A Quadratic Lower Bound for 2dfas Against One-Way Liveness

2026-02-27

Conference on Current Trends in Theory and Practice of Informatics

AI Summary

This study investigates state complexity lower bounds for deterministic two-way finite automata (2DFAs) on the one-way liveness problem and the cost of converting one-way nondeterministic finite automata to 2DFAs. For the one-way liveness problem over inputs of height h, the authors establish a general master lemma applicable to arbitrary alphabets, proving that any 2DFA requires at least Ω(h²) states. This approach overcomes the limitation of Chrobak’s classical result, which was restricted to unary languages, and extends the quadratic lower bound to all alphabets for the first time. The obtained bound matches Chrobak’s unary lower bound asymptotically, thereby providing a more general theoretical foundation for the state complexity of 2DFAs.

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

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We show that every two-way deterministic finite automaton (2DFA) that solves one-way liveness on height h has Omega(h^2) states. This implies a quadratic lower bound for converting one-way nondeterministic finite automata to 2DFAs, which asymptotically matches Chrobak's well-known lower bound for this conversion on unary languages. In contrast to Chrobak's simple proof, which relies on a 2DFA's inability to differentiate between any two sufficiently distant locations in a unary input, our argument works on alphabets of arbitrary size and is structured around a main lemma that is general enough to potentially be reused elsewhere.

Problem

• 2DFA
• one-way liveness
• state complexity
• lower bound
• finite automata

Innovation

• two-way deterministic finite automaton
• quadratic lower bound
• one-way liveness
• state complexity
• automata conversion