2-сжимающие слова и проблема реконструкции последовательности

Для данного слова свойство быть 2-сжимаемым (2-синхронизирующим) существенно зависит от свойств некоторого специального множества S его факторов. Мы изучаем возможность реконструкции 2-сжимающего (2-синхронизирующего) слова по этому множеству. Переходя от множества S ко множеству Xs его факторов длины три, мы показываем, что 2-сжимающее (2-синхронизирующее) слово является накрывающим для Xs

Experiments on synchronizing automata

This work is motivated by the Černý Conjecture – an old unsolved problem in the automata theory. We describe the results of the experiments on synchronizing automata, which have led us to two interesting results. The first one is that the size of an automaton alphabet may play an important role in the issue of synchronization: we have found a 5-state automaton over a 3-letter alphabet which attains the upper bound from the Černý Conjecture, while there is no such automaton (except Černý automaton $C_5$) over a binary alphabet. The second result emerging from the experiments is a theorem describing the dependencies between the automaton structure $S$ expressed in terms of the so-called merging system and the maximal length of all minimal synchronizing words for automata of type $S$

Experiments on Synchronizing Automata

This work is motivated by the ˇCern´y Conjecture – an old unsolved problem in the automata theory. We describe the results of the experiments on synchronizing automata, which have led us to two interesting results. The first one is that the size of an automaton alphabet may play an important role in the issue of synchronization: we have found a 5-state automaton over a 3-letter alphabet which attains the upper bound from the ˇCern´y Conjecture, while there is no such automaton (except ˇCern´y automaton C5) over a binary alphabet. The second result emerging from the experiments is a theorem describing the dependencies between the automaton structure S expressed in terms of the so-called merging system and the maximal length of all minimal synchronizing words for automata of type S

Image reducing words and subgroups of free groups

AbstractA word w over a finite alphabet Σ is said to be n-collapsing if for an arbitrary finite automaton A=〈Q,Σ−·−〉, the inequality |Q·w|⩽|Q|−n holds provided that |Q·u|⩽|Q|−n for some word u (depending on A). We give an algorithm to test whether a word is 2-collapsing. To this aim we associate to every word w a finite family of finitely generated subgroups in finitely generated free groups and prove that the property of being 2-collapsing reflects in the property that each of these subgroups has index at most 2 in the corresponding free group. We also find a similar characterization for the closely related class of so-called 2-synchronizing words