Quantifying over information change with common knowledge

Public announcement logic (PAL) extends multi-agent epistemic logic with dynamic operators modelling the effects of public communication. Allowing quantification over public announcements lets us reason about the existence of an announcement that reaches a certain epistemic goal. Two notable examples of logics of quantified announcements are arbitrary public announcement logic (APAL) and group announcement logic (GAL). While the notion of common knowledge plays an important role in PAL, and in particular in characterisations of epistemic states that an agent or a group of agents might make come about by performing public announcements, extensions of APAL and GAL with common knowledge still haven’t been studied in detail. That is what we do in this paper. In particular, we consider both conservative extensions, where the semantics of the quantifiers is not changed, as well as extensions where the scope of quantification also includes common knowledge formulas. We compare the expressivity of these extensions relative to each other and other connected logics, and provide sound and complete axiomatisations. Finally, we show how the completeness results can be used for other logics with quantification over information change.


Definition 2.4 A model M is a tuple (S, R, V)
, where S is a non-empty set of states, R ∶ A → 2 S×S gives an equivalence relation for each agent, and V ∶ P → 2 S is the valuation function. We will denote model M with a distinguished state s as M s . Whenever necessary, we refer to the elements of the tuple as S M , R M , and V M .
A model is called finite if S is finite. We call model N a submodel of M if S N ⊆ S M , and R N and V N are restrictions of R M and V M to S N . We will also write M X s = (S X , R X , V X ) , where X ⊆ S , s ∈ X , S X = X , R X (a) = R(a) ∩ (X × X) for all a ∈ A , and V X (p) = V(p) ∩ X for all p ∈ P.
It is assumed that for group announcements, agents know the formulas they announce. In the following, we write EL G = { ⋀ i∈G ◻ i i | for all i ∈ G, i ∈ EL} (with typical elements G ) to denote the set of all possible announcements by agents from group G. It is immediate from the semantics that common knowledge of a group consisting of a single agent is equivalent to the knowledge of that agent: In what follows, we will sometimes say -state to refer to a state in a given model that satisfies .
As discussed in the introduction, we now define the semantics of alternative variants of APAL and GAL extended with common knowledge, where the quantification also ranges over common knowledge formulas. Let Intuitively, ELC G is the set of possible group announcements by agents from G that may include common knowledge.

Definition 2.6 Let
be a model, p ∈ P , G ⊆ A , and , ∈ APALC X ∪ GALC X . The semantics of APALC X and GALC X is as in Definition 2.5 with the following modification: 40 19 For the case of Q-bisimulation where Q ⊂ P , Theorem 1 holds only for ∈ PALC that include propositional variables only from Q. The reason the result in this case cannot be extended to a language with quantified announcements is that the quantification is implicit, and hence can use propositional variables outside of Q. if B i+1 (s � , t � ) , then for all a ∈ A and u ∈ S M such that R M (a)(s � , u) , there is a v ∈ S N such that R N (a)(t � , v) and B i (u, v), , then for all a ∈ A and v ∈ S N such that R N (a)(t � , v) , there is a u ∈ S M such that R M (a)(s � , u) and B i (u, v).
It is a standard result that M s ⇆ n N t implies M s ⊧ if and only if N t ⊧ for ∈ EL with modal depth less or equal n (see, e.g, [20]). This does not hold if contains either a common knowledge modality or a quantified announcement. In the first case, common knowledge can access a state on an arbitrarily long distance from the origin. In the second case, quantified announcements are not restricted by any modal depth.
If n-bisimulation between M s and N t is restricted to Q ⊂ P , then we will write M s ⇆ n Q N t , and say that M s and N t are Q-n-bisimilar.

Definition 2.11
Let ∈ L 1 and ∈ L 2 . We say that and are equivalent, if for all M s : M s ⊧ if and only if M s ⊧ .

Definition 2.12
Let L 1 and L 2 be two languages. If for every ∈ L 1 there is an equivalent ∈ L 2 , we write L 1 ⩽ L 2 and say that L 2 is at least as expressive as L 1 . We write L 1 < L 2 if and only if L 1 ⩽ L 2 and L 2  L 1 , and we say that L 2 is strictly more expressive than L 1 . If L 1  L 2 and L 2  L 1 , we say that L 1 and L 2 are incomparable.

Sharing common knowledge
As one of the main purposes of communication is sharing information, in the context of quantified announcements it is quite natural to ask whether a set of agents can make some fact common knowledge among themselves and other agents. We now state a number of observations for GALC and APALC, but they do in fact all hold for APALC X and GALC X as well.
We start with showing that, in general, if a group of agents jointly knows , then it is not always the case that they can share this knowledge with another group in such a way that becomes commonly known among the members of the other group. A counter-example is the well known Moore sentence (see the extended discussion in the setting of EL in [24]): p is true and agent a does not know this. It is also the case that it is not always possible to share common knowledge of one group with some other group.

Proposition 2
There is a such that ▪ G → ⟨G⟩▪ H and ▪ G → ⟨!⟩▪ H are not valid for G ≠ H.
Proof Follows from Proposition 1 and ▪ {b} ↔ ◻ b . ◻ We have the next proposition as a corollary with ∶= p ∨ ¬p . Informally, the proposition says that it is not always possible for two groups of agents to exchange their common knowledge with one another. Interestingly, it is not always possible to make group knowledge common even among the members of the group.

Proposition 4
There is a such that ◻ G → ⟨G⟩▪ G and ◻ G → ⟨!⟩▪ G are not valid. Proof Let G = {a, b} and ∶= ◊ a (◊ a p ∧ ◊ b ◻ a ¬p) , and consider model M s in Fig. 1 All the negative results of this section should not come as a surprise. Target formulas in our proof contained modalities expressing that an agent does not know something. Achieving an epistemic goal that also requires someone to remain ignorant of some fact is quite tricky in the setting of public communication. Indeed, formulas with negated knowledge modalities are unstable in the sense that providing additional public information may make them false.
However, for many applications in AI and multi-agent systems, having a stable, easily verifiable epistemic goal is desirable. Examples of such applications include reading a blockchain ledger and alternating bit protocol. See more on this in [37]. It is known that formulas of the positive fragment remain true after public communication [38], and below we show that for positive formulas our intuitions regarding sharing common knowledge are indeed true.

Definition 3.1
The positive fragment of epistemic logic with common knowledge ELC + is defined by the following BNF: where p ∈ P , a ∈ A , and G ⊆ A . We call ELC + without ▪ G + the positive fragment of epistemic logic EL + .
The distinctive feature of positive formulas is that they are preserved under submodels, i.e. if + holds in a model, then + also holds in all submodels of the model in the same state of evaluation. In particular, this fact implies the following result.
Proof The proof for the case of common knowledge of the whole set of agents ▪ A + can be found in [38]. It is easily adapted to any G ⊆ A . ◻ Proposition 5 All of the following are valid for any + , + ∈ ELC + : We outline the general idea for proving all of the statements. First, note that formula ◻ G + is already in a form of a group announcement by G (also, for the case of common knowledge we have that ▪ G + → ◻ G + ). Moreover, ◻ G + is positive and holds in the current state of a model. These two facts, in conjunction with Lemma 1, yield The latter is equivalent to ⟨G⟩▪ G + by the semantics. Finally, Again, all the results above hold for APALC X and GALC X as well, substituting the corresponding modalities.

