**Proof.** For item 1:

For any LDL-SL formula *ϕ*, given any two CGSs G<sup>1</sup> and G<sup>2</sup> with G<sup>1</sup> ∼= G2, two states *w*<sup>1</sup> ∈ *W*<sup>1</sup> and *w*<sup>2</sup> ∈ *W*<sup>2</sup> with *w*<sup>1</sup> ∼ *w*2, two assignments *χ*<sup>1</sup> ∈ *Asg*(G1, *w*1), and *χ*<sup>2</sup> ∈ *Asg*(G, *w*2) with *χ*<sup>1</sup> ∼ *χ*2, here *f ree*(*ϕ*) ⊆ *dom*(*χ*1) = *dom*(*χ*2), we inductively show that

$$\mathcal{G}\_{1\prime}\chi\_1, \mathfrak{w}\_1 \vdash \varphi \quad \text{if and only if} \quad \mathcal{G}\_{2\prime}\chi\_2, \mathfrak{w}\_2 \vdash \varphi. \tag{25}$$

From the bisimulation definition and the inductive hypothesis, the cases of atoms and Boolean connectives are easy. As for the cases of existential quantification *x* and agent binding (*a*, *x*), the proofs are the same as those in [30]. Here we just show the case of **E***ψ*, here *ψ* is a path formula. G1, *χ*1, *w*<sup>1</sup> |= **E***ψ* iff there exists a *π* ∈ *out*(G1, *χ*1, *w*1) such that G1, *χ*1, *π*1, 0 |= *ψ*.

That means we should mutually show with state formulas by induction, i.e.,

$$\mathcal{G}\_{1}, \chi\_{1}, \pi\_{1}, i \vDash \psi \quad \text{if and only if} \quad \mathcal{G}\_{2}, \chi\_{2}, \pi\_{2}, i \vDash \psi \quad \text{(26)}$$

For the case *ψ* = *ϕ* : G1, *χ*1, *π*1, *i* |= *ϕ* iff G1, *χ*1,(*π*1)*<sup>i</sup>* |= *ϕ* iff G2, *χ*2,(*π*2)*<sup>i</sup>* |= *ϕ*<sup>2</sup> by induction iff G2, *χ*2, *π*2, *i* |= *ϕ* .

For the cases of Boolean connectives, these are easy from the definitions and the inductive hypothesis.

For the case *ψ* = *ρψ* : we need to show the following by induction,

$$
\mathcal{R}\_{\epsilon}(i,j) \in \mathcal{R}(\mathcal{G}\_1, \rho, \pi\_1, \chi\_1) \quad \text{if and only if} \quad (i,j) \in \mathcal{R}(\mathcal{G}\_2, \rho, \pi\_2, \chi\_2). \tag{27}
$$

For case *ρ* = Φ: (*i*, *j*) ∈ R(G1, Φ, *π*1, *χ*1) iff *j* = *i* + 1 and G1, *χ*1, *π*1, *i* |= Φ by definition iff *j* = *i* + 1 and G2, *χ*2, *π*2, *i* |= Φ.

For case *ρ* = *ψ*?: (*i*, *j*) ∈ R(G1, *ψ*?, *π*1, *χ*1) iff *j* = *i* and G1, *χ*1, *π*1, *i* |= *ψ* by definition iff *j* = *i* and G2, *χ*2, *π*2, *i* |= *ψ* by induction.

For case *ρ* = *ρ*<sup>1</sup> + *ρ*2: (*i*, *j*) ∈ R(G1, *ρ*<sup>1</sup> + *ρ*2, *π*1, *χ*1) iff (*i*, *j*) ∈ R(G1, *ρ*1, *π*1, *χ*1) or (*i*, *j*) ∈ R(G1, *ρ*2, *π*1, *χ*1) iff (*i*, *j*) ∈ R(G2, *ρ*1, *π*2, *χ*2) or (*i*, *j*) ∈ R(G2, *ρ*2, *π*2, *χ*2) by induction iff (*i*, *j*) ∈ R(G2, *ρ*<sup>1</sup> + *ρ*2, *π*2, *χ*2).

For case *ρ* = *ρ*1; *ρ*2: (*i*, *j*) ∈ R(G1, *ρ*1; *ρ*2, *π*1, *χ*1) iff there exists *k*, *i* ≤ *k* ≤ *j*, satisfying that (*i*, *k*) ∈ R(G1, *ρ*1, *π*1, *χ*1) and (*k*, *j*) ∈ R(G1, *ρ*2, *π*1, *χ*1) iff there exists k, *i* ≤ *k* ≤ *j*, satisfying that (*i*, *k*) ∈ R(G2, *ρ*1, *π*2, *χ*2) and (*k*, *j*) ∈ R(G2, *ρ*2, *π*2, *χ*2) by induction iff (*i*, *j*) ∈ R(G2, *ρ*1; *ρ*2, *π*2, *χ*2).

For case *ρ* = *ρ*∗ : (*i*, *j*) ∈ R(G1, *ρ*<sup>∗</sup> , *π*1, *χ*1) iff *j* = *i* or (*i*, *j*) ∈ R(G1, *ρ*1; *ρ*<sup>∗</sup> , *π*1, *χ*1) iff *j* = *i* or (*i*, *j*) ∈ R(G2, *ρ*2; *ρ*<sup>∗</sup> , *π*1, *χ*1) by induction iff (*i*, *j*) ∈ R(G2, *ρ*<sup>∗</sup> , *π*2, *χ*2).

Therefore, it implies that LDL-SL is indeed invariant under local isomorphism.

For item 2: by item 1 in Theorem 6, for any CGS G, it holds that G ∼= G*su*. So by item 1, each LDL-SL sentence *ϕ* is an invariant for CGS G and its state-unwinding G*su*.

For item 3: let the LDL-SL sentence *ϕ* be satisfiable. Therefore, there exists one CGS G |= *ϕ*, and by item 2, it holds that G*su* |= *ϕ*. Since G*su* is a tree model, this means that LDL-SL has the (unbounded) tree model property.
