Abstract
Public announcement logic (PAL) is an extension of epistemic logic with dynamic operators that model the effects of all agents simultaneously and publicly acquiring the same piece of information. One of the extensions of PAL, group announcement logic (GAL), allows quantification over (possibly joint) announcements made by agents. In GAL, it is possible to reason about what groups can achieve by making such announcements. It seems intuitive that this notion of coalitional ability should be closely related to the notion of distributed knowledge, the implicit knowledge of a group. Thus, we study the extension of GAL with distributed knowledge, and in particular possible interaction properties between GAL operators and distributed knowledge. The perhaps surprising result is that, in fact, there are no interaction properties, contrary to intuition. We make this claim precise by providing a sound and complete axiomatisation of GAL with distributed knowledge. We also consider several natural variants of GAL with distributed knowledge, as well as some other related logic, and compare their expressive power.
1 Introduction
There has recently been considerable interest in epistemic logics with quantifiers over informationchanging actions. See, for example, formal systems proposed in Ågotnes and van Ditmarsch (2008), Balbiani et al. (2008), Hales (2013), Bozzelli et al. (2014), van Ditmarsch et al. (2017), and a recent survey (van Ditmarsch 2020). Arguably the most studied formalisms of this kind are extensions of public announcement logic (PAL) (Plaza 2007) with quantification over announcements. The notable extensions are arbitrary public announcement logic (APAL) (Balbiani et al. 2008), group announcement logic (GAL) (Ågotnes et al. 2010), and coalition announcement logic (CAL) (Ågotnes and van Ditmarsch 2008).
APAL extends PAL with a modality that quantifies over all possible (truthful) announcements. In GAL, there are modalities for each group of agents G, and these modalities quantify over all possible joint announcements the group can make. Modalities of CAL are similar to those of GAL with the additional property that the agents outside of group G also make a simultaneous announcement. Thus GAL and CAL can be seen as logics of coalitional ability (Pauly 2002) in terms of epistemic consequences of publicly observable joint actions.
Distributed knowledge (Fagin et al. 1995) is a standard notion of group knowledge that captures the total, combined, knowledge in a group. While there has been a renewed interest in the dynamics of distributed knowledge (Wáng and Ågotnes 2013; Ågotnes and Alechina 2019; Ågotnes and Wáng 2017), extensions of logics of quantifed actions with distributed knowledge have not been studied in detail. In this paper we investigate some extensions of the latter type, with the focus on the interaction of group announcements and distributed knowledge.
In addition to filling the obvious gap in the literature, the main motivation for studying these two modalities is their apparent connectedness: distributed knowledge is often understood as a state of knowledge agents in a group have the ability to bring about if they share their individual knowledge (Fagin et al. 1995). Careful analysis of this intuition (van der Hoek and Meyer 1992; Roelofsen 2007; Ågotnes and Wáng 2017) shows that this strong relationship does not always hold. This fact only begs the question of what exactly the relationship between the two types of modalities is, particularly in the form of interaction axioms.
We start out by considering several intuitively plausible candidates for such interaction axioms, and show that none of them are actually valid. Then we show that in fact, contrary to intuition, there are no (nontrivial) interaction axioms at all: the axiom system obtained by the independent combination of axioms for epistemic logic with distributed knowledge and GAL is complete.
From the semantical perspective, adding distributed knowledge to GAL (and some other logics of quantified announcements) is not straightforward, since it allows for three meaningful extensions. The first extension, which we denote as \(\text {GALD}^{\textit{paD}}\), is the most conservative one. Note that in GAL, under the standard assumption, there is a public announcement operator for each formula in the language. In \(\text {GALD}^{\textit{paD}}\) we do not add any new public announcement operators. Syntactically, this means that formulas with distributed knowledge do not occur inside public announcement operators.
The next two extensions deal with the domain of quantification of group announcements. One of them, GALD, keeps the semantics of group announcement operators as in GAL, i.e. they quantify over the purely epistemic language.^{Footnote 1} Compared to \(\text {GALD}^{\textit{paD}}\), in GALD we have more public announcement operators since formulas with distributed knowledge are allowed to occur inside of them. Finally, in the third extension, \(\text {GALD}^{\textit{ga+D}}\), we change the semantics of group announcements to allow agents to announce formulas with distributed knowledge.^{Footnote 2} In order to get \(\text {GALD}^{\textit{ga+D}}\), we substitute classic group announcements in GALD with this new type of operators.
One of the reasons to consider the three variants is to understand the significance of distributed knowledge in the context of public communication. As it turns out, the relationship between the variants is quite surprising, as their relative expressive power is not strictly increasing. For example, the very conservative fragment \(\text {GALD}^{\textit{paD}}\) can capture some properties of models that cannot be captured by the seemingly more expressive \(\text {GALD}^{\textit{ga+D}}\). On the other hand, sound and complete axiomatisations of each of these three variants are quite similar.
Having dealt with group announcements and distributed knowledge, we also briefly consider two other logical variants. First, instead of the classic distributed knowledge, which falls short of capturing the intuition of ‘pooling knowledge together’, we discuss extending GAL with resolved distributed knowledge (Ågotnes and Wáng 2017) that is a better approximation of the intuition. Second, we consider extending CAL, which is quite different from GAL in many aspects, with distributed knowledge, and propose some preliminary results. These results indicate that, at least expressivitywise, there are some parallels between GAL and CAL.
The paper is organised as follows. In the next section we set the stage by defining the syntax and semantics of GAL with distributed knowledge, including the variants mentioned above, as well as some other background information and preliminary results. In Sect. 3 we look at some potential interaction axioms relating group announcements and group knowledge. In Sect. 4 we present a Hilbertstyle axiomatic system for group announcement logic with distributed knowledge, and show that it is sound and complete. In Sect. 5 we investigate the relative expressive power of \(\text {GALD}^{\textit{paD}}\), GALD, and \(\text {GALD}^{\textit{ga+D}}\). Operators for resolving distributed knowledge are discussed in Sect. 6, and CAL with distributed knowledge is considered in Sect. 7. We conclude in Sect. 8.
2 Syntax and semantics
In this section, we introduce the main logical languages and semantics we consider in the paper, together with some additional basic tools.
2.1 Languages
All languages in the paper are defined relative to a finite set of agents A and a countable set of propositional variables P. The distinctions between the following languages correspond partly to the subtle distinctions in semantics discussed in the introduction and will be clearer shortly.
Definition 1
(Languages) Languages \({\mathcal {L}}_{{\text {EL}}}\) of epistemic logic, \({\mathcal {L}}_{{\text {PAL}}}\) of public announcement logic, \({\mathcal {L}}_{{\text {GAL}}}\) of group announcement logic, and extensions thereof with distributed knowledge, are recursively defined by the following grammars:
where \(p \in P\), \(\psi \in {\mathcal {L}}_{{\text {GAL}}}\), \(a \in A\), \(G \subseteq A\), and all the usual abbreviations of propositional logic (e.g. \(\varphi \vee \psi \), \(\varphi \rightarrow \psi \), and \(\varphi \leftrightarrow \psi \)) and conventions for deleting parentheses hold. Duals are defined as \({\widehat{K}}_a \varphi := \lnot K_a \lnot \varphi \), \(\langle \varphi \rangle \psi := \lnot [\varphi ] \lnot \psi \), \(\langle G \rangle \varphi := \lnot [G] \lnot \varphi \), and \(\langle G \rangle ^{\vartriangle }\varphi := \lnot [G]^{\vartriangle }\lnot \varphi \).
The intuitive meaning of formulas is as follows: \(K_a \varphi \) means that agent a knows that \(\varphi \); \(D_G \varphi \) means that G has distributed knowledge of \(\varphi \) (\(\varphi \) is true in the set of states that all agents in G consider possible); \([\varphi ] \psi \) means that if \(\varphi \) is true, then after it is announced (and everyone’s knowledge updated by removing states not satisfying \(\varphi \)), \(\psi \) is true; \([G] \varphi \) and \([G]^{\vartriangle }\varphi \) mean that after any joint announcement by agents in G of formulas they know, \(\varphi \) is true.
The quantification in the latter modalities are intended to be over conjunctions of formulas of \({\mathcal {L}}_{{\text {EL}}}\) for [G] and \({\mathcal {L}}_{{\text {ELD}}}\) for \([G]^{\vartriangle }\). The different scope of quantification is the reason we distinguish syntactically between the group modality \([G]^{\vartriangle }\) of \({\mathcal {L}}_{\text {GALD}^{\textit{ga+D}}}\) and [G] of \({\mathcal {L}}_{{\text {GALD}}}\). The meaning of [G] is the same in \({\mathcal {L}}_{\text {GALD}^{\textit{paD}}}\) as in \({\mathcal {L}}_{{\text {GALD}}}\), but note that in the former distributed knowledge formulas are not allowed inside public announcement operators. As discussed in the introduction, the distinction is relevant because \({\mathcal {L}}_{\text {GALD}^{\textit{paD}}}\) has exactly the same set of public announcement modalities as \({\mathcal {L}}_{{\text {GAL}}}\).
2.2 Models and bisimulation
Definition 2
(Epistemic Model) An epistemic model M is a triple \((S, \sim , V)\), where S is a nonempty set of states, \(\sim : A \rightarrow 2^{S \times S}\) assigns to each agent an equivalence relation, and \(V:P \rightarrow 2^S\) is a valuation. For a group \(G \subseteq A\), \(\sim _G\) denotes \(\bigcap _{a \in G} \sim _a\). If necessary, we refer to the elements of the tuple as \(S^M\), \(\sim ^M\), and \(V^M\). A model M with a designated state \(s \in S\) is called a pointed model and denoted by \(M_s\).
Model M is called finite if S is finite. Also, we write \(M \subseteq N\) if \(S^M \subseteq S^N\), \(\sim ^M\) and \(V^M\) are results of restricting \(\sim ^N\) and \(V^N\) to \(S^M\), and call M a submodel of N.
Let \(M_s = (S, \sim , V)\), and \(X \subseteq S\) such that \(X \ne \emptyset \). An updated model \(M^X_s\) is \((S^X,\) \(\sim ^X,\) \(V^X)\), where \(s \in X\), \(S^X = X\), \(\sim ^X_a = \sim _a \cap \) \((X \times X)\) for all \(a \in A\), and \(V^X (p) = V(p) \cap X\).
Definition 3
(Collective Bisimulation) Let \(M = (S^M, \sim ^M, V^M)\) and \(N = (S^N,\) \(\sim ^N,\) \(V^N)\) be two models. A nonempty binary relation \(Z \subseteq S^M \times S^N\) is called a collective bisimulation if and only if for all \(s \in S^M\) and \(u \in S^N\) with \((s,u) \in Z\):

