Turn Complexity of Context-free Languages, Pushdown Automata and One-Counter Automata

2026-03-09

AI Summary

This study investigates the complexity and decidability of the number of “turns” in accepting computations of pushdown automata and one-counter automata when recognizing context-free languages. Employing tools from formal language and automata theory, along with constructive proofs and undecidability arguments, the work establishes that boundedness of turn complexity is undecidable and reveals a non-recursive trade-off in expressive power between the two automaton models. The main contributions include the construction of an infinite strict hierarchy of turn complexity classes bounded by distinct sublinear functions, and the demonstration that there exist languages requiring sublinear—but not constant—turn complexity, with growth rates slower than any given sublinear function, thereby refining the theoretical framework of finite-turn automata.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

A turn in a computation of a pushdown automaton is a switch from a phase in which the height of the pushdown store increases to a phase in which it decreases. Given a pushdown or one-counter automaton, we consider, for each string in its language, the minimum number of turns made in accepting computations. We prove that it cannot be decided if this number is bounded by any constants. Furthermore, we obtain a non-recursive trade-off between pushdown and one-counter automata accepting in a finite number of turns and finite-turn pushdown automata, that are defined requiring that the constant bound is satisfied by each accepting computation. We prove that there are languages accepted in a sublinear but not constant number of turns, with respect to the input length. Furthermore, there exists an infinite proper hierarchy of complexity classes, with the number of turns bounded by different sublinear functions. In addition, there is a language requiring a number of turns which is not constant but grows slower than each of the functions defining the above hierarchy.

Problem

• turn complexity
• context-free languages
• pushdown automata
• one-counter automata
• finite-turn

Innovation

• turn complexity
• pushdown automata
• one-counter automata
• sublinear hierarchy
• undecidability