Shortest synchronizing words from the Fair Coin component in the MOAP

Identify the shortest synchronizing word(s) that take the mixed-state presentation of the Mother of All 1-State and 2-State Processes from the Fair Coin ergodic component to each other ergodic component, i.e., the minimal words that collapse the mixed-state distribution to the target component’s recurrent states when starting in the Fair Coin component.

Background

Within the MOAP constructed from all one-state and two-state epsilon-machines, the Fair Coin is one of the ergodic components. Synchronizing words are those that drive the mixed-state distribution to a delta over a specific component’s causal states, thereby identifying the component from observations.

The mixed states plotted in the simplex show close approaches to vertices associated with individual components, suggesting that specific words can synchronize the model to those components. Determining shortest such words from the Fair Coin to each other component would clarify the accessibility and detectability of components within the MOAP.