For all \(p \in P\), \(s \in V^M(p)\) if and only if \(u \in V^N(p)\);

For all \(G \subseteq A\) and all \(t \in S^M\): if \(s \sim _G^M t\), then there is a \(v \in S^N\) such that \(u \sim _G^N v\) and \((t,v) \in Z\);

For all \(G \subseteq A\) and all \(v \in S^N\): if \(u \sim _G^N v\), then there is a \(t \in S^M\) such that \(s \sim _G^M t\) and \((t,v) \in Z\).
The notion of an individual bisimulation (or just bisimulation) is defined in exactly the same way as a collective bisimulation, except that the two last conditions are only required to hold for singleton groups G. Thus, a collective bisimulation is also an individual bisimulation, but not necessarily the other way around
If there is a bisimulation between models M and N linking states s and u, we say that \(M_s\) and \(N_u\) are bisimilar, and write \(M_s \rightleftarrows N_u\). For collective bisimulation we write \(M_s \rightleftarrows ^C N_u\), and say that \(M_s\) and \(N_u\) are collectively bisimilar.
If a (collective) bisimulation between \(M_s\) and \(N_u\) is over \(P \setminus \{p\}\) for some \(p \in P\), we say that \(M_s\) and \(N_u\) are (collectively) bisimilar except for p.
Definition 4
(Bisimulation contraction) Let \(M = (S, \sim , V)\) be an epistemic model. The bisimulation contraction of M is the model \(\Vert M\Vert = (\Vert S\Vert , \Vert \! \! \sim \! \! \Vert , \Vert V\Vert )\), where \(\Vert S\Vert = \{[ s ] \mid s \in S\}\) and \([ s ] = \{t \in S \mid M_s \leftrightarrows M_t\}\), \([ s ] \Vert \! \! \sim \! \! \Vert _a [ t ]\) if and only if \(\exists s^\prime \in [s]\), \(\exists t^\prime \in [t]\) such that \(s^\prime \sim _a t^\prime \) 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 known result that \(M_s \leftrightarrows \Vert M\Vert _{[s]}\) (Goranko and Otto 2007).
2.3 Semantics
Let \({\mathcal {L}}_{{\text {EL}}}^G = \{\bigwedge _{i \in G} K_i \psi _i \mid \psi _i \in {\mathcal {L}}_{{\text {EL}}}\}\) with typical elements \(\psi _G\) be the set of formulas describing individual knowledge of members of group G. Similarly, we fix \({\mathcal {L}}_{{\text {ELD}}}^G = \{\bigwedge _{i \in G} K_i \psi _i \mid \psi _i \in {\mathcal {L}}_{{\text {ELD}}}\}\).
Definition 5
(Semantics) Let \(M = (S, \sim , V)\), \(s \in S\), and \(M_s\) be a pointed epistemic model. The semantics is defined recursively as follows :
Whenever \(X = \{t \in S \mid M_t \models \psi \}\), we will write \(M_s^\psi \) for \(M^X_s\). Observe that in order to avoid circularity, the quantification in the definition of the semantics of both group announcement operators \([G]\varphi \) and \([G]^{\vartriangle }\varphi \) is restricted to formulas without group announcement operators.
Formula \(\varphi \) is valid if and only if for any pointed model \(M_s\) it holds that \(M_s \models \varphi \).
For convenience, let us also provide the semantics for the diamond versions of the announcement operators.
We will use GALD to refer to the logic with language \({\mathcal {L}}_{{\text {GALD}}}\) and semantics as given above, and so on for the other logical languages we consider.
The next proposition states that collective bisimulation implies modal equivalence.
Proposition 1
Let \(\varphi \in ({\mathcal {L}}_{{\text {GALD}}} \cup {\mathcal {L}}_{\text {GALD}^{\textit{ga+D}}} \cup {\mathcal {L}}_{\text {GALD}^{\textit{paD}}})\), and \(M_s\) and \(N_t\) be epistemic models. If \(M_s \rightleftarrows ^C N_t\), then \(M_s \models \varphi \) if and only if \(N_t \models \varphi \).
Proof
By an induction following the corresponding proof for \({\mathcal {L}}_{{\text {ELD}}}\) from Roelofsen (2007). \(\square \)
2.4 The positive fragment
Positive formulas can be considered as a particularly well behaved fragment of public announcement logic (van Ditmarsch and Kooi 2006). In particular, they remain true after an announcement.
Definition 6
(Positive Fragments) Positive fragments of languages of group announcement logic with distributed knowledge are defined as follows:
where \(p \in P\), \(\psi \in {\mathcal {L}}^+_{GAL}\), \(a \in A\), and \(G \subseteq A\). We will abbreviate \({\mathcal {L}}^+_{{{\text {GALD}}}} \cup {\mathcal {L}}^+_{\text {GALD}^{\textit{paD}}} \cup {\mathcal {L}}^+_{\text {GALD}^{\textit{ga+D}}}\) as \({\mathcal {L}}^+\).
Definition 7
(Preservation) A formula \(\varphi \) is preserved under submodels if and only if \(M_s \models \varphi \) implies \(N_s \models \varphi \) for any pointed models \(M_s\) and \(N_s\) such that \(N_s \subseteq M_s\).
In the following proposition we show that formulas of any of the positive fragments remain true under submodels. Particularly, this means that if a positive formula is true in a model, then no matter what agents announce, the formula will remain true (see more on positive formulas in the context of quantified announcements in van Ditmarsch et al. 2020).
Proposition 2
Formulas of \({\mathcal {L}}^+\) are preserved under submodels.
Proof
Let \(M = (S^M, \sim ^M, V^M)\) and \(N = (S^N, \sim ^N, V^N)\) be models such that \(s \in S^M\), \(s \in S^N\), and \(N_s \subseteq M_s\). Boolean cases, case \(K_a \varphi \), and case \([\lnot \psi ] \varphi \) are proved in van Ditmarsch and Kooi (2006), Proposition 8. We show the remaining two cases \(D_G \varphi \) and \([G]\varphi \).
Induction hypothesis. If \(M_s \models \varphi \), then \(N_s \models \varphi \).
Case \(D_G \varphi \). Let \(M_s \models D_G \varphi \). By the definition of semantics, this is equivalent to the fact that \(M_t \models \varphi \) for all \(t \in S^M\) such that \(s \sim _G t\). The latter implies \(M_t \models \varphi \) for every \(t \in S^N\) such that \(s \sim _G t\). By the induction hypothesis, we have that \(N_t \models \varphi \) for all \(t \in S^N\) such that \(s \sim _G t\), which is equivalent to \(N_s \models D_G \varphi \) by the semantics.
Case \([G] \varphi \). Assume towards a contradiction that \(M_s \models [G] \varphi \) and \(N_s \not \models [G] \varphi \). By the duality of group announcements, this is equivalent to \(N_s \models \langle G \rangle \lnot \varphi \), and by the definition of semantics, the latter is equivalent to \(N_s \models \langle \psi _G \rangle \lnot \varphi \) for some \(\psi _G \in {\mathcal {L}}_{{\text {ELD}}}\), which, in turn, equals to \(N_s \models \psi _G\) and \(N_s^{\psi _G} \not \models \varphi \). Now observe that \(N_s^{\psi _G} \subseteq N_s \subseteq M_s\). From that and the contraposition of the induction hypothesis, it follows that \(M_s \not \models \varphi \). However, \(M_s \models [G] \varphi \) implies that \(M_s \models [\bigwedge _{i \in G} K_i (p \vee \lnot p)] \varphi \). It is immediate that \(M_s \models [\bigwedge _{i \in G} K_i (p \vee \lnot p)] \varphi \) is equivalent to \(M_s \models \varphi \), which contradicts \(M_s \not \models \varphi \).
Case \([G]^{\vartriangle }\varphi \). Similar to \([G] \varphi \). \(\square \)
3 Ability, announcements, and group knowledge
Distributed knowledge is often described as potential individual (or even common) knowledge that members of a group can establish ‘through communication’ or by ‘pooling their knowledge together’. However, this intuition is in fact not correct (Ågotnes and Wáng 2017). For example, a group can have distributed knowledge of a formula of the form \(p \wedge \lnot K_a p\) (sometimes called a Moore sentenceHolliday and Icard III 2010), which can never become individual knowledge in a group that contains agent a (Ågotnes and Wáng 2017). Nevertheless, that doesn’t mean that there are no interaction properties between group announcements and group knowledge.
We consider some candidates in this section. Note that all the properties mentioned below hold in GALD, \(\text {GALD}^{\textit{paD}}\) and \(\text {GALD}^{\textit{ga+D}}\), with \(\langle G \rangle ^{\vartriangle }\varphi \) substituted for \(\langle G \rangle \varphi \), with the exception of Proposition 3.
It is known that the following potential axioms are not valid (Ågotnes et al. 2010):

