**Conjecture 1.** *The MCP for LDL-SL is non-elementary-complete.*

In addition, since ATL∗, BSL, and BSL[1G] can be linearly embedded into their corresponding star-free strategic logics ADL∗ *s f* , LDL-SL*s f* , and LDL-SL[1G]*s f* , respectively, MCPs for ATL∗ [5] and BSL[1G] [24] are 2EXPTIME-complete, and MCP for BSL is nonelementary-complete [24], then the following holds.

**Corollary 3.** *The MCPs for both ADL*∗ *s f and LDL-SL[1G]s f are 2EXPTIME-hard. The MCP for LDL-SLs f is non-elementary-hard.*

Although similar expressive power by Theorem 3, we do not know how to linearly translate star-free logics to the corresponding logics. For the time being, the upper bounds of these star-free strategic logics are not known.

The main complexity results about the MCPs are given in Table 1.


**Table 1.** Complexity of Model Checking for Strategic Logics.
