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We study synchronizing partial DFAs, which extend the classical concept ofsynchronizing complete DFAs and are a special case of synchronizing unambiguousNFAs. 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 uniquestate and is undefined for all other states. The class of strongly connectedpartial automata is precisely the class of automata recognized prefix codes.While in the general case the problem of checking whether a partial DFA issynchronizing is PSPACE-complete, we show that in the strongly connected casethis problem can be efficiently reduced to the same problem for a complete DFA.Using combinatorial, algebraic, and formal languages methods, we developtechniques that relate main synchronization problems for strongly connectedpartial DFAs to the same problems for complete DFAs. In particular, thisincludes the \v{C}ern\'{y} and the rank conjectures, the problem of finding areset word, and upper bounds on the length of the shortest reset words ofliteral automata of finite prefix codes. We conclude that solving fundamentalsynchronization problems is equally hard in both models, as an essentialimprovement of the results for one model implies an improvement for the other.

Synchronization of strongly connected partial DFAs and prefix codes