\(\langle G \rangle \varphi \rightarrow D_G \langle G \rangle \varphi \)

\(D_G \langle G \rangle \varphi \rightarrow \langle G \rangle D_G \varphi \)
It is also known that the following are valid:

\(\langle G \rangle D_G \varphi \rightarrow D_G \langle G \rangle \varphi \) (implied by Proposition 28 of Ågotnes et al. (2010) and the fact that knowledge de re implies knowledge de dicto)

\(D_G \langle G \rangle \varphi \rightarrow \langle G \rangle \varphi \) (distributed knowledge is veridical)
Consider weaker properties which involve ‘everybody knows’ operator \(E_G\), where \(E_G \varphi := \bigwedge _{i \in G} K_i \varphi \). These properties encapsulate the intuition that distributed knowledge can be made explicit through public communication. It is known that the following is not valid:

\(D_G \varphi \rightarrow \langle G \rangle E_G \varphi \) (take \(\varphi : = p \wedge \lnot K_a p\) where \(a \in G\), see Ågotnes and Wáng 2017)
The other direction also does not hold:
Fact 1
\(\langle G \rangle E_G \varphi \rightarrow D_G \varphi \) is not valid.
Proof
Let \(\varphi := K_b p \vee K_b \lnot p\) and \(\psi _{\{a,b\}}:= K_a (p \rightarrow K_b p)\), and consider Fig. 1.
We have \(M_s \models \langle \psi _{\{a,b\}} \rangle E_{\{a,b\}} \varphi \), which is equivalent to \(M_s \models \psi _{\{a,b\}}\) and \(M_s^{\psi _{\{a,b\}}} \models E_{\{a,b\}} \varphi \). On the other hand, it is easy to verify that \(M_s \not \models D_{\{a,b\}} \varphi \) as the only \(\sim _{\{a,b\}}\)accessible state is s itself, and \(M_s \not \models \varphi \). \(\square \)
In general, distributed knowledge of a group cannot be made known to members of the group via public communication. This is contrary to the intuition that distributed knowledge is a kind of knowledge that members can attain through public communication. Thus it is interesting to know what are the requirements on formulas and models so that this intuition would be true. We argue that positive formulas can be made known on bisimulation contracted models (this restriction is not surprising given analysis in van der Hoek and Meyer 1992).
Fact 2
\(D_G \varphi \rightarrow \langle G \rangle E_G \varphi \) with \(\varphi \in {\mathcal {L}}^+\) is valid on finite bisimulation contracted models.
Proof
Let \(M_s \models D_G \varphi \) for an arbitrary finite bisimulation contracted \(M_s\). Since distributed knowledge is veridical, the latter implies \(M_s \models \varphi \). Now let us a consider the maximally informative announcement by agents from G. Since \(M_s\) is finite and bisimulation contracted, each state in the model can be uniquely described by a characteristic formula. Moreover, disjunctions of these formulas correspond to sets of states. Agents from G can announce characteristic formulas that describe their equivalence classes and include s, i.e. \(\bigcap _{i \in G} [s]_i \) for all \(i \in G\) (see Galimullin et al. 2018; Alechina et al. 2021 for details). In the resulting model \(M_s^{\psi _G}\), relation \(\sim _{G}\) on set of states \(S^{\psi _G}\) is universal. Moreover, since \(M_s \models D_G \varphi \) and \(\varphi \) is preserved under submodels, we have that \(M_s^{\psi _G} \models E_G \varphi \), and, consequently, \(M_s \models \langle G \rangle E_G \varphi \). \(\square \)
The restriction to finite bisimulation contracted models is essential in the previous proposition.
Fact 3
\(D_G \varphi \rightarrow \langle G \rangle E_G \varphi \) with \(\varphi \in {\mathcal {L}}^+\) is not valid.
Proof
Consider the model in Fig. 2. It is easy to check that \(M_s \models D_{\{a,b\}} p\). In order to see that \(M_s \not \models \langle \{a,b\} \rangle E_{\{a,b\}} p\), observe that \(M_s \leftrightarrows M_v\) and \(M_t \leftrightarrows M_u\). Thus, any announcement by a that preserves \(\{s,t\}\) also preserves \(\{u,v\}\). The same holds for agent b and \(\{s,u\}\) and \(\{t,v\}\). \(\square \)
Finally, we consider the relation between \([G]\varphi \) and \([G]^{\vartriangle }\varphi \). Recall that [G] is the standard group announcement operator, where announcements by agents belong to a fragment of epistemic logic without distributed knowledge. On the other hand, in \([G]^{\vartriangle }\) agents’ announcements may include distributed knowledge. In Proposition 3 we show that if agents in group G cannot avoid \(\varphi \) by any announcement involving distributed knowledge, then they cannot avoid \(\varphi \) by announcing purely epistemic formulas. Also, we show that the other direction of this conditional is not valid, i.e. allowing agents to make announcements with distributed knowledge increases their ability to force certain submodels of a model.
Proposition 3
Consider a unified language \({\mathcal {L}} = {\mathcal {L}}_{{\text {GALD}}} \cup {\mathcal {L}}_{\text {GALD}^{\textit{ga+D}}}\). We have that \([G]^{\vartriangle }\varphi \rightarrow [G]\varphi \) is valid and \([G]\varphi \rightarrow [G]^{\vartriangle }\varphi \) is not valid.
Proof
Validity of \([G]^{\vartriangle }\varphi \rightarrow [G]\varphi \) follows from the fact that \({\mathcal {L}}_{{\text {EL}}} \subseteq {\mathcal {L}}_{{\text {ELD}}}\). To show that \([G]\varphi \rightarrow [G]^{\vartriangle }\varphi \) is not valid we present a counterexample in Fig. 3.
Also, consider two submodels of M depicted in Fig. 4.
Let \(\varphi :=\lnot p \wedge {\widehat{K}}_d (K_a \lnot p \wedge K_b \lnot p) \wedge {\widehat{K}}_a K_d p \wedge {\widehat{K}}_b K_d p\), and consider formula \([\{d\}]\lnot \varphi \in {\mathcal {L}}_{{\text {GALD}}}\). Observe that \(N_s \models \varphi \), \(M_s \not \models \varphi \), and \(O_s \not \models \varphi \). In model M all black states are bisimilar to each other, and thus d does not have an announcement such that it would remove some of the black states and leave the other. Thus, the only possible submodels of \(M_s\) that d can enforce are \(O_s\) and \(M_s\) itself, which implies that \(M_s \models [\{d\}] \lnot \varphi \). On the other hand, \(M_s \models \langle \{d\} \rangle ^{\vartriangle }\varphi \); in particular, \(M_s^{K_d \lnot D_{\{a,b\}} p} \models \varphi \) since \(K_d \lnot D_{\{a,b\}} p\) holds in every state but two rightmost black ones, yielding \(N_s\). \(\square \)
4 Proof system
In this section, we first provide an axiomatic system for GALD, and then prove that it is sound and complete. The system and the proof are then easily adapted to \(\text {GALD}^{\textit{paD}}\) and \(\text {GALD}^{\textit{ga+D}}\). The completeness proof is based on adaptions of definitions, results and proof techniques from Balbiani and van Ditmarsch (2015), Wáng and Ågotnes (2013).
