Elena V. Pribavkina

Synchronizing automata with finitely many minimal synchronizing words

A synchronizing word for a given synchronizing DFA is called minimal if none of its proper factors is synchronizing. We characterize the class of synchronizing automata having only finitely many minimal synchronizing words (the class of such automata is denoted by FG). Using this characterization we prove that any such automaton possesses a synchronizing word of length at most 3n-5. We also prove that checking whether a given DFA A is in FG is co-NP-hard and provide an algorithm for this problem which is exponential in the number of states A.

Recognizing synchronizing automata with finitely many minimal synchronizing words is PSPACE-complete

A deterministic finite-state automaton A is said to be synchronizing if there is a synchronizing word, i.e. a word that takes all the states of the automaton A to a particular one. We consider synchronizing automata whose language of synchronizing words is finitely generated as a two-sided ideal in Σ*. Answering a question stated in [1], here we prove that recognizing such automata is a PSPACE-complete problem.

On some properties of the language of 2-collapsing words

We present two new results on 2-collapsing words. First, we show that the language of all 2-collapsing words over 2 letters is not context-free. Second, we prove that the length of a 2-collapsing word over an arbitrary finite alphabet Σ is at least 2|Σ|2 thus improving the previously known lower bound |Σ|2+1.

STATE COMPLEXITY OF CODE OPERATORS

We consider five operators on a regular language. Each of them is a tool for constructing a code (respectively prefix, suffix, bifix, infix) and a hypercode out of a given regular language. We give the precise values of the (deterministic) state complexity of these operators: over a constant-size alphabet for the first four of them and over a quadratic-size alphabet for the hypercode operator.

Finitely Generated Synchronizing Automata

A synchronizing word w for a given synchronizing DFA is called minimal if no proper prefix or suffix of w is synchronizing. We characterize the class of synchronizing automata having finite language of minimal synchronizing words (such automata are called finitely generated ). Using this characterization we prove that any such automaton possesses a synchronizing word of length at most 3n *** 5. We also prove that checking whether a given DFA

On non-complete sets and Restivo's conjecture

\mathcal{A}

State complexity of prefix, suffix, bifix and infix operators on regular languages

is finitely generated is co-NP-hard, and provide an algorithm for this problem which is exponential in the number of states

ON SOME PROPERTIES OF THE LANGUAGE OF 2-COLLAPSING WORDS

\mathcal{A}.

Reset Thresholds of Automata with Two Cycle Lengths

A finite set S of words over the alphabet Σ is called noncomplete if Fact(S*) ≠ Σ*. A word w ∈ Σ* \ Fact(S*) is said to be uncompletable. We present a series of non-complete sets Sk whose minimal uncompletable words have length 5k2 - 17k + 13, where k ≥ 4 is the maximal length of words in Sk. This is an infinite series of counterexamples to Restivos conjecture, which states that any non-complete set possesses an uncompletable word of length at most 2k2.