Input-commutative series and control presentations

Exact statement

A formal series is input-commutative when its value depends only on the Parikh vector of the input word; a commutative-control presentation requires every control series, not only the output, to have this property.

StatusDefined and used consistently in the proofs

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

Context

The output of a polynomial automaton can ignore input order even when hidden controls retain it, so output commutativity and control commutativity must be distinguished.

Definitions

The Parikh vector records the number of occurrences of each input letter.
A series is input-commutative when words with the same Parikh vector receive the same value.

Proof or evidence

The distinction is used in the exact control-algebra reformulation and in the remaining open normalization question.

Problem and model →

Verification notes

The definitions and reversal convention were checked against Clemente 2025, Lemma 14: reversal preserves the Parikh vector and therefore input commutativity.

Limitations

This definition does not assert that a commutative output can always be represented by commutative controls.

Open work

Keep input commutativity distinct from commutativity of a product rule in later comparisons.