Decision trees for binary subword-closed languages

Abstract

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 $i\in \{1,\dots ,n\}$, 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.