Expressivity
In the previous section we did not find any explicit distinction between GALC and GALC X , since all the results were true for both. An interesting question, then, is whether there is any difference in expressive power between GALC and GALC X , and APALC and APALC X . In this section we show that not only are they different but, perhaps even more surprisingly, they are in fact incomparable. We also situate these languages within a wider context of logics based on EL.
We note that the real difference in expressivity between logics of quantified announcements with common knowledge and their extended versions is only visible on infinite models. Indeed, as we claim in the next theorem, both pairs APALC and APALC X , and GALC and GALC X , are equally expressive on finite models. Proof Left-to-right directions of both statements are immediate. Now assume that for some finite M s , we have M s ⊧ ⟨!⟩ X . Without loss of generality, we also assume that M s is bisimulation contracted. By the definition of semantics, we have that M s ⊧ ⟨ ⟩ for some ∈ ELC . Since M s is finite, S is also finite. It is known that in a finite model each state can be uniquely characterised (up to bisimulation) by a distinguishing formula from EL , i.e. a formula that is true only in this state (and all bisimilar states) [6,33]. Hence, we can construct an announcement that will have the same effect as : ∶=

Logics of quantified announcements with common knowledge relative to other logics
Before venturing into the problem of relative expressivity of APALC , APALC X , GALC , and GALC X , we compare the aforementioned logics to other logics discussed in the paper. We hope that this section will strengthen the reader's intuitions about quantified announcements, and highlight the crucial role of Q-bisimulation in the coming proofs. First of all, it is known from the literature that EL < ELC < PALC [31]. Now, we turn our attention to the logics with quantification over public announcements.

Proof
The proof is quite similar to those for PAL < GAL [2, Theorem 19] and PAL < APAL [9, Proposition 3.13]. We, however, provide some details here for completeness' sake.
First, we show that PALC < GALC (the proof PALC < GALC X is similar). That PALC ⩽ GALC follows trivially from the fact that PALC ⊂ GALC . To see that GALC  PALC , consider formula ⟨b⟩◻ a p , and assume towards a contradiction that there is an equivalent formula ∈ PALC . Since has a finite number of symbols, there must be a propositional variable q ∈ P that does not occur in . Now consider models M s and N s depicted in Fig. 2. It is clear that the two models are P ⧵ {q}-bisimilar, and thus they cannot be distinguished by . On the other hand, we have that M s ̸ ⊧ ⟨b⟩◻ a p , since all ◻ b that are true in s will also be true in t. This is not the case for model N s . Indeed, announcement of ◻ b q results in N ◻ b q s for which it holds that N ◻ b q s ⊧ ◻ a p . Hence, N s ⊧ ⟨b⟩◻ a p , and we have GALC  PALC. Now we argue that PALC < APALC (again, the proof PALC < APALC X is similar). The fact that PALC ⊂ APALC entails that PALC ⩽ APALC . Next, we consider formula ⟨!⟩(¬◻ a p ∧ ◊ b ◻ a ¬p) of APALC , and assume towards a contradiction that there is an equivalent ∈ PALC that does not contain atom q. Similarly to the previous case, we see that cannot distinguish M s and N s . To argue that M s ⊧ ⟨!⟩(¬◻ a p ∧ ◊ b ◻ a ¬p) it is enough to notice that the only two model updates available in M s are the trivial one (the model remains intact), and the one that removes state t. In both cases, formula ¬◻ a p ∧ ◊ b ◻ a ¬p is not satisfied. Contrary to that, N s ⊧ ⟨!⟩(¬◻ a p ∧ ◊ b ◻ a ¬p) . Indeed, consider announcement of formula ¬p ∨ q that results in model N ¬p∨q s . It is easy to check that N ¬p∨q s ⊧ ¬◻ a p ∧ ◊ b ◻ a ¬p , thus implying N s ⊧ ⟨!⟩(¬◻ a p ∧ ◊ b ◻ a ¬p) by the semantics, and APALC  PALC . ◻ In the proof of the next theorem we exploit the fact that a given formula with common knowledge modality can reach states on arbitrary distance from a given state. In other words, while modal depth of a given formula is some specific number n, presence of common knowledge modality forces us to consider states on distances greater than n. This is something we will have to take care of in proofs of Sect. 4.3. Proof In one direction, the proof is exactly like the proof of Theorem 3.
For the other direction, i.e. to see that ELC  GAL , consider ▪ {a,b} ¬p ∈ ELC and assume that there is an equivalent ∈ GAL . As is finite, it must have some finite modal depth n. Now, let us consider models M and N depicted in Fig. 3. Lengths of the models are n + 1 . It is easy to see that M s ⊧ ▪ {a,b} ¬p and N t ⊧ ▪ {a,b} ¬p To show that M s ⊧ if and only if N t ⊧ , we use the induction on the size of . Since the models are n-bisimilar, no EL formula of modal depth n can distinguish M s and N t .
Case ∶= [ ] and for some m < n , u and v, M u and N v are (n − m)-bisimilar, where m is a current number of a step in the induction, and u and v are states, where we may have ended up (e.g. after epistemic cases). There are two possible cases. First, update of M with preserves the path to the black state. Then, however, has a modal depth of at most (n − m) − 1 , while M u and N v are (n − m) − 1-bisimilar. Second, update with may not preserve the path to the black state. In this case the two models become bisimilar, and thus cannot be distinguished by any .
Cases ∶= [G] and ∶= ⟨!⟩ are like the previous one noting that in the first case we quantify over EL G and in the second case we quantify over EL . ◻ We have the following two theorems as corollaries, noting that ▪ {a,b} ¬p is also a formula of PALC , APALC , GALC , APALC X , and GALC X .

Formula games
One of the classic techniques for comparing expressive power of modal languages is by using games over models [39,Chapter 8]. Such games are usually played between two players, one of which tries to show that the two models are the same, and another one tries to demonstrate that the models are different. Moves in a game are determined by a given formula of a logic, and the number of moves by either player is bounded by the modal depth of the formula.
Formula games for GAL and coalition announcement logic [3,15] were originally introduced in [14] (see also [15,Chapter 7] for details and examples). Here we introduce formula games for logics of quantified announcements with common knowledge considered in the paper. where p ∈ P and G ⊆ A . If for formula ∈ NNF the outermost operator or the main connective are from the top line, then we say that is in universal negation normal form UNNF ; and if the outermost operator or the main connective are from the line below, then is in existential negation normal norm ENNF . We would also like to point out the absence of clause ⟨ ⟩ in the BNF. As it will become clear later, in Lemma 2, we can do without it.
Lemma 2 Every formula of APALC , APALC X , GALC , and GALC X can be equivalently rewritten to a formula in NNF .

