Arbitrary Public Announcement Logic with Memory

We introduce Arbitrary Public Announcement Logic with Memory (APALM), obtained by adding to the models a ‘memory’ of the initial states, representing the information before any communication took place (“the prior”), and adding to the syntax operators that can access this memory. We show that APALM is recursively axiomatizable (in contrast to the original Arbitrary Public Announcement Logic, for which the corresponding question is still open). We present a complete recursive axiomatization, that includes a natural finitary rule, and study this logic’s expressivity and the appropriate notion of bisimulation. We then examine Group Announcement Logic with Memory (GALM), the extension of APALM obtained by adding to its syntax group announcement operators, and provide a complete finitary axiomatization (again in contrast to the original Group Announcement Logic, for which the only known axiomatization is infinitary). We also show that, in the memory-enhanced context, there is a natural reduction of the so-called coalition announcement modality to group announcements (in contrast to the memory-free case, where this natural translation was shown to be invalid).


Introduction
Arbitrary Public Announcement Logic (APAL) and its relatives are natural extensions of Public Announcements Logic (PAL), involving the addition of operators ϕ and ♦ϕ, quantifying over public announcements [θ ]ϕ of some given type. These logics are of great interest both philosophically and from the point of view of applications. Motivations range from supporting an analysis of Fitch's paradox [31] by modeling notions of 'knowability' (expressible as ♦Kϕ), to determining the existence of communication protocols that achieve certain goals (cf. the famous Russian Card problem, given at a mathematical Olympiad [32]), and more generally to epistemic planning [16], and to inductive learnability in empirical science [10]. Many such extensions have been investigated, starting with the original APAL [6], and continuing with its variants CAL (Coalition Announcement Logic) [27], GAL (Group Announcement Logic) [1], Future Event Logic [37], FAPAL (Fully Arbitrary Public Announcement Logic) [41], APAL + (Positive Arbitrary Announcement Logic) [36], BAPAL (Boolean Arbitrary Public Announcement Logic) [35], and extensions of APAL and GAL with common knowledge [21] etc. Similar ideas for arbitrary-quantification modalities have been adopted in other contexts, see [23] for a modality quantifying over action models in Arbitrary Action Model Logic and [17] for a quantifier over the set of all refinements of a given model (a variation on a bisimulation quantifier) in Refinement Modal Logic (see [34] for a recent survey on dynamic epistemic logics with modalities that quantify over information change).
One problem with the above formalisms, with the exception of BAPAL 1 , is that it is not known whether they have finitary axiomatizations; only their infinitary axiomatizations have been established so far. It has already been proven that APAL and GAL are undecidable [4,20]. It is therefore not guaranteed that the validities of these logics are recursively enumerable. 2 The seminal paper on APAL [6] proved completeness using an infinitary rule, and then went on to claim that in theorem-proving 3 this rule can be replaced by the following finitary rule: from ψ → [θ][p]ϕ, infer ψ → [θ] ϕ, as long as the propositional variable p is "fresh". A similar method is adopted in the completeness proof of GAL in [1] and it was claimed that the infinitary rule used in the completeness proof could be replaced by the finitary rule 'from χ → [θ][ i∈G K i p i ]ψ infer χ → [θ][G]ψ', where p i 's are "fresh" and [G] is the group announcement operator, quantifying over updates with formulas of the form i∈G K i ϕ i . These are natural and [G]-introduction rules, similar to the introduction rule for universal quantifier in First Order Logic (FOL), and they are based on the intuition that variables that do not occur in a formula are irrelevant for its truth value, and thus can be taken to stand for any "arbitrary" formula (via some appropriate change of valuation). However, the soundness of the -introduction rule was later disproved via a counterexample by Kuijer [24]. Moreover, a slightly modified version of Kuijer's counterexample also proves that the aforementioned [G]-introduction rule for GAL is unsound. 4 The original intention behind APAL was to quantify over all possible announcements that can be expressed within its language, but clearly the quantification had to be restricted, in order to avoid circular definitions (with their potential Liar-like consequences for the semantics). 5 This also applies to GAL and its quantification over all possible group announcements. To the best of our knowledge, no recursive axiomatization of a variant of APAL or GAL whose range of quantification over announcements is at least as wide as the original APAL and GAL, respectively, is known. 6 Thus, a long-standing open question concerns finding 'quantificationally strong' versions of APAL and GAL for which there exist recursive axiomatizations.
Here, an APAL-type language is called 'quantificationally strong' if it includes at least the standard epistemic operators as well as an APAL-type modality that quantifies over public announcements of all formulas in the given language that do not contain the operator . For 'quantificationally strong' variants of GAL, this definition has of course to be appropriately modified by further restricting the scope of the GAL-type modality [G] to cover all announcements by group G that can be expressed by the formulas of the given language that do not contain the operator [G]. In this sense, APAL and GAL, as well as the logics APALM and GALM that we will introduce in this paper are quantificationally strong; however, none of the other recursively-axiomatized variants in the existing literature is.
We feel that only such strong variants of APAL, GAL etc. can be legitimately called logics of "arbitrary" public (or group) announcements. The qualification "arbitrary" is thus to be read as meaning that the only restriction imposed on the scope of is the one that follows naturally from the need to have a non-circular definition. Moreover, one can easily see that the scopes of and [G] in any such strong variant will be at least as wide as the scope of the corresponding operator in the original logics APAL or GAL, respectively.
In this paper, we solve these open questions for both APAL and GAL, starting with a diagnosis of the APAL counterexample, that leads naturally to our proposed solution. The framework for the 'quantificationally strong' version of GAL will be 4 The formulations of the aforementioned inference rules here are meant to give the reader a picture of the kind of finitary inference rules that the original APAL and GAL were proposed to have, which were initially claimed to be sound, but later proven to be unsound. The modalities that occur in these rules will of course be properly introduced in the relevant sections of the paper. 5 In [42], van Ditmarsch et al. develop a self-referential versions of APAL, called Fully Arbitrary Public Announcement Logic (F-APAL), where quantifies over all public announcements in the given language. However, the price to pay for this version is that the language becomes a proper class, beyond any cardinality, thus the computational complexity goes even further, dashing any hopes for a recursive axiomatization. 6 Recall that the only known finitary variant BAPAL allows to quantify over only purely propositional announcements. developed analogously, as an extension of the one for APAL. Due to the similar syntactic and semantic behaviours of the group announcement ([G]) and arbitrary announcement ( ) operators, most of our analysis of the latter also applies to the former.
Our diagnosis of Kuijer's counterexample is that it makes an essential use of a known undesirable feature of PAL and APAL, namely their lack of memory: the updated models "forget" the initial states. As a consequence, the expressivity of the APAL -modality reduces after any update. This is what invalidates the above rule. We fix this problem by adding to the models a memory of the initial epistemic situation W 0 , representing the information before any non-trivial communication took place ("the prior"). Since communication -gaining more information -deletes possibilities, the set W of currently possible states is a (possibly proper) subset of the set W 0 of initial states. On the syntactic side, we add an operator ϕ 0 saying that "ϕ was initially the case" (before all communication). To mark the initial states, we also need a constant 0, stating that "no non-trivial communication has taken place yet". Therefore, 0 will be true only in the initial epistemic situation. It is convenient, though maybe not absolutely necessary, to add a universal modality Uϕ that quantifies over all currently possible states. 7 In the resulting Arbitrary Public Announcement Logic with Memory (APALM), the arbitrary announcement operator quantifies over updates (not only of epistemic formulas but) of arbitrary formulas that do not contain the operator itself. 8 As a result, the range of is wider than in standard APAL, covering announcements that may refer to the initial situation (by the use of the operators 0 and ϕ 0 ) or to all currently possible states (by the use of Uϕ).
