We define the notion of $k$-safe infinitary series over idempotent ordered totally generalized product $\omega $-valuation monoids that satisfy specific properties. For each element $k$ of the underlying structure (different from the neutral elements of the additive, and the multiplicative operation) we determine two syntactic fragments of the weighted $LTL$ with the property that the semantics of the formulas in these fragments are $k$ -safe infinitary series. For specific idempotent ordered totally generalized product $\omega $-valuation monoids we provide algorithms that given a weighted B\"{u}chi automaton and a weighted $LTL$ formula in these fragments, decide whether the behavior of the automaton coincides with the semantics of the formula.

翻译：我们定义了在满足特定性质的幂等全序广义乘积$\omega$ -赋值幺半群上$k$ -安全无穷级数的概念。对于底层结构中的每个元素$k$（不同于加法与乘法运算的幺元），我们确定了带权$LTL$的两个语法片段，使得这些片段中公式的语义是$k$ -安全无穷级数。针对特定的幂等全序广义乘积$\omega$ -赋值幺半群，我们给出了算法：给定一个带权Büchi自动机以及这些片段中的一个带权$LTL$公式，判定该自动机的行为是否与公式的语义一致。