Abstract
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.
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1 Introduction
Quantified dynamic epistemic logics and common knowledge. Epistemic logic (EL) [23] is a multimodal logic where formulas \(\square _a \varphi\) mean ‘agent a knows \(\varphi\)’. Formulas of EL are interpreted on epistemic models that consist of states and equivalence relations between them for each agent. Such a logic allows us to reason not only about an agent’s knowledge of some basic facts, but about what other agents know as well.
While EL deals with individual knowledge of particular agents, there are also various kinds of group knowledge. A prime example of group knowledge is common knowledge which has played an important part in reasoning about knowledge in the multi-agent setting [13]. It has also been used in epistemic planning [26], machine learning [42], game theory [27], and so on. Epistemic logic (EL) with common knowledge (ELC) [13] extends the language of EL with common knowledge modalities \(\blacksquare _G \varphi\), where G is a subset of the set of all agents. Informally, \(\blacksquare_G \varphi\) is read as ‘everybody in G knows that \(\varphi\), everybody in G knows that everybody in G knows that \(\varphi\), and so on’. On the level of models this corresponds to truth in all states accessible by the reflexive transitive closure of relations for agents from G.
Both EL and ELC provide a static description of knowledge in a multi-agent system. Logics that are covered by the umbrella term dynamic epistemic logic (DEL) [39] study the effects of various epistemic events on the individual and group knowledge of agents. The prime example of such a logic is public announcement logic (PAL) [30] that models public communication. A public announcement is an event where all agents publicly and simultaneously receive the same true piece of information. Syntactically, PAL extends EL with construct \([\psi ]\varphi\) that say ‘after truthful public announcement of \(\psi\), \(\varphi\) is true’. From the model perspective, public announcement of \(\psi\) removes all the states from a model that do not satisfy \(\psi\). The interaction of epistemic events, in particular of public announcements, and common knowledge was studied in [11].
Aribitrary public announcement logic (APAL) [9] and group announcement logic (GAL) [2] are extensions of PAL with quantifiers over possible truthful announcements. APAL extends PAL with constructs of the form \(\langle ! \rangle \varphi\) that mean ‘after some public announcement, \(\varphi\) holds’. GAL has quantifiers with a more limited scope, with group announcement operators \(\langle G\rangle \varphi\) meaning that ‘after some (joint) announcement by agents from group G, \(\varphi\) is true’. A ‘joint announcement’ in this context means an announcement of a formula of the shape \(\bigwedge _{i \in G} \square _i \varphi _i\). In other words, each agent can announce something they know. GAL thus allows us to reason about the ability of an agent or a group of agents to achieve their epistemic goal by a joint public announcement.
Common knowledge plays a significant role in PAL, and in particular in characterisations of epistemic states that an agent or a group of agents might make come about by making public announcements. Investigating logics of quantified announcements (or any other quantified epistemic actions) with common knowledge is long overdue, and it was reiterated as an open question in a recent survey [32]. In this paper, we address this problem. First, we study the languages APALC and GALC obtained by extending APAL and GAL, respectively, with common knowledge without changing the semantics of any of the operators. This allows us to gain further insight into the standard APAL and GAL modalities. There is a subtlety here, however, in the scope of quantification. In both APAL and GAL the quantification is restricted to announcements in the purely epistemic language. The reason for this is, in addition to the fact that the quantification does not range over formulas with quantifiers in them to avoid circularity, that EL and PAL are equally expressive [30]. Thus quantifying over EL has the same effect as quantifying over PAL. Adding common knowledge changes the picture, since EL and ELC are not equally expressive. In this paper, in addition to the ‘conservative’ variants APALC and GALC, we also study variants of APAL and GAL with common knowledge where the quantification ranges over formulas of ELC, called APALC\(^X\) and GALC\(^X\) (for ‘eXtended semantics’), respectively. It turns out that the difference in scope of the quantifiers is significant and non-trivial.
Overview of the paper and main results. In Sect. 2 we introduce languages of the logics and the corresponding semantics. We investigate some intuitive potential properties of the interaction between quantified announcements and common knowledge in Sect. 3. In particular, we show that some of the immediate intuitions about sharing knowledge in a group and between groups are actually not correct. Then we specify a fragment of the language for which these intuitions indeed hold.
Section 4 is devoted to the study of the relative expressivity of the languages of GALC, GALC\(^X\), APALC, and APALC\(^X\) and situating these languages within a broader landscape of EL-based logics. We show that both pairs, APALC and APALC\(^X\), and GALC and GALC\(^X\), are in fact incomparable when it comes to the expressive power. The fact that GALC\(^X\) and APALC\(^X\) can express some properties of models that cannot be captured by GALC and APALC, respectively, perhaps follows intuition. The converse, however, may come across as unexpected. In the proof, we demonstrate that sometimes the existential quantification over announcements in GALC\(^X\) and APALC\(^X\) is ‘too powerful’ to notice a difference in models—even though the same announcement might not have the same effect in both models there is often another announcement in the scope of quantification that has the same effect.
In Sect. 5 we give sound and complete proof systems for APALC, GALC, APALC\(^X\), and GALC\(^X\). Like all existing complete systems for APAL and GAL, these are infinitary. A detailed proof is given for the case of GALC; the other cases follow by relatively simple modifications. Our treatment of common knowledge differs from the classic fixed-point approach. Since both APAL and GAL are already infinitary, we use a straightforward infinitary inference rule for common knowledge as well.
Our completeness proof is modular in its nature, meaning that the parts corresponding to common knowledge can be reused as is for other logics with quantification over information change. In Sect. 6 we show that the proof can be adapted to obtain axiomatisations and completeness results for two decidable restrictions of APALFootnote 1 extended with common knowledge, namely Boolean APAL [34] and Positive APAL [37]. A similar result can also be obtained for a variant of coalition announcement logic (CAL) [3, 15] that is called coalition and relativised group announcement logic (CoRGAL) [16] extended with common knowledge. Coalition announcement modalities \([\!\langle G \rangle \!]\varphi\) quantify over announcements by agents from G and simultaneous counter announcement by the agents outside of G. These constructs are read as ‘whatever agents from G announce, there is a simultaneous announcement by the agents from outside of G such that \(\varphi\) is true after the joint announcement’.
There are also logics that quantify over other types of information changing events (see [32]), for some of which only infinitary axiomatisations are known. We claim that our completeness proof can be used to show the completeness of their extensions with common knowledge. As an example, we consider arbitrary arrow update logic with common knowledge and indicate how to obtain its complete axiomatisation.
Finally, we conclude in Sect. 7 and discuss directions of further research.
2 Logics of quantified announcements with common knowldge
2.1 Syntax and semantics
Let us fix a finite set of agents A and a countable set of propositional variables P.
Definition 2.1
The language of arbitrary public announcement logic with common knowledge \(\mathcal {APALC}\), the language of group announcement logic with common knowledge \(\mathcal {GALC}\) and their extended versions \(\mathcal {APALC}^X\) and \(\mathcal {GALC}^X\) respectively, are inductively defined as
where \(p \in P\), \(a \in A\), and \(G \subseteq A\). Duals are defined as \(\Diamond _a \varphi := \lnot \square _a \lnot \varphi\), \(\langle \psi \rangle \varphi := \lnot [\psi ]\lnot \varphi\), \(\blacklozenge _G \varphi := \lnot \blacksquare _G \lnot \varphi\), \(\langle ! \rangle \varphi := \lnot [!] \lnot \varphi\), \(\langle ! \rangle ^X \varphi := \lnot [!]^X \lnot \varphi\), \(\langle G \rangle \varphi := \lnot [G] \lnot \varphi\) and \(\langle G \rangle ^X \varphi := \lnot [G]^X \lnot \varphi\).
Formula \(\square _a \varphi\) is read as ‘agent a knows \(\varphi\)’; \([\psi ]\varphi\) means that ‘after truthful public announcement of \(\psi\), \(\varphi\) will hold’; \(\blacksquare _G \varphi\) is read as ‘it is common knowledge among agents from group G that \(\varphi\)’; \([!]\varphi\) and \([!]^X\varphi\) are read as ‘after any truthful public announcement, \(\varphi\) holds’; \([G]\varphi\) and \([G]^X \varphi\) are read as ‘after any truthful public announcement by agents from group G, \(\varphi\) holds’;.
The fragment of \(\mathcal {GALC}\) without \([G]\varphi\) is called public announcement logic with common knowledge \(\mathcal {PALC}\); the latter without \([\varphi ]\varphi\) is epistemic logic with common knowledge \(\mathcal {ELC}\); \(\mathcal {PALC}\) and \(\mathcal {ELC}\) minus \(\blacksquare _G \varphi\) are, correspondingly, public announcement logic \(\mathcal {PAL}\) and epistemic logic \(\mathcal{E}\mathcal{L}\). Finally, fragments of \(\mathcal {GALC}\) and \(\mathcal {APALC}\) without \(\blacksquare _G \varphi\) are called group announcement logic \(\mathcal {GAL}\) and arbitrary public announcement logic \(\mathcal {APAL}\) respectively.
‘Everyone in group G knows \(\varphi\)’ is denoted by \(\square _G \varphi := \bigwedge _{i \in G} \square _i \varphi\), and \(\square ^n_G \varphi\) is defined inductively as \(\square ^0_G \varphi := \varphi\) and \(\square ^{n+1}_G \varphi := \square _G \square ^n_G \varphi\) for all natural numbers n. Expression \(\Diamond ^n_G \varphi\) is defined similarly by substituting diamonds instead of boxes.
Definition 2.2
Modal depth of \(\varphi \in \mathcal {APALC} \cup \mathcal {GALC} \cup \mathcal {APALC}^X \cup \mathcal {GALC}^X\) (denoted \(md (\varphi )\)) is defined inductively as
Definition 2.3
Let \(\varphi \in \mathcal {APALC} \cup \mathcal {GALC} \cup \mathcal {APALC}^X \cup \mathcal {GALC}^X\). The quantifier depth \(\delta _{\forall } (\varphi )\) of \(\varphi\) is defined inductively as
where \(Q \in \{[!], [G], [!]^X, [G]^X\}\).
Definition 2.4
A model M is a tuple (S, R, V), where S is a non-empty set of states, \(R: A \rightarrow 2^{S \times S}\) gives an equivalence relation for each agent, and \(V:P \rightarrow 2^S\) 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 \(S_N \subseteq S_M\), 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 \(X \subseteq S\), \(s\in X\), \(S^X = X\), \(R^X(a) = R(a) \cap (X \times X)\) for all \(a \in A\), and \(V^X(p) = V(p) \cap X\) for all \(p \in P\).
It is assumed that for group announcements, agents know the formulas they announce. In the following, we write \(\mathcal{E}\mathcal{L}^G = \{\bigwedge _{i \in G} \square _i \psi _i \mid \text {for all } i \in G, \psi _i \in \mathcal{E}\mathcal{L}\}\) (with typical elements \(\psi _G\)) to denote the set of all possible announcements by agents from group G.
Definition 2.5
Let \(M_s = (S, R, V)\) be a model, \(p \in P\), \(G \subseteq A\), and \(\varphi , \psi \in \mathcal {APALC} \cup \mathcal {GALC}\).
where \(M_s^\psi = (S^\psi , R^\psi , V^\psi )\) with \(S^\psi = \{s \in S \mid M_s \models \psi \}\), \(R^\psi (a)\) is the restriction of R(a) to \(S^\psi\) for all \(a \in A\), and \(V^\psi (p) = V(p) \cap S^\psi\) for all \(p \in P\).
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: \(\blacksquare _{\{a\}} \varphi \leftrightarrow \square _a \varphi\).
In what follows, we will sometimes say \(\varphi\)-state to refer to a state in a given model that satisfies \(\varphi\).
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 \(\mathcal {ELC}^G = \{\bigwedge _{i \in G} \square _i \psi _i \mid \text {for all } i \in G, \psi _i \in \mathcal {ELC}\}\). Intuitively, \(\mathcal {ELC}^G\) is the set of possible group announcements by agents from G that may include common knowledge.
Definition 2.6
Let \(M_s = (S, R, V)\) be a model, \(p \in P\), \(G \subseteq A\), and \(\varphi , \psi \in \mathcal {APALC}^X \cup \mathcal {GALC}^X\). The semantics of APALC\(^X\) and GALC\(^X\) is as in Definition 2.5 with the following modification:
Note that in a language with both types of operators, \([!]^X \varphi \rightarrow [!]\varphi\) and \([G ]^X \varphi \rightarrow [ G ] \varphi\) would be true in every model.
Definition 2.7
We call formula \(\varphi\) valid if and only if for all \(M_s\) it holds that \(M_s \models \varphi\).
For convenience, let us also provide the semantics for diamonds:
It is common in the literature [13, Chapter 2] to define common knowledge of group G via reflexive transitive closure of \(\bigcup _{a \in G} R(a)\). We denote such a relation by R(G). The corresponding definition of the semantics then looks like the following:
Both definitions of common knowledge, via \(\square ^n_G \varphi\) for all \(n \in \mathbb {N}\) and via R(G), are equivalent to each other, and we will use them interchangeably.
2.2 Bisimulation and expressivity
We will also use several notions of bisimulation.
Definition 2.8
Let Q be a set of propositional variables, and \(M = (S_M, R_M, V_M)\) and \(N = (S_N, R_N, V_N)\) be models. We say that M and N are Q-bisimilar (denoted \(M \leftrightarrows _Q N\)) if there is a non-empty relation \(B \subseteq S_M \times S_N\), called Q-bisimulation, such that for all B(s, t), the following conditions are satisfied:
- Atoms:
-
for all \(p \in Q\): \(s \in V_M(p)\) if and only if \(t \in V_N(p)\),
- Forth:
-
for all \(a \in A\) and \(u \in S_M\) such that \(R_M (a)(s,u)\), there is a \(v \in S_N\) such that \(R_N (a)(t, v)\) and B(u, v),
- Back:
-
for all \(a \in A\) and \(v \in S_N\) such that \(R_N(a)(t,v)\), there is a \(u \in S_M\) such that \(R_M(a)(s,u)\) and B(u, v).
