**6. Positive and Negative Properties for LDL-SL**

In this section, similar with those results about BSL in [30], we state negative/positive results about LDL-SL.

Firstly, as in [30], for LDL-SL, we introduce four basic definitions, including bisimilarity between two CGSs, local isomorphism between two CGSs, state-unwinding, and decision-unwinding.

**Definition 12** ([30])**.** *CGSs* <sup>G</sup><sup>1</sup> <sup>=</sup> *Act*1, *<sup>W</sup>*1, *<sup>λ</sup>*1, *<sup>τ</sup>*1, *<sup>w</sup>*<sup>0</sup> <sup>1</sup> *and* <sup>G</sup><sup>2</sup> <sup>=</sup> *Act*2, *<sup>W</sup>*1, *<sup>λ</sup>*2, *<sup>τ</sup>*2, *<sup>w</sup>*<sup>0</sup> <sup>2</sup> *are called bisimilar, denoted as* G<sup>1</sup> ∼ G2*, if and only if (1) there exists one relation* ∼ ⊆ *W*<sup>1</sup> × *W*2*, named as bisimulation relation, and (2) there exists a function <sup>f</sup>* : ∼ → <sup>2</sup>*Act*1×*Act*<sup>2</sup> *, named as bisimulation function, satisfying that:*

	- *(a) λ*1(*w*1) = *λ*2(*w*2)*;*
	- *(b) for each ac*<sup>1</sup> ∈ *Act*1*, there exists ac*<sup>2</sup> ∈ *Act*<sup>2</sup> *satisfying* (*ac*1, *ac*2) ∈ *f*(*w*1, *w*2)*;*
	- *(c) for each ac*<sup>2</sup> ∈ *Act*2*, there exists ac*<sup>1</sup> ∈ *Act*<sup>1</sup> *satisfying* (*ac*1, *ac*2) ∈ *f*(*w*1, *w*2)*;*

*(d) for each decision pair* (*d*1, *<sup>d</sup>*2) <sup>∈</sup> <sup>ˆ</sup> *f*(*w*1, *w*2)*, it holds that τ*1(*w*1, *d*1) ∼ *τ*(*w*2, *d*2)*. Here,* ˆ *<sup>f</sup>* : ∼ → <sup>2</sup>*Dc*1×*Dc*<sup>2</sup> *is the lifting of function f from actions to decisions, satisfying*

(*d*1, *<sup>d</sup>*2) <sup>∈</sup> <sup>ˆ</sup> *f*(*w*1, *w*2) *iff it holds that* (*d*1(*a*), *d*2(*a*)) ∈ *f*(*w*1, *w*2), ∀*a* ∈ *Ag*. (23)

Obviously, according to the definition of bisimulation relation, the bisimulation of two CGSs can imply the existence of a bismulation between two decisions in them.

**Proposition 1.** *Suppose that two concurrent game structures* <sup>G</sup><sup>1</sup> <sup>=</sup> *Act*1, *<sup>W</sup>*1, *<sup>λ</sup>*1, *<sup>τ</sup>*1, *<sup>w</sup>*<sup>0</sup> <sup>1</sup> *and* <sup>G</sup><sup>2</sup> <sup>=</sup> *Act*2, *<sup>W</sup>*1, *<sup>λ</sup>*2, *<sup>τ</sup>*2, *<sup>w</sup>*<sup>0</sup> <sup>2</sup> *are bisimilar with a bisimulation relation* ∼ *and a bisimulation relation f , for each state pair* (*w*1, *w*2) ∈ *W*<sup>1</sup> × *W*<sup>2</sup> *with w*<sup>1</sup> ∼ *w*2*, it holds that:*


Next, we define the notion of local isomorphism relation between two CGSs.

**Definition 13** ([30])**.** *Two CGSs* <sup>G</sup><sup>1</sup> <sup>=</sup> *Act*1, *<sup>W</sup>*1, *<sup>λ</sup>*1, *<sup>τ</sup>*1, *<sup>w</sup>*<sup>0</sup> <sup>1</sup> *and* <sup>G</sup><sup>2</sup> <sup>=</sup> *Act*2, *<sup>W</sup>*1, *<sup>λ</sup>*2, *<sup>τ</sup>*2, *<sup>w</sup>*<sup>0</sup> 2 *are locally isomorphic, denoted as* G<sup>1</sup> ∼= G2*, if and only if there exists a bisimulation relation* ∼ ⊆ *W*<sup>1</sup> × *W*<sup>2</sup> *between these two CGSs, satisfying that, for each state pair* (*w*1, *w*2) ∈ *W*<sup>1</sup> × *W*<sup>2</sup> *with w*<sup>1</sup> ∼ *w*<sup>2</sup>

$$\sim \cap \left( \{ \mathfrak{r}\_1(w\_1, d) : d \in Dc\_1 \} \times \{ \mathfrak{r}\_2(w\_2, d) : d \in Dc\_2 \} \right) \tag{24}$$

