Width Measures of Alternating Finite Automata - Keeler, Casey; Salomaa, Kai (Descriptional Complexity of Formal Systems, 2021)
We study the tree width, maximal existential width and maximal universal width of AFAs, which, roughly speaking, count the largest number of leaves, the largest number of existential choices, and the largest number of universal branches in a computation tree. We give polynomial-time algorithms deciding finiteness of an AFA’s tree width and (under certain conditions) the finiteness of an AFA’s maximal existential width. We also show that the language of any m-state AFA with finite maximal existential width can be recognized by O(m2) $$O(m^2)$$ m-state AFAs with no existential branching. Additionally, we give polynomial-time algorithms deciding the growth rate of an AFA’s tree width, and (under certain conditions) the growth rate of an AFA’s maximal universal width. Finally, we establish necessary and sufficient conditions for an AFA to have exponential tree width, as well as sufficient conditions for an AFA to have exponential maximal existential width or exponential maximal universal width.