Although APALM is inspired from APAL and designed specifically to avoid the aforementioned flaws that affect the soundness of the 'natural' axiomatization of APAL, the exact relationship between APAL and APALM is not easy to elucidate. This is because of the extreme sensitivity to context of the "arbitrary public announcement" operator: since its semantics quantifies over the remaining syntax, any addition to the syntax changes its meaning. One way to look at APALM is to say that we simply added new operators to the syntax, while essentially keeping "the same" semantics for the arbitrary announcement operator , namely in terms of quantifying over the announcement of all sentences in the language that do not contain itself. However, this does change the semantics of by enlarging the scope of the quantifier. Although the language of APALM is quantificationally stronger than that of APAL (in the sense that the quantificational scope of our APALM modality is at least as wide as the scope of the original APAL operator), the former is not 7 From an epistemic point of view, it would be more natural to replace U by an operator Ck that describes current common knowledge and quantifies only over currently possible states that are accessible by epistemic chains from the actual state. We chose to stick with U for simplicity. Two versions of the addition of Ck to APAL and to GAL are presented in [21]. 8 This restriction is necessary to produce a well-defined semantics that avoids Liar-like vicious circles. In standard APAL, the quantification restriction of ♦ is present in its semantics and with respect to formulas of the form θ ϕ and ♦ϕ. Formulas of the form ♦p ϕ are allowed in the syntax of APAL. APAL and APALM expressivities seem to be incomparable, and that would still be the case if we dropped the above restriction.
necessarily more expressive than the latter. In fact, we conjecture that the expressive power of the language of APALM is incomparable to the one of the language of APAL. Proving this claim is a highly non-trivial task, which remains an open question for now. We show that the original finitary rule proposed in [6] is sound for APALM and, moreover, it forms the basis of a complete recursive axiomatization. 9 It can therefore be applied to all the puzzles and examples that motivated APAL, with the difference that one can now also use the complete proof system to reason axiomatically about them. 10 Besides its technical advantages, APALM is valuable in its own respect. Maintaining a record of the initial situation in our models helps us to formalize updates that refer to the 'epistemic past' such as "what you said, I knew already" [30]. This may be useful in treating certain epistemic puzzles involving reference to past information states, e.g. "What you said did not surprise me" [25]. The more recent Cheryl's Birthday problem also contains an announcement of the form "At first I didn't know when Cheryl's birthday is, but now I know" (although in this particular puzzle the past-tense announcement is redundant and plays no role in the solution). 11 See [30] for more examples.
Note though that the 'memory' of APALM is very limited: our models do not remember the whole history of communication, but only the initial epistemic situation (before any communication). Correspondingly, in the syntax we do not have a 'yesterday' operator Y ϕ, referring to the previous state just before the last announcement as in [28], but only the operator ϕ 0 referring to the initial state. We think of this 'economy' of memory as a (positive) "feature, not a bug" of our logic: a detailed record of all history is simply not necessary for solving the problem at hand. In fact, keeping all the history and adding a Y ϕ operator would greatly complicate our task by invalidating some of the standard nice properties of PAL and APAL. 12 We will briefly return to this connection with the yesterday operator in Section 6.
So we opt for simplicity, enriching the models and language with just enough memory to recover the full expressivity of after updates, and thus establish the soundness of the -introduction rule. Such a limited-memory semantics is sufficient for our purposes, but it also has an intrinsic naturality and simplicity, similar to the 9 We use a slightly different version of this rule, which is easily seen to be equivalent to the original version in the presence of the usual PAL reduction axioms. 10 Note again that the recursively axiomatizable but quantificationally restricted variants BAPAL and APAL + of APAL are too weak to allow for quantification over announcements of ignorance, which are of particular importance for puzzles such as the Muddy Children. 11 Cheryl's Birthday problem was part of the 2015 Singapore and Asian Schools Math Olympiad, and became viral after it was posted on Facebook by Singapore TV presenter Kenneth Kong. 12 E.g. the standard Composition Axiom (stating that any sequence of announcements is equivalent to a single announcement) fails in the presence of the Y operator. As a consequence, a logic with full memory of all history would lose some of the appealing features of the APAL operator (e.g. its S4 character: ϕ → ϕ). Moreover, this would force us to distinguish between "knowability via one communication step" ♦K versus "knowability via a finite communication sequence" ♦ * K, leading to an unnecessarily complex logic. one encountered in some Bayesian models, with their distinction between 'prior' and 'posterior' (aka current) probabilities. 13 Having established the desired results for APALM, we also study a version of GAL with the same memory mechanism -Group Announcement Logic with Memory (GALM) -obtained by extending APALM with group announcement operators. In this logic, the group announcement operators [G]ϕ quantify over updates with formulas of the form i∈G K i ϕ i , thus, represents what a group of agents can bring about via simultaneous public announcements. In order to avoid Liar-like vicious circles, these updates can have occurrences of every component of the language but and [G] (see footnote 8). A related modality is the coalition announcement operator [G] ϕ, which says that the members of group G can bring about ϕ by simultaneous public announcements no matter what other public announcements are simultaneously made by the 'outsiders' (i.e. the agents not in G). The resulting logic was called Coalition Announcement Logic (CAL) [2]. In the memory-free context, it was shown in [19] that the natural and apparently 'obvious' translation of [G] ϕ into GAL is in fact invalid. 14 Moreover, no recursive axiomatization is known for CAL, or for any of its extensions. In contrast, in this paper we show that, in our memory-enhanced framework, the obvious translation is valid: the analogue of the coalition announcement modality is definable in GALM. Further, we provide a complete finitary axiomatization for GALM, thus proving that validities of both GALM and CALM (the memory-enhanced variant of CAL) are recursively enumerable. This is done by following the same steps as for APALM, and showing that the finitary [G]-introduction rule that was originally proposed for GAL in [1] is in fact sound for GALM. 15 On the technical side, our completeness proof involves an essential detour into an alternative semantics for APALM and GALM ('pseudo-models'), in the style of Subset Space Logics (SSL) [18,26]. This reveals deep connections between apparently very different formalisms. Moreover, this alternative semantics is of independent interest, giving us a more general setting for modeling knowability and learnability (see, e.g., [10,13,14]). Various combinations of PAL or APAL with subset space semantics have been investigated in the literature [9-11, 13, 38, 39, 44, 45], including a version of SSL with backward looking public announcement operators that refer to what was true before a public announcement [8]. Following the SSL-style, our pseudo-models come with a given family of admissible sets of worlds, which 13 In such models, only the 'prior' and the 'posterior' information states are taken to be relevant, while all the intermediary steps are forgotten. As a consequence, all the evidence gathered in between the initial and the current state can be compressed into one set E, called "the evidence" (rather than keeping a growing tail-sequence of all past evidence sets). Similarly, in our logic, all the past communication is compressed in its end-result, namely in the set W of current possibilities, which plays the same role as the evidence set E in Bayesian models. 14 But it is not known if another, valid translation exists. 15 We again use a slightly different version of this rule, which can easily be proven to be equivalent to the original version in the presence of the PAL reduction axioms. This choice is clearly cosmetic and made in order to simplify the soundness and completeness proofs. in our context represent "publicly announceable" (or communicable) propositions. 16 We interpret in pseudo-models as the so-called 'effort' modality of SSL, which quantifies over updates with announceable propositions (regardless of whether they are syntactically definable or not). The modality [G] on the other hand quantifies over updates with those announceable propositions that are known by some agents in G. The operator [G] is thus modelled as a restricted version of the effort modality. The finitary -introduction rule is obviously sound for the effort modality, because of its more 'semantic' character. Similarly, the finitary [G]-introduction rule is also sound for this effort-like group announcement operator. These observations, together with the important fact that our models for APALM and GALM (unlike original APAL models) can be seen as a special case of pseudo-models, lie at the core of our soundness and completeness proofs. 17 The paper is organized as follows. In Section 2, we introduce the syntax and semantics of APALM (Section 2.1); then discuss Kuijer's counterexample for the soundness of the finitary -introduction rule of the original APAL (Section 2.2); present a complete finitary axiomatization for APALM (Section 2.3); and define a notion of bisimulation appropriate for the language of APALM and prove expressivity results comparing fragments of this language (Section 2.4). Section 3 presents first the syntax and semantics of GAL and CAL, as well as their problems; then proceeds to introduce our group announcement logic with memory GALM, providing a complete finitary axiomatization, and showing that the coalition announcement modality is definable in this memory-enhanced framework. In Section 4 we prove soundness, and in Section 5 we prove completeness, for both APALM and GALM. Section 6 contains some concluding comments and ideas for future work.
For readability, most of the rather technical proofs are omitted from the main text and presented in the Appendix.

Arbitrary Public Announcement Logic with Memory
We start by introducing APALM, obtained by enriching the models of APAL with a record of the initial information states (representing the informational situation before any communication took place) and the language of APAL 18 with operators that can refer to this memory.

