Several concept learning problems can be regarded as special cases of half-space separation in abstract closure systems over finite ground sets. For the typical scenario that the closure system is implicitly given via a closure operator, we show that the half-space separation problem is NP-complete. As a first approach to overcome this negative result, we relax the problem to maximal closed set separation, give a generic greedy algorithm solving this problem with a linear number of closure operator calls, and show that this bound is sharp. For a second direction, we consider Kakutani closure systems and prove that they are algorithmically characterized by the greedy algorithm. As a first special case of the general problem setting, we consider Kakutani closure systems over graphs and give a sufficient condition for this kind of closure systems in terms of forbidden graph minors. For a second special case, we then focus on closure systems over finite lattices, give an improved adaptation of the generic greedy algorithm, and present an application concerning subsumption lattices.

翻译：一些概念学习问题可被视为在抽象封闭系统中对有限地面装置进行半空分离的特殊情况。对于封闭系统被关闭操作员暗含的典型假设,我们表明,半空隔离问题已经完成。作为克服这一负面结果的第一种办法,我们将问题放松到最大封闭式分离,以线性数量关闭操作员电话来解决这个问题,并显示这一约束是尖锐的。第二个方向,我们考虑卡库塔尼封闭系统,并证明它们具有贪婪算法的特点。作为一般问题设置的第一个特殊事例,我们认为卡库塔尼封闭系统超越了图表,并以禁止的未成年人图形形式为这种封闭系统提供了充分的条件。第二个特殊事例是,我们然后侧重于有限封闭系统,改进了对一般贪婪算法的适应,并提出了关于子吸附式拉特克的应用程序。