10 - Synchronization

Summary

The notion of synchronization for codes and automata refers to the ability of parsing an input into codewords with a limited amount of information. It addresses a more general situation than deciphering which is left-to-right oriented. The interest of synchronization lies in the possibility of recovering from errors by the specific nature of the involved decoders.

The chapter starts with the definition of synchronizing pairs, synchronizing words, and absorbing words. These notions have already been considered in Chapter 3 for prefix codes. Next, as for the deciphering delay, two notions of synchronization delay are introduced, the first related to the number of words involved, the second connected to local automata. We describe the connection between synchronization delay and the notions of circular codes and limited codes. Important results are the completion of rational uniformly synchronized codes and of locally parsable codes (Theorem 10.3.13 and Theorem 10.2.11).

In the final section, we give a necessary and sufficient condition to guarantee that a deterministic automaton can be transformed into a synchronizing one by modifying the labels of its edges (Theorem 10.4.2). This theorem has been conjectured over many years as the road coloring problem.

Synchronizing pairs

The section starts with the definition of synchronizing pairs, synchronizingwords, and constants. Relations among these objects are described. Constants are characterized by their rank. Next, synchronized codes are defined, and shown to coincide with codes of degree 1. Finally, absorbing words are introduced.