Proof
The proof is a straightforward 'pushing' of negations inside of the scope of operators. We use translation function t ∶ (APALC ∪ APALC X ∪ GALC ∪ GALC X ) → NNF that is defined as follows: ◻ Before we continue with formula games, we introduce a size relation that will be helpful in induction proofs of this section. We will also need an auxiliary lemma that states that a formula and its translation to NNF has the same quantifier depth and size.
Proof A proof is straightforward, and we show just one case as an example. Consider ¬[ ] . Size of this formula, according to Defini- Assuming by the induction hypothesis that s(t( )) = s( ) and s(t(¬ )) = s(¬ ) = s( ) + 1 , we get the desired equality. ◻ Now we are ready to define formula games that are played between the ∀-player (the universal player) and the ∃-player (the existential player) over a given model. Types and order of moves are determined by a given formula that the game is constructed for: the universal player moves if a current subformula is in UNNF , and the existential player moves if the current subformula is in ENNF .

Definition 4.3
Let some model M s and ∈ NNF be given, and suppose that M is the set The game is played between the ∀-player and the ∃-player, and a play consists of a sequence of vertices Δ, Δ 1 , … , Δ n . The play is built by the players such that for some edge (Δ m , Δ m+1 ) ∈ E if Δ m ∈ V ∀ , then the universal player chooses Δ m+1 , and if Δ m ∈ V ∃ , then the existential player chooses Δ m+1 . If either player is unable to move, i.e. they are in a ⊤-vertex or ⊥-vertex, then they lose the game.
The intuition behind edges of a game is that they show which moves the current player has. For example, if we are in vertex ⌜N t , ∧ ⌝ of a game, then the ∀-player can either choose to move to vertex ⌜N t , ⌝ or to vertex ⌜N t , ⌝ . If we are in vertex ⌜N t , ⧫ G ⌝ of the game, then the ∃-player can choose any state u of N reachable from t via R(G), thus letting the game to carry on in vertex ⌜N u , ⌝.
Of special interest are moves that correspond to public announcements and quantifiers. From vertex ⌜N t , [ ] ⌝ the existential player can move to a vertex ⌜N t , X, , ⌝ , where X is a subset of S N . From this position, the universal player can challenge the choice of the existential player in three different ways. First, she can check whether whether all states in the chosen subset satisfy . Second, the universal player can check whether whether all states outside of X are ¬ -states. The third option is to continue the game in a submodel N X t with the formula . All these choices of the universal player correspond to the semantics of public announcements.
Finally, the game positions with quantified announcements also follow the semantics. For example, in vertex ⌜N t , ⟨G⟩ ⌝ of the game, the existential player can choose any formula G ∈ EL G thus making a move to vertex ⌜N t , t( G ) ∧ [t( G )] ⌝ , where the universal player can either check that the chosen formula is indeed true, or let the ∃-player to carry on with announcement of the chosen formula.
In the next proposition we show that all plays of formula games are finite.

Proposition 6
Given formula ∈ NNF , model M s , and a game G M s , every play of the game is finite.

Proof The proof is by structural induction on .
Base Case: in the case of a propositional variable there is exactly one step in a play of the game.
Induction Hypothesis (IH): for all pointed submodels N t of M and for all such that < ∀ (Definition 4.2), plays of the game are finite. The propositional and epistemic cases are straightforward, so we omit them. Also note that it means that plays for epistemic formulas are finite.
Case ⌜N t , [ ] ⌝ : in this position of the game the existential player chooses a subset X of the set of states S N of the given model. Such a choice leads to one of the vertices ⌜N t , X, , ⌝ . There are three possible choices of the ∀-player from this vertex: and thus plays from ⌜N t , ⌝ and ⌜N X t , ⌝ are finite by the IH. Moreover, by Lemma 3 we have that ∀ (t(¬ )) = ∀ (¬ ) and s(t(¬ )) = s(¬ ) . It holds that t(¬ ) < ∀ [ ] , and thus plays from ⌜N u , t(¬ )⌝ are finite by the IH.
, and thus by the IH, we conclude that the play from this vertex is finite.
[G] X ⌝ , and ⌜N t , ⟨G⟩ X ⌝ are similar to the previous one. ◻ In the following proposition we state the relation between a formula being true in the current state of a model, and the existence of the winning strategy for the existential player in the corresponding game.

Proposition 7 The ∃-player has a winning strategy in a game G M s if and only if M s ⊧ .
Proof From right to left. Base Case: Assume that M s ⊧ p . Then the corresponding formula game consists only of one ∃-step from ⌜M s , p⌝ to ⌜M s , ⊤⌝ , and the latter is the winning vertex of the existential player (it it universal player's turn but they cannot move). The same argument holds for ¬p.
Induction Hypothesis (IH): Assume that for all pointed submodels N t of M and all for- Propositional and epistemic cases are straightforward.
where X can be an empty set. We have that for all u ∈ X : N u ⊧ and for all v ∈ Y : N v ⊧ t(¬ ) . By the IH this implies that ⌜N u , ⌝ and ⌜N v , t(¬ )⌝ are winning positions for the existential player for all u ∈ X and v ∈ Y . Hence, ⌜N t , X, , ⌝ is also a winning position for the ∃-player that she ⌝ is a winning position for the ∃-player. Hence, ⌜N t , X, , ⌝ is a winning position for the ∃-player that she can choose we can use the IH to conclude that the ∃-player can always choose a step in the game that corresponds to the winning position ⌜N t , t( ) ∧ t([ ] )⌝ . Thus, ⌜N t , ⟨!⟩ ⌝ is also a winning position for the existential player. Cases and ⟨G⟩ X are similar to the previous one.
From left to right. A similar argument as in the opposite direction for the contraposition: if M s ̸ ⊧ , then the ∀-player has a winning strategy in game G M s . ◻ To recapitulate, Proposition 7 states that if a formula is true in a model, then the existential player has a winning strategy. Alternatively, if the formula is false in a model, then the universal player has a winning strategy. We will use these facts in the next section, when we will let both players to play their winning strategies against each other.

APALC and GALC relative to APALC X and GALC X
Now we turn to the key question of the relative expressivity of APALC and APALC X , and of GALC and GALC X . We show in Theorem 7 that there are some properties of models that can be captured by the extended versions of the logics, and cannot be captured by the conservative versions.
We start by presenting two models, M and N in Fig. 4 that we will be used in the proof.
In both models, there are chains starting from s and t correspondingly of length n + 2 for each n ∈ ℕ . Chains end with boxed states. In model N there is also an infinite vertical chain starting from state u. Propositional variable p is true in s and t, and q is true in boxed states at the ends of finite chains.
Model M is constructed in such a way that the upper and lower parts of the model (relative to state s) are bisimilar. In particular, M s u ⇆ M s l , M n u ⇆ M n l and M n u m ⇆ M n l m for all n, m ∈ ℕ with m < n . This is not the case for model N, where the presence of the infinite vertical chain allows us to distinguish the upper and lower parts of the model. Indeed, take an arbitrary state n u m from the upper part. Formula ¬⧫ {b,c} q is false in N n u m , and it is satisfied in N u (or any other state of the infinite chain).
Next, we show that there are formulas of APALC X and GALC X that can distinguish M s and N t .

