# Probabilistic logics based on Riesz spaces

## TL;DR

This paper introduces Riesz modal logic, a real-valued probabilistic logic inspired by Riesz spaces, providing a duality theory, axiomatization, and characterization of probabilistic bisimulation, laying groundwork for future extensions.

## Contribution

It develops the first Riesz space-inspired probabilistic logic with a duality theory, axiomatization, and bisimulation characterization, connecting algebraic and coalgebraic frameworks.

## Key findings

- Established a sound and complete axiomatization.
- Proved the logic characterizes probabilistic bisimulation.
- Developed a duality theory linking algebra and coalgebra.

## Abstract

We introduce a novel real-valued endogenous logic for expressing properties of probabilistic transition systems called Riesz modal logic. The design of the syntax and semantics of this logic is directly inspired by the theory of Riesz spaces, a mature field of mathematics at the intersection of universal algebra and functional analysis. By using powerful results from this theory, we develop the duality theory of the Riesz modal logic in the form of an algebra-to-coalgebra correspondence. This has a number of consequences including: a sound and complete axiomatization, the proof that the logic characterizes probabilistic bisimulation and other convenient results such as completion theorems. This work is intended to be the basis for subsequent research on extensions of Riesz modal logic with fixed-point operators.