# Weighted Operator Precedence Languages

## TL;DR

This paper extends operator precedence languages with weights, introducing weighted automata and logic, and demonstrates their expressive power and foundational properties, bridging formal language theory and quantitative system modeling.

## Contribution

It introduces weighted operator precedence automata and establishes their equivalence with weighted MSO logic, expanding the theoretical framework of OPL.

## Key findings

- Weighted automata extend OPL with quantitative capabilities.
- Weighted OPA are equivalent to a weighted MSO logic for OPL.
- A Nivat-like decomposition theorem for weighted OPL is proved.

## Abstract

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 the original grammar family, in terms of an automata family and an MSO logic; furthermore they significantly generalize the well-known visibly pushdown languages (VPL). In another area of research, quantitative models of systems are also greatly in demand. 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 B\"uchi-like theorem, we show that weighted OPA are expressively equivalent to a weighted MSO-logic for OPL.