Last-event prefix extension for synchronized shuffle

Exact statement

For strong synchronization, solo terminal events are allowed only outside $\Gamma$ and paired terminal events only on equal letters in $\Gamma$ ; for arbitrary synchronization the same paired clause is added without suppressing solo events. Applying the usual backward $R_\varepsilon$ -recursion to these clauses defines the natural prefix automata studied here.

StatusPrecisely stated for strong and arbitrary synchronization; the weak operator uses a separate backward-memory definition

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

Context

The synchronized-shuffle paper supplies location and partial-derivative automata for strong, arbitrary, and weak operators, but the prefix target needed for this quotient question must be specified separately.

Definitions

Strong synchronization forces every letter in $\Gamma$ to be read by both components together.
Arbitrary synchronization permits a matching $\Gamma$ -event either jointly or one component at a time.

Proof or evidence

The terminal-event clauses are the backward counterparts of the published strong and arbitrary word operations, and reduce to the familiar pure-shuffle rule when no synchronized event is available.

Synchronized prefix convention →

Verification notes

The strong and arbitrary clauses are checked by the exact synchronized-shuffle constructors; the weak operator is intentionally routed through its separate reversal-dual receipt.

Limitations

This is an FPRD definition, not a construction attributed to the source authors.
The weak operator is excluded from this definition and treated by a separate proved reversal-dual construction.

Open work

Keep the strong/arbitrary convention separate from the proved weak backward receipt.