We say that \(M_s\) and \(N_t\) are Q-bisimilar and denote this by \({M_s \leftrightarrows _Q N_t}\) if there is a Q-bisimulation linking states s and t. Also, we omit subscripts Q if \(Q = P\).
Theorem 1
Given \(M_s\) and \(N_t\), if \(M_s \leftrightarrows N_t\), then for all \(\varphi \in \mathcal {APALC} \cup \mathcal {APALC}^X \cup \mathcal {GALC} \cup \mathcal {GALC}^X\) we have that \(M_s \models \varphi\) if and only if \(N_t \models \varphi\).
Proof
The proof is by induction on \(\varphi\). Propositional, boolean, and epistemic cases are as usual. The case of common knowledge is proven in [39, Theorem 8.35], and the case of public announcements follows from the corresponding result for action models [39, Theorem 6.21]. Finally, the cases of arbitrary and group announcements follow from the induction hypothesis and the fact that public announcements preserve bisimilarity. \(\square\)
For the case of Q-bisimulation where \(Q \subset P\), Theorem 1 holds only for \(\varphi \in \mathcal {PALC}\) 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.
Definition 2.9
Let \(M = (S, R, V)\) be a model. The bisimulation contraction of M is the model \(\Vert M\Vert = (\Vert S\Vert , \Vert R \Vert , \Vert V\Vert )\), where \(\Vert S\Vert = \{[ s ] \mid s \in S\}\) and \([ s ] = \{t \in S \mid M_s \leftrightarrows M_t\}\), \(\Vert R \Vert (a) ([s],[t])\) if and only if \(\exists s^{\prime} \in [s]\), \(\exists t^{\prime} \in [t]\) such that \(R(a) (s^{\prime} , t^{\prime} )\) in M, and \([s] \in \Vert V\Vert (p)\) if and only if \(\exists s' \in [s]\) such that \(s' \in V(p)\).
Intuitively, the bisimulation contraction is the most compact representation of a model. It is a classic result that \(M_s \leftrightarrows \Vert M\Vert _{[s]}\) [20].
Definition 2.10
Let \(n \in \mathbb {N}\), and \(M = (S_M, R_M, V_M)\) and \(N = (S_N, R_N, V_N)\) be models. We say that \(M_s\) and \(N_t\) are n-bisimilar (denoted \(M_s \leftrightarrows ^n N_t\)) if there exists a sequence of binary relations \(B_n \subseteq \ldots \subseteq B_0\) such that
- Relation:
-
\(B_n (s,t)\),
- Atoms:
-
if \(B_0(s^{\prime} ,t^{\prime} )\), then for all \(p \in P\): \(s^{\prime} \in V_M(p)\) if and only if \(t^{\prime} \in V_N(p)\),
- Forth:
-
if \(B_{i+1}(s^{\prime} ,t^{\prime} )\), then for all \(a \in A\) and \(u \in S_M\) such that \(R_M(a)(s^{\prime} ,u)\), there is a \(v \in S_N\) such that \(R_N (a) (t^{\prime} , v)\) and \(B_i (u,v)\),
- Back:
-
if \(B_{i+1}(s^{\prime} ,t^{\prime} )\), then for all \(a \in A\) and \(v \in S_N\) such that \(R_N(a)(t^{\prime} ,v)\), there is a \(u \in S_M\) such that \(R_M(a)(s^{\prime} ,u)\) and \(B_i (u,v)\).
It is a standard result that \(M_s \leftrightarrows ^n N_t\) implies \(M_s \models \varphi\) if and only if \(N_t \models \varphi\) for \(\varphi \in \mathcal{E}\mathcal{L}\) with modal depth less or equal n (see, e.g, [20]). This does not hold if \(\varphi\) 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 \(Q \subset P\), then we will write \(M_s \leftrightarrows ^n_Q N_t\), and say that \(M_s\) and \(N_t\) are Q-n-bisimilar.
Definition 2.11
Let \(\varphi \in \mathcal {L}_1\) and \(\psi \in \mathcal {L}_2\). We say that \(\varphi\) and \(\psi\) are equivalent, if for all \(M_s\): \(M_s \models \varphi\) if and only if \(M_s \models \psi\).
Definition 2.12
Let \(\mathcal {L}_1\) and \(\mathcal {L}_2\) be two languages. If for every \(\varphi \in \mathcal {L}_1\) there is an equivalent \(\psi \in \mathcal {L}_2\), we write \(\mathcal {L}_1 \leqslant \mathcal {L}_2\) and say that \(\mathcal {L}_2\) is at least as expressive as \(\mathcal {L}_1\). We write \(\mathcal {L}_1 < \mathcal {L}_2\) if and only if \(\mathcal {L}_1 \leqslant \mathcal {L}_2\) and \(\mathcal {L}_2 \not \leqslant \mathcal {L}_1\), and we say that \(\mathcal {L}_2\) is strictly more expressive than \(\mathcal {L}_1\). If \(\mathcal {L}_1 \not \leqslant \mathcal {L}_2\) and \(\mathcal {L}_2 \not \leqslant \mathcal {L}_1\), we say that \(\mathcal {L}_1\) and \(\mathcal {L}_2\) are incomparable.
3 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 \(\varphi\), then it is not always the case that they can share this knowledge with another group in such a way that \(\varphi\) 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.
Proposition 1
There is a \(\varphi\) such that \(\square _G \varphi \rightarrow \langle G \rangle \blacksquare _H \varphi\) and \(\square _G \varphi \rightarrow \langle ! \rangle \blacksquare _H \varphi\) are not valid for \(G \ne H\).
Proof
Let \(G = \{b\}\), \(H = \{a\}\), and \(\varphi := p \wedge \lnot \square _a p\). Consider model \(M^1_s\) in Fig. 1, where agent b’s relation is the identity. It is clear that \(M^1_s \models \varphi\), and hence \(M^1_s \models \square _b \varphi\). Moreover, there are only two possible ways to update \(M^1_s\): leave the model as it is, and remove state t (thus resulting in model \(M^2_s\) from the same Fig. 1). It is straightforward to verify that \(M^1_s \not \models \square _a \varphi\) and \(M^2_s \not \models \square _a \varphi\), thus resulting in \(M^1_s \not \models \langle \{b\} \rangle \square _a \varphi\) and \(M^1_s \not \models \langle ! \rangle \square _a \varphi\). \(\square\)
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 \(\varphi\) such that \(\blacksquare _G \varphi \rightarrow \langle G \rangle \blacksquare _H \varphi\) and \(\blacksquare _G \varphi \rightarrow \langle ! \rangle \blacksquare _H \varphi\) are not valid for \(G \ne H\).
Proof
Follows from Proposition 1 and \(\blacksquare _{\{b\}} \varphi \leftrightarrow \square _{b} \varphi\). \(\square\)
We have the next proposition as a corollary with \(\psi := p \vee \lnot p\). Informally, the proposition says that it is not always possible for two groups of agents to exchange their common knowledge with one another.
Proposition 3
There are \(\varphi\) and \(\psi\) such that \(\blacksquare _{G} \varphi \wedge \blacksquare _H \psi \rightarrow \langle G \cup H \rangle \blacksquare _{G \cup H} (\varphi \wedge \psi )\) and \(\blacksquare _{G} \varphi \wedge \blacksquare _H \psi \rightarrow \langle ! \rangle \blacksquare _{G \cup H} (\varphi \wedge \psi )\) are not valid.
Proof
Let \(G = \{b\}\), \(H = \{a\}\), \(\varphi := p \wedge \lnot \square _a p\), and \(\psi := p \vee \lnot p\). The proof is similar to the proof of Proposition 1 with \(\blacksquare _{\{a\}} \varphi \leftrightarrow \square _{a} \varphi\) and \(\blacksquare _{\{b\}} \varphi \leftrightarrow \square _{b} \varphi\). \(\square\)
Interestingly, it is not always possible to make group knowledge common even among the members of the group.
Proposition 4
There is a \(\varphi\) such that \(\square _G \varphi \rightarrow \langle G \rangle \blacksquare _G \varphi\) and \(\square _G \varphi \rightarrow \langle ! \rangle \blacksquare _G \varphi\) are not valid.
Proof
Let \(G = \{a,b\}\) and \(\varphi := \Diamond _a(\Diamond _a p \wedge \Diamond _b \square _a \lnot p)\), and consider model \(M_s\) in Fig. 1. Formula \(\varphi\) holds in states s and t of M.
It is easy to verify that \(M_s \models \square _{\{a,b\}} \varphi\), and at the same time \(M_s \not \models \blacksquare _{\{a,b\}} \varphi\) (the rightmost state of the model, u, does not satisfy \(\varphi\)). Now let us consider all updates of \(M_s\) depicted in Fig. 1. The reader can check that none of the updates satisfy \(\blacksquare _{\{a,b\}} \varphi\). Hence, \(M_s \not \models \langle G \rangle \blacksquare _G \varphi\) and \(M_s \not \models \langle ! \rangle \blacksquare _G \varphi\). \(\square\)
Model M and some of its submodels. Relation for agent a is depicted by dashed lines and b’s relation is shown by solid lines
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 \(\mathcal {ELC}^+\) is defined by the following BNF:
where \(p \in P\), \(a \in A\), and \(G \subseteq A\). We call \(\mathcal {ELC}^+\) without \(\blacksquare _G \varphi ^+\) the positive fragment of epistemic logic \(\mathcal{E}\mathcal{L}^+\).
The distinctive feature of positive formulas is that they are preserved under submodels, i.e. if \(\varphi ^+\) holds in a model, then \(\varphi ^+\) also holds in all submodels of the model in the same state of evaluation. In particular, this fact implies the following result.
Lemma 1
Let \(\varphi ^+ \in \mathcal {ELC}^+\), then \([ \varphi ^+ ] \blacksquare _G \varphi ^+\) is valid for any \(G \subseteq A\).
Proof
The proof for the case of common knowledge of the whole set of agents \(\blacksquare _A \varphi ^+\) can be found in [38]. It is easily adapted to any \(G \subseteq A\). \(\square\)
Proposition 5
All of the following are valid for any \(\phi ^+,\psi ^+ \in \mathcal {ELC}^+\):
-
1.
\(\square _G \varphi ^+ \rightarrow \langle G \rangle \blacksquare _H \varphi ^+\)
-
2.
\(\blacksquare _G \varphi ^+ \rightarrow \langle G \rangle \blacksquare _H \varphi ^+\)
-
3.
\(\blacksquare _{G} \varphi ^+ \wedge \blacksquare _H \psi ^+ \rightarrow \langle G \cup H \rangle \blacksquare _{G \cup H} (\varphi ^+ \wedge \psi ^+)\)
-
4.
\(\square _G \varphi ^+ \rightarrow \langle G \rangle \blacksquare _G \varphi ^+\)
-
5.
\(\square _G \varphi ^+ \rightarrow \langle ! \rangle \blacksquare _H \varphi ^+\)
-
6.
\(\blacksquare _G \varphi ^+ \rightarrow \langle ! \rangle \blacksquare _H \varphi ^+\)
-
7.
\(\blacksquare _{G} \varphi ^+ \wedge \blacksquare _H \psi ^+ \rightarrow \langle ! \rangle \blacksquare _{G \cup H} (\varphi ^+ \wedge \psi ^+)\)
-
8.
\(\square _G \varphi ^+ \rightarrow \langle ! \rangle \blacksquare _G \varphi ^+\)
Proof
We outline the general idea for proving all of the statements. First, note that formula \(\square _G \varphi ^+\) is already in a form of a group announcement by G (also, for the case of common knowledge we have that \(\blacksquare _G \varphi ^+ \rightarrow \square _G \varphi ^+\)). Moreover, \(\square _G \varphi ^+\) is positive and holds in the current state of a model. These two facts, in conjunction with Lemma 1, yield \(\square _G \varphi ^+ \wedge [ \square _G \varphi ^+ ] \blacksquare _G \square _G \varphi ^+\). The latter is equivalent to \(\langle \square _G \varphi ^+ \rangle \blacksquare _G \square _G \varphi ^+\) due to the validity of \(\psi \wedge [\psi ]\varphi \leftrightarrow \langle \psi \rangle \varphi\). Noting that \(\blacksquare _G \square _G \varphi ^+ \rightarrow \blacksquare _G \varphi ^+\) is valid, we have that \(\langle \square _G \varphi ^+ \rangle \blacksquare _G \square _G \varphi ^+\) implies \(\langle \square _G \varphi ^+ \rangle \blacksquare _G \varphi ^+\). The latter is equivalent to \(\langle G \rangle \blacksquare _G \varphi ^+\) by the semantics. Finally, \(\langle ! \rangle \blacksquare _G \varphi ^+\) is implied by \(\langle G \rangle \blacksquare _G \varphi ^+\). \(\square\)
Again, all the results above hold for APALC\(^X\) and GALC\(^X\) as well, substituting the corresponding modalities.
4 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 \(\mathcal {APALC}\) and \(\mathcal {APALC}^X\), and \(\mathcal {GALC}\) and \(\mathcal {GALC}^X\), are equally expressive on finite models.
Theorem 2
Let \(M_s = (S, R, V)\) be a finite model. Then \(M_s \models \langle ! \rangle \varphi\) if and only if \(M_s \models \langle ! \rangle ^X \varphi\), and \(M_s \models \langle G \rangle \varphi\) if and only if \(M_s \models \langle G \rangle ^X \varphi\).
Proof
Left-to-right directions of both statements are immediate. Now assume that for some finite \(M_s\), we have \(M_s \models \langle ! \rangle ^X \varphi\). Without loss of generality, we also assume that \(M_s\) is bisimulation contracted. By the definition of semantics, we have that \(M_s \models \langle \psi \rangle \varphi\) for some \(\psi \in \mathcal {ELC}\). Since \(M_s\) is finite, \(S^\psi\) 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 \(\mathcal{E}\mathcal{L}\), 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 \(\psi\): \(\chi := \bigvee _{t \in S^\psi } \delta _t\), where \(\delta _t\)’s are distinguishing formulas of states t in model M. Since \(S^\psi =S^\chi\), we have that \(M^\psi _s \models \varphi\) if and only if \(M^\chi _s \models \varphi\), which implies \(M_s \models \langle \chi \rangle \varphi\) and \(M_s \models \langle ! \rangle \varphi\). The same approach can be used for group announcements. \(\square\)
4.1 Logics of quantified announcements with common knowledge relative to other logics
Before venturing into the problem of relative expressivity of \(\mathcal {APALC}\), \(\mathcal {APALC}^X\), \(\mathcal {GALC}\), and \(\mathcal {GALC}^X\), 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 \(\mathcal{E}\mathcal{L}< \mathcal {ELC} < \mathcal {PALC}\) [31]. Now, we turn our attention to the logics with quantification over public announcements.
