Exponential complexity of finite-memory automaton translations

Determine whether the translation of a quasi-regular language accepted by a finite-memory automaton to its restriction over a finite alphabet can have exponential complexity.

Background

The paper studies the relationship between finite-memory automata over infinite alphabets and ordinary finite-state automata over finite alphabets. For a quasi-regular language L and a finite alphabet subset Σ′, the restriction L ∩ (Σ′)* is regular, but the authors note that representing this restriction by an ordinary finite-state automaton may require substantially more states than the original finite-memory automaton description.

The authors state that it has been conjectured that this translation may incur exponential blowup. They motivate the conjecture by observing that non-emptiness for deterministic finite-memory automata is NP-complete, whereas emptiness of the corresponding restriction to an alphabet whose size equals the number of registers can be decided in linear time.