二元子字封闭语言的决策树
Decision trees for binary subword-closed languages
摘要：
本文研究了字母表$\{0,1\}$上的任意子词封闭语言（二元子词封闭语言）。对于属于二进制子字封闭语言$L$的长度为$n$的词集$L(n)$，我们研究了决策树确定性和非确定性地解决识别和成员问题的深度。在识别问题的情况下，对于来自$L(n)$的给定单词，我们应该使用查询来识别它，对于某些$i\in \{1,\ldots ,n\}$，每个查询返回单词的第$i$个字母。在成员问题的情况下，对于长度为$n$的字母表$\{0,1\}$上的给定单词，我们应该使用相同的查询来识别它是否属于集合$L(n)$。随着$n$的增长，确定性地解决识别问题的决策树的最小深度要么由常数从上限定，要么以对数或线性方式增长。对于其他类型的树和问题（决策树非确定性地解决识别问题，以及决策树确定性地和非确定性地解决成员资格问题），随着$n$的增长，决策树的最小深度要么从上到下由常数限制，要么线性增长。我们研究了所考虑的四种类型的决策树的最小深度的联合行为，并描述了五个复杂的二进制子字封闭语言类。
英文摘要：
In this paper, we study arbitrary subword-closed languages over the alphabet $\{0,1\}$ (binary subword-closed languages). For the set of words $L(n)$ of the length $n$ belonging to a binary subword-closed language $L$, we investigate the depth of decision trees solving the recognition and the membership problems deterministically and nondeterministically. In the case of recognition problem, for a given word from $L(n)$, we should recognize it using queries each of which, for some $i\in \{1,\ldots ,n\}$, returns the $i$th letter of the word. In the case of membership problem, for a given word over the alphabet $\{0,1\}$ of the length $n$, we should recognize if it belongs to the set $L(n)$ using the same queries. With the growth of $n$, the minimum depth of decision trees solving the problem of recognition deterministically is either bounded from above by a constant, or grows as a logarithm, or linearly. For other types of trees and problems (decision trees solving the problem of recognition nondeterministically, and decision trees solving the membership problem deterministically and nondeterministically), with the growth of $n$, the minimum depth of decision trees is either bounded from above by a constant or grows linearly. We study joint behavior of minimum depths of the considered four types of decision trees and describe five complexity classes of binary subword-closed languages.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
