In this paper, we study arbitrary infinite binary information systems each of which consists of an infinite set called universe and an infinite set of two-valued functions (attributes) defined on the universe. We consider the notion of a problem over information system, which is described by a finite number of attributes and a mapping associating a decision to each tuple of attribute values. As algorithms for problem solving, we investigate deterministic and nondeterministic decision trees that use only attributes from the problem description. Nondeterministic decision trees are representations of decision rule systems that sometimes have less space complexity than the original rule systems. As time and space complexity, we study the depth and the number of nodes in the decision trees. In the worst case, with the growth of the number of attributes in the problem description, (i) the minimum depth of deterministic decision trees grows either as a logarithm or linearly, (ii) the minimum depth of nondeterministic decision trees either is bounded from above by a constant or grows linearly, (iii) the minimum number of nodes in deterministic decision trees has either polynomial or exponential growth, and (iv) the minimum number of nodes in nondeterministic decision trees has either polynomial or exponential growth. Based on these results, we divide the set of all infinite binary information systems into three complexity classes. This allows us to identify nontrivial relationships between deterministic decision trees and decision rules systems represented by nondeterministic decision trees. For each class, we study issues related to time-space trade-off for deterministic and nondeterministic decision trees.

翻译：本文研究任意无限二元信息系统，每个系统包含一个称为全集的无限集，以及定义在该全集上的无限个二值函数（属性）。我们考虑信息系统上的问题概念，该问题由有限个属性及一个映射描述，该映射将每个属性值元组对应至一个决策。作为问题求解算法，我们研究仅使用问题描述中属性的确定性与非确定性决策树。非确定性决策树是决策规则系统的表示形式，其空间复杂度有时低于原始规则系统。在时间与空间复杂度方面，我们研究决策树的深度与节点数量。在最坏情况下，随着问题描述中属性数量的增长：(i)确定性决策树的最小深度呈对数或线性增长；(ii)非确定性决策树的最小深度或受常数上界约束，或呈线性增长；(iii)确定性决策树的最小节点数量呈多项式或指数增长；(iv)非确定性决策树的最小节点数量呈多项式或指数增长。基于这些结果，我们将所有无限二元信息系统划分为三个复杂度类别，从而揭示确定性决策树与非确定性决策树所表示的决策规则系统之间的非平凡关系。针对每个类别，我们研究了确定性与非确定性决策树在时间-空间权衡方面的相关问题。