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 LTL[ℱ] , which extends LTL with quality operators.

In this work, we introduce and study SL[ℱ] , which enables the specification of quantitative strategic behaviors. The satisfaction value of an SL[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 SL[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 SL[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.

时序逻辑广泛用于规范计算机系统的持续行为。该领域的两个重要发展是传统时态逻辑的扩展，其模态能够在多代理系统中指定持续的战略行为，以及时态逻辑向定量设置的过渡，其中不同的满意度值使指定者能够将确定性或质量等概念形式化。在第一类中，SL（Strategy Logic）是描述战略行为最自然、最具表现力的逻辑之一。在第二类中，一个值得注意的逻辑是 LTL[ℱ]，它使用优质运算符扩展了 LTL 。

在这项工作中，我们介绍并研究了SL[ℱ] ，它可以规范量化战略行为。SL[ℱ] 公式的满意度值是 [0,1] 中的实数值，反映了潜在代理人的战略持续目标得到满足的“多少”或“有多好”。我们展示了 SL[ℱ] 在多智能体系统定量推理中的应用，展示了它如何表达和测量多智能体系统中的稳定性等概念，以及它如何概括一些模糊时序逻辑。我们还基于量化 CTL 的定量扩展，为 SL[ℱ] 提供模型检查算法^⋆ . 我们的算法为 Strategy Logic 的定量扩展提供了第一个可判定性结果。此外，它还可用于综合策略，最大限度地提高系统行为的质量。