Lemma 4
There are formulas 1 ∈ APALC X and 2 ∈ GALC X , such that M s ⊧ * and N t ⊧ * , where * ∈ {1, 2}. and ⟨!⟩ X ∈ APALC X . In order to see that N t ⊧ ⟨!⟩ X , consider the following announcement: Note that we use an announcement with q here, while q does not appear in . Also note that this announcement belongs to ELC . In model N, formula ¬⧫ {b,c} q is true only in states t, t u , t l , and all states of the infinite vertical chain including u.
We now argue that the result of updating N with the announcement is presented in Fig. 5.
First, pick any non-zero boxed state, i.e. let n * ∈ {n * | n ∈ ℕ ⧵ {0} and * ∈ {u, l}} . We have that N n * ̸ ⊧ ¬p → (◻ {b,c} q ∨ ◊ b p) as p is true only in the black state and thus cannot be reached by b, and there is always either a b-or c-arrow to a neighbour circle node with ¬q . Hence, N n * ̸ ⊧ c . Now consider state 0 l : it holds that N 0 l ⊧ q → ◻ a (⧫ {b,c} q ∨ ◊ b p) since there is an a-arrow to state u and N u ⊧ ¬⧫ {b,c} q ∧ ¬◊ b p . On the other hand, all a-arrows from 0 u lead to states where either ⧫ {b,c} q or ◊ b p hold: each reachable finite chain ends with a q-state, and from state t u there is a b-arrow to the p-state. It is left to check that Second, pick any circle state apart from t u and t l . To see that c} q (q is false in the current state) and N • ̸ ⊧ ◊ b p (as p is true only in the black state, which is not reachable via b from any white circle state apart from t u and t l ). So, N • ̸ ⊧ c . In both t u and t l , ◊ b p is true and hence the whole formula is true.
It is easy to check that O t ⊧ . Formula is constructed in such a way that it can only be satisfied by model O t (up to bisimulation). The first conjunct in checks the truth of p in the current state. The second conjunct specifies that there is a ¬p-state reachable in one b-step that is not a numbered state. Finally, the third conjunct ensures that there is a numbered state reachable in two steps, and no other 'deeper' states are available.
To argue that M s ⊧ ⟨!⟩ X , we recall that the upper and lower halves of model M (relative to state s) are bisimilar. Now assume towards a contradiction that there is a ∈ ELC such that M s ⊧ . In particular, we have that M s ⊧ ◊ b (¬p ∧ ◻ a ◊ b p) . By the semantics, this means that there is a state, either s u or s l (or both), such that ◻ a ◊ b p holds in that state. By the construction of M, the only way ◻ a ◊ b p can be satisfied in s u or s l is by removing all other a-reachable states. Since M s u ⇆ M s l , by Theorem 1 we have that M s u ⊧ ◻ a ◊ b p if and Finally, note that the same argument works for GALC X . Indeed, formula c belongs to ELC {c} , and thus we have that N t ⊧ ⟨{c}⟩ X . The fact that M s ⊧ ⟨{c}⟩ X again follows from the proof of M s ⊧ ⟨!⟩ X noting that the choice of was arbitrary. ◻ Now we are left to show that models M s and N t cannot be distinguished by none of the formulas of APALC or GALC . For the proof we will use formula games introduced in Sect. 4.2. First, we will assume towards a contradiction that there is a formula of APALC or GALC such that it is true in one model and false in the other. By Proposition 7 this means that the ∃-player has a winning strategy in one model, and the ∀-player has a winning strategy in the other model.
We will play two games simultaneously, one over M s and , and the other over N t and . In each game each player will play according to their winning strategies. Since games are finite by Proposition 6, we should end up in the situation, where one player has won in one model, and the other player has won in the other model. However, we will use the notion of Q-n-bisimulation to argue that at the final step both players in both models will be in states satisfying the same propositional variables, meaning that one of the winning strategies for one of the players is not winning at all. And this will yield the desired contradiction.
Proof In Lemma 4 we have seen that formulas ⟨!⟩ X ∈ APALC X and ⟨{c}⟩ X ∈ GALC X distinguish models M s and N t . Now assume towards a contradiction that there is a ∈ APALC ∪ GALC that is equivalent to either ⟨!⟩ X or ⟨{c}⟩ X accordingly. Without loss of generality, we also assume that ∈ NNF . Since has a finite number of symbols, there must be a q ∈ P such that q does not occur in .
Since is equivalent to ⟨!⟩ X or ⟨{c}⟩ X , we have that M s ̸ ⊧ and N t ⊧ . This means, by Proposition 7, that the ∀-player has a winning strategy in G M s , and the ∃-player has a winning strategy in H N t . Given n = md( ) , we consider the following relation B ⊆ S M × S N : It is clear that B is an P ⧵ {q}-2 n -bisimulation relation between M and N, where each state of one model is in relation to the corresponding state of the other model. As for the infinite chain, we put states on the chain in relation to states from chain (2 n ) l of M in such a way that M (2 n ) l k ⇆ 2 n N k if k < 2 n , and all k with k ⩾ 2 n are put into relation with state (2 n ) l of M. Now we show that after k steps of a game, all the remaining states are still P ⧵ {q} -(2 n − k)-bisimilar.
Base Case: Let = p for some p ∈ P ⧵ {q} . Since all states that are in relation B satisfy the same propositional variables from P ⧵ {q} , we have P ⧵ {q}-0-bisimilarity.
Induction Hypothesis (IH): After k steps of a game, for all states s ′ and t ′ from all sub-

