In this paper we introduce a weighted LTL over product {\omega}-valuation monoids that satisfy specific properties. We also introduce weighted generalized B\"uchi automata with {\epsilon}-transitions, as well as weighted B\"uchi automata with {\epsilon}-transitions over product {\omega}-valuation monoids and prove that these two models are expressively equivalent and also equivalent to weighted B\"uchi automata already introduced in the literature. We prove that every formula of a syntactic fragment of our logic can be effectively translated to a weighted generalized B\"uchi automaton with {\epsilon}-transitions. Finally, we prove that the number of states of the produced automaton is polynomial in the size of the corresponding formula.

中文翻译：

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{\omega}-估值幺半群上的加权 LTL 公式到加权 B\"uchi 自动机的翻译

在本文中，我们在满足特定属性的乘积 {\omega} 估值幺半群上引入了加权 LTL。我们还引入了带有 {\epsilon}-transitions 的加权广义 B\"uchi 自动机，以及带有 {\epsilon}-transitions over product {\omega}-valuation monoids 的加权 B\"uchi 自动机，并证明这两个模型是表达等价，也等价于文献中已经引入的加权 B\"uchi 自动机。我们证明了我们逻辑的句法片段的每个公式都可以有效地转换为具有 {\epsilon}-transitions 的加权广义 B\"uchi 自动机. 最后，我们证明了生成的自动机的状态数是对应公式大小的多项式。