Remarks on separating words

The separating words problem asks for the size of the smallest DFA needed to distinguish between two words of length <= n (by accepting one and rejecting the other). In this paper we survey what is known and unknown about the problem, consider some variations, and prove several new results

Separating Words from Every Start State with Horner Automata

We show that a well-known family of deterministic finite automata can be used to distinguish distinct binary strings of the same length from every start state. Further, we establish almost matching lower and upper bounds on the number of states of such automata necessary to achieve this type of separation. Our result improves the currently best known linear upper bound for arbitrary DFA.Comment: In Proceedings AFL 2023, arXiv:2309.0112

Lower bounds on words separation: Are there short identities in transformation semigroups?

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 ≤ 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 Tk. The known lower bound on Sep stems from the unary identity in Tk. We find the first series of identities in Tkwhich 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. © 2017, Australian National University. All rights reserved.Natural Sciences and Engineering Research Council of Canada: 16-01-00795Russian Foundation for Basic Research∗Supported by an NSERC Discovery grant †Partially supported by the grant 16-01-00795 of the Russian Foundation for Basic Research

Lower bounds on words separation: Are there short identities in transformation semigroups?

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 ≤ 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 Tk. The known lower bound on Sep stems from the unary identity in Tk. We find the first series of identities in Tkwhich 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. © 2017, Australian National University. All rights reserved.Natural Sciences and Engineering Research Council of Canada: 16-01-00795Russian Foundation for Basic Research∗Supported by an NSERC Discovery grant †Partially supported by the grant 16-01-00795 of the Russian Foundation for Basic Research

Separating the Words of a Language by Counting Factors

For a given language L, we study the languages X such that for all distinct words u; v is an element of L, there exists a word x is an element of X that appears a different number of times as a factor in u and in v. In particular, we are interested in the following question: For which languages L does there exist a finite language X satisfying the above condition? We answer this question for all regular languages and for all sets of factors of infinite words