**8. Conclusions and Future Work**

In this paper, we propose logic LDL-SL, an expressive new strategic logic based on linear dynamic logic, which can naturally express *ω*-regular properties. This logic is a branching-time extension of SL based on linear-time temporal logic. We show that LDL-SL is more expressive than SL, whose model-checking complexity is non-elementarycomplete. Moreover, based on LDL, we define similar fragments of LDL-SL, which are more expressive than corresponding strategic logics based on LTL. However, all these fragments have the same model checking complexity, i.e., are 2EXPTIME-complete. At the same time, we define star-free-like strategic logics, based on star-free regular expressions. We show that these logics have the same expressivity as those corresponding strategic logics based on LTL or CTL∗.

In short, based on LDL, we propose a new class of strategic logics. These logics have the same model-checking complexities as, but more expressivity than, current mainstream strategic logics. Furthermore, these logics can extend the application areas in multi-agent systems.

However, until now, the upper bounds of LDL-SL and its star-free fragments (ADL∗ *s f* , LDL-SL[1G]*s f* , and LDL-SL*s f*) are not known. In future, we will study the compact bounds of these logics. As in [43,44], we will consider concrete implementations about these model checking problems. In addition, here we just consider perfect recall strategies in multi-agent concurrent games with complete information. Next, we will further study these new proposed strategic logics under incomplete information [45–47], where the strategies of agents maybe memoryless or perfect recall [48]. In this paper, we present formal frameworks and show technical results; in the future, we will also present case studies or practical applications to illustrate these theories, such as information security [49], solving winning strategies [50], and voting protocol [51].

**Author Contributions:** Conceptualization, L.X. and S.G.; methodology, L.X. and S.G; validation, L.X. and S.G.; formal analysis, L.X.; investigation, L.X.; resources, L.X.; writing—original draft preparation, L.X.; writing—review and editing, L.X. and S.G.; project administration, S.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** Our deepest gratitude goes to the anonymous reviewers for their careful work and thoughtful suggestions that have helped improve this paper substantially.

**Conflicts of Interest:** The authors declare no conflict of interest.
