# On the trade-off between labels and weights in quantitative bisimulation

## TL;DR

This paper explores the expressiveness of various quantitative transition system models, revealing a trade-off between label information and weights, and establishing their fundamental differences and relationships.

## Contribution

It introduces a formal analysis of the trade-off between labels and weights in quantitative models, showing their fundamental distinctions and the limits of reduction.

## Key findings

- Weighted transition systems use only weights to encode information.
- Labelled transition systems rely solely on labels for information.
- The models form a spectrum with no further significant reductions possible.

## Abstract

Reductions for transition systems have been recently introduced as a uniform and principled method for comparing the expressiveness of system models with respect to a range of properties, especially bisimulations. In this paper we study the expressiveness (w.r.t. bisimulations) of models for quantitative computations such as weighted labelled transition systems (WLTSs), uniform labelled transition systems (ULTraSs), and state-to-function transition systems (FuTSs). We prove that there is a trade-off between labels and weights: at one extreme lays the class of (unlabelled) weighted transition systems where information is presented using weights only; at the other lays the class of labelled transition systems (LTSs) where information is shifted on labels. These categories of systems cannot be further reduced in any significant way and subsume all the aforementioned models.