Theorem 3
\(\mathcal {PALC} < \mathcal {GALC}\), \(\mathcal {PALC} < \mathcal {GALC}^X\), \(\mathcal {PALC} < \mathcal {APALC}\), and \(\mathcal {PALC} < \mathcal {APALC}^X\).
Proof
The proof is quite similar to those for \(\mathcal {PAL} < \mathcal {GAL}\) [2, Theorem 19] and \(\mathcal {PAL} < \mathcal {APAL}\) [9, Proposition 3.13]. We, however, provide some details here for completeness’ sake.
First, we show that \(\mathcal {PALC} < \mathcal {GALC}\) (the proof \(\mathcal {PALC} < \mathcal {GALC}^X\) is similar). That \(\mathcal {PALC} \leqslant \mathcal {GALC}\) follows trivially from the fact that \(\mathcal {PALC} \subset \mathcal {GALC}\). To see that \(\mathcal {GALC} \not \leqslant \mathcal {PALC}\), consider formula \(\langle b\rangle \square _a p\), and assume towards a contradiction that there is an equivalent formula \(\psi \in \mathcal {PALC}\). Since \(\psi\) has a finite number of symbols, there must be a propositional variable \(q \in P\) that does not occur in \(\psi\). Now consider models \(M_s\) and \(N_s\) depicted in Fig. 2. It is clear that the two models are \(P\setminus \{q\}\)-bisimilar, and thus they cannot be distinguished by \(\psi\). On the other hand, we have that \(M_s \not \models \langle b\rangle \square _a p\), since all \(\square _b \varphi\) that are true in s will also be true in t. This is not the case for model \(N_s\). Indeed, announcement of \(\square _b q\) results in \(N_s^{\square _b q}\) for which it holds that \(N_s^{\square _b q} \models \square _a p\). Hence, \(N_s \models \langle b\rangle \square _a p\), and we have \(\mathcal {GALC} \not \leqslant \mathcal {PALC}\).
Now we argue that \(\mathcal {PALC} < \mathcal {APALC}\) (again, the proof \(\mathcal {PALC} < \mathcal {APALC}^X\) is similar). The fact that \(\mathcal {PALC} \subset \mathcal {APALC}\) entails that \(\mathcal {PALC} \leqslant \mathcal {APALC}\). Next, we consider formula \(\langle ! \rangle (\lnot \square _a p \wedge \Diamond _b \square _a \lnot p)\) of \(\mathcal {APALC}\), and assume towards a contradiction that there is an equivalent \(\psi \in \mathcal {PALC}\) that does not contain atom q. Similarly to the previous case, we see that \(\psi\) cannot distinguish \(M_s\) and \(N_s\). To argue that \(M_s \not \models \langle ! \rangle (\lnot \square _a p \wedge \Diamond _b \square _a \lnot p)\) 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 \(\lnot \square _a p \wedge \Diamond _b \square _a \lnot p\) is not satisfied. Contrary to that, \(N_s \models \langle ! \rangle (\lnot \square _a p \wedge \Diamond _b \square _a \lnot p)\). Indeed, consider announcement of formula \(\lnot p \vee q\) that results in model \(N^{\lnot p \vee q}_s\). It is easy to check that \(N^{\lnot p \vee q}_s \models \lnot \square _a p \wedge \Diamond _b \square _a \lnot p\), thus implying \(N_s \models \langle ! \rangle (\lnot \square _a p \wedge \Diamond _b \square _a \lnot p)\) by the semantics, and \(\mathcal {APALC} \not \leqslant \mathcal {PALC}\). \(\square\)
Models M, N, and \(N^{\square _b q}\). Relation for agent a is depicted by dashed lines and b’s relation is shown by solid lines. Propositional variable p is true in black states, and propositional variable q is true in square states
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.
Theorem 4
Both \(\mathcal {ELC}\) and \(\mathcal {GAL}\), and \(\mathcal {ELC}\) and \(\mathcal {APAL}\), are incomparable.
Proof
In one direction, the proof is exactly like the proof of Theorem 3.
For the other direction, i.e. to see that \(\mathcal {ELC} \not \leqslant \mathcal {GAL}\), consider \(\blacksquare _{\{a,b\}} \lnot p \in \mathcal {ELC}\) and assume that there is an equivalent \(\psi \in \mathcal {GAL}\). As \(\psi\) 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 \(M_s \not \models \blacksquare _{\{a,b\}} \lnot p\) and \(N_t \models \blacksquare _{\{a,b\}} \lnot p\)
To show that \(M_s \models \psi\) if and only if \(N_t \models \psi\), we use the induction on the size of \(\psi\). Since the models are n-bisimilar, no \(\mathcal{E}\mathcal{L}\) formula of modal depth n can distinguish \(M_s\) and \(N_t\).
Case \(\psi := [\chi ]\tau\) 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 \(\chi\) preserves the path to the black state. Then, however, \(\tau\) has a modal depth of at most \((n-m)-1\), while \(M^\chi _u\) and \(N^\chi _v\) are \((n-m)-1\)-bisimilar. Second, update with \(\chi\) may not preserve the path to the black state. In this case the two models become bisimilar, and thus cannot be distinguished by any \(\tau\).
Cases \(\psi :=[G]\chi\) and \(\psi :=\langle ! \rangle \chi\) are like the previous one noting that in the first case we quantify over \(\mathcal{E}\mathcal{L}^G\) and in the second case we quantify over \(\mathcal{E}\mathcal{L}\). \(\square\)
Models M and N. Relation for agent a is depicted by dashed lines and b’s relation is shown by solid lines. Propositional variable p is true in the black state
We have the following two theorems as corollaries, noting that \(\blacksquare _{\{a,b\}} \lnot p\) is also a formula of \(\mathcal {PALC}\), \(\mathcal {APALC}\), \(\mathcal {GALC}\), \(\mathcal {APALC}^X\), and \(\mathcal {GALC}^X\).
Theorem 5
Both pairs \(\mathcal {PALC}\) and \(\mathcal {GAL}\), and \(\mathcal {PALC}\) and \(\mathcal {APAL}\), are incomparable.
Theorem 6
\(\mathcal {GAL} < \mathcal {GALC}\), \(\mathcal {GAL} < \mathcal {GALC}^X\), \(\mathcal {APAL} < \mathcal {APALC}\), and \(\mathcal {APAL} < \mathcal {APALC}^X\).
4.2 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.
Definition 4.1
The set of formulas in negation normal form \(\mathcal {NNF}\) is defined by the following BNF:
where \(p \in P\) and \(G \subseteq A\). If for formula \(\varphi \in \mathcal {NNF}\) the outermost operator or the main connective are from the top line, then we say that \(\varphi\) is in universal negation normal form \(\mathcal {UNNF}\); and if the outermost operator or the main connective are from the line below, then \(\varphi\) is in existential negation normal norm \(\mathcal {ENNF}\). We would also like to point out the absence of clause \(\langle \varphi \rangle \varphi\) in the BNF. As it will become clear later, in Lemma 2, we can do without it.
Lemma 2
Every formula of \(\mathcal {APALC}\), \(\mathcal {APALC}^X\), \(\mathcal {GALC}\), and \(\mathcal {GALC}^X\) can be equivalently rewritten to a formula in \(\mathcal {NNF}\).
Proof
The proof is a straightforward ‘pushing’ of negations inside of the scope of operators. We use translation function \(t: (\mathcal {APALC} \cup \mathcal {APALC}^X \cup \mathcal {GALC} \cup \mathcal {GALC}^X) \rightarrow \mathcal {NNF}\) that is defined as follows:
\(\square\)
Before we continue with formula games, we introduce a size relation that will be helpful in induction proofs of this section.
Definition 4.2
Let \(\varphi\) be a formula. The size \(s (\varphi )\) of \(\varphi\) is defined inductively as
In the definition, \(O \in \{\lnot , \square _a, \Diamond _a,\blacksquare _G, \blacklozenge _G, [!], \langle ! \rangle , [!]^X, \langle ! \rangle ^X, [G], \langle G \rangle , [G]^X, \langle G \rangle ^X\}\) and \(C \in \{\wedge , \vee \}\). We will write \(\varphi <^{\forall } \psi\) if and only if \(\delta _\forall (\varphi ) < \delta _\forall (\psi )\) (Definition 2.3), or, otherwise, \(\delta _\forall (\varphi ) = \delta _\forall (\psi )\) and \(s(\varphi ) < s(\psi )\).
We will also need an auxiliary lemma that states that a formula and its translation to NNF has the same quantifier depth and size.
Lemma 3
Let \(\varphi \in \mathcal {APALC} \cup \mathcal {APALC}^X \cup \mathcal {GALC} \cup \mathcal {GALC}^X\). Then \(\delta _\forall (\varphi ) = \delta _\forall (t(\varphi ))\) and \(s(\varphi ) = s(t(\varphi ))\).
Proof
A proof is straightforward, and we show just one case as an example. Consider \(\lnot [\psi ]\varphi\). Size of this formula, according to Definition 4.2, is \(s(\lnot [\psi ]\varphi ) = s([\psi ]\varphi ) + 1 = s(\psi ) + s(\varphi ) + 2\). Now, let us take the translation \(t(\lnot [\psi ]\varphi ) = t(\psi ) \wedge t([\psi ]\lnot \varphi )\). Size of the translation is \(s(t(\psi ) \wedge t([\psi ]\lnot \varphi )) = \text {max}(s(t(\psi )), s( t([\psi ]\lnot \varphi ))) = s( t([\psi ]\lnot \varphi ))= s( [t(\psi )]t(\lnot \varphi ))= s(t(\psi )) + s(t(\lnot \varphi )) + 1\). Assuming by the induction hypothesis that \(s(t(\psi )) = s(\psi )\) and \(s(t(\lnot \varphi )) = s(\lnot \varphi ) = s(\varphi ) + 1\), we get the desired equality. \(\square\)
Now we are ready to define formula games that are played between the \(\forall\)-player (the universal player) and the \(\exists\)-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 \(\mathcal {UNNF}\), and the existential player moves if the current subformula is in \(\mathcal {ENNF}\).
Definition 4.3
Let some model \(M_s\) and \(\varphi \in \mathcal {NNF}\) be given, and suppose that \(\mathcal {M}\) is the set of pointed submodels \(N_t^{X}\) of model \(M_s\), where \(X \subseteq S\) and \(s \in X\). A formula game for \(\varphi\) over \(M_s\) is a tuple \(\mathcal {G}^\varphi _{M_s} = (V_{\forall }, V_{\exists }, E, \Delta )\), where
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\(V_{\forall } = \{ \ulcorner N_t, \psi \urcorner \mid N_t \in \mathcal {M}, \psi \in \mathcal {UNNF}\} \cup \{\ulcorner N_t, X, \chi , \psi \urcorner \mid N_t \in \mathcal {M}, X \subseteq S, \chi \in \mathcal {NNF}, \psi \in \mathcal {NNF}\}\) is the set of vertices of the \(\forall\)-player,
-
\(V_{\exists } = \{ \ulcorner N_t, \psi \urcorner \mid N_t \in \mathcal {M}, \psi \in \mathcal {ENNF}\}\) is the set of vertices of the \(\exists\)-player,
-
\(E \subset (V_\forall \cup V_\exists ) \times (V_\forall \cup V_\exists )\) is the set of edges, where E is a union of the following sets
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\(\{(\ulcorner N_t, p \urcorner , \ulcorner N_t, \top \urcorner ),(\ulcorner N_t, \lnot q \urcorner , \ulcorner N_t, \top \urcorner ) \mid t \in V(p) \text { and } t \not \in V(q)\}\),
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\(\{(\ulcorner N_t, p \urcorner , \ulcorner N_t, \bot \urcorner ),(\ulcorner N_t, \lnot q \urcorner , \ulcorner N_t, \bot \urcorner ) \mid t \not \in V(p) \text { and } t \in V(q)\}\),
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\(\{(\ulcorner N_t, \psi \wedge \chi \urcorner , \ulcorner N_t, \psi \urcorner ), (\ulcorner N_t, \psi \wedge \chi \urcorner , \ulcorner N_t, \chi \urcorner ) \}\),
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\(\{(\ulcorner N_t, \psi \vee \chi \urcorner , \ulcorner N_t, \psi \urcorner ), (\ulcorner N_t, \psi \vee \chi \urcorner , \ulcorner N_t, \chi \urcorner \}\),
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\(\{(\ulcorner N_t, \square _a \psi \urcorner , \ulcorner N_u, \psi \urcorner ) \mid R(a)(t,u)\}\),
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\(\{(\ulcorner N_t, \lozenge _a \psi \urcorner , \ulcorner N_u, \psi \urcorner ) \mid R(a)(t,u)\}\),
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\(\{(\ulcorner N_t, \blacksquare _G \psi \urcorner , \ulcorner N_u, \psi \urcorner ) \mid R(G)(t,u)\}\),
-
\(\{(\ulcorner N_t, \blacklozenge _G \psi \urcorner , \ulcorner N_u, \psi \urcorner ) \mid R(G)(t,u)\}\),
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\(\{(\ulcorner N_t, [\chi ] \psi \urcorner , \ulcorner N_t, X, \chi , \psi \urcorner )\}\),
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\(\{(\ulcorner N_t, X, \chi , \psi \urcorner , \ulcorner N_u, \chi \urcorner ) \mid u \in X\}\),
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\(\{(\ulcorner N_t, X, \chi , \psi \urcorner , \ulcorner N_u, t(\lnot \chi ) \urcorner ) \mid u \in S \setminus X\}\),
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\(\{(\ulcorner N_t, X, \chi , \psi \urcorner , \ulcorner N_t^{X}, \psi \urcorner )\}\),
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\(\{(\ulcorner N_t, [!] \psi \urcorner , \ulcorner N_t, [t(\chi )] \psi \urcorner ) \mid \chi \in \mathcal{E}\mathcal{L}\}\),
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\(\{(\ulcorner N_t, \langle ! \rangle \psi \urcorner , \ulcorner N_t, t(\chi ) \wedge [t(\chi )] \psi \urcorner ) \mid \chi \in \mathcal{E}\mathcal{L}\}\),
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\(\{(\ulcorner N_t, [!]^X \psi \urcorner , \ulcorner N_t, [t(\chi )] \psi \urcorner ) \mid \chi \in \mathcal {ELC}\}\),
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\(\{(\ulcorner N_t, \langle ! \rangle ^X \psi \urcorner , \ulcorner N_t, t(\chi ) \wedge [t(\chi )] \psi \urcorner ) \mid \chi \in \mathcal {ELC}\}\),
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\(\{(\ulcorner N_t, [G] \psi \urcorner , \ulcorner N_t, [t(\chi _G)] \psi \urcorner ) \mid \chi _G \in \mathcal{E}\mathcal{L}^G\}\),
-
\(\{(\ulcorner N_t, \langle G \rangle \psi \urcorner , \ulcorner N_t, t(\chi _G) \wedge [t(\chi _G)] \psi \urcorner ) \mid \chi _G \in \mathcal{E}\mathcal{L}^G\}\),
-
\(\{(\ulcorner N_t, [G]^X \psi \urcorner , \ulcorner N_t, [t(\chi _G)] \psi \urcorner ) \mid \chi _G \in \mathcal {ELC}^G\}\),
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\(\{(\ulcorner N_t, \langle G \rangle ^X \psi \urcorner , \ulcorner N_t, t(\chi _G) \wedge [t(\chi _G)] \psi \urcorner ) \mid \chi _G \in \mathcal {ELC}^G\}\).
