Product-prefix and expression-prefix constructions need not commute

Exact statement

For ordinary component expressions, FPRD-SH-T11 gives a canonical left quotient from the Boolean product of their position automata to the Boolean product of their prefix automata. This target need not equal the natural prefix automaton of the combined Boolean-product expression: pure shuffle $a^*\amalg b^*$ is already a counterexample to that identification.

StatusExact boundary, now explained by the event-signature expansion theorem

External reviewNo documented external or specialist review of these FPRD results is recorded.

Context

Componentwise automaton reduction and the backward prefix recursion are different operations. The transfer theorem controls the first; shuffle homogeneity shows that the second may merge the asynchronous frontier differently.

Proof or evidence

FPRD-SH-T11 constructs the componentwise left quotient for every memoryless Boolean policy. FPRD-SH-X01 proves that the natural expression-level prefix automaton for a-star shuffled with b-star is not any quotient of its location automaton, so it cannot be identified with that canonical transferred quotient.

Two prefix constructions →

Verification notes

The boundary combines the proved componentwise transfer with the independently verified a-star shuffle b-star counterexample.

Limitations

This boundary reuses an existing sharp counterexample; the later event-signature theorems provide the new characterization.

Open work

Use FPRD-SH-T12 through FPRD-SH-T14 for the exact construction and policy criteria.