Some Properties Related to Reduct of Consistent Decision Systems

Reduct of decision systems is the topic that has been attracting the interest of many researchers in data mining and machine learning for more than two decades. So far, many algorithms for finding reduct of decision systems by rough set theory have been proposed. However, most of the proposed algorithms are heuristic algorithms that find one reduct with the best classification quality. The complete study of properties of reduct of decision systems is limited. In this paper, we discover equivalence properties of reduct of consistent decision systems related to a Sperner-system. As the result, the study of the family of reducts in a consistent decision system is the study of Sperner-systems.

Some computational problems related to normal forms

In the relational database theory the most desirable normal form is the Boyce-Codd normal form (BCNF). This paper investigates some computational problems concerning BCNF relation scheme and BCNF relations. We give an effective algorithm finding a BCNF relation r such that r represents a given BCNF relation scheme s (i.e., Kr=Ks, where Kr and Ks are sets of all minimal keys of r and s). This paper also gives an effective algorithm which from a given BCNF relation finds a BCNF relation scheme such that Kr=Ks. Based on these algorithms we prove that the time complexity of the problem that finds a BCNF relation r representing a given BCNF relation scheme s is exponential in the size of s and conversely, the complexity of finding a BCNF relation scheme s from a given BCNF relation r such that r represents s also is exponential in the number of attributes. We give a new characterization of the relations and the relation scheme that are uniquely determined by their minimal keys. It is known that these relations and the relation schemes are in the BCNF class. From this characterization we give a polynomial time algorithm deciding whether an arbitrary relation is uniquely determined by its set of all minimal keys. In the rest if this paper some new bounds of the size of minimal Armstrong relations for BCNF relation scheme are given. We show that given a Sperner system K and BCNF relation scheme s a set of minimal keys of which is K, the number of antikeys (maximal nonkeys) of K is polynomial in the number of attributes iff so is the size of minimal Armstrong relation of s.

On Saturated $k$-Sperner Systems

Given a set $X$, a collection $\mathcal{F}\subseteq\mathcal{P}(X)$ is said to be $k$-Sperner if it does not contain a chain of length $k+1$ under set inclusion and it is saturated if it is maximal with respect to this property. Gerbner et al. conjectured that, if $|X|$ is sufficiently large with respect to $k$, then the minimum size of a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ is $2^{k-1}$. We disprove this conjecture by showing that there exists $\varepsilon>0$ such that for every $k$ and $|X| \geq n_0(k)$ there exists a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ with cardinality at most $2^{(1-\varepsilon)k}$.A collection $\mathcal{F}\subseteq \mathcal{P}(X)$ is said to be an oversaturated $k$-Sperner system if, for every $S\in\mathcal{P}(X)\setminus\mathcal{F}$, $\mathcal{F}\cup\{S\}$ contains more chains of length $k+1$ than $\mathcal{F}$. Gerbner et al. proved that, if $|X|\geq k$, then the smallest such collection contains between $2^{k/2-1}$ and $O\left(\frac{\log{k}}{k}2^k\right)$ elements. We show that if $|X|\geq k^2+k$, then the lower bound is best possible, up to a polynomial factor.

On A Problem of Purdy Related to Sperner Systems

Purdy asked whether the following conjecture is trueConjecture. Let E be a set of 2n elements. If S={Sl, S2, …, St} is a Sperner system of E, i.e. for i≠j, i, j, =1, 2, …, t; and if(1)thenThe proof of the conjecture will be obtained using the following theorem of Katona (Acta Math. 15 (1964), 329-337):