4.1 Axiomatisation of group announcement logic with distributed knowledge
Similarly to the axiomatisation of GAL, the system we provide here is infinitary (it contains rules with infinitely many premises), and it is defined using necessity forms (Goldblatt 1982).
Definition 8
Necessity forms are defined by the following grammar:
where \(\varphi \in {\mathcal {L}}_{{\text {GALD}}}\). The result of substituting \(\varphi \) for \(\sharp \) in \(\eta \) is denoted by \(\eta (\varphi )\).
Observe that \(\sharp \) has a unique occurrence in \(\eta (\sharp )\).
Definition 9
The axiomatisation of GALD comprises axiom systems for EL (Fagin et al. 1995), PAL (van Ditmarsch et al. 2008), GAL (Ågotnes et al. 2010), and PALD (Wáng and Ågotnes 2013)
We denote by \(\mathbf {GALD}\) the smallest set that contains all instances of A0–A17 and is closed under R0–R3. Elements of \(\mathbf {GALD}\) are called theorems.
Lemma 1
Rule R3 is truthpreserving.
Proof
The proof is by induction on the construction of \(\eta \). Let \(M_s\) be a pointed epistemic model. We show only the case \(D_H \eta (\sharp )\), and other cases are similar.
Case \(D_H \eta (\sharp )\). Let \(M_s \models D_H \eta ([\psi _G] \varphi )\) for all \(\psi _G \in {\mathcal {L}}^G_{{\text {EL}}}\). By the semantics this means that for every \(\psi _G\), \(M_t \models \eta ([\psi _G] \varphi )\) for all t such that \(s \sim _H t\). Pick any t such that \(s \sim _H t\). By the induction hypothesis we have \(M_t \models \eta ([G] \varphi )\). Since t was arbitrary, \(M_t \models \eta ([G] \varphi )\) for all t such that \(s \sim _H t\). The latter is equivalent to \(M_s \models D_H \eta ([G] \varphi )\). \(\square \)
Theorem 1
The axiomatisation of GALD is sound.
Proof
Follows from the soundness of PALD (Wáng and Ågotnes 2013) GAL (Ågotnes et al. 2010) and Lemma 1. \(\square \)
4.2 Completeness
Following the technique from Wáng and Ågotnes (2013, 2015), we prove the completeness of GALD by making a detour through pre and pseudo models, where distributed knowledge operators are treated as classic knowledge modalities.
Definition 10
(Pre and pseudo models) An epistemic premodel is a tuple \({\mathcal {M}} = (S,\) \(\sim ,\) V), where \(\sim \) maps every agent a and every subset \(G \subseteq A\) to an element of \(2^{S \times S}\). A premodel is called a pseudo model (and is written \({\mathfrak {M}}\)) if for all a it holds that \(\sim _{\{a\}} = \sim _a\), and for all \(G, H \subseteq A\): if \(G \subseteq H\), then \(\sim _H \subseteq \sim _G\).
Next, we define theories that will be used for the construction of the canonical model.
Definition 11
(Theories) A set x of formulas of \({\mathcal {L}}_{{\text {GALD}}}\) is called a theory, if it contains all theorems and is closed under R0 and R3. A theory is consistent if for all \(\varphi \), \(\varphi \wedge \lnot \varphi \not \in x\). A theory is called maximal if for all \(\varphi \), either \(\varphi \in x\) or \(\lnot \varphi \in x\). \(\mathbf {GALD}\) is the smallest theory, and \({\mathcal {L}}_{{\text {GALD}}}\) is the largest theory.
Theories are not required to be closed under R1 and R2 since these rules of inference, unlike R0 and R3, preserve only validity and not truth.
Lemma 2
Let x be a theory, and \(\varphi , \psi \in {\mathcal {L}}_{{\text {GALD}}}\). The following are theories: \(x+\varphi = \{\psi \mid \varphi \rightarrow \psi \in x\}\), \(K_a x = \{\varphi \mid K_a \varphi \in x\}\), \(D_G x = \{\varphi \mid D_G \varphi \in x\}\), and \([\varphi ] x = \{\psi \mid [\varphi ] \psi \in x\}\).
Proof
Cases for \(x+\varphi \), \(K_a x\), \([\varphi ]x\) are proved in (Balbiani et al. 2008, Lemma 4.11). Here we argue that \(D_G x\) is a theory.
We need to show that \(D_G x\) contains \(\mathbf {GALD}\) and is closed under R0 and R3. Let \(\varphi \in \mathbf {GALD}\). Then we also have that \(D_G \varphi \in \mathbf {GALD}\) by the necessitation of \(D_G\), which is derivable in PALD (Wáng and Ågotnes 2013). Since x is a theory, and hence \(\mathbf {GALD} \subseteq x\), we have that \(D_G \varphi \in x\), and \(\varphi \in D_G x\). This establishes that \(\mathbf {GALD} \subseteq D_G x\).
Assume that \(\varphi \rightarrow \psi , \varphi \in D_G x\). By A5 and R0 this implies that \(D_G \psi \in x\), or, equivalently, \(\psi \in D_G x\).
Suppose that \(\eta ([\psi _H] \varphi ) \in D_G x\) for all \(\psi _H \in {\mathcal {L}}_{{\text {EL}}}^H\). This means that \(D_G \eta ([\psi _H] \varphi ) \in x\) for all \(\psi _H\), and from the fact that \(D_G \eta (\sharp )\) is a necessity form, we conclude by R3 that \(D_G \eta ([H] \varphi ) \in x\). Finally, by the definition of \(D_G x\) we yield \(\eta ([H] \varphi ) \in D_G x\). \(\square \)
Lemma 3
For all consistent theories x, \(\lnot \varphi \not \in x\) if and only if \(x + \varphi \) is consistent.
Lemma 4
(Theorem 2.5.2 of Goldblatt (1982)) Every consistent theory can be extended to a maximal consistent theory.
Definition 12
(Canonical pseudo model) The canonical pseudo model is the tuple \({\mathfrak {M}}^C =\) \((S^C, \sim ^C, V^C)\), where \(S^C = \{x \mid x \text { is maximal consistent theory}\}\), \(x \sim _a^C y\) if and only if \(K_a x \subseteq y\), \(x \sim _G^C y\) if and only if for all \(H \subseteq G\) it holds that \(D_H x \subseteq y\) , and \(V^C(p) = \{x \in S^C \mid p \in x\}\).
For the rest of the section, we employ the following strategy. First, we prove the truth lemma for the canonical pseudo model. Next, we unravel \({\mathfrak {M}}^C\) into the treelike premodel \({\mathcal {M}}^C\), which satisfies the same GALD formulas as \({\mathfrak {M}}^C\). After that, we fold \({\mathcal {M}}^C\) into the model \(M^C\). Folding is a truthpreserving operation, and hence we will be able to conclude the completeness of GALD.
Definition 13
(Size Relation) The \(\cdot \)size and ddepth of \(\varphi \in {\mathcal {L}}_{{\text {GALD}}}\) are defined as follows:
The binary relation \(<^{\cdot }_d\) between \(\varphi , \psi \in {\mathcal {L}}_{{\text {GALD}}}\) is defined as follows:
\(\varphi <^{\cdot }_d \psi \) iff \(d (\varphi ) < d (\psi )\), or \(d (\varphi ) = d (\psi )\) and \(\varphi  < \psi \). The relation is a wellfounded strict partial order between formulas. Note that for all \(\psi \in {\mathcal {L}}_{{\text {PALD}}}\) we have that \(d (\psi ) = 0\).
Lemma 5
Let \(\varphi , \chi \in {\mathcal {L}}_{{\text {GALD}}}\).