Syntax and Semantics of APALM
Let P rop be a countable set of propositional variables and AG = {1, . . . , n} be a finite set of agents. The language L of APALM (Arbitrary Public Announcement Logic with Memory) is defined by the grammar: where p ∈ P rop, i ∈ AG, and θ ∈ L −♦ is a formula in the sublanguage L −♦ obtained from L by removing the ♦ operator. Given a formula ϕ ∈ L, we denote by P ϕ the set of all propositional variables occurring in ϕ. We follow the standard rules for omission of the parentheses. We define ⊥ as ¬ . The propositional connectives ∨, →, and ↔ are defined as ϕ ∨ ψ := ¬(¬ϕ ∧ ¬ψ), ϕ → ψ := ¬(ϕ ∧ ¬ψ), and ϕ ↔ ψ := (ϕ → ψ) ∧ (ψ → ϕ). The dual modalities are defined asK i ϕ := ¬K i ¬ϕ, Eϕ := ¬U ¬ϕ, ϕ := ¬♦¬ϕ, and [θ]ϕ := ¬ θ ¬ϕ. 19 We read K i ϕ as "ϕ is known by agent i"; θ ϕ as "θ can be truthfully announced, and after this ϕ is true". U and E are the universal and existential modalities quantifying over all current possibilities: Uϕ says that "ϕ is true in all current alternatives of the actual state". ♦ϕ and ϕ are the (existential and universal) arbitrary announcement operators, quantifying over updates with formulas in L −♦ . We can read ϕ as "ϕ is stably true (under public announcements)", i.e., ϕ stays true no matter what (true) announcements are made. The constant 0, meaning that "no (non-trivial) announcements took place yet", holds only at the initial time. Similarly, the formula ϕ 0 means that "initially (prior to all communication), ϕ was true".
In some of our inductive proofs pertaining to APALM, we need a complexity measure on formulas that is different from the standard one based on subformula complexity. The standard notion requires only that formulas are more complex than their subformulas, while we also need that ♦ϕ is more complex than θ ϕ for all θ ∈ L −♦ . A similar complexity measure is also required for the language of Group Announcement Logic, denoted by L G , that we define in Section 3. In order to avoid repetitions, we prefer to present the relevant syntactic definition for both L and L G together in Appendix A.1, where the reader can find the definition of a (proper) subformula in L, ♦-depth of a formula, and a well-founded strict order < on formulas of L (similarly for the language of L G ). Here we only state the core lemma pertaining to this complexity measure that will be useful in our expressivity and completeness proofs.

Lemma 1
There exists a well-founded strict partial order < on L such that: Proof The proof is via easy arithmetic calculations following the definitions in Appendix A.1, restricted to language L. Note that, Definition 67 is redundant for the cases restricted to language L −♦ .

Definition 2 (Model, Initial Model, and Relativized Model)
non-empty sets of states, ∼ i ⊆ W 0 × W 0 are equivalence relations labeled by 'agents' i ∈ AG, and · : P rop → P(W 0 ) is a valuation function that maps every propositional variable p ∈ P rop to a set of states p ⊆ W 0 . W 0 is the initial domain, representing the initial informational situation before any communication took place; its elements are called initial states. In contrast, W is the current domain, and its elements are called current states. • For every model M = (W 0 , W, ∼ 1 , . . . , ∼ n , · ), we define its initial model For states w ∈ W and agents i, we will use the notation w i := {s ∈ W : w ∼ i s} to denote the equivalence class of w modulo ∼ i restricted to W .

Definition 3 (Semantics) Given a model
] M for every formula ϕ ∈ L as follows: What we study in this paper is information update via public announcements. But the models given in Definition 1 are too general for this purpose: their current domain W can be any subset of the initial domain W 0 . Our intended models (which we call "announcement models") will thus be a subclass of these models, in which the current domain comes from updating the initial domain with some public announcement. Given this definition and the semantics, the following fact is obvious: , then for all formulas ϕ ∈ L, we have: More generally, for all formulas ρ ∈ L −♦ , we have: Proof The first identity follows from the sequence of equalities: where we used first the semantics of ψ 0 and then the semantics of θ ϕ.
For the second identity, we use the first one to get the equalities:

An Analysis of Kuijer's Counterexample
The language of APAL is defined recursively as where p ∈ P rop. In APAL, quantifies only over updates with epistemic formulas. More precisely, the APAL semantics of is given by where L epi is the sublanguage generated from propositional variables p ∈ P rop using only the Boolean connectives ¬ and ∧, and the epistemic operators K i . The finitary -introduction rule proposed for the original APAL: Kuijer's counterexample [24] shows the unsoundness of this rule, by taking the formula γ := p ∧K b ¬p ∧K a K b p, and showing that [K b p] ¬γ → [q]¬γ is valid in APAL models, i.e., in multi-agent epistemic models with equivalence relations. (In fact, it is also valid in our a-models.) By the above -intro rule, the formula [K b p] ¬γ → ¬γ should also be valid. But this is contradicted by the model M in Fig. 2.
The premise Fig. 3a: indeed, the only way to falsify ¬γ in Fig. 3a would be by deleting the node u 2 while keeping (at least) node u 1  To see that the counterexample does not apply to APALM, notice that a-models keep track of the initial states. The a-model corresponding to our initial model M is the one drawn in Fig. 4 -where the initial states and current states are the same. Moreover, we can see that the unsoundness of -introduction rule for APAL has to do with its lack of memory, which leads to information loss after updates: while initially (in M) there were epistemic sentences (e.g. p ∨K a r) that could separate u 1 and u 2 , there are no such sentences after the update.
APALM solves this by keeping track of the initial states, and referring back to them, as in (p ∨K a r) 0 .  Table 1 presents a complete proof system APALM for our logic APALM (where recall that P ϕ is the set of propositional variables in ϕ).

Axiomatization
The notion of derivation, denoted by , in APALM is defined as usual. Thus, ϕ means ϕ is a theorem of APALM. For any set of formulas Γ ⊆ L and any ϕ ∈ L, we write Γ ϕ if there exists a finitely many formulas ϕ 1 , . . . , ϕ n ∈ Γ such that (ϕ 1 ∧ · · · ∧ ϕ n ) → ϕ. We say that Γ is APALM-consistent if Γ ⊥, and APALM-inconsistent otherwise. 20 We drop mention of the logic APALM when it is clear from the context. Intuitive Reading of the Axioms Parts (I) and (II) should be obvious. The axiom R[ ] says that updating with tautologies is redundant. The reduction laws that do not contain 0 , U or 0 are well-known PAL axioms. R U is the natural reduction law for the universal modality. The axiom R 0 says that the truth value of ϕ 0 formulas stays the same in time (because the superscript 0 serves as a time stamp), so they can be treated similarly to atoms. Ax 0 says that 0 was initially the case, and R 0 says that at any later stage (after any update) 0 can only be true if it was already true before the update and the update was trivial (universally true). Together, these two say that the constant 0 characterizes states where no non-trivial communication has occurred. Axiom 0-U is a synchronicity constraint: if no non-trivial communication has taken place yet, then this is the case in all the currently possible states. Axiom 0-eq says that initially, ϕ is equivalent to its initial correspondent ϕ 0 . The Equivalences with 0 express that 0 distributes over negation and over conjunction. Imp 0 says that if initially ϕ was stably true (under any further announcements), then ϕ is the case now. Taken together, the elimination axiom [!] -elim and introduction rule [!] -intro Table 1 The axiomatization APALM (Here, ϕ, ψ, χ ∈ L, while θ, ρ ∈ L −♦ .)

(I) Basic Axioms of system APALM:
(CPL) all classical propositional tautologies and Modus Ponens (S5 Ki ) a l l S5 axioms and rules for knowledge operator K i (S5 U ) a l l S5 axioms and rules for U operator

(II) Axioms and rules for dynamic modalities [!]
: Reduction laws: (III) Axioms and rules for , 0, and initial operator 0 : Equivalences with 0 : Implications with 0 : Elim-axiom and Intro-rule for : say that ϕ is a stable truth after an announcement θ iff ϕ stays true after any more informative announcement (of the form θ ∧ ρ). 21

Proposition 9
The following schemas and rules are derivable in APALM, for ϕ, ψ, χ ∈ L and θ ∈ L −♦ : 22 → ψ 6. all S4 axioms and rules for 14 We arrive now at one of the main results of our paper.
Theorem 10 (Soundness and Completeness of APALM) APALM validities are recursively enumerable. Indeed, the axiom system APALM in Table 1 is sound and complete wrt a-models.
Soundness is proved in Section 4, and completeness in Section 5.

