A quadratic lower bound for 2DFAs against one-way liveness

TL;DR

This paper establishes a quadratic lower bound on the number of states needed for two-way deterministic finite automata to solve one-way liveness, extending known bounds to arbitrary alphabets and providing a versatile proof approach.

Contribution

It introduces a new quadratic lower bound for 2DFAs solving one-way liveness on arbitrary alphabets, generalizing previous unary-specific results with a novel proof technique.

Findings

2DFA solving one-way liveness requires Omega(h^2) states

Lower bound applies to arbitrary alphabet sizes

Proof technique is potentially reusable in other contexts

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.