Reset Complexity of Ideal Languages

Complexity of Checking Whether Two Automata Are Synchronized by the Same Language

“…On the other hand, every regular ideal language L serves as the language of reset words for some automaton. For instance, the minimal automaton recognizing L is synchronized exactly by L [7]. Thus synchronizing automata can be considered as a special representation of an ideal language.…”

“…Thus synchronizing automata can be considered as a special representation of an ideal language. Effectiveness of such a representation was addressed in [7]. The reset complexity rc(L) of an ideal language L is the minimal possible number of states in a synchronizing automaton A such that Syn(A ) = L. Every such automaton A is called a minimal synchronizing automaton (for brevity, MSA).…”

“…The reset complexity rc(L) of an ideal language L is the minimal possible number of states in a synchronizing automaton A such that Syn(A ) = L. Every such automaton A is called a minimal synchronizing automaton (for brevity, MSA). Let sc(L) be the number of states in the minimal automaton recognizing L. For every ideal language L we have rc(L) ≤ sc(L) [7]. Moreover, there are languages L n for every n ≥ 3 such that rc(L n ) = n and sc(L n ) = 2 n − n [7].…”

“…Let sc(L) be the number of states in the minimal automaton recognizing L. For every ideal language L we have rc(L) ≤ sc(L) [7]. Moreover, there are languages L n for every n ≥ 3 such that rc(L n ) = n and sc(L n ) = 2 n − n [7]. Thus the representation of an ideal language by means of a synchronizing automaton can be exponentially more succinct than the "traditional" representation via the minimal automaton.…”

“…One of the obstacles is that an MSA is not uniquely defined. For instance, there is a language with at least two different MSAs [7].…”