1.
\(\varphi <^{\cdot }_d \lnot \varphi \),

2.
\(\varphi <^{\cdot }_d \varphi \wedge \psi \),

3.
\(\varphi <^{\cdot }_d K_a \varphi \),

4.
\(\varphi <^{\cdot }_dD_G \varphi \),

5.
\((\varphi \rightarrow p)<^{\cdot }_d [\varphi ]p\),

6.
\((\varphi \rightarrow \lnot [\varphi ] \psi ) <^{\cdot }_d[\varphi ]\lnot \psi \),

7.
\(([\varphi ] \psi \wedge [\varphi ] \chi ) <^{\cdot }_d [\varphi ] (\psi \wedge \chi )\),

8.
\([\varphi \wedge [\varphi ] \chi ] \psi <^{\cdot }_d [\varphi ][\chi ] \psi \),

9.
\((\varphi \rightarrow K_a [\varphi ] \psi ) <^{\cdot }_d [\varphi ] K_a \psi \),

10.
\((\varphi \rightarrow D_G [\varphi ] \psi ) <^{\cdot }_d [\varphi ] D_G \psi \),

11.
\([\psi _G] \varphi <^{\cdot }_d [G] \varphi \),

12.
\([\chi ] [\psi _G] \varphi <^{\cdot }_d [\chi ] [G] \varphi \).
Lemma 6
Let x be a theory. If \(D_G \varphi \not \in x\), then there is a maximal consistent theory y such that \(D_G x \subseteq y\) and \(\varphi \not \in y\).
Proof
Assume that \(D_G \varphi \not \in x\). This means that \(\varphi \not \in D_G x\), and hence \(D_G x + \lnot \varphi \) is a consistent theory by Lemma 3. By Lemma 4, \(D_G x + \lnot \varphi \) can be extended to a maximal consistent theory y. Since \(\lnot \varphi \in y\), by consistency we have that \(\varphi \not \in y\). \(\square \)
Lemma 7
Let x be a theory. If \(K_a \varphi \not \in x\), then there is a maximal consistent theory y such that \(K_a x \subseteq y\) and \(\varphi \not \in y\).
Proof
Similar to the proof of Lemma 6. \(\square \)
In the following, we use satisfaction with respect to pre and pseudo models. The definition of pre and pseudo semantics is exactly like the definition of normal semantics, where group relations \(\sim _G\) are treated as primitive relations. We will use the same symbol, \(\models \), for all three versions of satisfaction, since it is clear which one is employed from the font used for models (M, \({\mathcal {M}}\), or \({\mathfrak {M}}\)).
Lemma 8
For all formulas \(\varphi \) and maximal consistent theories x it holds that \({\mathfrak {M}}^C_x \models \varphi \) if and only if \(\varphi \in x\).
Proof
The proof is by induction on the size of \(\varphi \). Boolean cases are straightforward, and cases with public announcements are dealt with using A11–A16. Here we show only cases with distributed knowledge and group announcements.
Case \(D_G \varphi \). \((\Rightarrow )\): Let \({\mathfrak {M}}^C_x \models D_G \varphi \). By the semantics we have that for all \(y \in S^C\): \(x \sim _G^C y\) implies \({\mathfrak {M}}^C_y \models \varphi \). By the definition of the canonical pseudo model, axiom A10, Lemma 5, and the induction hypothesis, the latter is equivalent to the fact that for all \(y \in S^C\) and all \(H \subseteq G\): \(D_H x \subseteq y\) implies \(\varphi \in y\). In particular, for all \(y \in S^C\): \(D_G x \subseteq y\) implies \(\varphi \in y\). By the contraposition of Lemma 6 this implies that \(D_G \varphi \in x\).
\((\Leftarrow )\): Assume that \(D_G \varphi \in x\) and \(x \sim ^C_G y\) for some maximal consistent theory y. By A7 and R0 it holds that \(D_G D_G \varphi \in x\). By the definition of the canonical model, we have that \(D_G \varphi \in y\). Since y is a maximal consistent theory and thus contains \(D_G \varphi \rightarrow \varphi \), it holds that \(\varphi \in y\). Next, by the induction hypothesis we have that \({\mathfrak {M}}^C_y \models \varphi \). Since y was arbitrary, we have that \({\mathfrak {M}}^C_y \models \varphi \) for all y such that \(x \sim _G^C y\). The latter is equivalent to \({\mathfrak {M}}^C_x \models D_G \varphi \).
Case \([\varphi ] D_G \psi \). \({\mathfrak {M}}^C_x \models [\varphi ] D_G \psi \) if and only if \({\mathfrak {M}}^C_x \models \varphi \rightarrow D_G [\varphi ] \psi \) by the validity of A15. By Lemma 5 and the induction hypothesis, \({\mathfrak {M}}^C_x \models \varphi \rightarrow D_G [\varphi ] \psi \) if and only if \(\varphi \rightarrow D_G [\varphi ] \psi \in x\) if and only if \([\varphi ] D_G \psi \in x\) by A15.
Case \([\varphi ] [G] \psi \). \((\Rightarrow )\): Let \({\mathfrak {M}}^C_x \models [\varphi ] [G] \psi \). By the semantics, \({\mathfrak {M}}^C_x \models [\varphi ] [\psi _G] \psi \) for all \(\psi _G\). By Lemma 5 and the induction hypothesis, \([\varphi ] [\psi _G] \psi \in x\) for all \(\psi _G\). Note that \([\varphi ](\sharp )\) is a necessity form, hence, by R3, we have that \([\varphi ] [G] \psi \in x\).
\((\Leftarrow )\): Let \([\varphi ] [G] \psi \in x\). The distributivity rule for public announcements is derivable in PAL (van Ditmarsch et al. 2008, Proposition 4.46). Hence, by A17 and R0 it holds that \([\varphi ] [\psi _G] \psi \in x\). By Lemma 5 and the induction hypothesis we have that \({\mathfrak {M}}^C_x \models [\varphi ] [\psi _G] \psi \) for all \(\psi _G\). By the semantics, \({\mathfrak {M}}^C_x \models \varphi \) implies \(({\mathfrak {M}}^C_x)^\varphi \models [\psi _G] \psi \), for every \(\psi _G\). The latter is equivalent to the fact that \({\mathfrak {M}}^C_x \models \varphi \) implies \(({\mathfrak {M}}^C_x)^\varphi \models [G] \psi \), and thus \({\mathfrak {M}}^C_x \models [\varphi ][G] \psi \).
Case \([G] \varphi \). \((\Rightarrow )\): Let \({\mathfrak {M}}^C_x \models [G] \varphi \). By the semantics, this is equivalent to \({\mathfrak {M}}^C_x \models [\psi _G] \varphi \) for all \(\psi _G\). By Lemma 5 and the induction hypothesis, for every \(\psi _G\) we have that \([\psi _G] \varphi \in x\), and by R3, \([G] \varphi \in x\).
\((\Leftarrow )\): Let \([G] \varphi \in x\). By the validity of A17, \([\psi _G] \varphi \in x\) for all \(\psi _G\). By Lemma 5 and the induction hypothesis, \({\mathfrak {M}}^C_x \models [\psi _G] \varphi \) for every \(\psi _G\), which is equivalent to \({\mathfrak {M}}^C_x \models [G] \varphi \). \(\square \)
For the rest of the proof, we closely follow Wáng and Ågotnes (2013). Since most of the remaining part involves transformation of the canonical model, group announcement operators do not play a role here. Hence, we just present main points of the transformation, and particular details can be found in the cited literature.
The canonical pseudo model \({\mathfrak {M}}^C\) can be unravelled into the treelike canonical premodel \({\mathcal {M}}^C\). Such an operation preserves collective bisimulation.
Definition 14
(Folding) Let \({\mathcal {M}} = (S, \sim , V)\) be a premodel. The folding of \({\mathcal {M}}\) is the tuple \((S, \sim ^*, V)\), where for all \(a \in A\), \(\sim ^*_a\) is the transitive closure of \(\sim ^\cup _a = \sim _a \cup \bigcup \{\sim _G \mid a \in G\}\).
Folding of an unravelled treelike premodel yields an epistemic model.
Definition 15
(Transbisimulation) Let \(M = (S^M, \sim ^M, V^M)\) be a model and \({\mathcal {N}} = (S^{\mathcal {N}},\) \(\sim ^{\mathcal {N}},\) \(V^{\mathcal {N}})\) be a premodel. A nonempty binary relation \(Z \subseteq S^M \times S^{\mathcal {N}}\) is called a transbisimulation if and only if for all \(s \in S^M\) and \(u \in S^{\mathcal {N}}\) with \((s,u) \in Z\):

For all \(p \in P\), \(s \in V^M(p)\) if and only if \(u \in V^{\mathcal {N}}(p)\);

For all \(a \in A\) and all \(t \in S^M\): if \(s \sim _a^M t\) or \(s \sim _{\{a\}}^M t\), then there is a \(v \in S^{\mathcal {N}}\) such that \(u \sim _{\nu _0} \cdots \sim _{\nu _n} v\), where \(\nu _i\) is either a or G such that \(a \in G\), and \((t,v) \in Z\);

For all \(G \subseteq A\) such that \(G \geqslant 2\) and all \(t \in S^M\): if \(s \sim _G^M t\), then there is a \(v \in S^{\mathcal {N}}\) such that \(u \sim _{H} \cdots \sim _{I} v\) with \(G \subseteq H \cap \cdots \cap I\), and \((t,v) \in Z\);

