*4.1. LDL/LDLs f-Based Strategic Logics*

**Definition 9** (LDL-SL Formula)**.** *LDL-SL formulas are defined inductively as follows.*

*State formula ϕ* ::= *p* | ¬*ϕ* | *ϕ* ∧ *ϕ* | (*a*, *x*)*ϕ* | *xϕ* | *Eψ*;

*Path formula ψ* ::= *ϕ* | ¬*ψ* | *ψ* ∧ *ψ* | *ρψ*; *Path expression ρ* ::= Φ | *ψ*? | *ρ* + *ρ* | *ρ*; *ρ* | *ρ*∗, *where a* ∈ *Ag, x* ∈ *Var, p* ∈ *AP, and* Φ ∈ L(*AP*)*.*

In fact, LDL-SL is a logic that combines BSL with LDL. LDL-SL formula is defined recursively by three components: state formula, path formula, and path expression. Now we present the complete definition about the semantics of LDL-SL formula.

Given a CGS G, a state formula *ϕ*, a strategy assignment *χ*, and a state *w*, the relation G, *χ*, *w* |= *ϕ* is defined as follows.


Given a CGS G, a path formula *ψ*, a strategy assignment *χ*, a path *π* and some *i* ∈ N, the relation G, *χ*, *π*, *i* |= *ψ* is defined as follows.


The relation (*i*, *j*) ∈ R(G, *ρ*, *π*, *χ*) is defined as follows:


In the above, we omit G in R(G, *ρ*, *π*, *χ*) when there is no confusion. Intuitively, (*i*, *j*) ∈ R(G, *ρ*, *π*, *χ*) means that the sequence *πi*...*π<sup>j</sup>* is a legal execution of *ρ* under assignment *χ* in CGS G.

For two special path expressions, *ψ*?; *true* and its nondeterministic iteration (*ψ*?; *true*)∗, the following properties hold, where *ψ* is an LDL-SL path formula.

**Lemma 4.** *Given a CGS* G*, a path formula ψ, a path π, a strategy assignment χ, and i*, *j* ∈ N*,*

$$\mathcal{C}(i,j) \in \mathcal{R}(\mathcal{G}, \psi ?; true, \pi, \chi) \quad \text{if and only if} \quad j = i+1 \text{ and } \mathcal{G}, \chi, \pi, i \mid = \psi. \tag{13}$$

**Proof.** (*i*, *j*) ∈ R(G, *ψ*?; *true*, *π*, *χ*) iff there exists *k* with *i* ≤ *k* ≤ *j* such that (*i*, *k*) ∈ R(G, *ψ*?, *π*, *χ*) and (*k*, *j*) ∈ R(G, *true*, *π*, *χ*) iff there exists *k* with *i* ≤ *k* ≤ *j* such that *k* = *i* and G, *χ*, *π*, *k* |= *ψ* and *j* = *k* + 1 iff *j* = *i* + 1 and G, *χ*, *π*, *i* |= *ψ*.

**Corollary 2.** *Given a CGS* G*, a path formula ψ, a path π, a strategy assignment χ, and i*, *j* ∈ N*,*

$$\mathcal{R}(i,j) \in \mathcal{R}(\mathcal{G}, (\psi \mathcal{I}; \text{true})^\*, \pi, \chi) \text{ if and only if } j = i \text{ or } (\forall k. i \le k < j, \mathcal{G}, \chi, \pi, k \mid = \psi). \tag{14}$$

**Proof.** (*i*, *j*) ∈ R(G,(*ψ*?; *true*)∗, *π*, *χ*) iff *j* = *i* or there exists *k* (*i* ≤ *k* ≤ *j*) s.t. (*i*, *k*) ∈ R(G,(*ψ*?; *true*), *π*, *χ*) and (*k*, *j*) ∈ R(G,(*ψ*?; *true*)∗, *π*, *χ*) iff *j* = *i* or (G, *χ*, *π*, *i* |= *ψ* and (*i* + 1, *j*) ∈ R(G,(*ψ*?; *true*)∗, *π*, *χ*)) by Lemma 4 iff *j* = *i* or (G, *χ*, *π*, *i* |= *ψ*, G, *χ*, *π*, *i* + 1 |= *ψ* and (*i* + 2, *j*) ∈ R(G,(*ψ*?; *true*)∗, *π*, *χ*)) iff *j* = *i* or (G, *χ*, *π*, *i* |= *ψ*, G, *χ*, *π*, *i* + 1 |= *ψ*,..., and (*j* − 1, *j*) ∈ R(G,(*ψ*?; *true*)∗, *π*, *χ*)) iff *j* = *i* or (G, *χ*, *π*, *i* |= *ψ*, G, *χ*, *π*, *i* + 1 |= *ψ*,..., and G, *χ*, *π*, *j* − 1 |= *ψ*) repeatedly iff *j* = *i* or (∀*k*.*i* ≤ *k* < *j*, G, *χ*, *π*, *k* |= *ψ*).