Cases
= ∧ and = ∨ . In game G M s , the ∀-player makes a move from ⌜M � s � ∧ ⌝ to either ⌜M ′ s ′ , ⌝ or ⌜M ′ s ′ , ⌝ . The universal player makes the same choice (either or ) in game H N t . Since such a move does not change current states, we have Similarly for = ∨ and the ∃ -player.
Cases = ◻ a and = ◊ a . In game G M s , the ∀-player makes a move, according to her winning strategy, from ⌜M ′ The existence of such a t * follows from the IH and Definition 2.10. Note that the way we defined B specifies that if the player made a move in game H N t to state u on the infinite chain, then the move will be matched by a move to state (2 n ) l 0 in game G Since the ∃-player has a winning strategy in H N t , then she can choose a subset Observe that our construction of Y guarantees that for each state of X there is always a state of Y such that they are in relation B, and vice versa.
The ∀-player can now reply with one of three possible moves in both games. First, she can choose some state s * ∈ Y (resp. t * ∈ X ) to get to position ⌜M � s * , ⌝ (resp. ⌜N � t * , ⌝ ). That the bisimulation is preserved follows from the construction of Y and the IH. Similarly for the move of the universal player to position ⌜M � , the ∀-player makes a move, according to her winning strategy, from ⌜M � s � , [!] ⌝ to some ⌜M � s � , [t( )] ⌝ such that t( ) ∈ EL . It can be shown 2 [39,Theorem 8.15] that for each d ∈ ℕ , for all states s � ∈ S M � there is a state t � ∈ S N � (and vice versa) such that M ′ s ′ ⊧ iff N ′ t ′ ⊧ for all ∈ EL such that md( ) = d . This fact in conjunction with t( ) ∈ EL , entails that the universal player can choose the same formula in game H N t to move to a winning state ⌜N � t � , [t( )] ⌝ . Note that the modal depth of t( ) can exceed 2 n − k . In this case, the games are continued with the current IH, and if the number of moves in a game exceeds 2 n − k , then the game is continued with the assumption of P ⧵ {q} -0-bisimilarity. It is enough for our purposes, since we are interested only in up to 2 n moves.
The case of = ⟨!⟩ is similar with the existential player as the protagonist. Cases = [G] and = ⟨G⟩ are similar to the cases above substituting t( ) with t( G ) , and EL with EL G .
As a result of these two simultaneous games over formula and models M s and N t we end up in states in both games where the ∃-player (resp. the ∀-player) has a winning strategy. This contradicts the assumption that the ∀-player (resp. the ∃-player) has a winning strategy in one of the games, or, equivalently, it contradicts the fact that M s ̸ ⊧ iff N t ⊧ . ◻ Now we turn to the other direction of the expressivity relation. We use the same approach with formula games to show that, perhaps more surprisingly, there are some properties of models that can be expressed by APALC and GALC and cannot be expressed by APALC X and GALC X . Indeed, one may have expected that since quantifiers of APALC X and GALC X range over a strictly more expressive language than quantifiers of APALC and GALC ( ELC in the first case, and EL in the second case), then APALC X and GALC X would end up being more expressive than their non-extended siblings. We show that this is not the case.
We start with providing two models and arguing that there are formulas of APALC and GALC that can distinguish them. Consider models M and N in Fig. 6. In both of the models, there are vertical chains starting from s and t correspondingly of length n + 2 for each n ∈ ℕ . These finite chains have at their end a numbered boxed state where q is true. Both models also have infinite vertical chains starting from u and v correspondingly. For the infinite chains, there are no states where q holds. Propositional variable p ′ is true only on the infinite chain of model N in the black square state. We now argue that N t ⊧ ⟨!⟩ . Notice that p ′ holds on the infinite chain starting at state v. Since the quantification over announcements is implicit, we can use p ′ and q in announcements. Moreover, we can use announcement of arbitrary finite depth. Before giving the announcement that results in a model satisfying , we show how using p ′ we can specify a distinguishing formula for state v; such a formula will be true in v and nowhere else in the model. We can characterise states on the infinite chain by using their distance from p ′ . See Fig. 7 for the representation of the approach.
Thus the distinguishing formula for v is where ◊ m i stands for m alternating c-and b-diamonds, and ◻ m j stands for m alternating band c-boxes. Informally, the first conjunct means that state v is at most m + 1 steps away from the p ′ -state, the second conjunct specifies that the state is at least m + 1 steps away from the p ′ -state, and the third conjunct says that v is two steps away from the p-state. Now we can use v to provide the necessary formula. Consider the following announcement: That the result of updating N t with this formula is model O t that satisfies can be shown similarly to the proof of Lemma 4 with state u being substituted by state v, and ⧫ {b,c} ¬q being substituted with ¬ v . The argument for ⟨{c}⟩ ∈ GALC also follows noting that Before continuing with the expressivity proof, let us take another look at the two models. First, the reader can notice that they are P ⧵ {p � , q} -and P ⧵ {p � }-bisimilar, and hence they satisfy the same formulas of ELC that do not contain p ′ . Second, all states on finite chains can be distinguished from all states on infinite chains. To see this, we show how to construct distinguishing formulas for each state on finite chains.
We can use the (slightly modified) method from Fig. 7. First, we can construct a formula that is true only on a particular depth (number of steps from a p-state). For example, a formula that is true in all states that are exactly 4 steps away from a p-state is The reader can verify that this formula holds in states, e.g., 3 l 2 and 3 u 2 of both models. To distinguish upper states from lower states, we, in addition to 4 , need to use infinite chains. In model N t or any submodel thereof containing the p ′ -state, we can use formula v from the proof of Lemma 5. Thus, a formula that is true in all states that are exactly 4 steps away from a p-state and that are in the lower part of a models is The reader can check that N 3 l 2 ⊧ l 4 and N 3 u 2 ̸ ⊧ l 4 . In order to choose states in the upper part of the model, we just negate the last conjunct. Thus, Now we turn to distinguishing upper and lower parts of M s and its submodels. Prima facie, it seems enough to use formula ▪ {b,c} ¬q that is true only in state u of M. However, if we want to deal also with submodels of M s , it is not enough. Indeed, there may be some finite chains in some M ′ s ′ that do not have q-states at their ends, and that will thus satisfy ▪ {b,c} ¬q . Hence, suppose that in some M ′ s ′ there is an infinite chain, and only a finite number of finite chains do not have q-states. Among those finite chains we take the longest, and denote its length by d. Now, a formula that is true only in state u is where ◊ d+1 i stands for d + 1 alternating b-and c-diamonds. The first conjunct ensures that the formula is false on all chains with a q-state, and the second conjunct specifies that the formula is false on all chains with length less than d + 1 . Having defined u , we can define a formula that would be true in all states that are exactly n steps away from a p-state and that are in the lower (or upper) part of the model. Formulas for states 4 steps away would be like l 4 and u 4 for N t with v being substituted with u . Finally, for the construction of the formula that is true only in 3 l 2 , assume that l 6 and l 5 have been specified. Notice that 3 l 2 is the only state in the lower parts of our models that is at depth 4, one step away from l 5 and does not reach a state satisfying l 6 on its chain. Formally, The described method of constructing distinguishing formulas of particular states will be used in the proof of Theorem 8.
Proof According to Lemma 5, there are formulas ⟨!⟩ ∈ APALC and ⟨{c}⟩ ∈ GALC that distinguish models M s and N t . Now assume towards a contradiction that there is a ∈ APALC X ∪ GALC X that is equivalent to either ⟨!⟩ or ⟨{c}⟩ accordingly. Without loss of generality, we also assume that ∈ NNF . Since has a finite number of symbols, there must be q, p � ∈ P such that q and p ′ do not occur in . Moreover, let n = md( ).
Similarly to the proof of Theorem 7, we define a P ⧵ {q, p � }-m-bisimulation relation B ⊆ S M × S N : Relation B connects each state of M with the corresponding state of N. Again, similarly to the proof of Theorem 7, we play two games simultaneously: game G M s over M s and , and game H N t over N t and . We also assume towards a contradiction that the ∀-player has a winning strategy in G M s , and the ∃-player has a winning strategy in The proof for Boolean and epistemic cases, and the case of public announcements, follows the similar lines as the proof of Theorem 7, where the players play a move according to their winning strategy in one game, and play the corresponding move the other game. The crucial difference are the cases of quantified announcements.
Induction Hypothesis (IH): After k steps of a game, for all states s ′ and t ′ from all sub- , the ∀-player makes a move, according to her winning strategy, from Due to the construction of our models, we cannot guarantee that choosing t( ) in H N t will result in P ⧵ {q, p � }-(2 n − k)-bisimilar models, or, in other words, that it will also be a winning move for the universal player. However, as described earlier, we can construct a � ∈ ELC such that ⌜N � t � , [t( � )] ⌝ is a corresponding winning move in H N t . Construction of ′ depends on the way the original updates M ′ . In particular, presence of the infinite chain in M ′ and of p ′ -state in N ′ allows us to distinguish upper and lower parts of the models. Thus, we need to take care that if one is affected, so is the other.
First, if in game G M s the ∀-player chooses such a that updating M ′ with the formula does not affect the infinite chain, does not remove an infinite number of q-states, and does not contain p ′ , then she can make the same choice of in game H N t in position ⌜N � t � , [!] ⌝ . And vice versa for game H N t . Such an announcement does not affect the ability to distinguish upper and lower halves of both models, thus retaining P ⧵ {q, p � }-(2 n − k − 1) -bisimilarity.
Assume now that in game G M s formula contains p ′ . Since the valuation of p ′ in M is empty, we can get an equivalent ′ for game H N t by substituting p ′ in with ⊥ . This will ensure that updating M ′ with and updating N ′ with ′ results in P ⧵ {q, p � }-(2 n − k − 1) -bisimilar models.
Let updating M ′ with remove an infinite number of q-states. As a result, we cannot distinguish states on the infinite chain from states on finite chains without q-states. In particular, for formulas u with any d, there will a finite chain satisfying it. To model such an effect in N ′ , the ∀-player chooses � ∶= ∧ ¬p � , that removes the p ′ -state once being announced. As a result, we also lose the power to distinguish the upper and lower parts in N ′ . Moreover, since the p ′ -state is 2 n + 2 away from t, we have M � s � ⇆ 2 n −k−1 P⧵{q,p � } N � t � . Now consider game H N t and that, once being announced, removes an infinite number of q-states. Since the p ′ -state is still present in the updated model (N � ) , we need to retain the power to distinguish upper and lower parts in model M ′ . To this end, in game G M s the ∀ -player chooses ′ announcement which would remove a finite number of q-states in M ′ . It is enough to consider only first 2 n chains. Since the number of states to remove is finite, the is a distinguishing formula of state i l j to be removed. This will preserve the power to distinguish upper and lower parts of the model using formulas u for various d's, while also retaining P ⧵ {q, p � }-(2 n − k − 1)-bisimilarity.
Finally, let updating M ′ with in game G M s cut the infinite chain to some finite length. In the resulting updated model, each state i on now finite chain will be bisimilar to some state on a finite chain, thus making it impossible to distinguish upper and lower parts of the model. To simulate this in model N ′ in game H N t , the ∀-player can choose � ∶= ∧ ¬p � , thus making it impossible also in N ′ to distinguish upper and lower halves and maintaining the P ⧵ {q, p � }-(2 n − k − 1)-bisimilarity.
If in game H N t the choice of is such that in the resulting update (N � ) the infinite chain is cut, then we consider two cases. First, suppose that the chain was cut in such a way that i is the last state of the now finite chain, and that the p ′ -state is still in S (N � ) . Then we just need to cut a finite chain of length greater than 2 n (to maintain the P ⧵ {q, p � }-(2 n − k − 1) -bisimilarity) in model M ′ to the same length i. This can be done by the ∀-player choosing ⋀ ¬ l j , where l j are distinguishing formulas of states on the chosen finite chain. Second, if the chain was cut in such a way that i is the last state of the now finite chain, and that the p ′ -state is not in S (N � ) , then the infinite chain of M ′ should be cut to the same length. This can be done by the choice of ◊ i j u by the ∀-player, where ◊ i j is a stack of alternating b-and c-diamonds of the required size. In both models, the power to distinguish upper and lower parts will be gone, thus preserving the P ⧵ {q, p � }-(2 n − k − 1)-bisimilarity.
The case of = ⟨!⟩ can be shown by similar reasoning, substituting the ∀-player with the ∃-player. Cases = [G] X and = ⟨G⟩ X . The method of constructing announcements described in the previous case can be also used for group announcements. The only difference is that chosen announcements are prefixed with ◻ a for all a ∈ G . This is due to the fact that group announcements quantify over ELC G . If a group of agents cannot target a particular state, then they can announce a disjunction of formulas in their equivalence class. For example, agent b cannot announce a formula that will only be true 3 l 0 : such a formula would be prefixed with ◻ b and thus should also be satisfied in 3 l ) ∈ ELC {b} to target both 3 l 0 and 3 l 1 . As in the proof of Theorem 7, we play two simultaneous games over M s and N t that end up in states where the ∃-player (resp. the ∀-player) has a winning strategy. This contradicts the assumption that the ∀-player (resp. the ∃-player) has a winning strategy in the other model, or, equivalently, it contradicts the fact that M s ̸ ⊧ iff N t ⊧ . ◻

