Synchronization of strongly connected partial DFAs and prefix codes

TL;DR

This paper investigates the synchronization properties of strongly connected partial DFAs, showing they can be efficiently related to complete DFAs, and explores implications for key conjectures and reset word lengths.

Contribution

It establishes a reduction from synchronization problems in strongly connected partial DFAs to complete DFAs, linking their complexities and implications for major conjectures.

Findings

Synchronization problems are equally hard in both models.

Efficient reduction from partial to complete DFA synchronization.

Implications for the Černý and rank conjectures.

Abstract

We study synchronizing partial DFAs, which extend the classical concept of synchronizing complete DFAs and are a special case of synchronizing unambiguous NFAs. A partial DFA is called synchronizing if it has a word (called a \emph{reset word}) whose action brings a non-empty subset of states to a unique state and is undefined for all other states. The class of strongly connected partial automata is precisely the class of automata recognized prefix codes. While in the general case the problem of checking whether a partial DFA is synchronizing is PSPACE-complete, we show that in the strongly connected case this problem can be efficiently reduced to the same problem for a complete DFA. Using combinatorial, algebraic, and formal languages methods, we develop techniques that relate main synchronization problems for strongly connected partial DFAs to the same problems for complete DFAs. In particular, this includes the \v{C}ern\'{y} and the rank conjectures, the problem of finding a reset word, and upper bounds on the length of the shortest reset words of literal automata of finite prefix codes. We conclude that solving fundamental synchronization problems is equally hard in both models, as an essential improvement of the results for one model implies an improvement for the other.