Secondly, LDL*s f*-based Strategy Logic (abbr. LDL-SL*s f* is introduced).

**Definition 10** (LDL-SL*s f* Formula)**.** *The LDL-SLs f formulas are defined as follows:*

*State formula ϕ* ::= *p* | ¬*ϕ* | *ϕ* ∧ *ϕ* | *xϕ* | (*a*, *x*)*ϕ* | *Eψ Path formula ψ* ::= *ϕ* | ¬*ψ* | *ψ* ∧ *ψ* | *ρψ Star-free path expression ρ* ::= Φ | *ψ*? | *ρ* + *ρ* | *ρ*; *ρ* | *ρ where a* ∈ *Ag, x* ∈ *Var, p* ∈ *AP, and* Φ ∈ L(*AP*)*.*

For the semantics of star-free fragment, given a CGS G, a star-free path expression *ρ*, and a strategy assignment *χ*, for any *i* ≤ *j*,

(*i*, *j*) ∈ R(G, *ρ*, *π*, *χ*) if and only if (*i*, *j*) ∈ R/ (G, *ρ*, *π*, *χ*). (15)

*4.2. Fragments of LDL-SL and LDL-SLs f*

In this subsection, we consider fragments for both LDL-SL and LDL-SL*s f* , including SL-like, one-goal fragments, and ATL-like fragments.

Firstly, we consider the SL-like fragment BSL of LDL-SL.

Since LTL is a sublogic of LDL, then by Corollary 2 it is easily shown that BSL is a fragment of LDL-SL by induction and semantics definition. In the following, suppose a logic L∈{BSL, LDL-SL, ATL∗, ADL∗}, let <sup>L</sup>*<sup>s</sup>* (resp, <sup>L</sup>*p*) denote all the set of state (resp. path) formulas in L.

**Theorem 2.** *LDL-SL is strictly more expressive than BSL.*

**Proof.** Firstly, we define two functions *<sup>T</sup><sup>s</sup>* : *BSL<sup>s</sup>* <sup>→</sup> *LDL* <sup>−</sup> *SL<sup>s</sup>* and *<sup>T</sup><sup>p</sup>* : *BSL<sup>p</sup>* <sup>→</sup> *LDL* <sup>−</sup> *SL<sup>p</sup>* by induction of structures of state formulas and path formulas.


By induction, both *<sup>T</sup><sup>s</sup>* and *<sup>T</sup><sup>p</sup>* are well-defined; i.e., for any *<sup>ϕ</sup>* <sup>∈</sup> BSL*<sup>s</sup>* and *<sup>ψ</sup>* <sup>∈</sup> BSL*p*, *<sup>T</sup>s*(*ϕ*) <sup>∈</sup> LDL-SL*<sup>s</sup>* and *<sup>T</sup>p*(*ψ*) <sup>∈</sup> LDL-SL*p*.

Moreover, for any CGS G, a BSL state formula *ϕ*, a strategy assignment *χ*, and a state *w*, the following holds:

$$\mathcal{G}\_{\prime}\chi\_{\prime}w \left| = q \right. \quad \text{if and only if} \quad \mathcal{G}\_{\prime}\chi\_{\prime}w \left| = T^{\varepsilon}(q). \tag{16}$$

For any CGS G, a BSL path formula *ψ*, a strategy assignment *χ*, a path *π*, and some *i* ∈ N, the following holds:

$$\mathcal{G}\_{\prime}\chi\_{\prime}\pi\_{\prime}i \vartriangledown \psi \quad \text{if and only if} \quad \mathcal{G}\_{\prime}\chi\_{\prime}\pi\_{\prime}i \vartriangledown T^{p}(\psi). \tag{17}$$

We can show the above two mutually by induction.

It is easy to see that for the Boolean cases, the above two are obvious.

