Equivalence between linear-space recognition of L_q and linearly bounded automata

Determine whether, for a fixed rational q ∈ (0, 1/2) and a finite alphabet Σ, recognition of the language L_q = { x ∈ Σ* : Ne(x) < q|x| } in linear space is equivalent to recognition by a linearly bounded automaton (LBA).

Background

The languages L_q collect words whose exact non-deterministic automatic complexity is less than a linear threshold q|x|. The paper proves strong lower bounds on the computational power required to recognise these languages (e.g., non-context-freeness and certain circuit lower bounds).

The authors suggest that L_q may be recognisable in linear space and conjecture an equivalence with LBA recognition, hinting at a potential linear-space brute-force algorithm via encoding NFAs and known bounds on N(x). Verifying this conjecture would situate L_q within classical language classes tied to space complexity, notably the context-sensitive languages recognised by LBAs.