Decision trees for binary subword-closed languages

TL;DR

This paper analyzes the complexity of decision trees for recognizing and classifying binary subword-closed languages, revealing bounds on their depth and classifying languages into five complexity categories.

Contribution

It characterizes the growth of decision tree depths for various recognition problems in binary subword-closed languages and introduces five complexity classes.

Findings

Deterministic recognition depth is either constant, logarithmic, or linear.

Nondeterministic recognition and membership decision depths are either constant or linear.

Five complexity classes of binary subword-closed languages are described.

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