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We investigate the state complexity of the square operation on languages represented by deterministic, alternating, and Boolean automata. For each k such that 1 ≤ k ≤ n−2, we describe a binary language accepted by an n-state DFA with k final states meeting the upper bound n2 n − k2 n−1 on the state complexity of its square. We show that in the case of k = n − 1, the corresponding upper bound cannot be met. Using the DFA witness for square with 2 n states where half of them are final, we get the tight upper bounds on the complexity of the square operation on alternating and Boolean automata.

We investigate the state complexity of the square operation on languages represented by deterministic, alternating, and Boolean automata. For each [Formula: see text] such that [Formula: see text], we describe a binary language accepted by an [Formula: see text]-state deterministic finite automaton with [Formula: see text] final states meeting the upper bound [Formula: see text] on the state complexity of its square. We show that in the case of [Formula: see text], the corresponding upper bound cannot be met. Using the binary deterministic witness for square with [Formula: see text] states where half of them are final, we get the tight upper bounds [Formula: see text] and [Formula: see text] on the complexity of the square operation on alternating and Boolean automata, respectively.