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-
\(\Delta\) is the initial vertex \(\ulcorner M_s, \varphi \urcorner\).
The game is played between the \(\forall\)-player and the \(\exists\)-player, and a play consists of a sequence of vertices \(\Delta , \Delta _1, \ldots , \Delta _n\). The play is built by the players such that for some edge \((\Delta _m, \Delta _{m+1}) \in E\) if \(\Delta _m \in V_\forall\), then the universal player chooses \(\Delta _{m+1}\), and if \(\Delta _m \in V_\exists\), then the existential player chooses \(\Delta _{m+1}\). If either player is unable to move, i.e. they are in a \(\top\)-vertex or \(\bot\)-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 \(\ulcorner N_t, \psi \wedge \chi \urcorner\) of a game, then the \(\forall\)-player can either choose to move to vertex \(\ulcorner N_t, \psi \urcorner\) or to vertex \(\ulcorner N_t, \chi \urcorner\). If we are in vertex \(\ulcorner N_t, \blacklozenge _G \psi \urcorner\) of the game, then the \(\exists\)-player can choose any state u of N reachable from t via R(G), thus letting the game to carry on in vertex \(\ulcorner N_u, \psi \urcorner\).
Of special interest are moves that correspond to public announcements and quantifiers. From vertex \(\ulcorner N_t, [\chi ] \psi \urcorner\) the existential player can move to a vertex \(\ulcorner N_t, X, \chi , \psi \urcorner\), 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 \(X \subseteq \{u \in S_N \mid N_u \models \chi \}\), i.e. whether all states in the chosen subset satisfy \(\chi\). Second, the universal player can check whether \(S \setminus X \subseteq \{u \in S_N \mid N_u \models t(\lnot \chi )\}\), i.e. whether all states outside of X are \(\lnot \chi\)-states. The third option is to continue the game in a submodel \(N_t^X\) with the formula \(\psi\). 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 \(\ulcorner N_t, \langle G \rangle \psi \urcorner\) of the game, the existential player can choose any formula \(\chi _G \in \mathcal{E}\mathcal{L}^G\) thus making a move to vertex \(\ulcorner N_t, t(\chi _G) \wedge [t(\chi _G)] \psi \urcorner\), where the universal player can either check that the chosen formula is indeed true, or let the \(\exists\)-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 \(\varphi \in \mathcal {NNF}\), model \(M_s\), and a game \(\mathcal {G}^\varphi _{M_s}\), every play of the game is finite.
Proof
The proof is by structural induction on \(\varphi\).
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 \(\psi\) such that \(\psi <^\forall \varphi\) (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 \(\ulcorner N_t, [\chi ] \psi \urcorner\): 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 \(\ulcorner N_t, X, \chi , \psi \urcorner\). There are three possible choices of the \(\forall\)-player from this vertex: \(\ulcorner N_t, \chi \urcorner\), \(\ulcorner N_u, t(\lnot \chi ) \urcorner\), and \(\ulcorner N_t^{X}, \psi \urcorner\). Observe that \(\chi <^\forall [\chi ] \psi\) and \(\psi <^\forall [\chi ] \psi\), and thus plays from \(\ulcorner N_t, \chi \urcorner\) and \(\ulcorner N_t^{X}, \psi \urcorner\) are finite by the IH. Moreover, by Lemma 3 we have that \(\delta _\forall (t(\lnot \chi )) = \delta _\forall (\lnot \chi )\) and \(s(t(\lnot \chi )) = s(\lnot \chi )\). It holds that \(t(\lnot \chi ) <^\forall [\chi ] \psi\), and thus plays from \(\ulcorner N_u, t(\lnot \chi ) \urcorner\) are finite by the IH.
Case \(\ulcorner N_t, [!] \psi \urcorner\): there is just one step from this vertex to some \(\ulcorner N_t,\) \([t(\chi )] \psi \urcorner\) such that \(\psi \in \mathcal{E}\mathcal{L}\). Observe that \([!]\psi <^\forall [t(\chi )] \psi\), and thus by the IH, we conclude that the play from this vertex is finite.
Cases \(\ulcorner N_t, \langle ! \rangle \psi \urcorner\), \(\ulcorner N_t, [!]^X \psi \urcorner\), \(\ulcorner N_t, \langle ! \rangle ^X \psi \urcorner\), \(\ulcorner N_t, [G] \psi \urcorner\), \(\ulcorner N_t, \langle G \rangle \psi \urcorner\), \(\ulcorner N_t, [G]^X \psi \urcorner\), and \(\ulcorner N_t, \langle G \rangle ^X \psi \urcorner\) are similar to the previous one. \(\square\)
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 \(\exists\)-player has a winning strategy in a game \(\mathcal {G}^\varphi _{M_s}\) if and only if \(M_s \models \varphi\).
Proof
From right to left.
Base Case: Assume that \(M_s \models p\). Then the corresponding formula game consists only of one \(\exists\)-step from \(\ulcorner M_s, p \urcorner\) to \(\ulcorner M_s, \top \urcorner\), 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 \(\lnot p\).
Induction Hypothesis (IH): Assume that for all pointed submodels \(N_t\) of M and all formulas \(t(\psi )\) in NNF such that \(t(\psi ) <^\forall \varphi\), if \(N_t \models t(\psi )\), then \(\ulcorner N_t, t(\psi ) \urcorner\) is a winning position for the \(\exists\)-player.
Propositional and epistemic cases are straightforward.
Case \(N_t \models [\psi ] \chi\): by the semantics this is equivalent to \(N_t \models \lnot \psi\) or \(N_t^\psi \models \chi\). First, assume that \(N_t \models \lnot \psi\), and consider \(X = \{ u \in S_N \mid N_u \models \psi \}\) and \(Y = S_N\setminus X\), where X can be an empty set. We have that for all \(u \in X\): \(N_u \models \psi\) and for all \(v \in Y\): \(N_v \models t(\lnot \psi )\). By the IH this implies that \(\ulcorner N_u, \psi \urcorner\) and \(\ulcorner N_v, t(\lnot \psi ) \urcorner\) are winning positions for the existential player for all \(u \in X\) and \(v \in Y\). Hence, \(\ulcorner N_t, X, \psi , \chi \urcorner\) is also a winning position for the \(\exists\)-player that she can choose from \(\ulcorner N_t, [\psi ] \chi \urcorner\).
If \(N_t^\psi \models \chi\), then again we consider \(X = \{ u \in S_N \mid N_u \models \psi \}\) similarly to the case of \(N_t \models \lnot \psi\). Since \(\chi <^\forall [\psi ]\chi\), then by the IH we have that \(\ulcorner N_t^X, \chi \urcorner\) is a winning position for the \(\exists\)-player. Hence, \(\ulcorner N_t, X, \psi , \chi \urcorner\) is a winning position for the \(\exists\)-player that she can choose from \(\ulcorner N_t, [\psi ] \chi \urcorner\).
Case \(N_t \models \langle ! \rangle \psi\): by the semantics \(N_t \models \langle ! \rangle \psi\) is equivalent to \(\exists \chi \in \mathcal{E}\mathcal{L}\): \(N_t \models \langle \chi \rangle \psi\). The latter is equivalent to \(N_t \models t(\chi ) \wedge t([\chi ] \psi )\). Since \(t(\chi ) \wedge t([\chi ] \psi ) <^\forall \langle ! \rangle \psi\), we can use the IH to conclude that the \(\exists\)-player can always choose a step in the game that corresponds to the winning position \(\ulcorner N_t, t(\chi ) \wedge t([\chi ] \psi ) \urcorner\). Thus, \(\ulcorner N_t, \langle ! \rangle \psi \urcorner\) is also a winning position for the existential player.
Cases for \([ !] \psi\), \([!]^X \psi\), \(\langle ! \rangle ^X \psi\), \([G] \psi\), \(\langle G \rangle \psi\), \([G]^X \psi\), and \(\langle G \rangle ^X \psi\) are similar to the previous one.
From left to right. A similar argument as in the opposite direction for the contraposition: if \(M_s \not \models \varphi\), then the \(\forall\)-player has a winning strategy in game \(\mathcal {G}^\varphi _{M_s}\). \(\square\)
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.
4.3 APALC and GALC relative to APALC\(^X\) and GALC\(^X\)
Now we turn to the key question of the relative expressivity of \(\mathcal {APALC}\) and \(\mathcal {APALC}^X\), and of \(\mathcal {GALC}\) and \(\mathcal {GALC}^X\). 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.
Models M and N. Relation for agent a is depicted by dashed lines, relation b is shown by solid lines, and c’s relations are double lines. Propositional variable p is true in black states, and q is true in boxed states at the ends of chains
In both models, there are chains starting from s and t correspondingly of length \(n+2\) for each \(n \in \mathbb {N}\). 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, \(M_{s^u} \leftrightarrows M_{s^l}\), \(M_{n^u} \leftrightarrows M_{n^l}\) and \(M_{n^u_m} \leftrightarrows M_{n^l_m}\) for all \(n,m \in \mathbb {N}\) 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 \(n_m^u\) from the upper part. Formula \(\lnot \blacklozenge _{\{b,c\}} q\) is false in \(N_{n_m^u}\), and it is satisfied in \(N_u\) (or any other state of the infinite chain).
Next, we show that there are formulas of \(\mathcal {APALC}^X\) and \(\mathcal {GALC}^X\) that can distinguish \(M_s\) and \(N_t\).
Lemma 4
There are formulas \(\psi _1 \in \mathcal {APALC}^X\) and \(\psi _2 \in \mathcal {GALC}^X\), such that \(M_s \not \models \psi _*\) and \(N_t \models \psi _*\), where \(*\in \{1,2\}\).
Proof
Let
and \(\langle ! \rangle ^X \varphi \in \mathcal {APALC}^X\). In order to see that \(N_t \models \langle ! \rangle ^X \varphi\), consider the following announcement:
Note that we use an announcement with q here, while q does not appear in \(\varphi\). Also note that this announcement belongs to \(\mathcal {ELC}\). In model N, formula \(\lnot \blacklozenge _{\{b,c\}} q\) 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 \(n^* \in \{n^* \mid n \in \mathbb {N}\setminus \{0\} \text { and } * \in \{u, l\}\}\). We have that \(N_{n^*} \not \models \lnot p \rightarrow (\square _{\{b,c\}} q \vee \Diamond _b p)\) 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 \(\lnot q\). Hence, \(N_{n^*} \not \models \psi _c\). Now consider state \(0^l\): it holds that \(N_{0^l} \not \models q \rightarrow \square _a (\blacklozenge _{\{b,c\}} q \vee \Diamond _b p)\) since there is an a-arrow to state u and \(N_u \models \lnot \blacklozenge _{\{b,c\}} q \wedge \lnot \Diamond _b p\). On the other hand, all a-arrows from \(0^u\) lead to states where either \(\blacklozenge _{\{b,c\}} q\) or \(\Diamond _b p\) 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 \(N_{0^u} \models \lnot p \rightarrow (\square _{\{b,c\}} q \vee \Diamond _b p)\), and indeed \(N_{0^u} \models \square _{\{b,c\}} q\), and hence \(N_{0^u} \models \psi _c\).
Second, pick any circle state apart from \(t^u\) and \(t^l\). To see that \(N_\circ \not \models \lnot p \rightarrow (\square _{\{b,c\}} q \vee \Diamond _b p)\), notice that \(N_\circ \models \lnot p\), \(N_\circ \not \models \square _{\{b,c\}} q\) (q is false in the current state) and \(N_\circ \not \models \Diamond _b p\) (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, \(N_\circ \not \models \psi _c\). In both \(t^u\) and \(t^l\), \(\Diamond _b p\) is true and hence the whole formula is true. Finally, we have \(N_{\bullet } \models \psi _c\) vacuously, since \(N_{\bullet } \not \models \lnot p\) and \(N_{\bullet } \not \models q\). Thus, the result of updating N with \(\psi _c\) is bisimilar to the model O in Fig. 5.
