*7.2. Model-Checking for LDL-SL[1G]*

To give a model-checking algorithm for LDL-SL[1G], we adopt a similar approach proposed in [4], which is used to show that *SL*1*G*[F] model checking is 2EXPTIME-complete.

First, we introduce the concept of *concurrent multi-player parity game* (CMPG) P = (*Ag*, *Ac*, *S*,*s*0, p, Δ) [38], here *Ag* = {1, ··· , *n*}, Δ is a transition function, and p : *S* → *N* is a priority function. In a CMPG P, there are *n* agents playing concurrently with infinite rounds. Informally, in a CMPG, if there exists one strategy for agent 0, s.t., for any strategy for agent 1, there exists one strategy for agent 2, and so forth, which make all the induced plays satisfy the parity condition, and then the existential coalition wins; otherwise, the universal coalition wins.

In a CMPG, P = (*Ag*, *Ac*, *S*,*s*0, p, Δ), the winners of which can be determined in polynomial-time with respect to |*S*| and |*Ac*| and exponential-time with respect to |*Ag*| and max p [38].

#### **Theorem 11.** *The MCP for LDL-SL[1G] is 2EXPTIME-complete.*

**Proof.** Firstly, Hardness follows from the fact that the MCP for BSL[1G] is 2EXPTIMEcomplete [24]. Then, we consider the lower bound of LDL-SL[1G].

Consider a CGS G = *Ac*, *W*, *λ*, *τ*, *w*0 and a LDL-SL[1G] sentence *ϕ*. As in ADL∗, we present a labelling algorithm to solve LDL-SL[1G] model checking. Like in [4], here we just consider the case sentence ℘*ψ*, where quantifiers perfectly alternate between existential and universal *x*1[[*x*2]] ··· [[*xn*]], and *ψ* is an LDL formula. Now *ψ* can be interpreted over paths of the pointed Kripke model *M* = (*W*, *R*, *λ*, *w*0), where *R* = {(*w*1, *w*2)|∃*d* ∈ *AcAg*, *<sup>w</sup>*<sup>2</sup> <sup>=</sup> *<sup>τ</sup>*(*w*1, *<sup>d</sup>*)}.

In [21], for LDL formula *ψ*, construct an ABA B*<sup>ψ</sup>* with linearly many states in *ψ*, and then turn B*<sup>ψ</sup>* into an NBA A*<sup>ψ</sup>* of exponential size of |*ψ*| [32]. Combining A*<sup>ψ</sup>* with Kripke model *M*, we get a new NBA A*M*,*ψ*, which accepts exactly all the infinite paths *π* of *M* s.t. *π*, 0 |= *ψ*. Then, by [39], we convert A*M*,*<sup>ψ</sup>* into a deterministic parity automaton (DPA) <sup>A</sup>*M*,*<sup>ψ</sup>* = (*W*, *<sup>Q</sup>*, *<sup>q</sup>*0, *<sup>δ</sup>*, p) of size in 22*<sup>O</sup>*(|*ψ*|) and index bounded by 2*O*(|*ψ*|).

Now as in [4], combining CGS G with A*M*,*ψ*, we use the same approach to define the following CMPG P = (*Ag* , *Ac*, *S*,*s*0, p, Δ ), where *Ag* is a set of agents, one for every variable occurring in ℘; *S* = *W* × *Q*. Firstly, game P emulates a path *π* generated in G; secondly, if the generated path *π* in G is read, then the game emulates the execution of <sup>A</sup>*M*,*ψ*. Hence, each execution (*π*, *<sup>l</sup>*) <sup>∈</sup> *<sup>W</sup><sup>ω</sup>* <sup>×</sup> *<sup>Q</sup><sup>ω</sup>* in game <sup>P</sup> satisfies the parity condition determined by the p in G iff *π*, 0 |= *ψ*. In addition, because A*M*,*<sup>ψ</sup>* is deterministic, for each possible track *h* ∈ *Trk*(G), there is one unique partial path *lh* that makes the partial execution (*h*, *lh*) possible in P. This makes the strategies from *w*<sup>0</sup> in *Str*(G) one-to-one with the strategies from *s*<sup>0</sup> in *Str*(P). Then P has a winning strategy if and only if G, *w*<sup>0</sup> |= ℘*ψ*.

As for complexity, the size of P is *O*(|*W*|·|*Q*|), where *W* is the state space of G and <sup>|</sup>*Q*<sup>|</sup> <sup>=</sup> <sup>2</sup>2*O*(|*ψ*|) , i.e., doubly exponential in the size of *ψ*. Since A*M*,*<sup>ψ</sup>* results from one NGBW <sup>B</sup>*M*,*ψ*, whose size is 2*O*(|*ψ*|), transformed into a DPW, which needs another exponential in *ψ*. Moreover, since the transformation from an NGBW to a DPW just needs <sup>2</sup>*O*(|*ψ*|) priorities, so the number of priorities in <sup>P</sup> is 2*O*(|*ψ*|). Hence, the constructed CMPG P can be solved in time polynomial with respect to the size of the CGS model G and double exponential in formula |*ψ*|.

In fact, according to Theorem 11, since ADL∗ and BSL[1G] can both be linearly embedded into LDL-SL[1G], then the MCPs for both logics are in 2EXPTIME.
