Lower Bounds on Words Separation: Are There Short Identities in Transformation Semigroups?
| Autor: | Bulatov, Andrei A., Karpova, Olga, Shur, Arseny M., Startsev, Konstantin |
|---|---|
| Rok vydání: | 2016 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | The words separation problem, originally formulated by Goralcik and Koubek (1986), is stated as follows. Let $Sep(n)$ be the minimum number such that for any two words of length $\le n$ there is a deterministic finite automaton with $Sep(n)$ states, accepting exactly one of them. The problem is to find the asymptotics of the function $Sep$. This problem is inverse to finding the asymptotics of the length of the shortest identity in full transformation semigroups $T_k$. The known lower bound on $Sep$ stems from the unary identity in $T_k$. We find the first series of identities in $T_k$ which are shorter than the corresponding unary identity for infinitely many values of $k$, and thus slightly improve the lower bound on $Sep(n)$. Then we present some short positive identities in symmetric groups, improving the lower bound on separating words by permutational automata by a multiplicative constant. Finally, we present the results of computer search for short identities for small $k$. Comment: 12 pages, 2 figures, 2 tables |
| Databáze: | arXiv |
| Externí odkaz: |
|