Minimality and uniqueness of the DFAO computing base-b digits from numeration representations

Determine whether, for every integer base b ≥ 2 and for each quadratic irrational α (including the golden ratio φ) treated via an appropriate numeration system (Zeckendorf for φ, Pell for √2, and Ostrowski for general quadratic irrationals), the deterministic finite automata with output that, on input the numeration-system representation of b^n, compute the n-th digit to the right of the point in the base-b expansion of α are minimal (i.e., have the fewest possible states among all DFAO’s correct on inputs of the form b^n) and unique up to isomorphism.

Background

The paper constructs deterministic finite automata with output that, on input the numeration-system representation of b^n (Zeckendorf for φ, Pell for √2, and Ostrowski for general quadratic irrationals), compute the n-th digit to the right of the point in the base-b expansion of the target irrational. The authors demonstrate explicit constructions and, in several cases, use SAT solving to show minimality and sometimes uniqueness.

They observe that in certain bases there are multiple distinct automata with the same number of states producing the same outputs to high precision, raising the broader question of whether their constructed automata are always minimal and unique across bases and irrationals. This motivates a general open question about minimality and uniqueness in this framework.