APALC and APALC X relative to GALC and GALC X
In this section we explore the relative expressivity of arbitrary and group announcements with common knowledge when pitched against one another. The results here are obtained by adapting the corresponding results on the relative expressivity of APAL and GAL [2,14,15]. Thus, we present only sketches and general intuitions of the proofs pointing an interested reader to the cited literature for additional details. We start by claiming that the proof of Theorem 20 from [2] can be used to show that GALC and GALC X are not at least as expressive as APALC and APALC X .
First, the authors of [2] consider an APAL formula ⟨!⟩(◻ a p ∧ ¬◻ b ◻ a p) , and assume towards a contradiction that there is an equivalent formula of GAL not containing q.
Then, models M u and N u from Fig. 8 are considered, noting that M u ⊧ ⟨!⟩(◻ a p ∧ ¬◻ b ◻ a p) and N u ⊧ ⟨!⟩(◻ a p ∧ ¬◻ b ◻ a p) . In particular, announcement of p ∨ ¬q makes ◻ a p ∧ ¬◻ b ◻ a p true in N u (see Fig. 8 and model N p∨¬q ). Since p ∨ ¬q ∈ ELC , we also have that ⟨!⟩ X (◻ a p ∧ ¬◻ b ◻ a p) is a distinguishing formula for M u and N u . Moreover, ⟨!⟩(◻ a p ∧ ¬◻ b ◻ a p) ∈ APALC and ⟨!⟩ X (◻ a p ∧ ¬◻ b ◻ a p) ∈ APALC X , and hence M u and N u are distinguishable by formulas of APALC and APALC X .
The argument that cannot distinguish M u and N u goes by induction [2,Theorem 20]. For our goals, it is enough to notice that M u and N u are P ⧵ {q}-bisimilar and thus satisfy the same formulas of PALC that do not contain q. Moreover, cases for extended arbitrary and group announcements follow from the fact that M and N are finite, and thus by Theorem 2 satisfy [G] if and only if they satisfy [G] X .
The fact that GALC and GALC X are not at least as expressive as APALC and APALC X follows from the proof of GAL  CAL [14,15], where CAL is the language of coalition announcement logic defined by The semantics of coalition announcement modality [⟨G⟩] and its dual ⟨[G]⟩ is as follows: Informally, formula ⟨[G]⟩ means that agents from G have a joint announcements such that no matter what agents from outside of G announce at the same time, will hold. Similarly, [⟨G⟩] stands for the fact that whatever agents from G jointly announce, there is a counterannouncement by agents from outside of G such that will hold.
For the purposes at hand, we are interested in a special case of coalition announcements, namely announcement by the grand coalition A. In such a case, the semantics can be simplified to The proof in [14,15] starts off by presenting two classes of finite models, called A-chain models and B-chain models. Examples of chain models are depicted in Fig. 9. Without giving a formal definition, we just mention that chain models have a leftmost state that satisfies ¬p ∧ ◻ a ¬p , and the rightmost state that satisfies ◻ a p ∧ [A](◊ b ¬p → ◻ a ◊ b ¬p) . In short, the models are similar in their extremities and differ only in length (see Fig. 9 for reference). Whether a given pointed chain model is an A-chain model or a B-chain model depends which agent relation is the first one in the direction of the state satisfying ¬p ∧ ◻ a ¬p : a-relation or b-relation. For example, model M s from Fig. 9 is a B-model since b's relation is the first one among a and b in the direction of the ¬p ∧ ◻ a ¬p-state (leftmost state). On the other hand, M t and N s are A-models.
Next, it is shown in [14,15] that there is a formula of GAL such that for all M s , M s ⊧ if and only if M s is an A-model. Hence, the same formula also belongs to the language of GALC. We do not present the formula since it is a bit involved and not essential for our argument here. To get a corresponding distinguishing formula of GALC X , we first note that contains the following group announcement operators: ⟨c⟩ , [c] , and [{a, b, c}] . Since chain models are finite, by Theorem 2 we can equivalently substitute all occurrences of the abovementioned group announcements with ⟨c⟩ X , [c] X , and [{a, b, c}] X respectively.
After that, the authors of [14,15] use formula games for GAL and CAL to show that no formula of CAL can distinguish the classes of A-and B-chain models. The proof follows a similar approach as we used in proofs of Theorems 7 and 8 in this paper. In particular, it is assumed towards a contradiction that for all A-chain models M s there is a ∈ CAL such that M s ⊧ , and for all B-chain models N t it holds that N t ̸ ⊧ . The proof proceeds by playing simultaneous formula games over 2 md( ) -bisimilar pointed A-and B-models, and the invariant that after i game steps, models are still 2 md( ) − i-bisimilar, is maintained.
We can reuse the proof to show that formulas of APALC and APALC X do not distinguish A-and B-chain models. For the cases of ▪ G and ⧫ G , the current player chooses a G-reachable state in one model, and the corresponding state in the other model. By 'corresponding' we mean a state which lies on the same distance from the closest extremity, i.e. from the leftmost or the rightmost state depending on which one is closer. In such a way we ensure that games continue in 2 md( ) − i-bisimilar models.