It is easy to check that \(O_t \models \varphi\). Formula \(\varphi\) is constructed in such a way that it can only be satisfied by model \(O_t\) (up to bisimulation). The first conjunct in \(\varphi\) checks the truth of p in the current state. The second conjunct specifies that there is a \(\lnot p\)-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 \(M_s \not \models \langle ! \rangle ^X \varphi\), 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 \(\psi \in \mathcal {ELC}\) such that \(M_s^{\psi } \models \varphi\). In particular, we have that \(M^\psi _s \models \Diamond _b (\lnot p \wedge \square _a \Diamond _b p)\). By the semantics, this means that there is a state, either \(s^u\) or \(s^l\) (or both), such that \(\square _a \Diamond _b p\) holds in that state. By the construction of M, the only way \(\square _a \Diamond _b p\) can be satisfied in \(s^u\) or \(s^l\) is by removing all other a-reachable states. Since \(M_{s^u} \leftrightarrows M_{s^l}\), by Theorem 1 we have that \(M_{s^u} \models \square _a \Diamond _b p\) if and only if \(M_{s^l} \models \square _a \Diamond _b p\). But this contradicts the third conjunct of \(\varphi\). Since \(\psi\) was arbitrary, \(M_s \not \models \langle ! \rangle ^X \varphi\).
Finally, note that the same argument works for \(\mathcal {GALC}^X\). Indeed, formula \(\psi _c\) belongs to \(\mathcal {ELC}^{\{c\}}\), and thus we have that \(N_t \models \langle \{c\} \rangle ^X \varphi\). The fact that \(M_s \not \models \langle \{c\} \rangle ^X \varphi\) again follows from the proof of \(M_s \not \models \langle ! \rangle ^X \varphi\) noting that the choice of \(\psi\) was arbitrary. \(\square\)
Submodel O of model N
Now we are left to show that models \(M_s\) and \(N_t\) cannot be distinguished by none of the formulas of \(\mathcal {APALC}\) or \(\mathcal {GALC}\). 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 \(\psi\) of \(\mathcal {APALC}\) or \(\mathcal {GALC}\) such that it is true in one model and false in the other. By Proposition 7 this means that the \(\exists\)-player has a winning strategy in one model, and the \(\forall\)-player has a winning strategy in the other model.
We will play two games simultaneously, one over \(M_s\) and \(\varphi\), and the other over \(N_t\) and \(\varphi\). 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.
Theorem 7
\(\mathcal {APALC}^X \not \leqslant \mathcal {APALC}\) and \(\mathcal {GALC}^X \not \leqslant \mathcal {GALC}\).
Proof
In Lemma 4 we have seen that formulas \(\langle ! \rangle ^X \varphi \in \mathcal {APALC}^X\) and \(\langle \{c\} \rangle ^X \in \mathcal {GALC}^X\) distinguish models \(M_s\) and \(N_t\). Now assume towards a contradiction that there is a \(\psi \in \mathcal {APALC} \cup \mathcal {GALC}\) that is equivalent to either \(\langle ! \rangle ^X \varphi\) or \(\langle \{c\} \rangle ^X \varphi\) accordingly. Without loss of generality, we also assume that \(\psi \in \mathcal {NNF}\). Since \(\psi\) has a finite number of symbols, there must be a \(q \in P\) such that q does not occur in \(\psi\).
Since \(\psi\) is equivalent to \(\langle ! \rangle ^X \varphi\) or \(\langle \{c\} \rangle ^X \varphi\), we have that \(M_s \not \models \psi\) and \(N_t \models \psi\). This means, by Proposition 7, that the \(\forall\)-player has a winning strategy in \(\mathcal {G}_{M_s}^\psi\), and the \(\exists\)-player has a winning strategy in \(\mathcal {H}_{N_t}^\psi\). Given \(n = md(\psi )\), we consider the following relation \(B \subseteq S_M \times S_N\):
It is clear that B is an \(P\setminus \{q\}\)-\(2^n\)-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 \(M_{({2^n})^l_k} \leftrightarrows ^{2^n} N_{\omega _k}\) if \(k < {2^n}\), and all \(\omega _k\) with \(k \geqslant {2^n}\) 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 \(P\setminus \{q\}\)-\((2^{n}-k)\)-bisimilar.
Base Case: Let \(\psi = p\) for some \(p \in P\setminus \{q\}\). Since all states that are in relation B satisfy the same propositional variables from \(P\setminus \{q\}\), we have \(P\setminus \{q\}\)-0-bisimilarity.
Induction Hypothesis (IH): After k steps of a game, for all states \(s^{\prime}\) and \(t^{\prime}\) from all submodels \(M^{\prime}\) and \(N^{\prime}\), if \(B(s^{\prime} , t^{\prime} )\), then \(M^{\prime} _{s^{\prime} } \leftrightarrows ^{2^{n} -k }_{P\setminus \{q\}} N^{\prime} _{t^{\prime} }\).
Cases \(\psi = \chi \wedge \tau\) and \(\psi = \chi \vee \tau\). In game \(\mathcal {G}^\psi _{M_s}\), the \(\forall\)-player makes a move from \(\ulcorner M^{\prime} _{s^{\prime} } \chi \wedge \tau \urcorner\) to either \(\ulcorner M^{\prime} _{s^{\prime} }, \chi \urcorner\) or \(\ulcorner M^{\prime} _{s^{\prime} }, \tau \urcorner\). The universal player makes the same choice (either \(\chi\) or \(\tau\)) in game \(\mathcal {H}_{N_t}^\psi\). Since such a move does not change current states, we have \(M^{\prime} _{s^{\prime} } \leftrightarrows ^{2^{n-k}}_{P\setminus \{q\}} N^{\prime} _{t^{\prime} }\) by the IH, which implies \(M^{\prime} _{s^{\prime} } \leftrightarrows ^{2^{n}-k-1}_{P\setminus \{q\}} N^{\prime} _{t^{\prime} }\). Similarly for \(\psi = \chi \vee \tau\) and the \(\exists\)-player.
Cases \(\psi = \square _a \chi\) and \(\psi = \Diamond _a \chi\). In game \(\mathcal {G}^\psi _{M_s}\), the \(\forall\)-player makes a move, according to her winning strategy, from \(\ulcorner M^{\prime} _{s^{\prime} }, \square _a \chi \urcorner\) to some \(\ulcorner M^{\prime} _{s^*}, \chi \urcorner\) such that \(R_M(a)(s^{\prime} , s^*)\). A similar move is made in game \(\mathcal {H}_{N_t}^\psi\): from \(\ulcorner N^{\prime} _{t^{\prime} }, \square _a \chi \urcorner\) to some \(\ulcorner N^{\prime} _{t^*}, \chi \urcorner\) such that \(R_N(a)(t^{\prime} , t^*)\) and \(M^{\prime} _{s^*} \leftrightarrows ^{2^{n}-k-1}_{P\setminus \{q\}} N^{\prime} _{t^*}\). 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 \(\mathcal {H}_{N_t}^\psi\) to state u on the infinite chain, then the move will be matched by a move to state \(({2^n})_0^l\) in game \(\mathcal {G}^\psi _{M_s}\), i.e. the first state of chain \((2^n)^l\) in model M. Moves along the infinite chain are matched by moves along chain \((2^n)^l\), and all moves on the infinite chain beyond \(\omega _{2^n}\) are matched by the player choosing to stay in state \((2^n)^l\), which is the last state of the chain.
The similar reasoning applies to \(\psi = \Diamond _a \chi\) and the \(\exists\)-player.
Cases \(\psi = \blacksquare _G \chi\) and \(\psi = \blacklozenge _G \chi\). These are cases similar to the previous ones with substituting R(a) by R(G). Again, according to B, moves on the infinite chain are matched by moves on the chain of size \(2^n\).
Case \(\psi = [\chi ]\tau\). Since the \(\exists\)-player has a winning strategy in \(\mathcal {H}_{N_t}^\psi\), then she can choose a subset \(X \subseteq S^{N^{\prime} }\) such that \(\ulcorner N^{\prime} _{t^{\prime} }, X, \chi , \tau \urcorner\) is a winning position. At the same time, she chooses \(Y \subseteq S^{M^{\prime} }\) in game \(\mathcal {G}^\psi _{M_s}\), where \(Y = \{s^{\prime} \mid \exists t^{\prime} \in X: B(s^{\prime} , t^{\prime} )\}\). By the IH, for all \(s^{\prime} \in Y\) and all \(t^{\prime} \in X\), we have \(M^{\prime} _{s^{\prime} } \leftrightarrows ^{2^{n}-k}_{P\setminus \{q\}} N^{\prime} _{t^{\prime} }\), which implies \(M^{\prime} _{s^{\prime} } \leftrightarrows ^{2^{n}-k-1}_{P\setminus \{q\}} N^{\prime} _{t^{\prime} }\). 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 \(\forall\)-player can now reply with one of three possible moves in both games. First, she can choose some state \(s^*\in Y\) (resp. \(t^*\in X\)) to get to position \(\ulcorner M^{\prime} _{s^*}, \chi \urcorner\) (resp. \(\ulcorner N^{\prime} _{t^*}, \chi \urcorner\)). That the bisimulation is preserved follows from the construction of Y and the IH. Similarly for the move of the universal player to position \(\ulcorner M^{\prime} _{s^*}, t(\lnot \chi ) \urcorner\) (resp. \(\ulcorner N^{\prime} _{t^*}, t(\lnot \chi ) \urcorner\)). Finally, if the \(\forall\)-player chooses \(\ulcorner M^Y_{s^{\prime} }, \tau \urcorner\) (resp. \(\ulcorner N^X_{t^{\prime} }, \tau \urcorner\)) in game \(\mathcal {G}^\psi _{M_s}\) (resp. \(\mathcal {H}_{N_t}^\psi\)), then by the IH \(M^Y_{s^{\prime} } \leftrightarrows ^{2^{n}-k}_{P\setminus \{q\}} N^X_{t^{\prime} }\), which implies \(M^Y_{s^{\prime} } \leftrightarrows ^{2^{n}-k-1}_{P\setminus \{q\}} N^X_{t^{\prime} }\).
Cases \(\psi = [ ! ] \chi\) and \(\psi = \langle ! \rangle \chi\). In game \(\mathcal {G}^\psi _{M_s}\), the \(\forall\)-player makes a move, according to her winning strategy, from \(\ulcorner M^{\prime} _{s^{\prime} }, [!] \chi \urcorner\) to some \(\ulcorner M^{\prime} _{s^{\prime} }, [t(\tau )] \chi \urcorner\) such that \(t(\tau ) \in \mathcal{E}\mathcal{L}\). It can be shownFootnote 2 [39, Theorem 8.15] that for each \(d \in \mathbb {N}\), for all states \(s^{\prime} \in S^{M^{\prime} }\) there is a state \(t^{\prime} \in S^{N^{\prime} }\) (and vice versa) such that \(M^{\prime} _{s^{\prime} } \models \varphi\) iff \(N^{\prime} _{t^{\prime} } \models \varphi\) for all \(\varphi \in \mathcal{E}\mathcal{L}\) such that \(md(\varphi ) = d\). This fact in conjunction with \(t(\tau ) \in \mathcal{E}\mathcal{L}\), entails that the universal player can choose the same formula in game \(\mathcal {H}_{N_t}^\psi\) to move to a winning state \(\ulcorner N^{\prime} _{t^{\prime} }, [t(\tau )] \chi \urcorner\). Note that the modal depth of \(t(\tau )\) 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 \(P\setminus \{q\}\)-0-bisimilarity. It is enough for our purposes, since we are interested only in up to \(2^n\) moves. Hence, we still have \(M^{\prime} _{s^{\prime} } \leftrightarrows ^{2^{n-k}}_{P\setminus \{q\}} N^{\prime} _{t^{\prime} }\) by the IH, which implies \(M^{\prime} _{s^{\prime} } \leftrightarrows ^{2^{n}-k-1}_{P\setminus \{q\}} N^{\prime} _{t^{\prime} }\).
The case of \(\psi = \langle ! \rangle \chi\) is similar with the existential player as the protagonist.
Cases \(\psi = [G]\chi\) and \(\psi = \langle G \rangle \chi\) are similar to the cases above substituting \(t(\tau )\) with \(t(\tau _G)\), and \(\mathcal{E}\mathcal{L}\) with \(\mathcal{E}\mathcal{L}^G\).
As a result of these two simultaneous games over formula \(\psi\) and models \(M_s\) and \(N_t\) we end up in states in both games where the \(\exists\)-player (resp. the \(\forall\)-player) has a winning strategy. This contradicts the assumption that the \(\forall\)-player (resp. the \(\exists\)-player) has a winning strategy in one of the games, or, equivalently, it contradicts the fact that \(M_s \not \models \psi\) iff \(N_t \models \psi\). \(\square\)
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 \(\mathcal {APALC}\) and \(\mathcal {GALC}\) and cannot be expressed by \(\mathcal {APALC}^X\) and \(\mathcal {GALC}^X\). Indeed, one may have expected that since quantifiers of \(\mathcal {APALC}^X\) and \(\mathcal {GALC}^X\) range over a strictly more expressive language than quantifiers of \(\mathcal {APALC}\) and \(\mathcal {GALC}\) (\(\mathcal {ELC}\) in the first case, and \(\mathcal{E}\mathcal{L}\) in the second case), then \(\mathcal {APALC}^X\) and \(\mathcal {GALC}^X\) 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 \(\mathcal {APALC}\) and \(\mathcal {GALC}\) 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 \(n \in \mathbb {N}\). 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 \(p^{\prime}\) is true only on the infinite chain of model N in the black square state.
Models M and N. Relation for agent a is depicted by dashed lines, relation for b is shown by solid lines, and c’s relations are double lines. Propositional variable p is true in black circle states, q is true in boxed states at the ends of chains, and \(p^{\prime}\) is true in the black square state
Lemma 5
There are formulas \(\psi _1 \in \mathcal {APALC}\) and \(\psi _2 \in \mathcal {GALC}\), such that \(M_s \not \models \psi _*\) and \(N_t \models \psi _*\), where \(*\in \{1,2\}\).
