*Article* **Representation and Reasoning about Strategic Abilities with** *ω***-Regular Properties**

**Liping Xiong 1,\* and Sumei Guo 2,\***


**Abstract:** Specification and verification of coalitional strategic abilities have been an active research area in multi-agent systems, artificial intelligence, and game theory. Recently, many strategic logics, e.g., Strategy Logic (SL) and alternating-time temporal logic (ATL∗), have been proposed based on classical temporal logics, e.g., linear-time temporal logic (LTL) and computational tree logic (CTL∗), respectively. However, these logics cannot express general *ω*-regular properties, the need for which are considered compelling from practical applications, especially in industry. To remedy this problem, in this paper, based on linear dynamic logic (LDL), proposed by Moshe Y. Vardi, we propose LDLbased Strategy Logic (LDL-SL). Interpreted on concurrent game structures, LDL-SL extends SL, which contains existential/universal quantification operators about regular expressions. Here we adopt a branching-time version. This logic can express general *ω*-regular properties and describe more programmed constraints about individual/group strategies. Then we study three types of fragments (i.e., one-goal, ATL-like, star-free) of LDL-SL. Furthermore, we show that prevalent strategic logics based on LTL/CTL∗, such as SL/ATL∗, are exactly equivalent with those corresponding star-free strategic logics, where only star-free regular expressions are considered. Moreover, results show that reasoning complexity about the model-checking problems for these new logics, including one-goal and ATL-like fragments, is not harder than those of corresponding SL or ATL∗.

**Keywords:** strategic abilities; *ω*-regular properties; linear dynamic logic; strategic logics; model checking; concurrent game structure
