Lower Bounds on Avoiding Thresholds.

For a DFA, a word avoids a subset of states, if after reading that word the automaton cannot be in any state from the subset regardless of its initial state. A subset that admits an avoiding word is avoidable. The k-avoiding threshold of a DFA is the smallest number such that every avoidable subset of size k can be avoided with a word no longer than that number. We study the problem of determining the maximum possible k-avoiding thresholds. For every fixed k ≥ 1, we show a general construction of strongly connected DFAs with n states and the k-avoiding threshold in Θ(n). This meets the known upper bound for k ≥ 3. For k = 1 and k = 2, the known upper bounds are respectively in O(n2) and in O(n3). For k = 1, we show that 2n − 3 is attainable for every number of states n in the class of strongly connected synchronizing binary DFAs, which is supposed to be the best possible in the class of all DFAs for n ≥ 8. For k = 2, we show that the conjectured solution for k = 1 (an upper bound in O(n)) also implies a tight upper bound in O(n2) on 2-avoiding threshold. Finally, we discuss the possibility of using k-avoiding thresholds of synchronizing automata to improve upper bounds on the length of the shortest reset words. 2012 ACM Subject Classification Theory of computation → Formal languages and automata theory; Mathematics of computing → Discrete mathematics