Proof
Let \(\varphi\) be as in the proof of Theorem 7, and \(\langle ! \rangle \varphi \in \mathcal {APALC}\). Similarly to the proof of Theorem 7, formula \(\varphi\) is satisfied in model \(O_t\) from Fig. 5 and all models bisimilar to it. That \(M_s \not \models \langle ! \rangle \varphi\) can be shown similarly to \(M_s \not \models \langle ! \rangle ^X \varphi\) (see proof of Theorem 7) noting that although upper and lower halves of M are not bisimialr, they nevertheless satisfy the same formulas of \(\mathcal{E}\mathcal{L}\) [39, Theorem 8.15].
We now argue that \(N_t \models \langle ! \rangle \varphi\). Notice that \(p^{\prime}\) holds on the infinite chain starting at state v. Since the quantification over announcements is implicit, we can use \(p^{\prime}\) and q in announcements. Moreover, we can use announcement of arbitrary finite depth. Before giving the announcement that results in a model satisfying \(\varphi\), we show how using \(p^{\prime}\) 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 \(p^{\prime}\). See Fig. 7 for the representation of the approach.
Thus the distinguishing formula for v is
where \(\Diamond ^{m}_i\) stands for m alternating c- and b-diamonds, and \(\square ^{m}_j\) stands for m alternating b- and c-boxes. Informally, the first conjunct means that state v is at most \(m+1\) steps away from the \(p^{\prime}\)-state, the second conjunct specifies that the state is at least \(m+1\) steps away from the \(p^{\prime}\)-state, and the third conjunct says that v is two steps away from the p-state.
Now we can use \(\chi _v\) 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 \(\varphi\) can be shown similarly to the proof of Lemma 4 with state u being substituted by state v, and \(\blacklozenge _{\{b,c\}} \lnot q\) being substituted with \(\lnot \chi _v\). The argument for \(\langle \{c\} \rangle \varphi \in \mathcal {GALC}\) also follows noting that \(\psi _c \in \mathcal{E}\mathcal{L}^{\{c\}}\). \(\square\)
A segment of the infinite vertical chain from model N. A formula over or under a state means that the formula is true in the corresponding state
Before continuing with the expressivity proof, let us take another look at the two models. First, the reader can notice that they are \(P\setminus \{p^{\prime} ,q\}\)- and \(P\setminus \{p^{\prime} \}\)-bisimilar, and hence they satisfy the same formulas of \(\mathcal {ELC}\) that do not contain \(p^{\prime}\). 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 \(\chi _4\), need to use infinite chains. In model \(N_t\) or any submodel thereof containing the \(p^{\prime}\)-state, we can use formula \(\chi _v\) 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 \(N_{3^l_2} \models \chi _4^l\) and \(N_{3^u_2} \not \models \chi _4^l\). 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 \(\blacksquare _{\{b,c\}} \lnot q\) 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 \(M^{\prime} _{s^{\prime} }\) that do not have q-states at their ends, and that will thus satisfy \(\blacksquare _{\{b,c\}} \lnot q\). Hence, suppose that in some \(M^{\prime} _{s^{\prime} }\) 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 \(\Diamond _i^{d+1}\) 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 \(\chi _u\), 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 \(\chi ^l_4\) and \(\chi ^u_4\) for \(N_t\) with \(\chi _v\) being substituted with \(\chi _u\).
Finally, for the construction of the formula that is true only in \(3^l_2\), assume that \(\chi _6^l\) and \(\chi _5^l\) 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 \(\chi _5^l\) and does not reach a state satisfying \(\chi _6^l\) on its chain. Formally,
The described method of constructing distinguishing formulas of particular states will be used in the proof of Theorem 8.
Theorem 8
\(\mathcal {APALC} \not \leqslant \mathcal {APALC}^X\) and \(\mathcal {GALC} \not \leqslant \mathcal {GALC}^X\).
Proof
According to Lemma 5, there are formulas \(\langle ! \rangle \varphi \in \mathcal {APALC}\) and \(\langle \{c\} \rangle \varphi \in \mathcal {GALC}\) that distinguish models \(M_s\) and \(N_t\). Now assume towards a contradiction that there is a \(\psi \in \mathcal {APALC}^X \cup \mathcal {GALC}^X\) that is equivalent to either \(\langle ! \rangle \varphi\) or \(\langle \{c\} \rangle \varphi\) accordingly. Without loss of generality, we also assume that \(\psi \in \mathcal {NNF}\). Since \(\psi\) has a finite number of symbols, there must be \(q, p^{\prime} \in P\) such that q and \(p^{\prime}\) do not occur in \(\psi\). Moreover, let \(n=md(\psi )\).
Similarly to the proof of Theorem 7, we define a \(P\setminus \{q,p^{\prime} \}\)-m-bisimulation relation \(B \subseteq S_M \times S_N\):
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 \(\mathcal {G}_{M_s}^\psi\) over \(M_s\) and \(\psi\), and game \(\mathcal {H}_{N_t}^\psi\) over \(N_t\) and \(\psi\). We also assume towards a contradiction that the \(\forall\)-player has a winning strategy in \(\mathcal {G}_{M_s}^\psi\), and the \(\exists\)-player has a winning strategy in \(\mathcal {H}_{N_t}^\psi\).
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 \(s^{\prime}\) and \(t^{\prime}\) from all submodels \(M^{\prime}\) and \(N^{\prime}\), if \(B(s^{\prime} , t^{\prime} )\), then \(M^{\prime} _{s^{\prime} } \leftrightarrows ^{2^{n}-k}_{P\setminus \{q,p^{\prime} \}} N^{\prime} _{t^{\prime} }\).
Cases \(\psi = [!]^X\chi\) and \(\psi = \langle ! \rangle ^X \chi\). In game \(\mathcal {G}^\psi _{M_s}\), the \(\forall\)-player makes a move, according to her winning strategy, from \(\ulcorner M^{\prime} _{s^{\prime} }, [!] \chi \urcorner\) to some \(\ulcorner M^{\prime} _{s^{\prime} }, [t(\tau )] \chi \urcorner\) such that \(t(\tau ) \in \mathcal {ELC}\). Due to the construction of our models, we cannot guarantee that choosing \(t(\tau )\) in \(\mathcal {H}_{N_t}^\psi\) will result in \(P\setminus \{q,p^{\prime} \}\)-\((2^n-k)\)-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 \(\tau ^{\prime} \in \mathcal {ELC}\) such that \(\ulcorner N^{\prime} _{t^{\prime} }, [t(\tau ^{\prime} )] \chi \urcorner\) is a corresponding winning move in \(\mathcal {H}_{N_t}^\psi\). Construction of \(\tau ^{\prime}\) depends on the way the original \(\tau\) updates \(M^{\prime}\). In particular, presence of the infinite chain in \(M^{\prime}\) and of \(p^{\prime}\)-state in \(N^{\prime}\) 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 \(\mathcal {G}^\psi _{M_s}\) the \(\forall\)-player chooses such a \(\tau\) that updating \(M^{\prime}\) with the formula does not affect the infinite chain, does not remove an infinite number of q-states, and \(\tau\) does not contain \(p^{\prime}\), then she can make the same choice of \(\tau\) in game \(\mathcal {H}_{N_t}^\psi\) in position \(\ulcorner N^{\prime} _{t^{\prime} }, [!] \chi \urcorner\). And vice versa for game \(\mathcal {H}_{N_t}^\psi\). Such an announcement does not affect the ability to distinguish upper and lower halves of both models, thus retaining \(P\setminus \{q,p^{\prime} \}\)-\((2^{n}-k-1)\)-bisimilarity.
Assume now that in game \(\mathcal {G}^\psi _{M_s}\) formula \(\tau\) contains \(p^{\prime}\). Since the valuation of \(p^{\prime}\) in M is empty, we can get an equivalent \(\tau ^{\prime}\) for game \(\mathcal {H}_{N_t}^\psi\) by substituting \(p^{\prime}\) in \(\tau\) with \(\bot\). This will ensure that updating \(M^{\prime}\) with \(\tau\) and updating \(N^{\prime}\) with \(\tau ^{\prime}\) results in \(P\setminus \{q,p^{\prime} \}\)-\((2^{n}-k-1)\)-bisimilar models.
Let updating \(M^{\prime}\) with \(\tau\) 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 \(\chi _u\) with any d, there will a finite chain satisfying it. To model such an effect in \(N^{\prime}\), the \(\forall\)-player chooses \(\tau ^{\prime} := \tau \wedge \lnot p^{\prime}\), that removes the \(p^{\prime}\)-state once being announced. As a result, we also lose the power to distinguish the upper and lower parts in \(N^{\prime}\). Moreover, since the \(p^{\prime}\)-state is \(2^n+2\) away from t, we have \(M^{\prime} _{s^{\prime} } \leftrightarrows ^{2^{n}-k-1}_{P\setminus \{q,p^{\prime} \}} N^{\prime} _{t^{\prime} }\).
Now consider game \(\mathcal {H}_{N_t}^\psi\) and \(\tau\) that, once being announced, removes an infinite number of q-states. Since the \(p^{\prime}\)-state is still present in the updated model \((N^{\prime} )^\tau\), we need to retain the power to distinguish upper and lower parts in model \(M^{\prime}\). To this end, in game \(\mathcal {G}^\psi _{M_s}\) the \(\forall\)-player chooses \(\tau ^{\prime}\) announcement which would remove a finite number of q-states in \(M^{\prime}\). It is enough to consider only first \(2^{n}\) chains. Since the number of states to remove is finite, the universal player can choose \(\bigwedge \lnot \chi _{i^l_j}\), where \(\chi _{i^l_j}\) 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 \(\chi _u\) for various d’s, while also retaining \(P\setminus \{q,p^{\prime} \}\)-\((2^{n}-k-1)\)-bisimilarity.
Finally, let updating \(M^{\prime}\) with \(\tau\) in game \(\mathcal {G}^\psi _{M_s}\) cut the infinite chain to some finite length. In the resulting updated model, each state \(\omega _i\) 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 \(N^{\prime}\) in game \(\mathcal {H}_{N_t}^\psi\), the \(\forall\)-player can choose \(\tau ^{\prime} := \tau \wedge \lnot p^{\prime}\), thus making it impossible also in \(N^{\prime}\) to distinguish upper and lower halves and maintaining the \(P\setminus \{q,p^{\prime} \}\)-\((2^{n}-k-1)\)-bisimilarity.
If in game \(\mathcal {H}_{N_t}^\psi\) the choice of \(\tau\) is such that in the resulting update \((N^{\prime} )^\tau\) the infinite chain is cut, then we consider two cases. First, suppose that the chain was cut in such a way that \(\omega _i\) is the last state of the now finite chain, and that the \(p^{\prime}\)-state is still in \(S^{(N^{\prime} )^\tau }\). Then we just need to cut a finite chain of length greater than \(2^n\) (to maintain the \(P\setminus \{q,p^{\prime} \}\)-\((2^{n}-k-1)\)-bisimilarity) in model \(M^{\prime}\) to the same length i. This can be done by the \(\forall\)-player choosing \(\bigwedge \lnot \chi ^l_j\), where \(\chi ^l_j\) are distinguishing formulas of states on the chosen finite chain. Second, if the chain was cut in such a way that \(\omega _i\) is the last state of the now finite chain, and that the \(p^{\prime}\)-state is not in \(S^{(N^{\prime} )^\tau }\), then the infinite chain of \(M^{\prime}\) should be cut to the same length. This can be done by the choice of \(\Diamond ^i_j \chi _u\) by the \(\forall\)-player, where \(\Diamond ^i_j\) 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 \(P\setminus \{q,p^{\prime} \}\)-\((2^{n}-k-1)\)-bisimilarity.
The case of \(\psi = \langle ! \rangle \chi\) can be shown by similar reasoning, substituting the \(\forall\)-player with the \(\exists\)-player.
Cases \(\psi = [G]^X\chi\) and \(\psi = \langle G \rangle ^X \chi\). 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 \(\square _a\) for all \(a \in G\). This is due to the fact that group announcements quantify over \(\mathcal {ELC}^G\). 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 \(\square _b\) and thus should also be satisfied in \(3^l_1\). Instead, agent b can announce \(\square _b (\chi _{3^l_0} \vee \chi _{3^l_1}) \in \mathcal {ELC}^{\{b\}}\) 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 \(\exists\)-player (resp. the \(\forall\)-player) has a winning strategy. This contradicts the assumption that the \(\forall\)-player (resp. the \(\exists\)-player) has a winning strategy in the other model, or, equivalently, it contradicts the fact that \(M_s \not \models \psi\) iff \(N_t \models \psi\). \(\square\)
4.4 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 \(\mathcal {APAL}\) and \(\mathcal {GAL}\) [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 \(\mathcal {GALC}\) and \(\mathcal {GALC}^X\) are not at least as expressive as \(\mathcal {APALC}\) and \(\mathcal {APALC}^X\).
First, the authors of [2] consider an \(\mathcal {APAL}\) formula \(\langle ! \rangle (\square _a p \wedge \lnot \square _b \square _a p)\), and assume towards a contradiction that there is an equivalent formula \(\varphi\) of \(\mathcal {GAL}\) not containing q. Then, models \(M_u\) and \(N_u\) from Fig. 8 are considered, noting that \(M_u \not \models \langle ! \rangle (\square _a p \wedge \lnot \square _b \square _a p)\) and \(N_u \models \langle ! \rangle (\square _a p \wedge \lnot \square _b \square _a p)\). In particular, announcement of \(p \vee \lnot q\) makes \(\square _a p \wedge \lnot \square _b \square _a p\) true in \(N_u\) (see Fig. 8 and model \(N^{p \vee \lnot q}\)). Since \(p \vee \lnot q \in \mathcal {ELC}\), we also have that \(\langle ! \rangle ^X (\square _a p \wedge \lnot \square _b \square _a p)\) is a distinguishing formula for \(M_u\) and \(N_u\). Moreover, \(\langle ! \rangle (\square _a p \wedge \lnot \square _b \square _a p) \in \mathcal {APALC}\) and \(\langle ! \rangle ^X (\square _a p \wedge \lnot \square _b \square _a p) \in \mathcal {APALC}^X\), and hence \(M_u\) and \(N_u\) are distinguishable by formulas of \(\mathcal {APALC}\) and \(\mathcal {APALC}^X\).