For all \(\nu \) among a and G and all \(v \in S^{\mathcal {N}}\): if \(u \sim _{\nu }^{{\mathcal {N}}} v\), then there is a \(t \in S^M\) such that \(s \sim _\nu ^M t\), and \((t,v) \in Z\).
If there is a transbisimulation between model M and premodel \({\mathcal {N}}\) linking states s and u, we say that \(M_s\) and \({\mathcal {N}}_u\) are transbisimilar, and write \(M_s \rightleftarrows ^T {\mathcal {N}}_u\).
Folding preserves transbisimulation. Before stating the completeness, we need one more result.
Lemma 9
Given \(M_s\), \({\mathcal {M}}_t\), and \({\mathfrak {M}}_u\), if \(M_s \leftrightarrows ^T {\mathcal {M}}_t \leftrightarrows ^C {\mathfrak {M}}_u\), then for all \(\varphi \in {\mathcal {L}}_{{\text {GALD}}}\): \(M_s \models \varphi \) if and only if \({\mathcal {M}}_t \models \varphi \).
Proof
The proof is by induction on \(\varphi \). Boolean cases, cases for knowledge and distributed knowledge, and the case for public announcements are proved in Wáng and Ågotnes (2013), Lemma 26. We show the case of \([G]\psi \).
Assume that \(M_s \models [G] \psi \). By the semantics this is equivalent to the fact that \(M_s \models [\psi _G] \psi \) for all \(\psi _G\). By the induction hypothesis we have that \({\mathcal {M}}_t \models [\psi _G] \psi \) for every \(\psi _G\), which is equivalent to \({\mathcal {M}}_t \models [G] \psi \) by the semantics. \(\square \)
Finally, we have everything we need to prove the completeness of GALD.
Theorem 2
For all \(\varphi \in {\mathcal {L}}_{{\text {GALD}}}\), if \(\varphi \) is valid, then \(\varphi \in \mathbf {GALD}\).
Proof
Suppose towards a contradiction that \(\varphi \) is valid and \(\varphi \not \in \mathbf {GALD}\). Since \(\mathbf {GALD}\) is a consistent theory, by Lemma 3\(\mathbf {GALD}+ \lnot \varphi \) is a consistent theory. By Lemma 4, \(\mathbf {GALD}+ \lnot \varphi \) can be extended to a maximal consistent theory x such that \(\mathbf {GALD}+ \lnot \varphi \subseteq x\), and \(\lnot \varphi \in x\). By Lemma 8, we have that \({\mathfrak {M}}^C_x \not \models \varphi \). Next, the canonical pseudo model \({\mathfrak {M}}^C_x\) can be unravelled into the collectively bisimilar canonical premodel \({\mathcal {M}}^C_y\), and the latter can be folded into the transbisimilar canonical model \(M^C_z\). So, we have that \({\mathfrak {M}}^C_x \leftrightarrows ^C {\mathcal {M}}^C_y\) and \({\mathcal {M}}^C_y \leftrightarrows ^T M_z^C\). By Lemma 9, \({{\mathfrak {M}}^C_x \leftrightarrows ^C {\mathcal {M}}^C_y \leftrightarrows ^T M_z^C}\) imply the modal equivalence of \(M_z^C\) and \({\mathfrak {M}}^C_x\), and from \({\mathfrak {M}}^C_y \not \models \varphi \) we can infer that \(M_z^C \not \models \varphi \), which contradicts \(\varphi \) being a validity. \(\square \)
We can obtain corresponding proofs for \(\text {GALD}^{\textit{ga+D}}\) and \(\text {GALD}^{\textit{paD}}\) in the same way. For \(\text {GALD}^{\textit{ga+D}}\), it is enough to substitute \({\mathcal {L}}_{{\text {EL}}}^G\) with \({\mathcal {L}}_{{\text {ELD}}}^G\) in A17 and R3, and treat every \(\psi _G\) in the proof as a formula from \({\mathcal {L}}_{{\text {ELD}}}^G\). For \(\text {GALD}^{\textit{paD}}\), each public announcement in the axiomatisation, apart from cases A17 and R3, becomes some \(\psi \in {\mathcal {L}}_{{\text {GAL}}}\), and the proof follows.
Theorem 3
\(\text {GALD}^{\textit{paD}}\) and \(\text {GALD}^{\textit{ga+D}}\) are sound and complete.
5 Expressivity
Having three different versions of group announcement logic with distributed knowledge, it is very intriguing to analyse how they stand against each other in terms of expressivity.
Definition 16
(Expressivity) Let \({\mathcal {L}}_1\) and \({\mathcal {L}}_2\) be two languages. We say that \({\mathcal {L}}_1\) is at least as expressive as \({\mathcal {L}}_2\) (\({\mathcal {L}}_2 \leqslant {\mathcal {L}}_1\)) if and only if for all \(\varphi \in {\mathcal {L}}_2\) there is an equivalent \(\psi \in {\mathcal {L}}_1\). If \({\mathcal {L}}_1\) is not at least as expressive as \({\mathcal {L}}_2\), we write \({\mathcal {L}}_2 \not \leqslant {\mathcal {L}}_1\). If \({\mathcal {L}}_2 \leqslant {\mathcal {L}}_1\) and \({\mathcal {L}}_1 \not \leqslant {\mathcal {L}}_2\), we write \({\mathcal {L}}_2 < {\mathcal {L}}_1\), and we also write \({\mathcal {L}}_2 \equiv {\mathcal {L}}_1\) if \({\mathcal {L}}_2 \leqslant {\mathcal {L}}_1\) and \({\mathcal {L}}_1 \leqslant {\mathcal {L}}_2\). We will also abuse the notation and write L instead of \({\mathcal {L}}_{{\text {L}}}\).
It is known that PALD \(\equiv \) ELD (Wáng and Ågotnes 2013) since each formula of PALD can be translated into an equivalent formula of ELD using the reduction axioms of PALD. That all logics of group announcements with distributed knowledge are strictly more expressive than PALD comes as no surprise.
Proposition 4
\({\text {PALD}} < \text {GALD}^{\textit{paD}}\), \({\text {PALD}} < {\text {GALD}}\), and \({\text {PALD}} < \text {GALD}^{\textit{ga+D}}\)
Proof
The proof is similar to the one for \({\text {PAL}} < {\text {GAL}}\) (Ågotnes et al. 2010, Theorem 19). \(\square \)
As \({\mathcal {L}}_{\text {GALD}^{\textit{paD}}}\) is a syntactic fragment of \({\mathcal {L}}_{{\text {GALD}}}\), it is immediate that GALD is at least as expressive as \(\text {GALD}^{\textit{paD}}\).
Proposition 5
\(\text {GALD}^{\textit{paD}}\leqslant {\text {GALD}}\)
In the proof of the next proposition we exploit the fact that \(\text {GALD}^{\textit{paD}}\) can only witness a difference between two bisimilar (but not necessarily collectively bisimilar) models via K and D ‘steps’, which may be futile if models are large enough, i.e. if they exceed the modal depth of a formula. GALD and \(\text {GALD}^{\textit{ga+D}}\), on the other hand, can witness the difference via a public announcement of a PALD formula.
Proposition 6
\({\text {GALD}} \not \leqslant \text {GALD}^{\textit{paD}}\) and \(\text {GALD}^{\textit{ga+D}}\not \leqslant \text {GALD}^{\textit{paD}}\)
Proof
Let \(\varphi := p \wedge {\widehat{K}}_a (K_b \lnot p \wedge K_c \lnot p) \wedge {\widehat{K}}_a ({\widehat{K}}_b (p \wedge K_a p) \wedge {\widehat{K}}_c (p \wedge K_a p))\), \(\chi := \lnot K_a D_{\{b,c\}} p\), and consider a GALD formula \([\chi ] \langle \{a,b,c\} \rangle \varphi \). Assume towards a contradiction that there is an equivalent \(\text {GALD}^{\textit{paD}}\) formula \(\psi \), and \(\psi  = n\).
Consider models \(M_s\) and \(N_t\) (Figs. 5, 6). For both models, the size of the lower chain is \(n+1\) and the size of the upper chain is either \(n+2\), in the case of \(M_s\), or \(n+4\), in the case of \(N_t\).
These models are bisimilar, and hence they agree on formulas of GAL. Structurally, every model is almost symmetric, and the only difference are bits on the right: the corresponding states are not collectively bisimilar. Formula \(\varphi \) describes the configuration depicted in Fig. 7, i.e. it is false in \(M_s\) and \(N_t\).
Let us argue that \(M_s \models [\chi ] \langle \{a,b,c\} \rangle \varphi \). Formula \(\chi \) is true in every state of the model, and hence the announcement of it has no effect, and the agents can make \(\varphi \) true (note that the intersection of agents’ relations is the identity). The existence of the corresponding group announcement follows from the fact that each state in \(M_s\) can be uniquely specified by an epistemic formula. The upper rightmost state is the only one satisfying \(K_a p\). The one next to it would be the only one satisfying \(\lnot p \wedge {\widehat{K}}_b K_a p \wedge {\widehat{K}}_c K_a p\), and so on.
On the other hand, we have that \(N_t \not \models [\chi ] \langle \{a,b,c\} \rangle \varphi \). The announcement of \(\chi \) removes two rightmost upper black states in the model, and the resulting updated model, \(N_t^{\chi }\), is fully symmetric and upper and lower halves of the model become bisimilar. In order to get a further update of \(N_t^{\chi }\) that will be bisimilar to the model in Fig. 7, the agents should preserve states u and v in the upper half, and delete the corresponding ‘mirror’ states in the lower half. However, since the upper and lower halves of \(N_t^{\chi }\) are bisimilar, there is no announcement that will be true in u and v, and false in the ‘mirror’ states. Hence, the configuration depicted in Fig. 7 is unattainable.
To see that \(M_s \models \psi \) if and only if \(N_t \models \psi \), it is enough to notice the following two things. First, since \(M_s\) and \(N_t\) are bisimilar and distributed knowledge operators do not occur in public announcements, denotations of \([G]\psi ^\prime \) and \([\chi ] \psi ^\prime \) coincide on both models. Second, since the models are sufficiently large (the lengths of upper and lower chains are \(n+2\), or \(n+4\) in the case of \(N_t\), and \(n+1\) correspondingly) and \(\psi  = n\), no sequence of nested K and D modalities occurring in \(\psi \) can reach the states that are not collectively bisimilar (upper rightmost states).
Note that formula \([\chi ] \langle \{a,b,c\}\rangle ^{\vartriangle }\varphi \) is a formula of \(\text {GALD}^{\textit{ga+D}}\), and upper and lower halves of \(N_t^{\chi }\) are collectively bisimilar. Hence the same proof applies. \(\square \)
Interestingly, \(\text {GALD}^{\textit{ga+D}}\) is not more expressive than GALD and even \(\text {GALD}^{\textit{paD}}\). In the proof of this result, we use the fact that two models that are bisimilar except for some propositional variable q, can be distinguished by GALD and \(\text {GALD}^{\textit{paD}}\) as group announcements would implicitly quantify over formulas with q. At the same time, this new q is only true in the states that \(\text {GALD}^{\textit{ga+D}}\) could already distinguish, and thus the set of all possible submodels a \(\text {GALD}^{\textit{ga+D}}\) formula can enforce would coincide for both models.
Proposition 7
\({\text {GALD}} \not \leqslant \text {GALD}^{\textit{ga+D}}\) and \(\text {GALD}^{\textit{paD}}\not \leqslant \text {GALD}^{\textit{ga+D}}\).
Proof
Let us consider formula \([\{c\}] \lnot \langle \{ a,b,c\} \rangle \varphi \in {\mathcal {L}}_{{\text {GALD}}}\), where \(\varphi := {\widehat{K}}_c (K_a \lnot p \wedge K_b \lnot p) \wedge {\widehat{K}}_c ({\widehat{K}}_a K_c p \wedge {\widehat{K}}_b K_c p)\). Assume that there is an equivalent formula \(\psi \in {\mathcal {L}}_{\text {GALD}^{\textit{ga+D}}}\). Since the size of \(\psi \) is finite, we may assume that there is a propositional variable q that does not appear in \(\psi \). Also, consider two models \(M_s\) and \(N_t\) depicted in Fig. 8. Models \(M_s\) and \(N_t\) are almost identical with the only difference that q is false in all states of \(M_s\) and true in the crossedout states of \(N_t\). A model that satisfies \(\varphi \) is depicted in Fig. 9. At the same time, neither \(M_s\) nor \(N_t\) satisfy \(\varphi \).
