In this paper, we give a Nivat-like characterization for weighted alternating automata over commutative semirings (WAFA). To this purpose we prove that weighted alternating can be characterized as the concatenation of weighted finite tree automata (WFTA) and a specific class of tree homomorphism. We show that the class of series recognized by weighted alternating automata is closed under inverses of homomorphisms, but not under homomorphisms. 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. Using the corresponding result for polynomial automata, we are able to prove that the ZERONESS problem for weighted alternating automata with the rational numbers as weights is decidable.

中文翻译：

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交换半环上加权交替自动机的 Nivat 定理

在本文中，我们对交换半环（WAFA）上的加权交替自动机给出了类似 Nivat 的表征。为此，我们证明加权交替可以表征为加权有限树自动机（WFTA）和特定类别的树同态的串联。我们证明了加权交替自动机识别的级数类在同态的逆下是封闭的，但在同态下不是。我们给出了加权交替自动机的逻辑特征，它对树使用加权 M​​SO 逻辑。最后，我们研究了加权交替自动机和多项式自动机之间的强联系。使用多项式自动机的相应结果，我们能够证明以有理数作为权重的加权交替自动机的零问题是可判定的。