Models M, N, and \(N^{p \vee \lnot q}\). Relation for agent a is depicted by dashed lines and b’s relation is shown by solid lines. Propositional variable p is true in black states, and propositional variable q is true in square states
The argument that \(\varphi\) 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 \(P \setminus \{q\}\)-bisimilar and thus satisfy the same formulas of \(\mathcal {PALC}\) 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 \([G]\chi\) if and only if they satisfy \([G]^X \chi\).
Theorem 9
\(\mathcal {APALC} \not \leqslant \mathcal {GALC}\), \(\mathcal {APALC} \not \leqslant \mathcal {GALC}^X\), \(\mathcal {APALC}^X \not \leqslant \mathcal {GALC}\), and \(\mathcal {APALC} \not \leqslant \mathcal {GALC}^X\).
The fact that \(\mathcal {GALC}\) and \(\mathcal {GALC}^X\) are not at least as expressive as \(\mathcal {APALC}\) and \(\mathcal {APALC}^X\) follows from the proof of \(\mathcal {GAL} \not \leqslant \mathcal {CAL}\) [14, 15], where \(\mathcal {CAL}\) is the language of coalition announcement logic defined by
The semantics of coalition announcement modality \([ \! \langle G \rangle \! ]\varphi\) and its dual \(\langle \! [ G ] \! \rangle \varphi\) is as follows:
Informally, formula \(\langle \! [ G ] \! \rangle \varphi\) means that agents from G have a joint announcements such that no matter what agents from outside of G announce at the same time, \(\varphi\) will hold. Similarly, \([ \! \langle G \rangle \! ]\varphi\) stands for the fact that whatever agents from G jointly announce, there is a counter-announcement by agents from outside of G such that \(\varphi\) 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 \(\lnot p \wedge \square _a \lnot p\), and the rightmost state that satisfies \(\square _a p \wedge [A](\Diamond _b \lnot p \rightarrow \square _a \Diamond _b \lnot p)\). In short, the models are similar in their extremities and differ only in length (see Fig. 9 for reference).
Chain models M and N. Relation for agent a is depicted by dashed lines and b’s relation is shown by solid lines, and c’s relations are double lines. Propositional variable p is true in black states
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 \(\lnot p \wedge \square _a \lnot p\): 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 \(\lnot p \wedge \square _a \lnot p\)-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 \(\varphi\) such that for all \(M_s\), \(M_s \models \varphi\) 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 \(\varphi\) contains the following group announcement operators: \(\langle c \rangle \chi\), \([c]\chi\), and \([\{a,b,c\}] \chi\). Since chain models are finite, by Theorem 2 we can equivalently substitute all occurrences of the abovementioned group announcements with \(\langle c \rangle ^X \chi\), \([c]^X\chi\), and \([\{a,b,c\}]^X \chi\) respectively.
After that, the authors of [14, 15] use formula games for \(\mathcal {GAL}\) and \(\mathcal {CAL}\) to show that no formula of \(\mathcal {CAL}\) 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 \(\varphi \in \mathcal {CAL}\) such that \(M_s \models \varphi\), and for all B-chain models \(N_t\) it holds that \(N_t \not \models \varphi\). The proof proceeds by playing simultaneous formula games over \(2^{md(\varphi )}\)-bisimilar pointed A- and B-models, and the invariant that after i game steps, models are still \(2^{md(\varphi )} - i\)-bisimilar, is maintained.
We can reuse the proof to show that formulas of \(\mathcal {APALC}\) and \(\mathcal {APALC}^X\) do not distinguish A- and B-chain models. For the cases of \(\blacksquare _G \chi\) and \(\blacklozenge _G \chi\), 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 \(2^{md(\varphi )} - i\)-bisimilar models.
Cases \([!]\chi\) and \(\langle ! \rangle \chi\) follow from cases \(\langle \! [ \{a,b,c\} ] \! \rangle \chi\) and \([ \! \langle \{a,b,c\} \rangle \! ] \chi\) from [14, 15] noting that the intersection of a-, b-, and c-relations in chain models is an identity relation. This implies that the grand coalition \(\{a,b,c\}\) can force any submodel of a given chain model with their announcements. Thus, coalition announcement for chain models \(\langle \! [ \{a,b,c\} ] \! \rangle \chi\) is equivalent to \(\langle ! \rangle \chi\) and \([ \! \langle \{a,b,c\} \rangle \! ] \chi\) is equivalent to \([!]\chi\). Since chain models are finite, we also have the result for \(\langle ! \rangle ^X \chi\) and \([!]^X\chi\) by Theorem 2.
Theorem 10
\(\mathcal {GALC} \not \leqslant \mathcal {APALC}\), \(\mathcal {GALC} \not \leqslant \mathcal {APALC}^X\), \(\mathcal {GALC}^X \not \leqslant \mathcal {APALC}\) and \(\mathcal {GALC}^X \not \leqslant \mathcal {APALC}^X\).
5 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 \(\varphi \in \mathcal {GALC}\), \(a \in A\), \(G \subseteq A\), and \(\sharp \not \in P\). The set of necessity forms [19] is defined recursively below:
We will denote the result of replacing of \(\sharp\) with \(\varphi\) in a necessity form \(\eta (\sharp )\) as \(\eta (\varphi )\).
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. \(\square\)
Necessitation rules for common knowledge and group announcements are derivable in GALC.
Lemma 7
Rules ‘From \(\varphi\), infer \(\blacksquare _G \varphi\)’ and ‘From \(\varphi\), infer \([G] \varphi\)’ are derivable in GALC.
Proof
Given \(\varphi\), we can use \(NK\) to derive \(\square ^n_G \varphi\) for each n. Since formulas \(\square ^n_G \varphi\) are in the necessity form, we can apply \(IC\) to infer \(\blacksquare _G \varphi\). Similarly, we can infer \([\psi _G] \varphi\) for each \(\psi _G \in \mathcal{E}\mathcal{L}^G\) using rule \(NA\). After that, application of \(IG\) results in \([G]\varphi\). \(\square\)
Theorem 11
GALC is sound.
Proof
Due to the soundness of GAL, Lemma 6, and the validity of (A9). \(\square\)
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 \(\varphi \in \mathcal {GALC}\), we will use the following measure.
Definition 5.3
Let \(\varphi \in \mathcal {GALC}\). The \(\blacksquare\)-depth \(\delta _{\blacksquare } (\varphi )\) of \(\varphi\) is defined similarly to the quantifier depth \(\delta _{\forall }\) (Definition 2.3) with the following exceptions:
The complexity \(c(\varphi )\) of \(\varphi\) is
Let \(\varphi , \psi \in \mathcal {GALC}\). We have that \(\varphi <^{\forall }_{\blacksquare } \psi\) if and only if \(\delta _\forall (\varphi ) < \delta _\forall (\psi )\), or, otherwise, \(\delta _\forall (\varphi ) = \delta _\forall (\psi )\), and either \(\delta _{\blacksquare } (\varphi ) < \delta _{\blacksquare } (\psi )\), or \(\delta _{\blacksquare } (\varphi ) = \delta _{\blacksquare } (\psi )\) and \(c(\varphi ) < c(\psi )\). Relation \(<^{\forall }_{\blacksquare }\) is a well-founded partial order.
Lemma 8
Let \(\varphi , \psi , \chi \in \mathcal {GALC}\) and \(G \subseteq A\). The following inequalities hold:
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 \(\varphi \in \mathcal {GALC}\) such that \(\varphi , \lnot \varphi \in x\). Theory x is maximal if any theory y such that \(x \subset y\) is inconsistent.
Lemma 9
Theory x is maximal if and only if for all \(\varphi \in \mathcal {GALC}\) we have that either \(\varphi \in x\) or \(\lnot \varphi \in x\).
Proof
Let x be a maximal theory, and assume towards a contradiction that there is a \(\varphi \in \mathcal {GALC}\) such that neither \(\varphi \in x\) nor \(\lnot \varphi \in x\). Then theory \(x \cup \{\varphi \}\) is consistent, and \(x \subset x \cup \{\varphi \}\), which contradicts the definition of maximality.
In the other direction, let us for all \(\varphi \in \mathcal {GALC}\) have that either \(\varphi \in x\) or \(\lnot \varphi \in x\), and x be not maximal. This implies that there is a consistent y such that \(x \subset y\), and in particular that there is a \(\psi\) such that \(\psi \not \in x\) and \(\psi \in y\). Since y is consistent, \(\lnot \psi \not \in y\), and hence \(\lnot \psi \not \in x\). We now have that \(\psi \not \in x\) and \(\lnot \psi \not \in x\) that contradicts our assumption. \(\square\)
Lemma 10
Let \(\varphi , \psi \in \mathcal {GALC}\), if x is a theory, then \(x + \varphi := \{\chi \mid \varphi \rightarrow \chi \in x\}\), \(\square _a x:= \{\chi \mid \square _a \chi \in x\}\), and \([\psi ]x := \{\chi \mid [\psi ]\chi \in x\}\) are theories as well. Also, \(x + \varphi\) is consistent if and only if \(\lnot \varphi \not \in x\).
Proof
An extension of the proof of Lemma 4.11 in [9], where common knowledge cases are dealt with using (A9) and \(IC\). \(\square\)
Lemma 11
For all theories x and all \(\varphi \in \mathcal {GALC}\), it holds that \(x \subseteq x + \varphi\).
Proof
Let us for some \(\psi \in \mathcal {GALC}\) have that \(\psi \in x\). Since x is a theory and thus contains all the instances of propositional tautologies, \(\psi \rightarrow (\varphi \rightarrow \psi ) \in x\). As x is closed under \(MP\), \(\varphi \rightarrow \psi \in x\), and, by Lemma 10, \(\psi \in x + \varphi\). \(\square\)
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 \(x \subseteq y\).
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 \(\{\varphi _0, \varphi _1, \ldots \}\) be an enumeration of formulas of \(\mathcal {GALC}\), and let \(y_0 = x\). Assume that for some \(n \ge 0\), \(x \subseteq y_n\) and \(y_n\) is a consistent theory. If \(\lnot \varphi _n \not \in y_n\), then \(y_{n+1} = y_n + \varphi _n\). Otherwise, there are three cases to consider.
First, if \(\lnot \varphi _n \in y_n\) and \(\varphi _n\) is not of either the form \(\eta (\blacksquare _G \psi )\) or the form \(\eta ([G] \psi )\), then \(y_{n+1} = y_n\). Second, if \(\lnot \varphi _n \in y_n\) and \(\varphi _n\) is of the form \(\eta (\blacksquare _G \psi )\), then \(y_{n+1} = y_n + \lnot \eta (\square _G^n \psi )\), where \(\lnot \eta (\square _G^n \psi )\) is the first formula in the enumeration such that \(\eta (\square _G^n \psi ) \not \in y_n\). Third, if \(\lnot \varphi _n \in y_n\) and \(\varphi _n\) is of the form \(\eta ([G] \psi )\), then \(y_{n+1} = y_n + \lnot \eta ([\psi _G] \psi )\), where \(\lnot \eta ([\psi _G] \psi )\) is the first formula in the enumeration such that \(\eta ([\psi _G] \psi ) \not \in y_n\).
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 \(x \subseteq y_{n+1}\), it is relatively straightforward to show that \(y = \bigcup _{n=0}^{\infty } y_n\) is a maximal consistent theory such that \(x \subseteq y\). \(\square\)
Now we are ready to define the canonical model, where states are maximal consistent theories.
Definition 5.5
We call model \(\mathfrak {M = (S, R, V)}\), where \(\mathfrak {S}\) \(=\) \(\{x \mid x\) is a maximal consistent theory\(\}\), \(\mathfrak {R} (a) = \{(x,y) \mid \square _a x \subseteq y\}\), and \(\mathfrak {V}(p) = \{x \mid p \in x\}\), the canonical model.
Next, we prove the truth lemma.
Lemma 13
For all maximal consistent theories x and \(\varphi \in \mathcal {GALC}\), \(\varphi \in x\) if and only if \(\mathfrak {M}_x \models \varphi\).
Proof
Proofs for boolean, epistemic, some of public announcement cases are quite similar to those in [10, Lemma 11], and can be shown using the axioms of GALC and Lemma 8. We show here only the cases that include group announcements and common knowledge.
Induction hypothesis (IH): For all maximal consistent theories y and formulas \(\psi \in \mathcal {GALC}\), if \(\psi <^{\forall }_{\blacksquare } \varphi\), then \(\psi \in y\) iff \(\mathfrak {M}_y \models \psi\).
Case \(\varphi = [\chi ] \blacksquare _G \psi\). (\(\Rightarrow\)): Suppose that \([\chi ] \blacksquare _G \psi \in x\). Since x contains all theorems of GALC, we have for all \(n \in \mathbb {N}\), \([\chi ]( \blacksquare _G \psi \rightarrow \square ^n_G \psi ) \in x\) and \([\chi ]( \blacksquare _G \psi \rightarrow \square ^n_G \psi ) \rightarrow ([\chi ] \blacksquare _G \psi \rightarrow [\chi ]\square ^n_G \psi ) \in x\) (Proposition 4.46.3 of [39]). Using \(MP\) twice, we get \([\chi ] \square ^n_G \psi \in x\) for all \(n \in \mathbb {N}\). By the IH, this is equivalent to \(\forall n \in \mathbb {N}: \mathfrak {M}_x \models [\chi ] \square ^n_G \psi\). The latter is equivalent to the fact that \(\mathfrak {M}_x \models \chi\) implies \(\mathfrak {M}_x^\chi \models \square ^n_G \psi\) for all n. By the semantics of common knowledge we have that \(\mathfrak {M}_x \models \chi\) implies \(\mathfrak {M}_x^\chi \models \blacksquare _G \psi\), and the latter is \(\mathfrak {M}_x \models [\chi ] \blacksquare _G \psi\) by the semantics of public announcements.