Expressivity: Comparisons and Bisimulation
To compare APALM and its fragments with basic epistemic logic (and its extension with the universal modality), consider the static fragment L −♦, ! , obtained from L by removing both ♦ operator and the dynamic modality ϕ ψ; as well as the presentonly fragment L −♦, ! ,0,ϕ 0 , obtained by removing operators 0 and ϕ 0 from L −♦, ! ; and finally the epistemic fragment L epi , obtained by removing U from L −♦, ! ,0,ϕ 0 .

Proposition 11
The fragment L −♦ is co-expressive with the static fragment L −♦, ! . In fact, every formula ϕ ∈ L −♦ is provably equivalent to some formula ψ ∈ L −♦, ! (by using APALM reduction laws, given in Table 1 to eliminate dynamic modalities, as in standard PAL).
Proof Use step-by-step the reduction axioms (given in Table 1, Section 2.3), as a rewriting process, and prove termination by <-induction on ϕ by using Lemma 1.

Proposition 12
The static fragment L −♦, ! (and hence, also L −♦ ) is strictly more expressive than the present-only fragment L −♦, ! ,0,ϕ 0 , which in turn is more expressive than the epistemic fragment L epi . In fact, each of the operators 0 and ϕ 0 independently increase the expressivity of L −♦, ! ,0,ϕ 0 .
Proof Consider the a-model in Fig. 5a: while u 1 and u 2 are indistinguishable for L −♦, ! ,0,ϕ 0 , the sentence (p ∨K a r) 0 distinguishes the two. This shows that L −♦, ! ,0 is strictly more expressive than L −♦, ! ,0,ϕ 0 . To see that L −♦, ! ,ϕ 0 is strictly more expressive than L −♦, ! ,0,ϕ 0 , we just need to consider two a-models ) such that W ⊂ W 0 . As both models have the same underlying current models, they make the same formulas of L −♦, ! ,0,ϕ 0 true at the same states in W . However, only the latter makes 0 true (at every state) since it is an initial model. Finally, note that the present-only fragment, L −♦, ! ,0,ϕ 0 , is precisely the extension of L epi with the universal modality and it is well-known that L epi is strictly less expressive than its extension with the universal modality (see, e.g., [15,Chapter 7.1]). Thus, L −♦, ! ,0,ϕ 0 is more expressive than L epi .
Kuijer's counterexample presented in Section 2.2 shows that the standard epistemic bisimulation is not appropriate for APALM. In the following, we define an appropriate notion of bisimulation for APALM that leads to modal invariance and Hennessy-Milner property. We then also compare the expressive power of L ♦ and the static fragment L −♦, ! in Proposition 19, whose proof uses the notion of APALM bisimulation.

Definition 13 (Total/APALM Bisimulation)
• A total bisimulation between epistemic models (W, ∼ 1 , . . . , ∼ n , · ) and (W , ∼ 1 , . . . , ∼ n , · ) is a non-empty binary relation B ⊆ W × W such that 1. if sBs , then s ∈ p iff s ∈ p for all p ∈ P rop, 2. if sBs and s ∼ i t, then there exists t ∈ W such that tBt and s ∼ i t (the forth condition), 3. if sBs and s ∼ i t , then there exists t ∈ W such that tBt and s ∼ i t (the back condition), and 4. for every s ∈ W there exists some s ∈ W with sBs ; and dually, for every s ∈ W there exists some s ∈ W with sBs .
is a total bisimulation B (as defined above) between the corresponding initial epistemic models M initial 1 and M initial 2 , with the property that: if s 1 Bs 2 , then s 1 ∈ W 1 iff s 2 ∈ W 2 . Two current states s 1 ∈ W 1 and s 2 ∈ W 2 are APALM-bisimilar if there exists an APALM bisimulation B between the underlying a-models such that s 1 Bs 2 .
Since a-models are always of the form M = M 0 |[[θ ]] for some θ ∈ L −♦ , we have a characterization of APALM-bisimulation only in terms of the initial models as stated in Proposition 16. To prove Propositions 16 and 18, we need the following auxiliary Lemmas 14 and 15.

Lemma 14
Let B be a total epistemic bisimulation between initial epistemic models M initial 1 and M initial 2 (or equivalently, an APALM-bisimulation between initial amodels M 0 1 and M 0 2 ), and let s 1 ∈ W 0 1 , s 2 ∈ W 0 2 be two initial states such that s 1 Bs 2 . Then we have Proof By Proposition 11, it is enough to prove the claim for all formulas α ∈ L −♦, ! . Let B be an APALM bisimulation between initial a-models M 0 1 and M 0 2 . The proof is by subformula induction on α, using the following induction hypothesis (IH): for Case α := β ∧ γ and α := ¬β follow straightforwardly by the semantics and IH.
In the following sequence of equivalencies, we make repeated use of the semantic clauses in Definition 3.

Lemma 15
Let B be a total epistemic bisimulation between initial epistemic models M initial 1 and M initial 2 (or equivalently, an APALM-bisimulation between initial amodels M 0 1 and M 0 2 ), and let s 1 ∈ W 0 1 , s 2 ∈ W 0 2 be two initial states such that s 1 Bs 2 .
Then, for all ϕ ∈ L, we have Proof Let B be an APALM bisimulation between initial a-models M 0 1 and M 0 2 . The proof goes by <-induction on ϕ, using Lemma 1. We assume the following induction hypothesis: for all ψ < ϕ in L and all states s 1 ∈ W 0 1 , s 2 ∈ W 0 2 with s 1 Bs 2 , we have: Base cases ϕ := p, ϕ := , and ϕ := 0 follow directly from Lemma 14 and the fact that the formulas α p, α , and α 0 are in L −♦ .
In the following sequence of equivalencies, we make repeated use of the semantic clauses in Definition 3. Case . Cases ϕ := K i ψ and ϕ := Uψ follow similarly as in Lemma 14. We spell out here only the case ϕ := Uψ. We have two sub-cases: . The other direction is similar. For the case ϕ := K i ψ, we also use the back and forth conditions of B.
The following are equivalent: 1. B is an APALM bisimulation between M 1 and M 2 ; 2. B is a total epistemic bisimulation between M initial 1 and M initial 2 (or equivalently, an APALM bisimulation between M 0 1 and M 0 2 ), and Proof (1) → (2): Let B be an APALM bisimulation between M 1 and M 2 . Then it is obvious (from the definition) that B is also a total bisimulation between M initial 1 and M initial 2 . Since M 1 and M 2 are a-models, there must exist . We need to verify that M 1 and M 2 are APALM-bisimilar. For this we just need to verify the property that if s 1 Bs 2 , then s 1 ∈ W 1 holds iff s 2 ∈ W 2 holds. Suppose s 1 Bs 2 and let s 1 By the totality of the bisimulation B, there must exist some s 2 ∈ W 0 2 with s 1 Bs 2 . By Lemma 14, The converse is analogous.
So, to check for APALM-bisimilarity, it is enough to check for total bisimilarity between the initial models and for both models being updates with the same formula.
Next, we verify that this is indeed the appropriate notion of bisimulation.

