We introduce a weighted linear dynamic logic (weighted LDL for short) and show the expressive equivalence of its formulas to weighted rational expressions. This adds a new characterization for recognizable series to the fundamental Schützenberger theorem. Surprisingly, the equivalence does not require any restriction to our weighted LDL. Our results hold over arbitrary (resp. totally complete) semirings for finite (resp. infinite) words. As a consequence, the equivalence problem for weighted LDL formulas over fields is decidable in doubly exponential time. In contrast to classical logics, we show that our weighted LDL is expressively incomparable to weighted LTL for finite words. We determine a fragment of the weighted LTL such that series over finite and infinite words definable by LTL formulas in this fragment are definable also by weighted LDL formulas. This is an extended version of [17].

我们引入了加权线性动态逻辑（简称加权LDL）并证明了其公式与加权有理表达式的表达等价性。这为基本的舒岑伯格定理添加了可识别级数的新特征。令人惊讶的是，这种等价性不需要对我们的加权 LDL 进行任何限制。我们的结果适用于有限（或无限）单词的任意（或完全完整）半环。因此，域上加权 LDL 公式的等价问题可以在双指数时间内判定。与经典逻辑相反，我们表明对于有限词，我们的加权 LDL 在表达上与加权 LTL 无法相比。我们确定加权 LTL 的片段，使得可由该片段中的 LTL 公式定义的有限和无限字的级数也可由加权 LDL 公式定义。这是[17]的扩展版本。