# Making Metric Temporal Logic Rational

## TL;DR

This paper introduces an extension of Metric Temporal Logic with rational expressions, proving its decidability and establishing a connection to timed automata, advancing the theoretical understanding of temporal logic expressiveness.

## Contribution

It defines and analyzes the decidability of the rational extension of MTL, introduces a fragment with elementary decidability, and links it to automata theory.

## Key findings

- Decidable satisfiability for the extended logic via reduction to MTL.
- Identification of a fragment with elementary decidability.
- Establishment of a tight automaton-logic correspondence.

## Abstract

We study an extension of $\mtl$ in pointwise time with rational expression guarded modality $\reg_I(\re)$ where $\re$ is a rational expression over subformulae. We study the decidability and expressiveness of this extension ($\mtl$+$\varphi \ureg_{I, \re} \varphi$+$\reg_{I,\re}\varphi$), called $\regmtl$, as well as its fragment $\sfmtl$ where only star-free rational expressions are allowed. Using the technique of temporal projections, we show that $\regmtl$ has decidable satisfiability by giving an equisatisfiable reduction to $\mtl$. We also identify a subclass $\mitl+\ureg$ of $\regmtl$ for which our equi-satisfiable reduction gives rise to formulae of $\mitl$, yielding elementary decidability. As our second main result, we show a tight automaton-logic connection between $\sfmtl$ and partially ordered (or very weak) 1-clock alternating timed automata.