Corollary 17 APALM formulas are invariant under APALM-bisimulation
Proof Let B be some APALM-bisimulation relation between a-models . By the same proposition, B is a total epistemic bisimulation between the initial epistemic models M initial 1 and M initial 2 . Thus, for every formula ϕ, we have the sequence of equivalences: be a-models with W 0 1 and W 0 2 finite. Then, s 1 ∈ W 1 and s 2 ∈ W 2 satisfy the same APALM formulas iff they are APALM-bisimilar.
Proof We only need to prove the left-to-right direction. Let s 1 ∈ W 1 and s 2 ∈ W 2 such that for all ϕ ∈ L, The opposite direction is analogous. We then show that the modal equivalence relation in W 0 1 × W 0 2 between the models M 0 1 and M 0 2 is an APALM bisimulation. We thus need to show the following: , contradicting the assumption that s 1 in M 0 1 and s 2 in M 0 2 satisfy the same APALM formulas. The second clause follows similarly. • (Valuation) This follows immediately from modal equivalence.
for all ϕ ∈ L and w 1 ∼ i w 1 . Suppose, toward contradiction, that for no element w 2 ∈ W 0 2 with w 2 ∼ i w 2 , M 0 1 , w 1 and M 0 2 , w 2 satisfy the same APALM formulas. Since W 0 2 is finite, the set ∼ i (w 2 ) = {t ∈ W 0 2 : w 2 ∼ i t} is finite, thus, we can write ∼ i (w 2 ) = {t 1 , . . . , t k }. As in the proof of (Totality), the assumption implies that for all t j with w 2 ∼ i t j , there exists ψ j ∈ L such that , contradicting the assumption that M 0 1 , w 1 and M 0 2 , w 2 satisfy the same APALM formulas. Back condition for ∼ i follows analogously.
We have therefore proven that the modal equivalence relation in W 0 1 ×W 0 2 between the models M 0 1 and M 0 2 is an APALM bisimulation between M 0 1 and M 0 2 . By Proposition 16, it suffices to further prove that , we can take θ := θ 1 . Then As a last result in this section, we compare the expressive powers of L and L −♦ .
Proposition 19 L is strictly more expressive than L −♦ and, therefore, than the static fragment L −♦, ! .
Proof By Proposition 11, it suffices to show that L is strictly more expressive than L −♦, ! . Wlog, we assume that AG = {a, b}. The proof follows by a similar argument as in [6, Proposition 3.13] via contradiction: suppose that L and L −♦, ! are equally expressive for a-models, i.e., for all ϕ ∈ L there exists ψ ∈ L −♦, ! such that |= ϕ ↔ ψ. Consider the formula ♦(K a p ∧ ¬K b K a p) in L. By the assumption, there must be ψ ∈ L −♦, ! such that |= ♦(K a p ∧ ¬K b K a p) ↔ ψ. To reach the desired contradiction, we now construct two a-models M and M (similar to the ones in the proof in [6]) which agree on ψ at the actual world but disagree on ♦(K a p ∧ ¬K b K a p). For this argument it is crucial to observe that any such ψ contains only finitely many propositional variables. As we have countably infinitely many propositional variables, there is a propositional variable q that does not occur in ψ (that is also different from p). Without loss of generality, suppose ψ is built using only one variable p. Consider the a-models M = (W, W, ∼ a , ∼ b , · ) and M = (W , W , ∼ a , ∼ b , · ) given in Fig. 6. It is easy to see that both M and M are initial a-models. Moreover, they are also APALM-bisimilar with respect to the language of APALM constructed from using only the propositional variable p and agents a and b. In particular, the corresponding bisimulation relation is All the expressivity results of this section are summarized by the diagram in Fig. 7.

Group Announcement Logic with Memory
In this section we turn our focus on the Group Announcement Logic (GAL), and we propose a memory-enhanced version GALM. As in the case of APALM, the addition (to both models and logic) of a memory of the initial states helps to provide a recursive axiomatization, by re-establishing the soundness of the natural GAL inference rule (already proposed in [1], but later shown to be unsound). Moreover, the same move makes possible the reduction of the related Coalition Announcement Logic CAL (or more precisely, its memory-enhanced version CALM) to a fragment of GALM, via an intuitively 'obvious' equivalence (-which was proved to be invalid on memory-free models, but becomes valid in the memory-enhanced version).

GAL, CAL and their problems
Like APAL, Group Announcement Logic GAL (first introduced in [1]) is also an extension of PAL, involving group announcement operators [G]ϕ and G ϕ (instead of the arbitrary announcement operators ϕ and ♦ψ). More precisely, the language of GAL is defined recursively as The group announcement operator can be seen as a restricted version of the arbitrary public announcement operator in the sense that it quantifies only over updates with formulas of the form i∈G K i θ i , where θ i ∈ L epi and i ∈ G ⊆ AG. Ågotnes  This operator intends to capture communication among a group of agents and what a coalition can bring about via public announcements. While GAL seems to provide more adequate tools than APAL to treat puzzles involving epistemic dialogues, the axiomatization of GAL presented in [1] has a similar shape as the one for APAL in [6]. To recall, [1] proves completeness of GAL also by using an infinitary rule and claims that it is replaceable in theorem-proving by the finitary rule where p i ∈ P ϕ ∪P ψ ∪P θ . However, Kuijer's counterexample presented in Section 2.2 constitutes a counterexample also for the soundness of this rule. Consider again the formula γ := p ∧K b ¬p ∧K a K b p and let G = {a}. We first show that . In other words, the model in Fig. 8

Coalition modality. A related operator is the coalition announcement modality
[G] ϕ that lies at the core of Coalition Announcement Logic (CAL), introduced in [2]: this is a coalition logic in the style of [27], but where the actions that agents can perform are restricted to public announcements. CAL is simply the extension of multi-agent epistemic logic with such coalition announcement modalities.Ågotnes et al. [2] interpret [G] ϕ on epistemic models M = (W, ∼ 1 , . . . , ∼ n , · ) as The semantics of group and coalition announcement operators suggests at the first sight that the latter might be defined in terms of the former as [G] ϕ ↔ G [AG − G]ϕ. However, this 'obvious' equivalence is proved to be invalid in [19].
Our diagnosis for the failure of this rather intuitive equivalence is the same as our explanation for the unsoundness of the finitary -introduction rule for APAL or GAL: the models' lack of memory is the reason for the non-equivalence between the coalition announcement operator [G] ϕ (which expresses coalition G's ability to bring about ϕ by a joint announcement against any simultaneous joint announcement by the anti-coalition AG − G) and the expression G [AG − G]ϕ (which captures a similar ability of group G against any subsequent joint announcement by the anticoalition).
As in the case of APAL, the same lack of memory leads also to difficulties in obtaining a recursive axiomatization for GAL, CAL and related logics. To the best of our knowledge, there are no known recursive axiomatizations for GAL, CAL etc. 23 In fact, the same state of affair applies to any logic that contains coalition announcement operators [2][3][4]33].

A principled solution: GALM
In this section, we develop a Group Announcement Logic with Memory (GALM), obtained by extending the syntax of APALM with group announcement operators interpreted on a-models. 24 Moreover, we show that the memory-enhanced version (CALM) of CAL is indeed embedded in GALM, via the natural analogue of the above-mentioned equivalence. Finally, we give a complete recursive axiomatization of GALM.
The language L G of GALM is defined recursively as where p ∈ P rop, i ∈ AG, θ ∈ L −♦ , and G ⊆ AG. Note that L −♦ is the same as before, namely, it is the set of sentences in L G that do not include ♦ or G . In the context of GALM, the elements of L −♦ are called ♦, G -free formulas. The dual modality for this new operator is defined as [G]ϕ := ¬ G ¬ϕ. G ϕ and [G]ϕ are the (existential and universal, respectively) group announcement operators, quantifying over updates with formulas of the form i∈G K i θ i , where θ i ∈ L −♦ and i ∈ G. This restricted quantification over L −♦ captures the assumption that each agent can announce only the (♦ and G -free) propositions she knows and nothing else. Analogous to the reading of , we read [G]ϕ as "ϕ is stably true under group G's public announcements", i.e., "ϕ stays true no matter what group G truthfully announces". We introduce the following abbreviation of relativized knowledge for notational convenience: The following lemma will be useful in the completeness proof.

Lemma 20
There exists a well-founded strict partial order < on L G , such that: Proof Similar to Lemma 1.
The language L G is interpreted on the same models introduced in Definition 2.
Definition 21 Given a model M = (W 0 , W, ∼ 1 , . . . , ∼ n , · ), the semantics for L G is defined recursively as in Definition 3 with the following additional clause for G :

Observation 3 Note that we have
The GALM analogue of Observation 2 is again a straightforward consequence of the semantics: As for APALM, the intended models for GALM are the announcement models (amodels, introduced in Definition 6). So GALM validities are defined with respect to a-models, as in Definition 6. It should also be obvious that the analogue of Lemma 7 still holds for GALM: is an a-model and θ ∈ L −♦ is a formula such that W = [[θ]] M 0 , then for all formulas ϕ ∈ L G and all formulas ρ ∈ L −♦ , we have: The proof is exactly the same as the proof of Lemma 7. This result can be used to prove that in the memory-enhanced environment of GALM, the (memoryenhanced) coalition announcement modalities [G] ϕ are in fact definable using group announcement modalities (in contrast to the situation in memory-free GAL): be an a-model, and θ ∈ L −♦ be a formula such that W = [[θ]] M 0 . For every group G ⊆ AG and every formula ϕ ∈ L G , we have: Proof For the right-to-left implication: suppose there exists a set of formula {ψ i : i ∈ G} ⊆ L −♦ satisfying the property in the right-hand side of the above Proposition, i.e.: Let {ψ j : j ∈ AG − G} ⊆ L −♦ be some arbitrary set of formulas in L −♦ , indexed by agents in AG − G. By applying the above claim (from the right-hand side of the Proposition) to the set {[ i∈G K i ψ i ]ψ j : j ∈ AG − G}, we obtain that Applying to this Proposition 9.3, then Axiom (R K i ), then Axiom (R ! ) and finally Proposition 9.3 again, we obtain that

For the left-to-right implication: suppose that we have w ∈ [[ G [AG − G]ϕ]] M , By the semantics, there exists a set of formula {ψ
Let {ψ j : j ∈ AG − G} ⊆ L −♦ be any arbitrary set of formulas in L −♦ , indexed by agents in AG − G. Applying the semantics of [AG − G]ϕ in M|[[ i∈G K i ψ i ]] to the set {( θ K j ψ j ) 0 : j ∈ AG − G}, the last displayed formula implies that: But by Lemma 23, we have

By Observation 4, the above facts imply that w ∈ [[ i∈G K i ψ i ∧ [ m∈AG K m ψ m ]
ϕ]] M , as desired.
So, in contrast to the situation in the memory-free case of Coalition Announcement Logic, the memory-enhanced version is essentially a fragment of GALM.
We move now to the main result of this section. Table 2.