For case *ϕ* = (*a*, *x*)*ϕ* : <sup>G</sup>, *<sup>χ</sup>*, *<sup>w</sup>* <sup>|</sup><sup>=</sup> *<sup>T</sup>s*((*a*, *<sup>x</sup>*)*<sup>ϕ</sup>* ) iff <sup>G</sup>, *<sup>χ</sup>*, *<sup>w</sup>* <sup>|</sup>= (*a*, *<sup>x</sup>*)*Ts*(*<sup>ϕ</sup>* ) by definition of *<sup>T</sup><sup>s</sup>* iff <sup>G</sup>, *<sup>χ</sup>*[*<sup>a</sup>* → *<sup>χ</sup>*(*x*)], *<sup>w</sup>* <sup>|</sup><sup>=</sup> *<sup>T</sup>s*(*<sup>ϕ</sup>* ) by semantics definition iff G, *χ*[*a* → *χ*(*x*)], *w* |= *ϕ* by induction iff G, *χ*, *w* |= (*a*, *x*)*ϕ* by semantics definition.

For case *ϕ* = *xϕ* : <sup>G</sup>, *<sup>χ</sup>*, *<sup>w</sup>* <sup>|</sup><sup>=</sup> *<sup>T</sup>s*(*x<sup>ϕ</sup>* ) iff <sup>G</sup>, *<sup>χ</sup>*, *<sup>w</sup>* <sup>|</sup><sup>=</sup> *xTs*(*<sup>ϕ</sup>* ) by definition of *<sup>T</sup><sup>s</sup>* iff <sup>∃</sup>*<sup>g</sup>* <sup>∈</sup> *Str*(G), <sup>G</sup>, *<sup>χ</sup>*[*<sup>x</sup>* → *<sup>g</sup>*], *<sup>w</sup>* <sup>|</sup><sup>=</sup> *<sup>T</sup>s*(*<sup>ϕ</sup>* ) iff ∃*g* ∈ *Str*(G), G, *χ*[*x* → *g*], *w* |= *ϕ* iff G, *χ*, *w* |= *xϕ* .

For case *<sup>ϕ</sup>* <sup>=</sup> **<sup>E</sup>***ψ*: <sup>G</sup>, *<sup>χ</sup>*, *<sup>w</sup>* <sup>|</sup><sup>=</sup> *<sup>T</sup>s*(**E***ψ*) iff <sup>G</sup>, *<sup>χ</sup>*, *<sup>w</sup>* <sup>|</sup><sup>=</sup> **<sup>E</sup>**(*Tp*(*ψ*)) by definition of *<sup>T</sup><sup>s</sup>* iff <sup>∃</sup>*<sup>π</sup>* <sup>∈</sup> *out*(G, *<sup>χ</sup>*, *<sup>w</sup>*) s.t. <sup>G</sup>, *<sup>χ</sup>*, *<sup>π</sup>*, 0 <sup>|</sup><sup>=</sup> *<sup>T</sup>p*(*ψ*) iff <sup>∃</sup>*<sup>π</sup>* <sup>∈</sup> *out*(G, *<sup>χ</sup>*, *<sup>w</sup>*) s.t. <sup>G</sup>, *<sup>χ</sup>*, *<sup>π</sup>*, 0 <sup>|</sup><sup>=</sup> *<sup>ψ</sup>* iff G, *χ*, *w* |= **E***ψ*.

For case *<sup>ψ</sup>* <sup>=</sup> *<sup>ϕ</sup>* <sup>∈</sup> BSL*<sup>s</sup>* : <sup>G</sup>, *<sup>χ</sup>*, *<sup>π</sup>*, *<sup>i</sup>* <sup>|</sup><sup>=</sup> *<sup>T</sup>p*(*ϕ*) iff <sup>G</sup>, *<sup>χ</sup>*, *<sup>π</sup>*, *<sup>i</sup>* <sup>|</sup><sup>=</sup> *<sup>T</sup>s*(*ϕ*) by definition of *<sup>T</sup><sup>p</sup>* iff <sup>G</sup>, *<sup>χ</sup>*, *<sup>π</sup><sup>i</sup>* <sup>|</sup><sup>=</sup> *<sup>T</sup>s*(*ϕ*) iff <sup>G</sup>, *<sup>χ</sup>*, *<sup>π</sup>*(*i*) <sup>|</sup><sup>=</sup> *<sup>ϕ</sup>* iff <sup>G</sup>, *<sup>χ</sup>*, *<sup>π</sup>*, *<sup>i</sup>* <sup>|</sup><sup>=</sup> *<sup>ϕ</sup>*.