Proof system
In this section we start with the presentation of a proof system of GALC and a detailed completeness proof for it. We then discuss how both are modified to get corresponding results for GALC X , APALC, and APALC X .
Let us first introduce an auxiliary notion.
Definition 5.1 Let ∈ GALC , a ∈ A , G ⊆ A , and ♯ ∉ P . The set of necessity forms [19] is defined recursively below: We will denote the result of replacing of ♯ with in a necessity form (♯) as ( ).

Definition 5.2
The proof system of GALC is the following extension of the proof system of GAL [2]: We call GALC the minimal set that contains axioms A0-A10 and is closed under MP, NK, NA, IC, and IG.
Like existing complete systems of APAL and GAL [9,40], this proof system of GALC is infinitary as it has inference rules that require an infinite number of premises. Note that one of them is the infinitary rule for common knowledge, which is less standard than the usual fixed point approach (see, for example, [11], and also [22] for an alternative axiomatisation of ELC). In an already infinitary system, this treatment is both more intuitive and technically simpler. The infinitary approach to common knowledge has also been discussed in [5], where the authors consider a corresponding Gentzen-type system.

Lemma 6 IC and IG are truth preserving.
Proof The proof is a straightforward induction on necessity forms with the application of the definition of semantics. ◻ Necessitation rules for common knowledge and group announcements are derivable in GALC. Proof Due to the soundness of GAL, Lemma 6, and the validity of (A9). ◻

Lemma 7 Rules 'From
In order to prove the completeness, we adapt the completeness proof of APAL from [8][9][10].
Whenever we will use induction on the formula structure of some ∈ GALC , we will use the following measure.
Our completeness proof is based on the canonical model construction. We will use theories as states in the canonical model.

Definition 5.4 A set x is called a theory if it contains all theorems and is closed under MP ,
IC , and IG . The smallest theory is GALC. Theory x is consistent if there is no ∈ GALC such that , ¬ ∈ x . Theory x is maximal if any theory y such that x ⊂ y is inconsistent.

Lemma 9 Theory x is maximal if and only if for all
Proof Let x be a maximal theory, and assume towards a contradiction that there is a ∈ GALC such that neither ∈ x nor ¬ ∈ x . Then theory x ∪ { } is consistent, and x ⊂ x ∪ { } , which contradicts the definition of maximality.
In the other direction, let us for all ∈ GALC have that either ∈ x or ¬ ∈ x , and x be not maximal. This implies that there is a consistent y such that x ⊂ y , and in particular that there is a such that ∉ x and ∈ y . Since y is consistent, ¬ ∉ y , and hence ¬ ∉ x . We now have that ∉ x and ¬ ∉ x that contradicts our assumption. Proof An extension of the proof of Lemma 4.11 in [9], where common knowledge cases are dealt with using (A9) and IC . ◻

Lemma 11
For all theories x and all ∈ GALC , it holds that x ⊆ x + .
Proof Let us for some ∈ GALC have that ∈ x . Since x is a theory and thus contains all the instances of propositional tautologies, → ( → ) ∈ x . As x is closed under MP , → ∈ x , and, by Lemma 10, ∈ x + . ◻ Next, we prove the Lindenbaum lemma.

Lemma 12
If x is a consistent theory, then it can be extended to a maximal consistent theory y such that x ⊆ y.
Proof The proof is a variation of the Lindenbaum Lemma for APAL [9,Lemma 4.12]. We give here a sketch of an extended proof. Let { 0 , 1 , …} be an enumeration of formulas of GALC , and let y 0 = x . Assume that for some n ≥ 0 , x ⊆ y n and y n is a consistent theory. If ¬ n ∉ y n , then y n+1 = y n + n . Otherwise, there are three cases to consider.
First, if ¬ n ∈ y n and n is not of either the form (▪ G ) or the form ([G] ) , then y n+1 = y n . Second, if ¬ n ∈ y n and n is of the form (▪ G ) , then y n+1 = y n + ¬ (◻ n G ) , where ¬ (◻ n G ) is the first formula in the enumeration such that (◻ n G ) ∉ y n . Third, if ¬ n ∈ y n and n is of the form ([G] ) , then y n+1 = y n + ¬ ([ G ] ) , where ¬ ([ G ] ) is the first formula in the enumeration such that ([ G ] ) ∉ y n .
In all these cases it is clear that y n+1 is consistent. Also, using the inductive construction of y n+1 , the fact that x ⊆ y n+1 , it is relatively straightforward to show that y = ⋃ ∞ n=0 y n is a maximal consistent theory such that x ⊆ y . ◻ Now we are ready to define the canonical model, where states are maximal consistent theories.
Observe that this formula is in a necessity form. Hence, we conclude, by rule IG , that [ ][G] ∈ x. Case = ▪ G . ( ⇒ ): Assume that ▪ G ∈ x . By (A9), ∀n ∈ ℕ ∶ ◻ n G ∈ x , which is equivalent, by the IH, to ∀n ∈ ℕ ∶ x ⊧ ◻ n G . This is equivalent to x ⊧ ▪ G by the semantics.
(⇐ ): Assume that x ⊧ ▪ G . By the semantics, this is equivalent to . This is equivalent to . By the semantics, this is equivalent to Finally, we can prove the completeness of GALC.
Proof Assume towards a contradiction that is valid and ∉ GALC . Since GALC is a consistent theory, it follows that GALC + ¬ is a consistent theory as well. By Lemma 5, there is a maximal consistent theory x such that GALC + ¬ ⊆ x . By Lemma 11, ¬ ∈ GALC + ¬ , and hence ¬ ∈ x . Since x is a maximal consistent theory, it follows that ∉ x . According to Lemma 13, ∉ x is equivalent to x ̸ ⊧ , which contradicts being valid. ◻ The proof system of GALC X is the same as in Definition 5.2 with following differences: The completeness proof is exactly as for GALC, with each [G] replaced by [G] X and EL G replaced by ELC G .
Theorem 13 GALC X is sound and complete.
The axiomatisation of APALC is the same as the proof system of GALC with the following differences: Again, the completeness proof is exactly the same as for GALC, replacing [G] with [!] and each EL G with EL.