Theorem 25 (Soundness and Completeness of GALM) GALM validities are recursively enumerable. In fact, the sound and complete axiomatization GALM wrt a-models is obtained by extending APALM with the axiom and rule given in
The axiom and rule in Table 2  [G]-intro say that ϕ is a stable truth under group G's announcements after an announcement θ iff ϕ stays true after any more informative announcement from the group G (of the form θ ∧ i∈G K θ i ρ i ).

Soundness via Pseudo-model Semantics
As GALM is an extension of APALM, we present the soundness and completeness proofs directly for the former. The same results for APALM are obtained following similar steps.
To start with, note that even the soundness of our axiomatic systems is not a trivial matter. As we saw from Kuijer's counterexample, the analogues of our finitary and [G]-introduction rules were not sound for APAL and GAL, respectively. To prove their soundness on a-models, we need a detour into an equivalent semantics, in the style of Subset Space Logics (SSL) [18,26]: pseudo-models. 25 We first introduce an auxiliary notion: 'pre-models' are just SSL models, coming with a given family A of "admissible sets" of worlds (which can be thought of as the communicable propositions). We interpret in these structures as the so-called "effort modality" of SSL, which quantifies over updates with admissible propositions in A. Analogously, G quantifies over updates with conjunctions of those admissible propositions in the scope of an epistemic operator labeled by an agent in G. Our 'pseudo-models' are pre-models with additional closure conditions (saying that the family of admissible sets includes the valuations and is closed under complement, intersection, and epistemic operators). These conditions imply that every set definable by a ♦, G -free formula 26 is admissible, and this ensures the soundness of our -elimination and [G]-elimination axioms on pseudo-models. As for the soundness of the long-problematic and [G]-introduction rules on (both preand) pseudo-models, this is due to the fact that both the effort modality and [G] operator interpreted on pseudo-models have a more 'robust' range than the arbitrary announcement versions of them: they quantify over admissible sets, regardless of whether these sets are syntactically definable or not. Soundness with respect to our amodels then follows from the observation that they (in contrast to the original APAL models) are in fact equivalent to a special case of pseudo-models: the "standard" ones (in which the admissible sets in A are exactly the sets definable by ♦, G -free formulas).

Definition 26 (Pre-model) A pre-model is a tuple
where W 0 is the initial domain, ∼ i are equivalence relations on W 0 , · : P rop → P(W 0 ) is a valuation map, and A ⊆ P(W 0 ) is a family of subsets of the initial domain, called admissible sets (representing the propositions that can be publicly announced). 25 A more direct soundness proof on a-models is in principle possible, but would require at least as much work as our detour. Unlike in standard EL, PAL or DEL, the meaning of an APALM formula (and therefore of a GALM formula) depends, not only on the valuation of the atoms occurring in it, but also on the family A of all sets definable by L −♦ -formulas. The move from models to pseudo-models makes explicit this dependence on the family A, while also relaxing the demands on A (which is no longer required to be exactly the family of L −♦ -definable sets), and thus makes the soundness proof both simpler and more transparent. Since we will need pseudo-models for our completeness proof anyway, we see no added value in trying to give a more direct soundness proof. 26 ♦, G -free formulas are the sentences in L −♦ . Given a set A ⊆ W 0 and a state w ∈ A, we use the notation w A i := {s ∈ A : w ∼ i s} to denote the restriction to A of w's equivalence class modulo ∼ i . We also introduce the following abbreviation for the semantic counterpart of relativized knowledge: Observation 5 Note that, for all w ∈ A, we have Observation 5.1 shows that our proposed semantics of on pre-models fits with the semantics of the effort modality in SSL [18,26]. The proof of Observation 5.3 is similar to that of Proposition 22. (W 0 , A, ∼ 1 , . . . , ∼ n , · ), satisfying the following closure conditions: 1. p ∈ A, for all p ∈ P rop,

Lemma 29 Given a pseudo-model
Proof First note that by Definition 28.5 and Boolean operations of sets we have: Then, by Definition 28. (3)(4)(5) and A, B ∈ A, we obtain

Proposition 30 Given a pseudo-model
Proof The proof is by subformula induction on θ. The base cases and the inductive cases for the Booleans are immediate (using the conditions in Definition 28).

Lemma 32 Let
By the semantics of the initial operator on pseudo-models, we obtain

Proposition 33
The system GALM is sound wrt pseudo-models. Therefore, the system APALM is also sound wrt pseudo-models.  (W 0 , A, ∼ 1 , . . . , ∼ n , · ) such that p := B and q = q for any q = p ∈ P rop. In order to use Lemma 32 we must show that M is a pseudo-model. For this we only need to verify that M satisfies the closure conditions given in Definition 28. First note that p := B ∈ A by the construction of M , so p ∈ A. For every q = p, since q = q and q ∈ A we have q ∈ A. Since A is the same for both M and M , and M is a pseudo-model, the rest of the closure conditions are already satisfied for M . Therefore M is a pseudo-model. Now continuing with our soundness proof, since p / ∈ P χ ∪ P θ ∪ P ϕ , by Lemma The latter means that there exists a pseudo model -intro. First note that for every q = p i , since q = q and q ∈ A, we have q ∈ A. Moreover, since for every i ∈ G, p i = B i ∈ A, we conclude that M satisfies Definition 28.1. Since A is the same for both M and M , and M is a pseudo model, the rest of the closure conditions are satisfied already. Therefore M is a pseudo model. Now continuing with our soundness proof, given that

Proposition 35
Every standard pre-model is a pseudo-model.
We need to show that M satisfies the closure conditions given in Definition 28. Conditions (1) and (2) are immediate.
Equivalence between the standard pseudo-models and announcement models. For Proposition 38 only, we use the notation [[ϕ]] P S A to refer to pseudo-model semantics (as in Definition 27) and use [[ϕ]] M to refer to the semantics on a-models (as in Definition 21).
The proof of Proposition 38 needs the following lemmas.

Lemma 36
The sentence Proof We only prove the direction from left-to-right since the direction from rightto-left is an instance of the T-axiom for K i . Let M = (W 0 , A, ∼ 1 , . . . , ∼ n , · ) be a pseudo-model, A ∈ A, and w ∈ A such that w ∈ [[(K i (ϕ → ψ)) 0 ]] A . This means, by the semantics of 0 , that w ∈ [ As v has been chosen arbitrarily from w A i , we obtain that M = (W 0 , A, ∼ 1 , . . . , ∼ n , · ) be a standard pseudo-model, A ∈ A and ϕ ∈ L G . Then we have the following:

Lemma 37 Let
Then, by the semantics of ♦ in pseudo-models, there exists some by Lemma 31 and the semantics. By the semantics of 0 and Lemma 36, we obtain Thus, for θ i : Since M is a standard pseudo-model, we know that B i := [[θ i ]] A ∈ A for every i ∈ G and by our initial

Corollary 39 Validity on standard pseudo-models coincides with validity on the amodels.
Proof This is a straightforward consequence of Proposition 38.

Corollary 40
The system GALM is sound wrt a-models. Moreover, the system APALM is sound wrt a-models.
Proof Follows immediately from Proposition 33 and Corollary 39.
It is important to note that the equivalence between standard pseudo-models and a-models (given by Proposition 38 above, and underlying our soundness result) is not trivial (the proof is in Appendix A.4). It relies in particular on the equivalence between the effort modality and the arbitrary announcement operator (see Lemma 37.1), and on the equivalence between the purely syntactic and purely semantic descriptions of the group announcement operator [G] on standard pseudo models (see Lemma 37.2). In turn, the equivalences between these operators hold only because our models and language retain the memory of the initial situation, i.e., having W 0 in a-models and in pseudo-models, and having the operators 0 and 0 in the language L G . Note that the most important steps in the proof of Lemma 37 make necessary use of the operator 0 . Hence, a similar equivalence of models fails for the original, memory lacking, APAL and GAL.

