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\begin{document}

\title*{Expected Size of Random Fuzzy Concept Lattices

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\titlerunning{Random Fuzzy Concept Lattices} %for an abbreviated version of your contribution title if the original one is too long

\author{Radim Belohlavek and Richard Emilion}

%\authorrunning{Name of First Author and Name of Second Author} %If there are more than two authors, please, abbreviate the authors' list using 'et al'

\institute{Radim Belohlavek \at Palacky University, Olomuc, Czech Republic, \email{radim.belohlavek@inf.upol.cz}

\and Richard Emilion \at Institut Denis Poisson, University of Orl\'eans, France, \email{richard.emilion1@univ-orleans.fr}}

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\abstract{In our many-valued logic context, truth degrees form the chain $L=\{0, \frac{1}{q},\dots,\frac{q-1}{q}, 1\}$, for a positive integer $q$, with a many-valued implication denoted $\to$. Given a probability space $(\Omega,\mathcal{F},P)$, a {\it random fuzzy context} is a triplet $\langle X,Y,I\rangle$, where $X = \{1,2,\dots,n\}$ is a set of $n$ objects, $Y=\{1,2,\dots,m\}$ is a set of $m$ attributes, and $I$ is a $L$-valued random matrix indexed by $X \times Y$, assigning to each $(x,y) \in X \times Y$ the {\it random} degree $I(x,y)(\omega) \in L, \omega \in \Omega,$ to which object $x$ has attribute $y$. Following \cite{Bel:FRS,Pol:FB}, we define

\begin{align}

\forall A \in L^X, \forall y \in Y,\; A^{\uparrow}(y)(\omega) &=

\textstyle{\bigwedge}_{x \in X}(A(x) \rightarrow I(x,y)(\omega)), \\

\forall B \in L^Y, \forall x \in X, \; B^{\downarrow}(x)(\omega) &=

\textstyle{\bigwedge}_{y \in Y}(B(y) \rightarrow I(x,y)(\omega)), \\

\mathcal{B}(X,Y,I)(\omega) &=\{\langle A,B \rangle \in {L}^X \times {L}^Y \,|\, A^\uparrow(\omega)=B, B^\downarrow(\omega)=A\}.

\end{align}

According to \cite{Bel:FRS,Bel:Clofl}, $\mathcal{B}(X,Y,I)(\omega)$ is the fuzzy concept lattice w.r.t. the context $(X,Y, I(.,.)(\omega))$ which is lattice-isomorphic to the lattice of closed fuzzy sets:

$$

\mathrm{Ext}(I)(\omega)= \{A\in L^X \mid A^{\uparrow\downarrow}(\omega) = A\}.

$$

Assuming that the entries of $I$ are independent and that the entries in column $y$ are i.i.d. with distribution $p_y$, only depending on $y$, our main results concern the computation of

$P(A^{\uparrow\downarrow} = A)$ (resp. of $P(\langle A,B\rangle\ \in \mathcal{B}(X,Y,I))$ which yields directly the expected size of $\mathcal{B}(X,Y,I)$, generalizing the results obtained in the binary case ($q =1$) \cite{Lhote, EmLe:Srglncfi, Klim}.

}

\keywords{random, fuzzy, concept, lattice}

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\begin{thebibliography}{99.}%

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\bibitem{Bel:FRS}

Belohlavek, R.: Fuzzy Relational Systems: Foundations and Principles.

Kluwer, New York (2002)

\bibitem{Pol:FB}

Pollandt, S.: Fuzzy Begriffe. Springer, Berlin (1997)

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\bibitem{Bel:Clofl}

Belohlavek, R., Concept lattices and order in fuzzy logic.

Ann. Pure Appl. Logic \textbf{128}(1--3): 277--298 (2004)

\bibitem{Lhote}

Lhote et al.: Average Number of Frequent (Closed) Patterns in Bernoulli and Markovian Databases.

In: \emph{Fifth IEEE International Conference on Data Mining} (ICDM'05), Houston, Texas, 713--716 (2005)

\bibitem{EmLe:Srglncfi}

Emilion, R., L\'evy G.: Size of random Galois lattices and number of closed frequent itemsets.

Discrete Appl. Math. \textbf{157}(13): 2945--2957 (2009)

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\bibitem{Klim}

Klimushkin et al.: Approaches to the Selection of Relevant Concepts in the Case of Noisy Data. In: Kwuida L., Serkaya B. (Eds.) Formal Concept Analysis (IFCA 2010). LNCS vol. 5986, pp. 255-266. Springer, Berlin, Heidelberg.

\url{https://doi.org/10.1007/978-3-642-11928-6\textunderscore18}

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\end{thebibliography}

\end{document}