Theorem 14 APALC is sound and complete.
Finally, the proof system and the completeness of APALC X can be obtained from those of APALC in the same way as for GALC X .
As with APAL and GAL, extending PAPAL with common knowledge can be done in (at least) two meaningful ways: we can add common knowledge to the language but leave the semantics of [!] PAPAL intact, or we can also extend the quantification to a larger fragment with common knowledge. The resulting logic is PAPAL with common knowledge, and we will denote the former variant as PAPALC and the latter variant as PAPALC X . The semantics of the quantifier in PAPALC X is the following: The axiomatisation of PAPALC is yet again a variation of the proof system for GALC with substitutions: To get the axiomatisation of PAPALC X it is enough to change EL + to ELC + , and

Coalition announcements
The results for GALC and APALC go hand-in-hand with each other due to the fact that the underlying logics are relatively similar. So far we have omitted from the discussion, however, an interesting cousin of GAL and APAL, coalition announcement logic (CAL) [3,15]. CAL extends PAL with the modality [⟨G⟩] , meaning 'whatever agents from group G announce, there is a simultaneous counter-announcement by the agents from outside of the group such that holds in the resulting model'. It is clear that modalities [⟨G⟩] are game-theoretical at heart and formalise -effectivity. Thus, CAL has a gametheoretic flavour to it and is reminiscent of coalition logic [28], alternating-time temporal logic [7], and game logic [29].
Providing a sound and complete axiomatisation of CAL is an open problem. Hence we will discuss an extension with common knowledge of a related logic with coalition announcement-coalition and relativised group announcement logic (CoRGAL) [16]. Compared to the language of CAL, the language of CoRGAL have additional constructs [G, ] that are called relativised group announcements, and that mean 'given true announcement , whatever agents from group G announce at the same time, they cannot avoid '. The double quantification of CAL modalities seems to be one of the reasons why finding an axiomatisation of CAL is hard. Relativised group announcements allow to split the double quantification and treat coalition's announcements and the anti-coaltion's response separately. In other words, formulas within modalities [G, ] act as a kind of memory that stores announcements by a coalition.
Formally, the semantics of coalition modalities and relativised group announcements is as follows: One of such logic is arbitrary arrow update logic (AAUL) [40] that extends modal logic K with dynamic arrow updates. Compared to public announcements, arrow updates, as is hinted in the name, focus on arrows rather than states. Informally, an arrow update is a set of triples ( , a, ) that mean that in the updated model a-arrows between -states and -states will be preserved. Arrows that do not satisfy any of the triples in the update operator are deleted from a model. Since arrow updates delete arrows, equivalence relations between states are not guaranteed to be preserved, unlike in PAL.
Formally, the language of AAUL extends the language of modal logic K with constructs [U] and [↕] . The former means that 'after arrow update U, is true', and the latter is read as 'after any arrow update, is true'. The semantics of the new operators is as follows: where AUL is a fragment of AAUL that does not contain [↕] , and M U = (S, R U , V) with R U (a) = {R(a)(s, t) | ∃( , a, ) ∈ U ∶ M s ⊧ and M t ⊧ } . Note that R in M is not necessarily an equivalence.
The reader can see that the axiomatisation of AAUL is quite similar in form to the proof system of APAL (we present only the part that includes the arbitrary arrow update modality): Arbitrary arrow update logic with common knowledge (AAULC) was first proposed in [25], where the author showed that the logic is not finitely axiomatisable. The way it was presented, [↕] quantified over AUL with common knowledge. In order to obtain a proof system for AAULC it is enough to add axiom (A9) and rule IC from Sect. 5 to the axiomatisation of AAULC from [40]. The completeness can be shown by combining the proofs for the completeness of AAUL and GALC, e.g., we would require a theory to be closed under MP , IC , and (R4).

Discussion
We studied common knowledge in the context of quantification over information change. In particular, we presented extensions of APAL and GAL with the common knowledge modality, both conservative and with the extended semantics. The extensions are called APALC, APALC X , GALC, and GALC X . According to the conservative semantics, the semantics of group and arbitrary announcement modalities is exactly the same as in in APAL and GAL, quantifying over formulas of epistemic logic. Extended semantics allowed group and arbitrary announcements to quantify over a larger set of formulas, namely epistemic logic with common knowledge. We observed that difference in the semantics matters: with the extended semantics we can express properties we cannot express with the conservative semantics, and (perhaps more surprisingly) vice versa. This echoes the results iff M s ⊧ [U] for each U ∈ AUL, for GAL extended with distributed knowledge [1,18]. A current expressivity map of GALC , APALC , and other connected logics is shown in Fig. 10. Moreover, we presented sound and complete axiomatisations of GALC, GALC X , APALC and APALC X . We also showed that our proof of the completeness of the axiomatisations can be used to obtain axiomatisations of other logics with quantification over information change and show their completeness.
Throughout the paper we sidestepped one particular fact that deals with public announcements and common knowledge. For the semantics of usual APAL and GAL, there is no difference whether quantification is over formulas of EL or formulas of PAL . This is a trivial corollary of the fact that both languages are equally expressive [30]. The same, however, cannot be said about the extensions of EL and PAL with common knowledge-ELC and PALC correspondingly. In particular, PALC is strictly more expressive than ELC [39,Theorem 8.48]. Thus, there is yet another way to extend APAL and GAL with common knowledge, i.e. to allow quantification over formulas of PALC . We can call the resulting logics APALC XX and GALC XX with the semantics being as follows: Fig. 10 Overview of the expressivity results. An arrow from L 1 to L 2 means L 1 ⩽ L 2 . If there is no symmetric arrow, then L 1 < L 2 . This relation is transitive, and we omit transitive arrows in the figure. An arrow from L 1 to L 2 is crossed-out, if L 1  L 2 . Dashed arrows depict results known from literature, and solid arrows show the results proven in this paper. All languages in the rounded rectangle are pairwise incomparable While we can yet again reuse our completeness proof to obtain sound and complete axiomatisations of APALC XX and GALC XX , their relative expressivity is left as an open question. Perhaps more intriguing open problem is specifying the exact relationship between APALC XX and APALC X , and GALC XX and GALC X .
In the same vein, it is worthwhile to investigate expressivities of other logic with quantification over information change mentioned in the article, e.g., coalition announcement logic with common knowledge, positive APAL with common knowledge, or arbitrary arrow update logic with common knowledge.