Decision Trees for Binary Subword-Closed Languages.

Area of Science:

• Theoretical Computer Science
• Formal Languages and Automata Theory
• Computational Complexity

Background:

• Subword-closed languages are a class of formal languages with specific structural properties.
• Decision trees are fundamental data structures used for solving recognition and membership problems.
• Understanding the complexity of these problems is crucial for efficient language processing.

Purpose of the Study:

• To investigate the minimum depth of decision trees for recognition and membership problems in binary subword-closed languages.
• To analyze the deterministic and nondeterministic complexities of these problems.
• To classify binary subword-closed languages based on the growth of decision tree depths.

Main Methods:

• Analysis of decision tree depths for deterministic and nondeterministic recognition.
• Analysis of decision tree depths for deterministic and nondeterministic membership testing.
• Comparison of growth rates of minimum tree depths with respect to word length 'n'.
• Identification of complexity classes based on joint behavior of tree depths.

Main Results:

• Deterministic recognition tree depth is constant, logarithmic, or linear in 'n'.
• Nondeterministic recognition and both deterministic/nondeterministic membership tree depths are constant or linear in 'n'.
• Five distinct complexity classes of binary subword-closed languages are identified based on these depth behaviors.

Conclusions:

• The depth of decision trees for these problems exhibits predictable growth patterns.
• These patterns allow for a robust classification of binary subword-closed languages into complexity classes.
• The findings contribute to a deeper understanding of the computational complexity of formal languages.