This paper connects the classes of weighted alternating finite automata (WAFA), weighted finite tree automata (WFTA), and polynomial automata (PA). First, we investigate the use of trees in the run semantics for weighted alternating automata and prove that the behavior of a weighted alternating automaton can be characterized as the composition of the behavior of a weighted finite tree automaton and a specific tree homomorphism, if weights are taken from a commutative semiring. Based on this, we give a Nivat-like characterization for weighted alternating automata. Moreover, we show that the class of series recognized by weighted alternating automata is closed under inverses of homomorphisms, but not under homomorphisms. Additionally, we give a logical characterization of weighted alternating automata, which uses weighted MSO logic for trees. Finally, we investigate the strong connection between weighted alternating automata and polynomial automata. We prove: A weighted language is recognized by a weighted alternating automaton iff its reversal in recognized by a polynomial automaton. Using the corresponding result for polynomial automata, we are able to prove that the ZERONESS problem for weighted alternating automata with weights taken from the rational numbers decidable.

翻译：本文建立了加权交替有限自动机(WAFA)、加权有限树自动机(WFTA)与多项式自动机(PA)三个类之间的联系。首先，我们研究了加权交替自动机运行语义中树的使用，并证明：当权重取自交换半环时，加权交替自动机的行为可表征为加权有限树自动机行为与特定树同态的复合。基于此，我们给出了加权交替自动机的Nivat型刻画。进一步，我们证明了加权交替自动机识别的级数类在同态逆映射下封闭，但在同态映射下不封闭。此外，我们给出了加权交替自动机的逻辑刻画，该刻画使用了树上的加权MSO逻辑。最后，我们深入探究了加权交替自动机与多项式自动机之间的强关联。我们证明了：一个加权语言被加权交替自动机识别当且仅当其反转被多项式自动机识别。利用多项式自动机的相应结果，我们证明了权重取自有理数的加权交替自动机的零问题(ZERONESS problem)是可判定的。