To see that \(M_s \models [\{c\}] \lnot \langle \{a,b,c\} \rangle \varphi \) it is enough to notice that in \(M_s\) all states in the upper part of the model are bisimilar to the corresponding states in the lower part (with \(s_1\) and \(s_2\) being bisimilar to the rightmost lower black state, and \(s_3\) and \(s_4\) being bisimilar to the penultimate lower white state). Hence, all the updates of \(M_s\) by announcements of c and consecutive announcements of \(\{a,b,c\}\) would remove states symmetrically from the upper and lower parts of the model, thus precluding obtaining a model bisimilar to \(O_u\) and satisfying \(\varphi \).
On the other hand, even though q does not occur in \([\{c\}] \lnot \langle \{a,b,c\} \rangle \varphi \), we still implicitly quantify over c and \(\{a,b,c\}\)announcements that may contain q. In particular, \(N_t^{K_c \lnot q} \models \langle \{a,b,d\} \rangle \varphi \) as \(N_t^{K_c \lnot q}\) (which is \(N_t\) without the crossedout states) has two states uniquely distinguishable by \(\lnot p \wedge K_a \lnot p \wedge K_b \lnot p\) (these states are \(t_3\) and \(t_4\)). The state to the left (the upper black state) can be distinguished by \(p \wedge {\widehat{K}}_c (\lnot p \wedge K_a \lnot p \wedge K_b \lnot p)\). In such a fashion we can assign a unique formula to each state of \(N_t^{K_c \lnot q}\), and, as the intersection of \(a\), \(b\), and \(c\)relations is the identity, \(\{ a,b,c \}\) can force any submodel of \(N_t^{K_c \lnot q}\) including the one isomorphic to \(O_u\). Finally, from \(N_t^{K_c \lnot q} \models \langle \{a,b,c\} \rangle \varphi \) and \(N_t \models K_c \lnot q\) we infer that \(N_t \models \langle \{c\} \rangle \langle \{a,b,c\} \rangle \varphi \).
That \(M_s \models \psi \) if and only if \(N_t \models \psi \) can be shown by a straightforward induction on the complexity of \(\psi \). Since M and N are collectively bisimilar except for q, propositional, epistemic, and public announcement cases are trivial. For \([G]^{\vartriangle }\psi ^\prime \) we need to show that for all \(\psi _G\) such that \(M_{s^\prime } \models \psi _G\) there is a formula \(\psi ^\prime _G\) such that \(N_{t^\prime } \models \psi ^\prime _G\), \(M^{\psi _G}_{s^\prime }\) and \(N^{\psi _G^\prime }_{t^\prime }\) are collectively bisimilar except for q, and vice versa. Notice that the only case when the denotation of some \(\psi _G\) does not coincide on M and N is when \(\psi _G\) contains q. So let us assume that \(\psi _G\) contains q. If \(M_{s^\prime } \models \psi _G\), then there is an equivalent formula \(\psi _G^\prime \) with the same denotation such that \(N_{t^\prime } \models \psi ^\prime _G\). This formula is exactly like \(\psi _G\) with all q’s substituted with \((p \wedge \lnot p)\) (since q has the empty denotation in M). On the other hand, if \(N_{t^\prime } \models \psi _G\), then there is \(\psi ^\prime _G\) where all q’s are replaced with \(D_{\{a,b\}} p \wedge K_c p\) (true only in \(s_1\) and \(s_2\), and \(t_1\) and \(t_2\)), and \(M_{s^\prime } \models \psi _G^\prime \). Since denotations of \(\psi _G\) and \(\psi _G^\prime \) coincide in both cases, the resulting updates are collectively bisimilar except for q. \(\square \)
The overview of the relative expressivity of group announcement logics with distributed knowledge is presented in Fig. 10.
We leave as an open problem whether \(\text {GALD}^{\textit{ga+D}}\not \leqslant {\text {GALD}}\), and conjecture that it is indeed the case.
6 Resolving distributed knowledge
In Sect. 3 we mentioned that distributed knowledge does not capture the property of a group of agents ‘pooling their knowledge together’. The notion of resolved distributed knowledge (Ågotnes and Wáng 2017) was introduced as a better formalisation of this intuition.
Resolution modalities are dynamic modalities that are used to express what is true after a group has actually shared what they know with each other. More precisely, they model the result of the publicly observable event that they privately share all their knowledge with each other. This is in contrast to standard distributed knowledge modalities which are static.
Taking into account both the group and dynamic aspects of resolved distributed knowledge, we discuss the extension of GAL with resolution modalities.
Definition 17
(Resolution) Let \(M = (S, \sim , V)\) be an epistemic model. A global Gresolved update of M is the model \(M^G = (S^G, \sim ^G, V^G)\), where \(S^G=S\), \(V^G=V\), and
Observe that according to the definition, \(\sim _a^{\{a\}} = \sim _a\), and thus \(M^{\{a\}}\) is the same as M.
For an example, consider model M (Fig. 11) and the result of resolution relative to group \(\{a,b\}\) (model \(M^{\{a,b\}}\)). Informally, if two states are distinguished by any agent (meaning that there is no corresponding arrow between the states) from a group, then they will be distinguished by all agents from the group after the resolved update.
Definition 18
(Language) The language of group announcement logic with resolved distributed knowledge is defined by the following grammar:
where \(p \in P\), \(a \in A\), and \(G \subseteq A\).
The semantics is defined exactly like for GAL, with the following additional clause for the resolution modalities:
Since resolved distributed knowledge models private communication, it does not coincide with group announcements, which are public. Indeed, a group of agents may have a goal to inform not only the members of the group but other agents as well of some fact. Such a goal can be achieved via public announcements, but not necessarily via private communication. And vice versa, a group’s epistemic goal may include not only informing each other of some fact, but also leaving the outsiders unaware of the truthfulness of the fact. In such scenario, group announcements fall short, while private communication allows the group to achieve their goal. We demonstrate this in Fact 4.
Fact 4
\(\langle G \rangle \varphi \rightarrow R_G \varphi \) and \(R_G \varphi \rightarrow \langle G \rangle \varphi \) are not valid.
Proof
For the first formula, consider \(\varphi := K_a p\) and models \(M_s\) and \(M_s^{K_b p}\) in Fig. 12. From the fact that \(M_s^{K_b p} \models \varphi \) it follows that \(M_s \models \langle \{b\} \rangle \varphi \). At the same time, the bresolved update of \(M_s\) leaves the model intact, i.e. \(M_s^{\{b\}}\) is exactly the same as \(M_s\). Hence, from the fact that \(M_s^{\{b\}} \not \models K_a p\) it follows that \(M_s \not \models R_{\{b\}} K_a p\).
For the second formula, consider \(\varphi := K_a p \wedge \lnot K_b p\) and models \(N_u\) and \(N_u^{\{a,c\}}\) in Fig. 12. From the fact that \(N_u^{\{a,c\}} \models \varphi \) it follows that \(N_u \models R_{\{a,c\}} \varphi \). On the other hand, due to the fact that public announcements remove states from a model, there is no truthful update of \(N_u\) that would satisfy \(\varphi \). \(\square \)
Even if we require the target formula to be positive, neither resolution implies ability, nor ability implies resolution. In the proof of the previous proposition, the counterexample for \(\langle G \rangle \varphi \rightarrow R_G \varphi \) used positive formula \(K_a p\).
Fact 5
\(R_G \varphi \rightarrow \langle G \rangle \varphi \) with \(\varphi \in {\mathcal {L}}^+\) is not valid.
Proof
Let \(\varphi := K_a p\), and consider model \(M_s\) from Fig. 2 . The \(\{a,b\}\)resolved update of \(M_s\), \(M_s^{\{a,b\}}\), is model \(M_s\) without any nonreflexive arrows between its states. It is clear that \(M_s^{\{a,b\}} \models \varphi \). On the other hand, \(M_s \not \models \langle \{a,b\} \rangle K_a p\) due to the fact that \(M_s \leftrightarrows M_v\), \(M_t \leftrightarrows M_u\), and by the argument similar to the one in the proof of Fact 3. \(\square \)
A perhaps surprising consequence of the proof of this proposition, where \(G = A\), is that semiprivate communication between all agents does not imply the possibility of equivalent public communication between all agents. Formally, \(R_A \varphi \rightarrow \langle A \rangle \varphi \) is not valid even for positive \(\varphi \).
Some results on the expressivity of logics with distributed knowledge and resolution are shown in Ågotnes and Wáng (2017). Relative expressivity of GALR and versions of GALD is an open question. Here we present some preliminary results.
Proposition 8
GALR \(\not \leqslant \) \(\text {GALD}^{\textit{paD}}\) and GALR \(\not \leqslant \text {GALD}^{\textit{ga+D}}\).
Proof
For the first item, the proof is similar to the one for Proposition 6 with formula \([\chi ] \langle \{a,b,c\} \rangle \varphi \) being substituted with \(R_{\{b,c\}} \langle \{a,b,c\} \rangle \varphi \). Note that \(R_{\{b,c\}}\) has no effect on \(M_s\) since b and c relations always occur together. On the other hand, \(N_t\) update with \(R_{\{b,c\}}\) makes the rightmost upper black states disconnected from the rest of the model, and thus in \(N_t^{\{b,c\}}\) the upper and lower halves are bisimilar. The remaining proof is the same as for Proposition 6. The proof of GALR \(\not \leqslant \text {GALD}^{\textit{ga+D}}\) is exactly like the one of Proposition 7 with \([\{c\}] \lnot \langle \{ a,b,c\} \rangle \varphi \in {\mathcal {L}}_{{\text {GALR}}}\). \(\square \)
7 Coalition announcement logic
Group announcement modalities \(\langle G \rangle \) intuitively refer to coalitional ability: \(\langle G \rangle \varphi \) holds if the group, or coalition, G has the ability to make \(\varphi \) true by making some joint public announcement. General coalitional ability modalities have been extensively studied. Two prime examples of logics with such modalities are coalition logic (CL) (Pauly 2002) and alternatingtime temporal logic (Alur et al. 2002). However, these logics typically formalise a stronger notion of coalitional ability, wellknown from game theory, namely that the coalition can perform some joint action such that no matter what the other agents do, \(\varphi \) will be true.
Coalition announcement logic (CAL) (Ågotnes and van Ditmarsch 2008; Galimullin 2019) can be considered as either a restriction of the set of actions in CL to public announcements, or as a variant of GAL, where the agents outside of a group also participate in the joint announcement. CAL extends PAL with formulas \(\langle \! [ G ] \! \rangle \varphi \) meaning that G can make a joint announcement such that no matter what the remaining agents outside of G announce at the same time, \(\varphi \) will be true in the resulting updated model. Thus, in CAL agents outside of the group G may prevent the group from reaching its epistemic goal.
Formally, the language of CAL is the same as GAL except that the \(\langle G \rangle \) modalities are replaced by \(\langle \! [ G ] \! \rangle \) (the dual diamond is taken as primary here instead of the box), with the following semantics:
We can now extend CAL with distributed knowledge operators into the logic CALD, in the same way as for GAL and GALD.
It turns out that the CALD counterparts of most of the observations we made about GALD still hold. This is not immediately obvious, since the semantics is significantly different and, moreover, since the the expressive power of CAL and GAL is different (French et al. 2019). In particular, we have the following (most of these points can be shown in the same way as for GALD, and hence we omit the proofs):

