Temporal logics
are extensively used for the specification of on-going behaviors of computer systems. Two significant developments in this area are the extension of traditional temporal logics with modalities that enable the specification of on-going
strategic
behaviors in multi-agent systems, and the transition of temporal logics to a
quantitative
setting, where different satisfaction values enable the specifier to formalize concepts such as certainty or quality. In the first class, SL (
Strategy Logic
) is one of the most natural and expressive logics describing strategic behaviors. In the second class, a notable logic is
\(\mathrm{LTL}[{\mathcal {F}}] \)
, which extends LTL with
quality operators
.
In this work we introduce and study
\(\mathrm{SL}[{\mathcal {F}}] \)
, which enables the specification of quantitative strategic behaviors. The satisfaction value of an
\(\mathrm{SL}[{\mathcal {F}}] \)
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
\(\mathrm{SL}[{\mathcal {F}}] \)
in quantitative reasoning about multi-agent systems, showing how it can express and measure concepts like stability in multi-agent systems, and how it generalizes some fuzzy temporal logics. We also provide a model-checking algorithm for
\(\mathrm{SL}[{\mathcal {F}}] \)
, based on a quantitative extension of Quantified CTL
⋆
. Our algorithm provides the first decidability result for a quantitative extension of Strategy Logic. In addition, it can be used for synthesizing strategies that maximize the quality of the systems’ behavior.