*is bijective between the successors of w*<sup>1</sup> *and those of w*2*.*

Now we extend the definition of locally isomorphic to tracks, paths, strategies, and assignments as follows.

**Definition 14.** *Let* ∼ *(resp. f) be a bisimulation relation (resp. function) between two CGSs* <sup>G</sup><sup>1</sup> <sup>=</sup> *Act*1, *<sup>W</sup>*1, *<sup>λ</sup>*1, *<sup>τ</sup>*1, *<sup>w</sup>*<sup>0</sup> <sup>1</sup> *and* <sup>G</sup><sup>2</sup> <sup>=</sup> *Act*2, *<sup>W</sup>*1, *<sup>λ</sup>*2, *<sup>τ</sup>*2, *<sup>w</sup>*<sup>0</sup> 2*.*


In Definition 14, obviously, if *χ*<sup>1</sup> ∼ *χ*<sup>2</sup> and *g*<sup>1</sup> ∼ *g*2, then *χ*1[*x* → *g*1] ∼ *χ*2[*x* → *g*2]. Further, if ∀*i* ∈ {1, 2}, *χ<sup>i</sup>* is a complete *wi*-total assignment, and *w*<sup>1</sup> ∼ *w*2, then it holds that *<sup>π</sup>*<sup>1</sup> <sup>∼</sup> *<sup>π</sup>*<sup>2</sup> and (*χ*1)(*<sup>π</sup>*1)≤*<sup>k</sup>* <sup>∼</sup> (*χ*2)(*<sup>π</sup>*2)≤*<sup>k</sup>* , ∀*k* ∈ N, where *π<sup>i</sup>* is the (*χi*, *wi*)-play.

To show whether LDL-SL has tree model properties, consider two unwinding forms of concurrent game structures; one is about state-unwinding, and another is about decision-unwinding.

**Definition 15** ([30])**.** *Given a CGS* <sup>G</sup> <sup>=</sup> *Act*, *<sup>W</sup>*, *<sup>λ</sup>*, *<sup>τ</sup>*, *<sup>w</sup>*0*, the state-unwinding of* <sup>G</sup> *is the new CGS* G*su* = *Ac*, *Wsu*, *λsu*, *τsu*,  *, where*


From Definition 15, the state-unwinding <sup>G</sup>*su* of a CGS <sup>G</sup> <sup>=</sup> *Act*, *<sup>W</sup>*, *<sup>λ</sup>*, *<sup>τ</sup>*, *<sup>w</sup>*0 is a tree, whose direction set is just the set *W* of states in G.

**Definition 16** ([30])**.** *Given a CGS* <sup>G</sup> <sup>=</sup> *Act*, *<sup>W</sup>*, *<sup>λ</sup>*, *<sup>τ</sup>*, *<sup>w</sup>*0*, the decision-unwinding of* <sup>G</sup> *is the new CGS* G*du* = *Act*, *Wdu*, *λdu*, *τdu*,  *, where*


From Definition 16, the decision-unwinding <sup>G</sup>*du* of a CGS <sup>G</sup> <sup>=</sup> *Act*, *<sup>W</sup>*, *<sup>λ</sup>*, *<sup>τ</sup>*, *<sup>w</sup>*0 is a tree, whose direction set is just the set *Dc* (i.e., *ActAg*) in <sup>G</sup>.

**Theorem 6** ([30])**.** *Given a CGS* G*, the following properties hold:*


We note that any CGS G just has a unique associated state-unwinding G*su* and a unique associated decision-unwinding G*du*.

For BSL logic, the following negative properties hold.

**Theorem 7** ([30])**.** *Four negative properties for BSL:*


These negative results can be extended into LDL-SL.

**Theorem 8.** *Four negative properties for LDL-SL:*


**Proof.** By Theorems 2 and 7, these results are the same as those for BSL.

Similar with those positive properties for BSL [30], the following properties also hold for LDL-SL.
