Maximal pattern complexity, dual system and pattern recognition

For a family Ω of sets in R and a finite subset S of R, let pΩ(S) be the number of distinct sets of the form S ∩ ω for all ω ∈ Ω. The maximum pattern complexity pΩ(k) is the maximum of pΩ(S) among S with #S = k. The S attaining the maximum is considered as the most effective sampling to distinguish the sets in Ω. We obtain the exact values or at least the order of pΩ(k) in k for various classes Ω. We also discuss the dual problem in the case that #Ω = ∞, that is, consider the partition of R generated by a finite family T ⊂ Ω. The number of elements in the partition is written as pR2(T ) and p ∗ R2(k) is the maximum of pR2(T ) among T with #T = k. Here, p ∗ Ω(k) = p ∗ R2(k) does not hold in general. For the general setting that Ω is an infinite subset of A, where A is a finite alphabet, Σ is an arbitrary infinite set, and pΩ(k) = max#S=k #Ω|S , it is known that the entropy h(Ω) := lim k→∞ log pΩ(k)/k exists and takes value in {log 1, log 2, · · · , log#A}. In this paper, we prove that the entropy h(Σ) of the dual system coincides with h(Ω).