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Axiomatizations of quasi-polynomial functions on bounded chains Couceiro, Miguel ; Marichal, Jean-Luc in Aequationes Mathematicae (2009), 78(1-2), 195-213 Two emergent properties in aggregation theory are investigated, namely horizontal maxitivity and comonotonic maxitivity (as well as their dual counterparts) which are commonly defined by means of certain ... [more ▼] Two emergent properties in aggregation theory are investigated, namely horizontal maxitivity and comonotonic maxitivity (as well as their dual counterparts) which are commonly defined by means of certain functional equations. We completely describe the function classes axiomatized by each of these properties, up to weak versions of monotonicity in the cases of horizontal maxitivity and minitivity. While studying the classes axiomatized by combinations of these properties, we introduce the concept of quasi-polynomial function which appears as a natural extension of the well-established notion of polynomial function. We give further axiomatizations for this class both in terms of functional equations and natural relaxations of homogeneity and median decomposability. As noteworthy particular cases, we investigate those subclasses of quasi-term functions and quasi-weighted maximum and minimum functions, and provide characterizations accordingly. [less ▲] Detailed reference viewed: 70 (4 UL)Characterizations of discrete Sugeno integrals as lattice polynomial functions Couceiro, Miguel ; Marichal, Jean-Luc in Bodenhofer, Ulrich; De Baets, Bernard; Klement, Erich Peter (Eds.) et al Proc. 30th Linz Seminar on Fuzzy Set Theory (LINZ 2009): The Legacy of 30 Seminars – Where Do We Stand and Where Do We Go? (2009, February) We survey recent characterizations of the class of lattice polynomial functions and of the subclass of discrete Sugeno integrals defined on bounded chains. Detailed reference viewed: 39 (1 UL)From discrete Sugeno integrals to generalized lattice polynomial functions: axiomatizations and representations (invited lecture) Couceiro, Miguel ; Marichal, Jean-Luc in González, Manuel; Mayor, Gaspar; Suner, Jaume (Eds.) et al Proc. 5th Int. Summer School on Aggregation Operators and their Applications (AGOP 2009) (2009) Two emergent properties in aggregation theory are investigated, namely horizontal maxitivity and comonotonic maxitivity (as well as their dual counterparts) which are commonly defined by means of certain ... [more ▼] Two emergent properties in aggregation theory are investigated, namely horizontal maxitivity and comonotonic maxitivity (as well as their dual counterparts) which are commonly defined by means of certain functional equations. We present complete descriptions of the function classes axiomatized by each of these properties, up to weak versions of monotonicity, in the cases of horizontal maxitivity and minitivity. While studying the classes axiomatized by combinations of these properties, we introduce the concept of quasi-polynomial function which appears as a natural extension of the well-established notion of polynomial function. We present further axiomatizations for this class both in terms of functional equations and natural relaxations of homogeneity and median decomposability. [less ▲] Detailed reference viewed: 37 (1 UL)Quasi-polynomial functions on bounded chains Couceiro, Miguel ; Marichal, Jean-Luc in Carvalho, J. P.; Dubois, D.; Kaymak, U. (Eds.) et al Proc. of 2009 Int. Fuzzy Systems Assoc. World Congress and 2009 Int. Conf. of the Eur. Soc. for Fuzzy Logic and Technology (IFSA-EUSFLAT 2009 Joint Conference) (2009) Two emergent properties in aggregation theory are investigated, namely horizontal maxitivity and comonotonic maxitivity (as well as their dual counterparts) which are commonly defined by means of certain ... [more ▼] Two emergent properties in aggregation theory are investigated, namely horizontal maxitivity and comonotonic maxitivity (as well as their dual counterparts) which are commonly defined by means of certain functional equations. We present complete descriptions of the function classes axiomatized by each of these properties, up to weak versions of monotonicity, in the cases of horizontal maxitivity and minitivity. While studying the classes axiomatized by combinations of these properties, we introduce the concept of quasipolynomial function which appears as a natural extension of the well-established notion of polynomial function. We present further axiomatizations for this class both in terms of functional equations and natural relaxations of homogeneity and median decomposability. As noteworthy particular cases, we investigate those subclasses of quasi-term functions and quasi-weighted maximum and minimum functions, and present characterizations accordingly. [less ▲] Detailed reference viewed: 53 (0 UL)Polynomial functions on bounded chains Couceiro, Miguel ; Marichal, Jean-Luc in Carvalho, J.-P.; Dubois, D.; Kaymak, U. (Eds.) et al Proc. of 2009 Int. Fuzzy Systems Assoc. World Congress and 2009 Int. Conf. of the Eur. Soc. for Fuzzy Logic and Technology (IFSA-EUSFLAT 2009 Joint Conference) (2009) We are interested in representations and characterizations of lattice polynomial functions $f\colon L^n\to L$, where $L$ is a given bounded distributive lattice. In an earlier paper [4,5], we investigated ... [more ▼] We are interested in representations and characterizations of lattice polynomial functions $f\colon L^n\to L$, where $L$ is a given bounded distributive lattice. In an earlier paper [4,5], 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 [4,5] and by considering further conditions, namely comonotonic minitivity and maxitivity. [less ▲] Detailed reference viewed: 53 (0 UL)Generalizations of Świerczkowski’s lemma and the arity gap of finite functions Couceiro, Miguel ; Lehtonen, Erkko in Discrete Mathematics (2009), 309(20), 5905-5912 Świerczkowski’s lemma–as it is usually formulated–asserts that if f:Aⁿ→A is an operation on a finite set A, n≥4, and every operation obtained from f by identifying a pair of variables is a projection ... [more ▼] Świerczkowski’s