For case *ψ* = *ψ* : <sup>G</sup>, *<sup>χ</sup>*, *<sup>π</sup>*, *<sup>i</sup>* <sup>|</sup><sup>=</sup> *<sup>T</sup>p*(*<sup>ψ</sup>* ) iff <sup>G</sup>, *<sup>χ</sup>*, *<sup>π</sup>*, *<sup>i</sup>* <sup>|</sup><sup>=</sup> *trueTp*(*<sup>ψ</sup>* ) by definition of *<sup>T</sup><sup>p</sup>* iff <sup>G</sup>, *<sup>χ</sup>*, *<sup>π</sup>*, *<sup>i</sup>* <sup>+</sup> <sup>1</sup> <sup>|</sup><sup>=</sup> *<sup>T</sup>p*(*<sup>ψ</sup>* ) iff G, *χ*, *π*, *i* + 1 |= *ψ* iff G, *χ*, *π*, *i* |= *ψ* .

For case *ψ* = *ψ* : <sup>G</sup>, *<sup>χ</sup>*, *<sup>π</sup>*, *<sup>i</sup>* <sup>|</sup><sup>=</sup> *<sup>T</sup>p*(*<sup>ψ</sup>* ) iff <sup>G</sup>, *<sup>χ</sup>*, *<sup>π</sup>*, *<sup>i</sup>* <sup>|</sup><sup>=</sup> *true*∗*Tp*(*<sup>ψ</sup>* ) by definition of *<sup>T</sup><sup>p</sup>* iff there exists *<sup>j</sup>* <sup>≥</sup> *<sup>i</sup>*, <sup>G</sup>, *<sup>χ</sup>*, *<sup>π</sup>*, *<sup>j</sup>* <sup>|</sup><sup>=</sup> *<sup>T</sup>p*(*<sup>ψ</sup>* ) iff there exists *j* ≥ *i*, G, *χ*, *π*, *j* |= *ψ* iff G, *χ*, *π*, *i* |= *ψ* .

For case *<sup>ψ</sup>* <sup>=</sup> *<sup>ψ</sup>*1U*ψ*2: <sup>G</sup>, *<sup>χ</sup>*, *<sup>π</sup>*, *<sup>i</sup>* <sup>|</sup><sup>=</sup> *<sup>T</sup>p*(*ψ*1U*ψ*2) iff <sup>G</sup>, *<sup>χ</sup>*, *<sup>π</sup>*, *<sup>i</sup>* <sup>|</sup><sup>=</sup> (*Tp*(*ψ*1)?; *true*)∗*Tp*(*ψ*2) by definition of *<sup>T</sup><sup>p</sup>* iff there exists *<sup>j</sup>* with (*i*, *<sup>j</sup>*) ∈ R(G,(*Tp*(*ψ*1)?; *true*)∗, *<sup>π</sup>*, *<sup>χ</sup>*), such that <sup>G</sup>, *<sup>χ</sup>*, *<sup>π</sup>*, *<sup>i</sup>* <sup>|</sup><sup>=</sup> *<sup>T</sup>p*(*ψ*2) iff there exists *<sup>j</sup>* with *<sup>j</sup>* <sup>=</sup> *<sup>i</sup>* or (∀*k*, *<sup>i</sup>* <sup>≤</sup> *<sup>k</sup>* <sup>&</sup>lt; *<sup>j</sup>*, satisfying that <sup>G</sup>, *<sup>χ</sup>*, *<sup>π</sup>*, *<sup>k</sup>* <sup>|</sup><sup>=</sup> *<sup>T</sup>p*(*ψ*1)), such that <sup>G</sup>, *<sup>χ</sup>*, *<sup>π</sup>*, *<sup>i</sup>* <sup>|</sup><sup>=</sup> *<sup>T</sup>p*(*ψ*2) by semantics definition and Corollary <sup>2</sup> iff there exists *j* with *j* = *i* or (∀*k*, *i* ≤ *k* < *j*, satisfying that G, *χ*, *π*, *k* |= *ψ*1), such that G, *χ*, *π*, *i* |= *ψ*<sup>2</sup> by induction iff G, *χ*, *π*, *i* |= *ψ*1U*ψ*2.

Secondly, according to a well-known property *even*(*q*) "a proposition *q* has to be true in each even state of one sequence" cannot be expressed in LTL [14], which can be expressed in LDL by [(*true*; *true*)∗]*q*. Considering those CGSs with only one agent, LDL-SL formula *x*(*a*, *x*)**E**[(*true*; *true*)∗]*q* cannot be expressed by any BSL formula.

