FPRD LabShare an approach

← Claims

Proposition · exact reformulation

FPRD-CPA-C02

Finite control-algebra equivalence

Exact statement

A commutative polynomial-automaton series has an equivalent presentation whose control series are all commutative if and only if it belongs to a finitely generated, derivative-stable Hadamard algebra consisting entirely of commutative series.

StatusProved by a written two-way construction; internally audited
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The desired normal form is a chronology-erasure statement: the output already ignores input order, while hidden controls may still remember it.

Definitions

  • A series is commutative when its value depends only on the Parikh vector of the input word.
  • A finite commutative control algebra is a finitely generated Hadamard algebra, closed under every one-letter left derivative, all of whose series are commutative.

Hypotheses and scope

  • The input series is recognised by a polynomial automaton over the rationals.
  • The Hadamard presentation is used through the published equivalence up to input reversal; commutativity is reversal-invariant.

Proof or evidence

Forward: commutative controls generate a finite derivative-stable Hadamard algebra contained in the commutative series. Reverse: choose finite algebra generators, express each one-letter derivative polynomially in them, and use those polynomials as automaton transitions; add the target series as the distinguished control if needed.

Control-algebra equivalence and proof →

Verification notes

The FPRD proof audit rechecked both construction directions, closure of commutative series under Hadamard polynomials and derivatives, and the role of reversal. No circular use of the desired normal form was found.

Limitations

  • The equivalence reformulates the open problem; it does not prove that every commutative output admits such an algebra.
  • No novelty or external-review claim is made.

Open work

Seek an author or specialist check of the presentation convention and reversal step before publication-ready status.