\(\langle \! [ G ] \! \rangle \varphi \rightarrow D_G \langle \! [ G ] \! \rangle \varphi \) is not valid.

\(D_G \langle \! [ G ] \! \rangle \varphi \rightarrow \langle \! [ G ] \! \rangle D_G \varphi \) is valid.

\(D_G \langle \! [ G ] \! \rangle \varphi \rightarrow \langle \! [ G ] \! \rangle \varphi \) is valid.

\(\langle \! [ G ] \! \rangle D_G \varphi \rightarrow D_G \langle \! [ G ] \! \rangle \varphi \) is valid.

\(D_G \varphi \rightarrow \langle \! [ G ] \! \rangle E_G \varphi \) is not valid.

\(\langle \! [ G ] \! \rangle E_G \varphi \rightarrow D_G \varphi \) is not valid.

\(D_G \varphi \rightarrow \langle \! [ G ] \! \rangle E_G \varphi \) with \(\varphi \in {\mathcal {L}}^+\) is valid on finite bisimulation contracted models.

\(D_G \varphi \rightarrow \langle \! [ G ] \! \rangle E_G \varphi \) with \(\varphi \in {\mathcal {L}}^+\) is not valid in general.
We can consider variants of CALD similarly to what we did for GALD, i.e. by allowing quantification over distributed knowledge formulas, or disallowing formulas with distributed knowledge in public announcement operators. Let the resulting logics be \(\text {CALD}^{\textit{ga+D}}\) and \(\text {CALD}^{\textit{paD}}\), respectively. We get the same relative expressivity results as for GALD:
Proposition 9

\(\text {CALD}^{\textit{paD}}< {\text {CALD}}\)

\(\text {CALD}^{\textit{ga+D}}\not \leqslant \text {CALD}^{\textit{paD}}\)

\({\text {CALD}} \not \leqslant \text {CALD}^{\textit{ga+D}}\)

\(\text {CALD}^{\textit{paD}}\not \leqslant \text {CALD}^{\textit{ga+D}}\)
Proof
(Sketch) For the two first points the proof is exactly like for Proposition 6. Note that G is either the grand or the empty coalition here, so the semantics of the group/coalition announcement operators coincide. For the two latter points, the proof is similar to that of Proposition 7, except that we use \(\lnot \langle \! [ \{a,b,c\} ] \! \rangle \varphi \) as the distinguishing formula. In the first model, the upper and the lower halves are bisimilar, and the agents cannot do anything to force an interesting submodel. In the second model, we have special states with q and can construct an announcement by the grand coalition to force the interesting submodel. The reasoning that no \(\text {CALD}^{\textit{ga+D}}\) formula can distinguish the models is the same as for \(\text {GALD}^{\textit{ga+D}}\), i.e., for each announcement with q there is an equivalent one without q. \(\square \)
8 Conclusions and future work
In this paper we studied the interaction between group announcements and distributed knowledge. In particular, we considered extensions of GAL with distributed knowledge modalities. We looked at the following three semantic variants of the language, ordered by the relative closeness to GAL:

\(\text {GALD}^{\textit{paD}}\): semantics of group announcement operators is identical to GAL, the same set of public announcement operators as in GAL;

GALD: semantics of group announcement operators is identical to GAL, public announcement operators include formulas with distributed knowledge;

\(\text {GALD}^{\textit{ga+D}}\): group announcement operators quantify over formulas that may contain distributed knowledge, public announcement operators include formulas with distributed knowledge.
While the three languages have different expressive power, they, perhaps surprisingly, have similar sound and complete axiomatisations. The corresponding proof systems are obtained by combining the axioms and rules of GAL and PALD, showing that there are no nontrivial interaction axioms. We find this result interesting as it is contrary to the intuitions about distributed knowledge and the ability of groups.
The relationship between the scope of quantification and expressive power turned out to be not obvious or trivial. In particular, broadening the scope of quantification does not necessarily lead to the increase in expressive power: there are properties that \(\text {GALD}^{\textit{ga+D}}\) cannot express, and GALD and \(\text {GALD}^{\textit{paD}}\) can. On the other hand, GALD is more expressive than \(\text {GALD}^{\textit{paD}}\), which is as expected. Whether there are some properties expressible in \(\text {GALD}^{\textit{ga+D}}\) and not expressible in GALD is an open question.
In special cases (the positive fragments of the languages, bisimulation contracted models) the operators interact more in line with the intuition (see Fact 2). Note that the formula in Fact 2 is valid on the class of finite bisimulation contracted models, but not valid on the class of all models or even on the class of finite models (see Fact 3). This means that the logic has a different axiomatisation on the class of finite bisimulation contracted models. We find that interesting, because this is typically not the case for other epistemic logics, with or without distributed knowledge. We leave a complete axiomatisation of a logic for this class of models for future research.
We also briefly studied two related logics. The first is GAL extended with the closely related dynamic version of distributed knowledge, namely resolved distributed knowledge. We showed some expressivity results relating GAL with resolution and versions of GALD, however the full expressivity picture is yet unclear. Moreover, a complete axiomatisation of the logic is an open problem. Finally, there are also variants of the logic which are left to future work to explore, such as \(\text {GALR}^{\textit{paD}}\) and \(\text {GALR}^{\textit{ga+D}}\) (with the obvious meaning). The second logic related closely to GALD is CAL with distributed knowledge. Recall that in CAL agents outside of a group also make a simultaneous joint announcement that can preclude the group from reaching its epistemic goals. We argued that the expressivity landscape for CALD is similar to that of GALD. Finding a complete axiomatisation of CALD seems hard, since there are no known axiomatisations of CAL (although there is an axiomatisation of an extended version of CAL, see Galimullin 2021).
GALD and the related logics we have studied here are the first step towards enriching logics of quantified actions (like Balbiani et al. 2008; Hales 2013; Bozzelli et al. 2014; van Ditmarsch et al. 2017) with distributed knowledge modalities. Particularly, we believe that the completeness and expressivity results for APAL with distributed knowledge can be obtained via a straightforward adaptation of the corresponding proofs presented in this paper. Another avenue of further research is investigating logics of quantified actions with other types of group knowledge like common knowledge (Fagin et al. 1995) and relativised common knowledge (van Benthem et al. 2006). So far, only APAL and GAL with common knowledge were studied in Galimullin and \(\text{\AA} \)gotnes (2021).
Notes
The quantification range of group announcements does not include formulas with group announcements to avoid circularity. We also exclude public announcements for simplicity, since they do not add expressivity to epistemic logic (Plaza 2007).
The point made above regarding public announcements still holds: epistemic logic with distributed knowledge and public announcements is equally expressive as the one without public announcements (Wáng and Ågotnes 2013).
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Acknowledgements
We would like to thank the anonymous reviewers of both the 7th LORI and the current special issue for their helpful suggestions and constructive criticism.
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This is an extended version of the LORI paper (Galimullin et al. 2019). Compared to the latter, we consider two new ways of extending GAL with distributed knowledge, and expand Sects. 3, 4 and 6. Sections 5 and 7 are entirely new.
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Ågotnes, T., Alechina, N. & Galimullin, R. Logics with Group Announcements and Distributed Knowledge: Completeness and Expressive Power. J of Log Lang and Inf 31, 141–166 (2022). https://doi.org/10.1007/s10849022093550
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DOI: https://doi.org/10.1007/s10849022093550
Keywords
 Group announcement logic
 Distributed knowledge
 Public announcement logic
 Dynamic epistemic logic