(\(\Leftarrow\)): Assume that \(\mathfrak {M}_x \models [\chi ] \blacksquare _G \psi\). By the semantics, this is equivalent to the fact that \(\mathfrak {M}_x \models \chi\) implies \(\mathfrak {M}_x^\chi \models \blacksquare _G \psi\). By the semantics of common knowledge, the latter is \(\forall n \in \mathbb {N}: \mathfrak {M}_x^\chi \models \square ^n_G \psi\). We can ‘fold’ the public announcement back: \(\forall n \in \mathbb {N}: \mathfrak {M}_x \models [\chi ] \square ^n_G \psi\). By the IH, \(\forall n \in \mathbb {N}: [\chi ] \square ^n_G \psi \in x\). Observe that this formula is in a necessity form. Hence, we conclude, by rule \(IC\), that \([\chi ]\blacksquare _G \psi \in x\).
Case \(\varphi = [\chi ] [G] \tau\). (\(\Rightarrow\)): Suppose that \([\chi ] [G] \tau \in x\). Since x contains all theorems of GALC, we have for all \(\psi _G \in \mathcal{E}\mathcal{L}^G\), \([\chi ]( [G] \tau \rightarrow [\psi _G] \tau ) \in x\) and \([\chi ]( [G] \tau \rightarrow [\psi _G] \tau ) \rightarrow ([\chi ][G] \tau \rightarrow [\chi ][\psi _G] \tau ) \in x\) (Proposition 4.46.3 of [39]). Using \(MP\) twice, we get \([\chi ] [\psi _G] \tau \in x\) for all \(\psi _G \in \mathcal{E}\mathcal{L}^G\). By the IH, this is equivalent to \(\forall \psi _G \in \mathcal{E}\mathcal{L}^G: \mathfrak {M}_x \models [\chi ] [\psi _G] \tau\). The latter is equivalent to the fact that \(\mathfrak {M}_x \models \chi\) implies \(\mathfrak {M}_x^\chi \models [\psi _G] \tau\) for all \(\psi _G \in \mathcal{E}\mathcal{L}^G\). By the semantics of group announcements we have that \(\mathfrak {M}_x \models \chi\) implies \(\mathfrak {M}_x^\chi \models [G] \tau\), and the latter is \(\mathfrak {M}_x \models [\chi ] [G] \tau\) by the semantics of public announcements.
(\(\Leftarrow\)): Assume that \(\mathfrak {M}_x \models [\chi ] [G] \tau\). By the semantics, this is equivalent to the fact that \(\mathfrak {M}_x \models \chi\) implies \(\mathfrak {M}_x^\chi \models [G] \tau\). By the semantics of group announcements, the latter is \(\forall \psi _G \in \mathcal{E}\mathcal{L}^G: \mathfrak {M}_x^\chi \models [\psi _G] \tau\). We can ‘fold’ the public announcement back: \(\forall \psi _G \in \mathcal{E}\mathcal{L}^G: \mathfrak {M}_x \models [\chi ] [\psi _G] \tau\). By the IH, \(\forall \psi _G \in \mathcal{E}\mathcal{L}^G: [\chi ] [\psi _G] \tau \in x\). Observe that this formula is in a necessity form. Hence, we conclude, by rule \(IG\), that \([\chi ][G] \tau \in x\).
Case \(\varphi = \blacksquare _G \psi\). (\(\Rightarrow\)): Assume that \(\blacksquare _G \psi \in x\). By (A9), \(\forall n \in \mathbb {N}: \square ^n_G \psi \in x\), which is equivalent, by the IH, to \(\forall n \in \mathbb {N}: \mathfrak {M}_x \models \square ^n_G \psi\). This is equivalent to \(\mathfrak {M}_x \models \blacksquare _G \psi\) by the semantics.
(\(\Leftarrow\)): Assume that \(\mathfrak {M}_x \models \blacksquare _G \varphi\). By the semantics, this is equivalent to \(\forall n \in \mathbb {N}: \mathfrak {M}_x \models \square ^n_G \varphi\). Furthermore, by the IH, we have \(\forall n \in \mathbb {N}: \square ^n_G \varphi \in x\). Since x is closed under \(IC\), we have \(\blacksquare _G \varphi \in x\).
Case \(\varphi = [G] \chi\). (\(\Rightarrow\)): Assume that \([G] \chi \in x\). By (A10), \(\forall \psi _G \in \mathcal{E}\mathcal{L}^G: [\psi _G] \chi \in x\), which is equivalent, by the IH, to \(\forall \psi _G \in \mathcal{E}\mathcal{L}^G: \mathfrak {M}_x \models [\psi _G] \chi\). This is equivalent to \(\mathfrak {M}_x \models [G] \chi\) by the semantics.
(\(\Leftarrow\)): Assume that \(\mathfrak {M}_x \models [G] \chi\). By the semantics, this is equivalent to \(\forall \psi _G \in \mathcal{E}\mathcal{L}^G: \mathfrak {M}_x \models [\psi _G] \varphi\). Furthermore, by the IH, we have \(\forall \psi _G \in \mathcal{E}\mathcal{L}^G: [\psi _G] \varphi \in x\). Since x is closed under \(IG\), we can infer that \([G] \chi \in x\). \(\square\)
Finally, we can prove the completeness of GALC.
Theorem 12
For all \(\varphi \in \mathcal {GALC}\), if \(\varphi\) is valid, then \(\varphi \in \text {GALC}\).
Proof
Assume towards a contradiction that \(\varphi\) is valid and \(\varphi \not \in \text {GALC}\). Since GALC is a consistent theory, it follows that \(\text {GALC} + \lnot \varphi\) is a consistent theory as well. By Lemma 5, there is a maximal consistent theory x such that \(\text {GALC} + \lnot \varphi \subseteq x\). By Lemma 11, \(\lnot \varphi \in \text {GALC} + \lnot \varphi\), and hence \(\lnot \varphi \in x\). Since x is a maximal consistent theory, it follows that \(\varphi \not \in x\). According to Lemma 13, \(\varphi \not \in x\) is equivalent to \(\mathfrak {M}_x \not \models \varphi\), which contradicts \(\varphi\) being valid. \(\square\)
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 \(\mathcal{E}\mathcal{L}^G\) replaced by \(\mathcal {ELC}^G\).
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 \(\mathcal{E}\mathcal{L}^G\) with \(\mathcal{E}\mathcal{L}\).
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\(^X\).
Theorem 15
APALC\(^X\) is sound and complete.
6 Common knowledge in quantification over information change
The way we dealt with common knowledge in Sect. 5 to get the completeness results is quite idiosyncratic. As the reader may have already observed, we treated common knowledge as an infinitary modality. Moreover, the proof systems we provided did not require any specific interaction axioms for common knowledge. Thus the proof can be used to establish completeness of extensions of various other logics of quantified information change.
6.1 Boolean and positive announcements
The undecidability result for APAL [4] spurred the quest for finding decidable fragments of the logic. In particular, the question was whether more modest versions of quantification lead to decidability. It was answered positively for (at least) two versions of APAL: the one, where [!] ranges over Boolean formulas, and the one, where [!] ranges over positive PAL formulas.
The language and the semantics of Boolean APAL (BAPAL) [34] are quite similar to those of APAL with the only difference in the interpretation of \([!]\varphi\):
where \(\mathcal{P}\mathcal{L}\) is the language of propositional logic.
BAPAL is quite a unique logic in the family of the logics of quantified announcements, since it has a finitary axiomatisation, it is decidable [36], and yet lacks the finite model property. Contrast this to the undecidability [4] and the lack of finite model property [35] for the standard logics of quantified announcements.
Alongside the finitary axiomatisation of BAPAL, the authors of [34] also provide an infinitary one, and using the latter we can give an axiomatisation of BAPAL with common knowledge (BAPALC).
The proof system of (the infinitary version of) BAPALC is the same as in Definition 5.2 with the following differences:
Theorem 16
BAPALC is sound and complete.
The completeness proof follows the one in Sect. 5 with [G] being substituted with \([!]_{\text {BAPAL}}\), and \(\mathcal{E}\mathcal{L}^G\) being replaced by \(\mathcal{P}\mathcal{L}\).
Positive APAL (PAPAL) [37], similarly to BAPAL, restricts the range of quantification of arbitrary public announcement operators:
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 \([!]_{\text {PAPAL}} \varphi\) 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 \({\mathcal{EL}}^{+}\) to \({\mathcal{ELC}}^{+}\), and \([!]_{\text {PAPAL}}\) to \([!]_{\text {PAPAL}}^X\) in \((A10)^{\prime}\) and \(IG ^{\prime}\).
Theorem 17
Both PAPALC and PAPALC\(^X\) are sound and complete.
For both systems completeness can be shown in a similar fashion to the completeness of GALC with [G] being substituted with \([!]_{\text {PAPAL}}\) (with \([!]_{\text {PAPAL}}^X\) for PAPALC\(^X\)), and with each \(\mathcal{E}\mathcal{L}^G\) replaced by \({\mathcal{EL}}^{+}\) (by \({\mathcal{ELC}}^{+}\) for PAPALC\(^X\)).
6.2 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 \([ \! \langle G \rangle \! ]\varphi\), meaning ‘whatever agents from group G announce, there is a simultaneous counter-announcement by the agents from outside of the group such that \(\varphi\) holds in the resulting model’. It is clear that modalities \([\!\langle G \rangle \!]\varphi\) are game-theoretical at heart and formalise \(\beta\)-effectivity. Thus, CAL has a game-theoretic 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 \([G, \chi ] \varphi\) that are called relativised group announcements, and that mean ‘given true announcement \(\chi\), whatever agents from group G announce at the same time, they cannot avoid \(\varphi\)’. 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 \(\chi\) within modalities \([G, \chi ] \varphi\) 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:
The axiomatisation of CoRGAL is an extension of the proof system of PAL with the following axioms and rules of inference:
Extending the proof system with axiom (A9) and rule \(IC\) from Definition 5.2 results in the axiomatisation of CoRGAL with common knowledge (CoRGALC). A similar logic where agents are allowed to make announcements involving common knowledge is denoted with CoRGALC\(^X\), and its proof system can be obtained from the axiomatisation of CoRGALC with the following changes: \([G, \chi ]^X \varphi\) replaces \([G, \chi ] \varphi\), \([\!\langle G \rangle \! ]^X \varphi\) replaces \([\!\langle G \rangle \! ] \varphi\), and \(\mathcal{E}\mathcal{L}^G\) is substituted by \(\mathcal {ELC}^G\), where
Theorem 18
Both CoRGALC and CoRGALC\(^X\) are sound and complete.
Completeness in both cases can be shown by combining the corresponding proofs from [16] and Sect. 5 of the current paper.
6.3 Beyond announcements
Quantification over public announcements is quite well-studied, and one wonders whether similar results could be obtained for other DELs with quantification. As it turns out, quantifying over other modes of information change may yield unexpected results. For example, action model logic [39], which allows us to reason about many other types of information changing scenarios apart from public announcement, e.g. private announcements, cheating, gossip, suspicion, etc., once being extended with quantification over action models, is as expressive as epistemic logic [21]. This trivially leads to the fact that such a logic is, for example, decidable.
The fact that action model logic with quantification over action models is as expressive as epistemic logic is due to the existence of so-called ‘reduction axioms’ that allow one to translate any formula of the former into an equivalent formula of the latter. The same technique has been employed to show completeness of axiomatisations of other logics with quantification over information change, for example refinement modal logic [12] and arbitrary arrow update model logic [41].
However, there also logics with quantification that have only infinitary known axiomatisations. Since, as a rule, completeness proofs for such logics are based on the completeness proof for APAL [10], we can use our proof from Sect. 5 to deal with the extensions of such logics with common knowledge.
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 \((\varphi , a, \psi )\) that mean that in the updated model a-arrows between \(\varphi\)-states and \(\psi\)-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 \([U]\varphi\) and \([\updownarrow ] \varphi\). The former means that ‘after arrow update U, \(\varphi\) is true’, and the latter is read as ‘after any arrow update, \(\varphi\) is true’. The semantics of the new operators is as follows:
where \(\mathcal {AUL}\) is a fragment of \(\mathcal {AAUL}\) that does not contain \([\updownarrow ]\), and \(M^U = (S, R^U, V)\) with \(R^U(a) = \{R(a)(s,t) \mid \exists (\varphi , a, \psi ) \in U: M_s \models \varphi \text { and } M_t \models \psi \}\). 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, \([\updownarrow ]\) quantified over \(\mathcal {AUL}\) 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).
Theorem 19
AAULC is sound and complete.
7 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 for GAL extended with distributed knowledge [1, 18]. A current expressivity map of \(\mathcal {GALC}\), \(\mathcal {APALC}\), and other connected logics is shown in Fig. 10.
Overview of the expressivity results. An arrow from \(\mathcal {L}_1\) to \(\mathcal {L}_2\) means \(\mathcal {L}_1 \leqslant \mathcal {L}_2\). If there is no symmetric arrow, then \(\mathcal {L}_1 < \mathcal {L}_2\). This relation is transitive, and we omit transitive arrows in the figure. An arrow from \(\mathcal {L}_1\) to \(\mathcal {L}_2\) is crossed-out, if \(\mathcal {L}_1 \not \leqslant \mathcal {L}_2\). 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
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 \(\mathcal{E}\mathcal{L}\) or formulas of \(\mathcal {PAL}\). 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 \(\mathcal{E}\mathcal{L}\) and \(\mathcal {PAL}\) with common knowledge—\(\mathcal {ELC}\) and \(\mathcal {PALC}\) correspondingly. In particular, \(\mathcal {PALC}\) is strictly more expressive than \(\mathcal {ELC}\) [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 \(\mathcal {PALC}\). We can call the resulting logics APALC\(^{XX}\) and GALC\(^{XX}\) with the semantics being as follows:
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.
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Acknowledgements
We would like to thank the anonymous reviewers of this paper. Their detailed suggestions helped us to significantly improve both readability and technical accuracy of our proofs.
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Ågotnes, T., Galimullin, R. Quantifying over information change with common knowledge. Auton Agent Multi-Agent Syst 37, 19 (2023). https://doi.org/10.1007/s10458-023-09601-0
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DOI: https://doi.org/10.1007/s10458-023-09601-0