Hence we have shown that LDL-SL is more expressively than BSL.

Secondly, we consider a one-goal fragment LDL-SL[1G] and an ATL-like fragment ADL∗ of LDL-SL.

The syntax of LDL-SL[1G] is the same as that of LDL-SL, except for state formulas:

$$\text{State formula } \varphi ::= p \mid \neg \varphi \mid \varphi \land \varphi \mid \mathbf{E} \psi \mid \varphi \flat \varphi,\tag{18}$$

where *p* ∈ *AP*, and ℘ is a closed combination of a quantification/binding prefix.

The following is ATL-like fragment ADL∗ of LDL-SL, of which the path formulas are different from those of ATL∗.

**Definition 11** (ADL∗ Syntax)**.** *The syntax of ADL*∗ *is defined as follows:*

*State formula ϕ* ::= *p* | ¬*ϕ* | *ϕ* ∧ *ϕ* | *Aψ Path formula ψ* ::= *ϕ* | ¬*ψ* | *ψ* ∧ *ψ* | *ρψ*

*Regular expression ρ* ::= Φ | *ψ*? | *ρ* + *ρ* | *ρ*; *ρ* | *ρ*∗,

*where p* ∈ *AP, A* ⊆ *Ag, and* Φ ∈ L(*AP*)*.*

By the following lemma, any ATL∗ formula can be expressed in ADL∗.

**Lemma 5.** *Any ATL*∗ *formula can be linearly encoded by one ADL*∗ *formula.*

**Proof.** Define two translation functions *<sup>T</sup><sup>s</sup>* : ATL∗*<sup>s</sup>* <sup>→</sup> ADL∗*<sup>s</sup>* , *<sup>T</sup><sup>p</sup>* : ATL<sup>∗</sup> *<sup>p</sup>* <sup>→</sup> ADL<sup>∗</sup> *<sup>p</sup>* :


Here to show this lemma, similarly with those in Theorem 2, the only different case is *ϕ* = *Aψ*. Given a CGS G, a state *w*, a state formula *ϕ*,

• for the case *<sup>ϕ</sup>* <sup>=</sup> *Aψ*: <sup>G</sup>, *<sup>w</sup>* <sup>|</sup><sup>=</sup> *<sup>T</sup>s*(*Aψ*) iff <sup>G</sup>, *<sup>w</sup>* <sup>|</sup><sup>=</sup> *ATp*(*ψ*) by definition of *<sup>T</sup><sup>s</sup>* iff there exist collective strategies *gA* s.t. for each *<sup>π</sup>* <sup>∈</sup> *out*(*w*, *gA*), <sup>G</sup>, *<sup>π</sup>*, 0 <sup>|</sup><sup>=</sup> *<sup>T</sup>p*(*ψ*) by semantics iff there exist collective strategies *gA* s.t. for each *π* ∈ *out*(*w*, *gA*), G, *π*, 0 |= *ψ* by induction iff G, *w* |= *Aψ* by semantics.

Obviously, for any *<sup>ϕ</sup>* <sup>∈</sup> ATL∗*<sup>s</sup>* , the size of *<sup>T</sup>s*(*ϕ*) is *<sup>O</sup>*(|*ϕ*|).

Thirdly, we consider one-goal fragment LDL-SL[1G]*s f* and ATL-like fragment ADL<sup>∗</sup> *s f* of LDL-SL*s f* . The syntax of LDL-SL[1G]*s f* is the same as that of LDL-SL[1G] except for regular expressions:

$$\text{Regular expression } \rho ::= \Phi \mid \text{ } \psi ? \mid \rho + \rho \mid \rho; \rho \mid \overline{\rho}. \tag{19}$$

where Φ ∈ L(*AP*), and *ψ* is a path formula in LDL-SL[1G]*s f* .

The syntax of ADL∗ *s f* is the same that of ADL<sup>∗</sup> except for regular expressions,

$$\text{Regular expression } \rho ::= \Phi \mid \psi ? \mid \rho + \rho \mid \rho; \rho \mid \overline{\rho}. \tag{20}$$

where Φ ∈ L(*AP*) and *ψ* is a path formula in ADL<sup>∗</sup> *s f* .

Here we consider three kinds of fragments of LDL-SL: one-goal fragment, star-free, and ATL-like. The semantics of these logics are the same as those of LDL-SL and LDL-SL*s f* , respectively.