lemma–as it is usually formulated–asserts that if f:Aⁿ→A is an operation on a finite set A, n≥4, and every operation obtained from f by identifying a pair of variables is a projection, then f is a semiprojection. We generalize this lemma in various ways. First, it is extended to B-valued functions on A instead of operations on A and to essentially at most unary functions instead of projections. Then we characterize the arity gap of functions of small arities in terms of quasi-arity, which in turn provides a further generalization of Świerczkowski’s lemma. Moreover, we explicitly classify all pseudo-Boolean functions according to their arity gap. Finally, we present a general characterization of the arity gaps of B-valued functions on arbitrary finite sets A. [less ▲] Detailed reference viewed: 68 (1 UL)On the arity gap of finite functions: results and applications Couceiro, Miguel ; Lehtonen, Erkko in Boudabbous, Youssef; Zaguia, Nejib (Eds.) Proceedings of the First International Conference on Relations, Orders and Graphs: Interaction with Computer Science (ROGICS '08) (2008) Let A be a finite set and B an arbitrary set with at least two elements. The arity gap of a function f:Aⁿ→B is the minimum decrease in the number of essential variables when essential variables of f are ... [more ▼] Let A be a finite set and B an arbitrary set with at least two elements. The arity gap of a function f:Aⁿ→B is the minimum decrease in the number of essential variables when essential variables of f are identified. A non-trivial fact is that the arity gap of such B-valued functions on A is at most |A|. Even less trivial to verify is the fact that the arity gap of B-valued functions on A with more than |A| essential variables is at most 2. These facts ask for a classification of B-valued functions on A in terms of their arity gap. In this paper, we survey what is known about this problem. We present a general characterization of the arity gap of B-valued functions on A and provide explicit classifications of the arity gap of Boolean and pseudo-Boolean functions. Moreover, we reveal unsettled questions related to this topic, and discuss links and possible applications of some results to other subjects of research. [less ▲] Detailed reference viewed: 44 (1 UL)On the effect of variable identification on the essential arity of functions on finite sets Couceiro, Miguel ; Lehtonen, Erkko in International Journal of Foundations of Computer Science (2007), 18(5), 975-986 We show that every function of several variables on a finite set of k elements with n > k essential variables has a variable identification minor with at least n − k essential variables. This is a ... [more ▼] We show that every function of several variables on a finite set of k elements with n > k essential variables has a variable identification minor with at least n − k essential variables. This is a generalization of a theorem of Salomaa on the essential variables of Boolean functions. We also strengthen Salomaa's theorem by characterizing all the Boolean functions f having a variable identification minor that has just one essential variable less than f. [less ▲] Detailed reference viewed: 61 (0 UL)Composition of Post classes and normal forms of Boolean functions Couceiro, Miguel ; ; Lehtonen, Erkko in Discrete Mathematics (2006), 306(24), 3223-3243 The class composition CK of Boolean clones, being the set of composite functions f(g1,...,gn) with f∈C, g1,...,gn∈K, is investigated. This composition CK is either the join C∨K in the Post Lattice or it ... [more ▼] The class composition CK of Boolean clones, being the set of composite functions f(g1,...,gn) with f∈C, g1,...,gn∈K, is investigated. This composition CK is either the join C∨K in the Post Lattice or it is not a clone, and all pairs of clones C,K are classified accordingly. Factorizations of the clone Ω of all Boolean functions as a composition of minimal clones are described and seen to correspond to normal form representations of Boolean functions. The median normal form, arising from the factorization of Ω with the clone SM of self-dual monotone functions as the leftmost composition factor, is compared in terms of complexity with the well-known DNF, CNF, and Zhegalkin (Reed–Muller) polynomial representations, and it is shown to provide a more efficient normal form representation. [less ▲] Detailed reference viewed: 79 (0 UL)On the complexity of representing sets of vertices in the n-cube Couceiro, Miguel ; ; Lehtonen, Erkko in Tsitouras, C.; Simos, T. E.; Psihoyios, G. (Eds.) ICNAAM 2005 (2005) Representations of an arbitrary set of vertices of an n-dimensional cube in terms of convex sets are compared with respect to their complexity. The notion of complexity used in the representations is ... [more ▼] Representations of an arbitrary set of vertices of an n-dimensional cube in terms of convex sets are compared with respect to their complexity. The notion of complexity used in the representations is based on the union, symmetric difference and ternary median operations. Convexity of a set of vertices refers to Hamming distance or, equivalently, to the intersection of cubes with supporting hyperplanes. [less ▲] Detailed reference viewed: 26 (1 UL)On compositions of clones of Boolean functions Couceiro, Miguel ; ; Lehtonen, Erkko in Simos, T.; Maroulis, G. (Eds.) International Conference of Computational Methods in Sciences and Engineering 2004 (ICCMSE 2004) (2004) We present some compositions of clones of Boolean functions that imply factorizations of ?, the clone of all Boolean functions, into minimal clones. These can be interpreted as representation theorems ... [more ▼] We present some compositions of clones of Boolean functions that imply factorizations of ?, the clone of all Boolean functions, into minimal clones. These can be interpreted as representation theorems, providing representations of Boolean functions analogous to the disjunctive normal form, the conjunctive normal form, and the Zhegalkin polynomial representations. [less ▲] Detailed reference viewed: 71 (0 UL) |
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