Completeness
In this section we prove the completeness of GALM and APALM. First, we show completeness with respect to pseudo-models, via an innovative modification of the standard canonical model construction. This is based on a method previously used in [11], that makes an essential use of the finitary and [G]-introduction rules, by requiring our canonical theories T to be (not only maximally consistent, but also) "witnessed". Roughly speaking, a theory T is witnessed if: every ♦ϕ occurring in every "existential context" in T is witnessed by some atomic formula p, meaning that p ϕ occurs in the same existential context in T , and for every G ϕ occurring in every "existential context" in T is witnessed by some formula ∧ i∈G K i p i , meaning that ∧ i∈G K i p i ϕ occurs in the same existential context in T . Our canonical premodel will consist of all initial, maximally consistent, witnessed theories (where a theory is 'initial' if it contains the formula 0). A Truth Lemma is proved, as usual. Completeness for (both pseudo-models and) a-models follows from the observation that our canonical pre-model is standard, hence it is (a standard pseudo-model, and thus) equivalent to a genuine a-model.
We now proceed with the details. The appropriate notion of "existential context" is represented by possibility forms, in the following sense. Example:

Definition 41 (Necessity forms and possibility forms) For any finite string s ∈
Definition 42 (Theories: witnessed, initial, maximal) A theory Γ is a consistent set of formulas in L G wrt the axiomatization of GALM, that is, A maximal theory is a theory Γ that is maximal with respect to ⊆ among all theories; in other words, Γ cannot be extended to another theory. A witnessed theory is a theory Γ such that, for every s ∈ NF and ϕ ∈ L G , (1) A maximal witnessed theory Γ is a witnessed theory that is not a proper subset of any witnessed theory. A maximal witnessed initial theory Γ is a maximal witnessed theory such that 0 ∈ Γ .

Lemma 43
For every s ∈ NF , there exist formulas θ ∈ L −♦ and ψ ∈ L G , with P ψ ∪ P θ ⊆ P s , such that for all ϕ ∈ L G , we have Proof See Appendix A.5

Lemma 44
The following rules are derivable in GALM: Proof The proof is standard. We prove only item (5): suppose GALM ⊆ Γ . This means that there is a sentence ψ ∈ L G such that ψ ∈ GALM but ψ ∈ Γ . The former means that ψ, thus, Γ ψ. Items (2) and (1)  Proof Let {Γ i } i∈N be an increasing chain of theories with Γ i ⊆ Γ i+1 and suppose, toward contradiction, that n∈N Γ n is not a theory, i.e., suppose that n∈N Γ n ⊥. This means that there exists a finite ⊆ n∈N Γ n such that ⊥. Then, since n∈N Γ n is a union of an increasing chain of theories, there is some m ∈ N such that ⊆ Γ m . Therefore, Γ m ⊥ contradicting the fact that Γ m is a theory. Hence, n∈N Γ n is a theory.
Proof Observe that, by axiom (  To define our canonical pseudo-model, we first put, for all maximal witnessed theories T , S and for every i ∈ AG: T ∼ U S iff ∀ϕ ∈ L G (Uϕ ∈ T implies ϕ ∈ S) , and

Lemma 51 For every
Proof Let i ∈ AG, let T and S be maximal witnessed theories such that T ∼ i S. Towards contradiction, suppose that T ∼ U S is not the case. From the former we have that ∀ϕ∈L G (K i ϕ∈T implies ϕ∈S). From the latter, we have that there is ψ∈L G such that Uψ∈T and ψ ∈S. Since Uψ→K i ψ and T is a maximal witnessed theory, Uψ→K i ψ∈T . Therefore K i ψ∈T and ψ ∈S, contradicting that T ∼ i S.
Definition 52 (Canonical Pre-Model) Given a maximal witnessed initial theory T 0 , the canonical pre-model for T 0 is a tuple M c = (W c , A c , ∼ c 1 , . . . , ∼ c n , · c ) such that: As usual, it is easy to see (given the S5 axioms for K i and for U ) that ∼ U and ∼ c i are equivalence relations.
To prove that the canonical pre-model is indeed a pseudo-model, we first need to prove the Truth Lemma. For that we need the following lemmas.
Lemma 53 (Existence Lemma for ∼ U ) Let T be a maximal witnessed theory, α ∈ L −♦ , and ϕ ∈ L G such that α ∈ T and U [α]ϕ ∈ T . Then, there is a maximal witnessed theory S such that T ∼ U S, α ∈ S and [α]ϕ ∈ S.
Proof Let α ∈ L −♦ and ϕ ∈ L G such that α ∈ T and U [α]ϕ ∈ T . The latter implies Proof If ϕ = W c , suppose Uϕ = W c . The latter means that there is a T ∈ W c such that Uϕ ∈ T . Then, by Lemma 53 (when α := ), there is a maximal witnessed theory S such that T ∼ U S and ϕ ∈ S. Since T 0 ∼ U T ∼ U S and ∼ U is transitive, we have T 0 ∼ U S, thus, S ∈ W c . Therefore, ϕ = W c , contradicting the initial assumption. If ϕ = W c , then there is a T ∈ W c such that ϕ ∈ T . Since T ∼ U S for all S ∈ W c , we obtain by the definition of ∼ U that Uϕ ∈ S for all S ∈ W c . Therefore, Uϕ = ∅.
Lemma 55 (Existence Lemma for ∼ i ) Let T be a maximal witnessed theory, α ∈ L −♦ , and ϕ ∈ L G such that α ∈ T and K i [α]ϕ ∈ T . Then, there is a maximal witnessed theory S such that T ∼ i S, α ∈ S, and [α]ϕ ∈ S.
Proof Let α ∈ L −♦ and ϕ ∈ L G such that α ∈ T and K i [α]ϕ ∈ T . The latter implies that {ψ ∈ L G : . Then, by axiom ([!] -elim) and the fact that T is maximal, we conclude that T ∈ α ♦ψ.
(⇐) Suppose T ∈ α ♦ψ, i.e., α ♦ψ ∈ T . Then, since T is a maximal witnessed theory, there is p ∈ P rop such that α p ψ ∈ T . By Lemma 20.2 we know that p ψ < ♦ψ. First we need to show that K α i θ i = K α i θ i . Note that K α i θ i = K i ( α → θ i ) = K i ( α → θ i ) and K α i θ i = K i (α → θ i ) . For (⊆): Let T ∈ K α i θ i , then for all S ∼ i T , S ∈ α → θ i . Therefore T ∈ K i (α → θ i ) and so T ∈ K α i θ i . For (⊇): Let T ∈ K α i θ i , this means that K i (α → θ i ) ∈ T . Thus for all S ∼ i T , α → θ i ∈ S. Therefore T ∈ K α i θ i . Using this, it is easy to see tha t . We then obtain that . Proposition 3.2 and the reduction axioms (R K i ) and (R p ), it is easy to see that the formula α i∈G K i p i ↔ α ∧ i∈G K α i p i is derivable in GALM. Therefore,

Corollary 60
The canonical pre-model M c is standard (and hence a pseudo-model).
Corollary 62 GALM is complete with respect to standard pseudo models.
Proof Let ϕ be a consistent formula. By Lemma 61, {0, ♦ϕ} is an initial theory. Then, by Extension Lemma (Lemma 50), there is a maximal witnessed initial theory T 0 such that {0, ♦ϕ} ⊆ T 0 . We can then construct the canonical pseudo-model M c for T 0 . Since ♦ϕ ∈ T 0 and T 0 is witnessed, there exists p ∈ P rop such that p ϕ ∈ T 0 . By Truth Lemma (applied to α := p), we get T 0 ∈ [[ϕ]] p . Hence, ϕ is satisfied at T 0 in the set p ∈ A c .
Theorem 63 APALM is complete with respect to standard pseudo models.
The completeness proof for APALM with respect to standard pseudo models is obtained by following the same steps in the completeness proof of GALM without the parts required for the operator G . This involves, for example, defining the witnessed theories only with respect to ♦ and modifying the auxiliary lemmas accordingly. This proof is presented in the earlier, shorter version [12] of this paper.
Corollary 64 GALM is complete with respect to a-models. Moreover, APALM is complete with respect to a-models.
Proof GALM completeness follows immediately from Corollaries 62 and 39. APALM completeness follows from Theorem 63 and Corollary 39.

