Technical Report No. 2011-577 State Complexity of Star and Quotient Operation for Unranked Tree Automata

We consider the state complexity of extensions of the Kleene star and quotient operations to unranked tree languages. Due to the nature of the tree structure, there are two distinct ways to define the star operation for trees, we call these operations, respectively, bottom-up and top-down star. We show that (n+ 3 2 )2 states are sufficient and necessary in the worst case to recognize the bottom-up star of a tree language recognized by an n-state deterministic unranked tree automaton. The bound is of a different order than the known state complexity result 3 2 · 2 for the Kleene star operation for automata on strings. On the other hand, for the top-down star we obtain a tight state complexity bound that coincides with the corresponding result for automata on strings. The bottom-quotient and top-quotient operations are extensions of the left and right quotient to trees. We establish tight state complexity bounds for both variants of quotient. The precise worst-case state complexity of bottom-quotient is shown to be (n+1)2 − 1, which differs by the multiplicative factor n+ 1 from the corresponding result 2 − 1 for ordinary finite automata.