An overview is given of results achieved by F. Matúš on probabilistic conditional independence (CI). First, his axiomatic characterizations of stochastic functional dependence and unconditional independence are recalled. Then his elegant proof of discrete probabilistic representability of a matroid based on its linear representability over a finite field is recalled. It is explained that this result was a basis of his methodology for constructing a probabilistic representation of a given abstract CI structure. His embedding of matroids into (augmented) abstract CI structures is recalled and his contribution to the theory of semigraphoids is mentioned as well. Finally, his results on the characterization of probabilistic CI structures induced by four discrete random variables and by four regular Gaussian random variables are recalled. Partial probabilistic representability by binary random variables is also mentioned., Milan Studený., and Obsahuje bibliografické odkazy
A secret sharing scheme is ideal if the size of each share is equal to the size of the secret. Brickell and Davenport showed that the access structure of an ideal secret sharing scheme is determined by a matroid. Namely, the minimal authorized subsets of an ideal secret sharing scheme are in correspondence with the circuits of a matroid containing a fixed point. In this case, we say that the access structure is a matroid port. It is known that, for an access structure, being a matroid port is not a sufficient condition to admit an ideal secret sharing scheme. In this work we present a linear secret sharing scheme construction for ports of matroids of rank 3 in which the size of each share is at most nn times the size of the secret. Using the previously known secret sharing constructions, the size of each share was O(n2/logn) the size of the secret. Our construction is extended to ports of matroids of any rank k≥2, obtaining secret sharing schemes in which the size of each share is at most nk-2 times the size of the secret. This work is complemented by presenting lower bounds: There exist matroid ports that require (Fq,ℓ)-linear secret schemes with total information ratio Ω(2n/2/ℓn3/4logq)., Oriol Farràs., and Obsahuje bibliografické odkazy