In the last years renewed investigation of operator precedence languages (OPL) led to discover important properties thereof: OPL are closed with respect to all major operations, are characterized, besides by the original grammar family, in terms of an automata family (OPA) and an MSO logic; furthermore they significantly generalize the well-known visibly pushdown languages (VPL). A different area of research investigates quantitative evaluations of formal languages by adding weights to strings. In this paper, we lay the foundation to marry these two research fields. We introduce weighted operator precedence automata and show how they are both strict extensions of OPA and weighted visibly pushdown automata. We prove a Nivat-like result which shows that quantitative OPL can be described by unweighted OPA and very particular weighted OPA. In a Büchi-like theorem, we show that weighted OPA are expressively equivalent to a weighted MSO-logic for OPL.

在过去几年中，对运算符优先语言 (OPL) 的重新调查导致发现了其重要属性：OPL 对所有主要操作都是封闭的，除了原始语法系列之外，还具有自动机系列 (OPA) 和MSO 逻辑；此外，它们显着地概括了众所周知的可见下推语言 (VPL)。另一个研究领域通过向字符串添加权重来研究形式语言的定量评估。在本文中，我们奠定了将这两个研究领域结合起来的基础。我们介绍了加权运算符优先自动机，并展示了它们如何既是 OPA 的严格扩展又是加权明显下推自动机。我们证明了一个类似 Nivat 的结果，该结果表明定量 OPL 可以用未加权的 OPA 和非常特殊的加权 OPA 来描述。