**5. Expressivity Relations among Fragments of LDL-SL and LDL-SL***s f*

In this section, we study the expressivity relations among mentioned fragments of LDL-SL and LDL-SL*s f* . Firstly, we give the following definitions about the expressive power between two logics.

Logic *L*<sup>1</sup> is *at least as expressive as* logic *L*2, denoted as *L*<sup>2</sup> ⊆ *L*1, if given a model *M*, for any formula *ϕ* in *L*2, there exists a formula *ϕ* in *L*1, satisfying that *M* |= *ϕ* iff *M* |= *ϕ* . *L*<sup>1</sup> is *strictly more expressive than L*2, denoted as *L*<sup>2</sup> - *L*1, if *L*<sup>2</sup> ⊆ *L*1, but *L*<sup>1</sup> ⊆ *L*<sup>2</sup> does not hold. *L*<sup>1</sup> has *the same expressive power as L*2, denoted as *L*<sup>1</sup> ≡ *L*2, if *L*<sup>1</sup> ⊆ *L*<sup>2</sup> and *L*<sup>2</sup> ⊆ *L*1. *L*<sup>1</sup> and *L*<sup>2</sup> are incomparable if neither *L*<sup>2</sup> ⊆ *L*<sup>1</sup> nor *L*<sup>1</sup> ⊆ *L*2.

According to Theorem 1, star-free type strategic logics have the same expressive power as their corresponding strategic logics based on LTL or CTL∗.

**Theorem 3.** *Star-free strategic logics have the same expressive power as their corresponding strategic logics whose underlying logic is LTL or ATL*∗*.*


**Proof.** By applying Lemma 2 that LDL*s f* is equivalent with LTL, these results can be shown by induction of the structures of formulas similarly. Here, we just sketch the ideas of proofs as follows.

In order to show that ADL∗ *s f* ⊆ ATL∗, by induction hypothesis, we just consider the case *ϕ* = *Aψ*, which is an ADL<sup>∗</sup> *s f* formula. Suppose for each maximal state subformulas *ϕ* in *ϕ*, by induction, there is an ATL∗ formula equivalent with *ϕ* . If we use a new atom *p<sup>ϕ</sup>* to replace it, then make *ψ* be equivalent with a pure LDL formula. By Lemma 2, replace *ψ* with one LTL formula; and further replace those new atoms *p<sup>ϕ</sup>* with original ATL<sup>∗</sup> state formulas. Hence the resulting formula is an ATL∗ state formula, equivalent with *ϕ*.

Similarly, we can show that LDL-SL*s f* ⊆ BSL and LDL-SL[1G]*s f* ⊆ BSL[1G].

For item 1: In order to show ATL<sup>∗</sup> ⊆ ADL<sup>∗</sup> *s f* , define two functions *<sup>T</sup><sup>s</sup>* and *<sup>T</sup><sup>p</sup>* similarly with those in Lemma 5 except the following two cases in *Tp*.

$$T^p(\diamond \psi) = \langle \overline{false} \rangle T^p(\psi), \\ T^p(\psi\_1 \mathcal{U} \psi\_2) = T^p(\psi\_2) \lor \langle \overline{false}; -T^p(\psi\_1) \lor \overline{false}; \text{true} \rangle T^p(\psi\_2) \tag{21}$$

Here, the proof for case *<sup>ψ</sup>* <sup>=</sup> *ψ*<sup>1</sup> or *<sup>ψ</sup>* <sup>=</sup> *<sup>ψ</sup>*1U*ψ*<sup>2</sup> about <sup>G</sup>, *<sup>π</sup>*, *<sup>i</sup>* <sup>|</sup><sup>=</sup> *<sup>T</sup>p*(*ψ*) iff <sup>G</sup>, *<sup>π</sup>*, *<sup>i</sup>* <sup>|</sup><sup>=</sup> *<sup>ψ</sup>* is the same as that of Lemma 2.

Similarly to Item 1, and by Theorem 2, BSL ⊆ LDL-SL*s f* and BSL[1G] ⊆ LDL-SL[1G]*s f* can be shown.

**Theorem 4.** *The following fragments are incomparable.*


**Proof.** Here, we just sketch the ideas of proofs.

For item (1), we consider the following formulas:


where *ϕ*<sup>1</sup> is a BSL formula, but it cannot be expressed in LDL-SL[1G]; = conversely, *ϕ*<sup>2</sup> is a LDL-SL[1G] formula, but it cannot be expressed in BSL.

