel ; Marichal, Jean-Luc in Galichet, Sylvie; Montero, Javier; Mauris, Gilles (Eds.) Proceedings of the 7th conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011) and LFA-2011 (2011) Three important properties in aggregation theory are investigated, namely horizontal min-additivity, horizontal max-additivity, and comonotonic additivity, which are defined by certain relaxations of the ... [more ▼] Three important properties in aggregation theory are investigated, namely horizontal min-additivity, horizontal max-additivity, and comonotonic additivity, which are defined by certain relaxations of the Cauchy functional equation in several variables. We show that these properties are equivalent and we completely describe the functions characterized by them. By adding some regularity conditions, the latter functions coincide with the Lovász extensions vanishing at the origin, which subsume the discrete Choquet integrals. We also propose a simultaneous generalization of horizontal min-additivity and horizontal max-additivity, called horizontal median-additivity, and we describe the corresponding function class. Additional conditions then reduce this class to that of symmetric Lovász extensions, which includes the discrete symmetric Choquet integrals. [less ▲] Detailed reference viewed: 89 (1 UL)Quasi-polynomial functions over bounded distributive lattices Couceiro, Miguel ; Marichal, Jean-Luc in Aequationes Mathematicae (2010), 80(3), 319-334 In [6] the authors introduced the notion of quasi-polynomial function as being a mapping $f\colon X^n\to X$ defined and valued on a bounded chain $X$ and which can be factorized as $f(x_1,\ldots,x_n)=p ... [more ▼] In [6] the authors introduced the notion of quasi-polynomial function as being a mapping $f\colon X^n\to X$ defined and valued on a bounded chain $X$ and which can be factorized as $f(x_1,\ldots,x_n)=p(\varphi(x_1),\ldots,\varphi(x_n))$, where $p$ is a polynomial function (i.e., a combination of variables and constants using the chain operations $\wedge$ and $\vee$) and $\varphi$ is an order-preserving map. In the current paper we study this notion in the more general setting where the underlying domain and codomain sets are, possibly different, bounded distributive lattices, and where the inner function is not necessarily order-preserving. These functions appear naturally within the scope of decision making under uncertainty since, as shown in this paper, they subsume overall preference functionals associated with Sugeno integrals whose variables are transformed by a given utility function. To axiomatize the class of quasi-polynomial functions, we propose several generalizations of well-established properties in aggregation theory, as well as show that some of the characterizations given in [6] still hold in this general setting. Moreover, we investigate the so-called transformed polynomial functions (essentially, compositions of unary mappings with polynomial functions) and show that, under certain conditions, they reduce to quasi-polynomial functions. [less ▲] Detailed reference viewed: 108 (2 UL)Explicit descriptions of associative Sugeno integrals Couceiro, Miguel ; Marichal, Jean-Luc in Hüllermeier, Eyke; Kruse, Rudolf; Hoffmann, Frank (Eds.) Information Processing and Management of Uncertainty in Knowledge-Based Systems: 13th International Conference, IPMU 2010, Dortmund, Germany, June 28–July 2, 2010. Proceedings, Part I (2010, June 17) The associativity property, usually defined for binary functions, can be generalized to functions of a given fixed arity $n\geq 1$ as well as to functions of multiple arities. In this paper, we ... [more ▼] The associativity property, usually defined for binary functions, can be generalized to functions of a given fixed arity $n\geq 1$ as well as to functions of multiple arities. In this paper, we investigate these two generalizations in the case of Sugeno integrals over bounded distributive lattices and present explicit descriptions of the corresponding associative functions. We also show that, in this case, both generalizations of associativity are essentially the same. [less ▲] Detailed reference viewed: 86 (1 UL)On two generalizations of associativity Couceiro, Miguel ; Marichal, Jean-Luc Scientific Conference (2010, June) Detailed reference viewed: 41 (12 UL)Characterizations of discrete Sugeno integrals as polynomial functions over distributive lattices Couceiro, Miguel ; Marichal, Jean-Luc in Fuzzy Sets & Systems (2010), 161(5), 694-707 We discuss several characterizations of discrete Sugeno integrals over bounded distributive lattices as particular cases of lattice polynomial functions, that is, functions which can be represented in the ... [more ▼] We discuss several characterizations of discrete Sugeno integrals over bounded distributive lattices as particular cases of lattice polynomial functions, that is, functions which can be represented in the language of bounded lattices using variables and constants. We also consider the subclass of term functions as well as the classes of symmetric polynomial functions and weighted infimum and supremum functions, and present their characterizations, accordingly. Moreover, we discuss normal form representations of these functions. [less ▲] Detailed reference viewed: 93 (7 UL)Representations and characterizations of polynomial functions on chains Couceiro, Miguel ; Marichal, Jean-Luc in Journal of Multiple-Valued Logic & Soft Computing (2010), 16(1-2), 65-86 We are interested in representations and characterizations of lattice polynomial functions f: L^n --> L, where L is a given bounded distributive lattice. In companion papers [5,6], we investigated certain ... [more ▼] We are interested in representations and characterizations of lattice polynomial functions f: L^n --> L, where L is a given bounded distributive lattice. In companion papers [5,6], we investigated certain representations and provided various characterizations of these functions both as solutions of certain functional equations and in terms of necessary and sufficient conditions. In the present paper, we investigate these representations and characterizations in the special case when L is a chain, i.e., a totally ordered lattice. More precisely, we discuss representations of lattice polynomial functions given in terms of standard simplices and we present new axiomatizations of these functions by relaxing some of the conditions given in [5,6] and by considering further conditions, namely comonotonic minitivity and maxitivity. [less ▲] Detailed reference viewed: 50 (5 UL)The arity gap of polynomial functions over bounded distributive lattices Couceiro, Miguel ; Lehtonen, Erkko in 40th IEEE International Symposium on Multiple-Valued Logic (ISMVL 2010) (2010) Let A and B be arbitrary sets with at least two elements. The arity gap of a function f : An→ B is the minimum decrease in its essential arity when essential arguments of f are identified. In this paper ... [more ▼] Let A and B be arbitrary sets with at least two elements. The arity gap of a function f : An→ B is the minimum decrease in its essential arity when essential arguments of f are identified. In this paper we study the arity gap of polynomial functions over bounded distributive lattices and present a complete classification of such functions in terms of their arity gap. To this extent, we present a characterization of the essential arguments of polynomial functions, which we then use to show that almost all lattice polynomial functions have arity gap 1, with the exception of truncated median functions, whose arity gap is 2. [less ▲] Detailed reference viewed: 58 (0 UL)Classes of operations closed under permutation, cylindrification and composition Couceiro, Miguel ; Lehtonen, Erkko in 40th IEEE International Symposium on Multiple-Valued Logic (ISMVL 2010) (2010) We describe the classes of operations closed under permutation of variables, addition of dummy variables and composition in terms of a preservation relation between operations and certain systems of ... [more ▼] We describe the classes of operations closed under permutation of variables, addition of dummy variables and composition in terms of a preservation relation between operations and certain systems of multisets. [less ▲] Detailed reference viewed: 66 (0 UL)Explicit descriptions of bisymmetric Sugeno integrals Couceiro, Miguel ; Lehtonen, Erkko in Hüllermeier, Eyke; Kruse, Rudolf; Hoffmann, Frank (Eds.) Computational Intelligence for Knowledge-Based Systems Design (2010) We provide sufficient conditions for a Sugeno integral to be bisymmetric. We explicitly describe bisymmetric Sugeno integrals over chains. Detailed reference viewed: 67 (1 UL) |
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