# Reasoning about Quality and Fuzziness of Strategic Behaviours

## TL;DR

This paper introduces FSL, a quantitative extension of Strategy Logic that assigns real-valued satisfaction levels to strategic behaviors in multi-agent systems, enabling nuanced reasoning about quality and fuzziness.

## Contribution

It presents FSL, a novel logic combining strategic reasoning with quantitative satisfaction, and provides a model-checking algorithm for this expressive framework.

## Key findings

- FSL can express stability concepts in multi-agent systems.
- FSL generalizes fuzzy temporal logics.
- A model-checking algorithm for FSL is developed.

## Abstract

Temporal logics are extensively used for the specification of on-going behaviours of reactive systems. Two significant developments in this area are the extension of traditional temporal logics with modalities that enable the specification of on-going strategic behaviours in multi-agent systems, and the transition of temporal logics to a quantitative setting, where different satisfaction values enable the specifier to formalise concepts such as certainty or quality. We introduce and study FSL---a quantitative extension of SL (Strategy Logic), one of the most natural and expressive logics describing strategic behaviours. The satisfaction value of an FSL formula is a real value in [0,1], reflecting `how much' or `how well' the strategic on-going objectives of the underlying agents are satisfied. We demonstrate the applications of FSL in quantitative reasoning about multi-agent systems, by showing how it can express concepts of stability in multi-agent systems, and how it generalises some fuzzy temporal logics. We also provide a model-checking algorithm for our logic, based on a quantitative extension of Quantified CTL*.