In order to show that *ϕ*<sup>2</sup> cannot be expressed in BSL, we consider all the CGSs with just one agent and an action. So in these CGSs, each BSL sentence is equivalent with one CTL<sup>∗</sup> state formula. Suppose *ϕ* is a CTL<sup>∗</sup> state formula with *m* temporal operators; then, consider the following two CGSs with just one agent and an action—see Figure 1. In G1, *p* holds in all states, and in G2, *p* does not hold only in state *w*2*m*<sup>+</sup>1. Due to unique path starting from the initial state, we can see that *ϕ* is equivalent with an LTL formula *ψ* under each G*i*, *i* ∈ {1, 2}. Then by the following theorem given by Wolper,

*Theorem 4.1 ([14]) Given an atomic proposition q, any LTL formula f*(*q*) *containing m temporal operators has the same truth value on all sequences such as <sup>q</sup>k*(¬*q*)*qω, here k* > *m and f*(*q*) *is a LTL formula containing only atomic q.*

It holds that G<sup>1</sup> |= *ψ* iff G<sup>2</sup> |= *ψ*. However, G<sup>1</sup> |= *ϕ*2, but G<sup>2</sup> |= *ϕ*2. Therefore, *ϕ*<sup>2</sup> cannot be expressed in BSL.

**Figure 1.** Two CGSs for *ϕ*2: the top is G<sup>1</sup> and the bottom is G2.

For item (2), we consider the following two formulas:


Here, *ϕ*<sup>3</sup> is a BSL[1G] formula, but it cannot be expressed in ADL∗; conversely, *ϕ*<sup>4</sup> is a ADL∗ formula, but cannot be expressed in BSL[1G].

Like in [24], consider two concurrent game structures CGSs with *AP* = {*p*} and *Ag* = {*a*, *b*, *c*}, G<sup>1</sup> = *Ac*1, *W*1, *λ*1, *τ*1, *w*0, and G<sup>2</sup> = *Ac*2, *W*2, *λ*2, *τ*2, *w*0, where *Ac*<sup>1</sup> = {0, 1}, *Ac*<sup>2</sup> = {0, 1, 2}, *W*<sup>1</sup> = *W*<sup>2</sup> = {*w*0, *w*1, *w*2}, *λ*<sup>1</sup> = *λ*2, and *D*<sup>1</sup> = {00∗, 11∗}, *D*<sup>2</sup> = {211, 202, 200, 00∗, 11∗, 12∗}. *λ*1(*w*0) = *λ*1(*w*2) = ∅, *λ*1(*w*1) = {*p*}. ∀*d* ∈ *Di*, *τi*(*w*0, *d*) = *w*1; ∀*d* ∈ *Dci* \ *Di*, *τ*(*w*0, *d*) = *w*2; ∀*d* ∈ *Dci*, *w* ∈ {*w*1, *w*2}, *τi*(*w*, *d*) = *w*, here *<sup>i</sup>* ∈ {1, 2} and *Dci* <sup>=</sup> *AcAg <sup>i</sup>* . We can show that G<sup>1</sup> |= *ϕ*3, but G<sup>2</sup> |= *ϕ*3. Inspired by the approach in [24], it can be shown that any ADL<sup>∗</sup> formula cannot distinguish between G<sup>1</sup> and G2.

In order to show that *ϕ*<sup>1</sup> cannot be expressed in LDL-SL[1G], we can adopt the same two CGSs like for *ϕ*<sup>3</sup> here. The proof that *ϕ*<sup>4</sup> cannot be expressed in BSL[1G] is similar with that for *ϕ*2.

**Theorem 5.** *Inclusion relations among existing strategic logics:*


**Proof.** By Lemma 1, the star-free logic ADL∗ *s f* (resp. LDL-SL[1G]*s f*) is less expressive than ADL∗. (resp. LDL-SL[1G]). One-goal fragment LDL-SL[1G] is obviously less expressive than LDL-SL, due to the restriction about the alternations about strategy variables and agent bindings. Furthermore, the ATL-like fragment ADL∗ of LDL-SL is less than one-goal fragment LDL-SL[1G] of LDL-SL, since the coalition operators *A* can be specified by the ℘ prefix.

According to Theorems 3–5, as well as CL - ATL - ATL∗ - BSL[1G] - SL, we can obtain an expressivity graph; see Figure 2.

**Figure 2.** Expressivity Graph.

Here, coalition logic (CL) [29] is a logic, which just has coalition operators without temporal operators.

$$
\varphi ::= p \mid \neg \varphi \mid \; \varphi \land \; \varphi \mid \; \langle \! \langle A \rangle \rangle \, \varphi \,. \tag{22}
$$