Conclusions and Future Work
This paper solves the open question of finding a 'quantificationally strong' variant of APAL and GAL that is recursively axiomatizable. Here, by 'quantificationally strong' version we mean a language that allows quantification over all the public announcements (or group knowledge announcements in the case of GAL) that are expressible by -free formulas (and [G]-free formulas in the case of GAL) in the given language. Our system APALM is inspired by our analysis of Kuijer's counterexample [24], which lead us to add to APAL a 'memory' of the initial situation. We then used similar methods to obtain a recursive axiomatization for the memoryenhanced variant GALM of GAL. The soundness and completeness proofs crucially rely on a Subset Space-like semantics and on the equivalence between the effort modality and the arbitrary announcement modality (and on the equivalence between their [G] counterparts), thus revealing the strong link between these two formalisms.
We note again that the problems with finding a recursive axiomatization apply to many other variants of APAL and GAL. To the best of our knowledge, there is no complete axiomatization for Coalition Announcement Logic (CAL) introduced in [2], though an extension of it (subsuming both GAL and CAL) was completely axiomatized in [22] using infinitary rules. But no recursive axiomatization is known for GAL, CAL, or any of their extensions [2][3][4]33]. In contrast, in this paper we showed that the memory-enhanced version of CAL is embeddable in the (recursively axiomatized) GALM. We believe that our methods could also be used to provide a direct recursive axiomatization of CALM (the memory-enhanced variant of CAL), but we leave this for future work. Another open question is to elucidate whether GALM and CALM are equally expressive. In the memory-free case, [19] provided a counterexample: there exists a property expressible in GAL that is not expressible in CAL. Is that still the case for the memory-enhanced versions? We leave this problem for future research as well.
A further comment is on the connection of our logic with the yesterday operator. The limited form of memory provided by APALM is in fact enough to 'simulate' the yesterday operator Y ϕ on any given model, by using context-dependent formulas. For instance, the dialogue in Cheryl's birthday puzzle (Albert: "I don't know when Cheryl's birthday is, but I know that Bernard doesn't know it either"; Bernard: "At first I didn't know when Cheryl's birthday is, but I know now"; Albert: "Now I also know"), can be simulated by the following sequence of announcements 27 : first, the formula 0∧¬K a c∧K a ¬K b c is announced (where 0 marks the fact that this is the first announcement), then (¬K b c) 0 ∧ K b c is announced, and finally K a c is announced.
For another example: if instead we change the story so that the third announcement (by Albert) is "I knew you knew it (just before you said so)", then the last step of this alternative scenario corresponds to announcing the formula ([0 ∧ ¬K a c ∧ K a ¬K b c]K a K b c) 0 (saying that, just after the first announcement but before the second, Albert knew that Bernard knew the birthday). This shows how the logic can simulate the use of any (iterated) Y 's in concrete examples, although at the cost of repeating the relevant part of history inside the announcement in order to mark the exact time when the announced formula was meant to be true. Therefore, APALM combines in a way the expressivity and advantages of APAL with some of the expressivity of TPAL (the extension of PAL with the yesterday operator, introduced in [28,29]), without any of "defects" of either of them: unlike APAL, it has a natural, straightforward recursive axiomatization, with intuitive axioms and rules; unlike TPAL, its semantics is not computationally much more demanding than the one of basic epistemic logic: instead of keeping track of a growing and unbounded number of past Kripke models, APALM keeps only the initial model and the current one. Nevertheless, a more systematic treatment of the yesterday operator on (a version of) our announcement models and its connection to arbitrary and group announcements deserves a closer look. Yet another line of further work concerns other meta-logical properties, such as decidability and complexity, of APALM and GALM.

Proof of Proposition 22
The proof is by <-induction on ϕ, using Lemma 20 and the following induction hypothesis (IH): for all ψ < ϕ and all models M = (W 0 , W, ∼ 1 , . . . , ∼ n , · ), we have [[ψ]] ⊆ W . The base cases ϕ := p, ϕ := , and ϕ := 0 are straightforward by the semantics given in Defn. 3 for some θ ∈ L −♦ , so M A is an a-model.  let us check the witnessing condition for G . Let G ⊆ AG, s ∈ NF , and ψ ∈ L G and suppose that s G ψ is consistent with T Γ . The pair (s, G ψ) appears in the above enumeration of all pairs, thus (s, G ψ) := (s m , ϕ m ) for some m ∈ N. Hence, s G ψ := s m ϕ m . Then, since s G ψ is consistent with T Γ and Γ m ⊆ T Γ , we know that s G ψ is in particular consistent with Γ m . Therefore, by the above construction, s i∈G K i p i ψ ∈ Γ m+1 for some {p i : i ∈ G} ⊆ P rop such that Γ + m is consistent with s i∈G K i p i ψ. Thus, as T Γ is consistent and Γ m+1 ⊆ T Γ , we have that s i∈G K i p i ψ is also consistent with T Γ . Hence, we conclude that T Γ is witnessed. Finally, T Γ is also maximal by construction: otherwise there would be a witness theory T such that T Γ T . This implies that there exists ϕ ∈ L G with ϕ ∈ T but ϕ ∈ T Γ . Then, by the construction of T Γ , we obtain Γ i ¬ϕ for all i ∈ N. Therefore, since T Γ ⊆ T , we have T ¬ϕ. Hence, since ϕ ∈ T , we conclude T ⊥ (contradicting T being consistent).

Proof of Lemma 50 (Extension Lemma)
Let θ ∈ L G and assume that {0, θ} is a theory. Moreover, let γ 0 , γ 1 , . . . , γ n , . . . an enumeration of all pairs of the form (s n , ϕ n ) consisting of any s n ∈ NF , and every formula ϕ n ∈ L G of the form ϕ n := ♦ψ or ϕ n := G ψ with ψ ∈ L G . We will recursively construct a chain of initial theories Γ 0 ⊆ · · · ⊆ Γ n ⊆ . . . such that 1. Γ 0 = {0, θ}, 2. P n := {p ∈ P : p occurs in Γ n } is finite for every n ∈ N, and 3. for every γ n := (s n , ϕ n ) with s n ∈ NF and ϕ n ∈ L G , if Γ n ¬ s n ϕ n where ϕ n := ♦ψ then there is p m "fresh" such that s n p m ψ ∈ Γ n+1 , and, if Γ n ¬ s n ϕ n where ϕ n := G ψ for some G ⊆ AG then there is {p m i : i ∈ G} where p m i is "fresh" for every i ∈ G such that s n i∈G K i p m i ψ ∈ Γ n+1 . Otherwise we will define Γ n+1 = Γ n .
For every γ n , let P (n) := {p ∈ P | p occurs either in s n or ϕ n }. Clearly every P (n) is always finite. We now construct an increasing chain of initial theories recursively. We set Γ 0 := {0, θ}, and let if Γ n ¬ s n ϕ n and ϕ n := ♦ψ Γ n ∪ { s n i∈G K i p m i ψ} if Γ n ¬ s n ϕ n and ϕ n := G ψ Γ n otherwise, where m, m i are, in each case, the least natural number greater than the indices in P n ∪ P (n), i.e., p m , p m i for all i ∈ G are fresh in each case (since P n ∪ P (n) is finite and P rop is countably infinite, we always have enough fresh propositional variables). We now show that Γ := n∈N Γ n is an initial witnessed theory. First show that Γ is a theory. By Lemma 47, it suffices to show by induction that every Γ n is a theory. We are given that Γ 0 is a theory. For the inductive step suppose Γ n is consistent but Γ n+1 is not. Hence, Γ n = Γ n+1 and moreover Γ n+1 ⊥.Then, Γ n+1 = Γ n ∪ { s n p m ψ} (when ϕ n := ♦ψ) or Γ n+1 = Γ n ∪ { s n i∈G K i p m i ψ} (when ϕ n := G ψ). Here we will only check the latter case since the former case is analogous. Since Γ n+1 = Γ n ∪ { s n i∈G K i p m i ψ} we have Γ n [s n ][ i∈G K i p m i ]¬ψ. Therefore there exists {θ 1 , . . . , θ k } ⊆ Γ n such that {θ 1 , . . . , θ k } [s n ][ i∈G K i p m i ]¬ψ. Let θ =