Lower Bounds on Cardinality of Reducts for Decision Tables from Closed Classes
Azimkhon Ostonov, Mikhail Moshkov
TL;DR
The paper addresses lower bounds on the size of reducts for decision tables drawn from closed classes closed under attribute removal and decision changes, under the assumption of unbounded rows. It introduces the growth parameter $I(C)$ and analyzes $N_C(n)$ to derive reduct-size bounds, revealing two regimes: when $I(C)<\infty$, $R(T) \ge {\rm cl}(T)^{1/I(C)}/k^{2}$, and when $I(C)=+\infty$, $R(T) \ge \log_{k}{\rm cl}(T)$, with examples showing tighter bounds in specific closed-class families derived from information systems. The paper provides concrete instances (e.g., $R(T) \ge {\rm cl}(T)^{1/2}/4$ for Tab$(U_P)$ and $R(T) \ge \log_{3}{\rm cl}(T)$ for Tab$(U(m))$) illustrating the two regimes and their practical implications for feature selection and compression in rough-set frameworks. By clarifying when stronger lower bounds hold, the work informs complexity and design considerations in decision-table based reasoning and data analysis.
Abstract
In this paper, we consider classes of decision tables closed under removal of attributes (columns) and changing of decisions attached to rows. For decision tables from closed classes, we study lower bounds on the minimum cardinality of reducts, which are minimal sets of attributes that allow us to recognize, for a given row, the decision attached to it. We assume that the number of rows in decision tables from the closed class is not bounded from above by a constant. We divide the set of such closed classes into two families. In one family, only standard lower bounds $Ω(\log $ ${\rm cl}(T))$ on the minimum cardinality of reducts for decision tables hold, where ${\rm cl}(T)$ is the number of decision classes in the table $T$. In another family, these bounds can be essentially tightened up to $Ω({\rm cl}(T)^{1/q})$ for some natural $q$.