Commutative controls in the weighted case

Exact statement

For a commutative rational series, a finite basis chosen from its derivative span yields an equivalent weighted automaton whose control series are all commutative.

StatusReconstructed from the classical finite derivative-space theorem

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

Automata Exchange states that the answer is yes for weighted automata. The FPRD note reconstructs the mechanism as finite linear observability.

Hypotheses and scope

The series is rational and commutative.
Controls are allowed to be a finite derivative basis with linear transitions.

Proof or evidence

The left derivatives span a finite-dimensional vector space. Choosing a basis from the derivatives makes every basis control commutative, and one-letter differentiation acts linearly on that basis.

Weighted-case reconstruction →

Verification notes

The reconstruction was checked against the finite derivative-space characterization in Berstel--Reutenauer, Chapter 1, pp. 3--28, DOI 10.1017/CBO9780511760860.002.

Limitations

This is the linear weighted case only.
It does not provide a finite polynomial recovery theorem in the nonlinear case.

Open work

Seek an independent specialist check of the classical derivative-basis reconstruction.