Quantifying over Boolean announcements

Various extensions of public announcement logic have been proposed with quantification over announcements. The best-known extension is called arbitrary public announcement logic, APAL. It contains a primitive language construct Box phi intuitively expressing that"after every public announcement of a formula, formula phi is true". The logic APAL is undecidable and it has an infinitary axiomatization. Now consider restricting the APAL quantification to public announcements of Boolean formulas only, such that Box phi intuitively expresses that"after every public announcement of a Boolean formula, formula phi is true". This logic can therefore called Boolean arbitrary public announcement logic, BAPAL. The logic BAPAL is the subject of this work. Unlike APAL it has a finitary axiomatization. Also, BAPAL is not at least as expressive as APAL. A further claim that BAPAL is decidable is deferred to a companion paper.

cards being dealt over a finite set of players, single states in such models can be uniquely identified with a card deal. Therefore, public announcements restricting such models correspond to Booleans. For example, let there be three cards 0, 1, 2 and three players Alice, Bob, and Eve, and suppose that Alice announces (truthfully) that she holds card 0. This corresponds to the public announcement of (some elementary Boolean representation of) two card deals, namely the one wherein, additionally, Bob holds 1 and Eve holds 2, and the one wherein Bob holds 2 and Eve holds 1.
As another example, consider multi-agent planning for publicly observable sensing actions under uncertainty [Lev96,vDHL11,BA11, CHL + 16]: given multiple agents that are uncertain about a number of system parameters (lights, switches, temperature settings) they may be informed, or they may be informing each other, about their observations of the state of the light. Or they may be planning to make such observations, and contingent on the outcome of such observations take further action.
We close the introduction with an outline of the content of this work. In Section 2 we define the logical language and semantics of BAPAL and in the subsequent Section 3 we give various results for this semantics that will be used in later sections. Section 4 is on the expressivity of BAPAL. Section 5 presents the complete axiomatization.

Boolean arbitrary public announcement logic
Given are a countable (finite or countably infinite) set of agents A and a countably infinite set of propositional variables P (a.k.a. atoms, or variables).
2.1. Syntax. We start with defining the logical language and some crucial syntactic notions.
Definition 2.1 (Language). The language of Boolean arbitrary public announcement logic is defined as follows, where a ∈ A and p ∈ P .
We also distinguish the language L el of epistemic logic (without the constructs [ϕ]ϕ and 2ϕ) and the language L pl of propositional logic (without additionally the construct K a ϕ), also known as the Booleans. Booleans are denoted ϕ 0 , ψ 0 , etc.
The set of propositional variables that occur in a given formula ϕ is denoted var (ϕ) (where one that does not occur in ϕ is called a fresh variable), its modal depth d(ϕ) is the maximum nesting of K a modalities, and its quantifier depth D(ϕ) is the maximum nesting of 2 modalities. These notions are inductively defined as follows. Arbitrary announcement normal form is a syntactic restriction of L bapal that pairs all public announcements with arbitrary Boolean announcement operators. It plays a role in the decidability proof. We will show that any formula in L bapal is equivalent to one in L aanf .
Definition 2.2 (Arbitrary announcement normal form). The language fragment L aanf is defined by the following syntax, where a ∈ A and p ∈ P .
We now define necessity forms and possibility forms. Necessity forms are used in derivation rules in the proof system.
By induction on the necessity form ψ( ) the reader may easily verify that each ψ( ) contains a unique occurrence of the symbol . If ψ( ) is a necessity form and ϕ ∈ L bapal , then ψ(ϕ) is ψ( )[ϕ/ ] (the substitution of in ψ( ) by ϕ), where we note that ψ(ϕ) ∈ L bapal . A possibility form is the dual of a necessity form. They are therefore defined as: Similarly to above, notation ψ{ϕ} means that is substituted by ϕ in ψ{ }.

2.2.
Structures. We consider the following structures and structural notions in this work.
Definition 2.4 (Model). An (epistemic) model M = (S, ∼, V ) consists of a non-empty domain S (or D(M )) of states (or 'worlds'), an accessibility function ∼: A → P(S × S), where each ∼ a is an equivalence relation, and a valuation V : P → P(S), where each V (p) represents the set of states where p is true. For s ∈ S, a pair (M, s), for which we write M s , is a pointed (epistemic) model.
We will abuse the language and also call M s a model. We will occasionally use the following disambiguating notation: if M is a model, S M is its domain, ∼ M a the accessibility relation for an agent a, and V M its valuation.
Definition 2.5 (Bisimulation). Let M = (S, ∼, V ) and M = (S , ∼ , V ) be epistemic models. A non-empty relation R ⊆ S × S is a bisimulation if for every (s, s ) ∈ R, p ∈ P , and a ∈ A the conditions atoms, forth and back hold.
• forth: for every t ∼ a s there exists t ∼ a s such that (t, t ) ∈ R.
• back: for every t ∼ a s there exists t ∼ a s such that (t, t ) ∈ R. If there exists a bisimulation R between M and M such that (s, s ) ∈ R, then M s and M s are bisimilar, notation M s ↔ M s (or R : M s ↔ M s , to be explicit about the bisimulation). Let Q ⊆ P . A relation R between M and M satisfying atoms for all p ∈ Q, and forth and back, is a Q-bisimulation (a bisimulation restricted to Q). The notation for Q-restricted bisimilarity is ↔ Q .
The notion of n-bisimulation, for n ∈ N, is given by defining relations R 0 ⊇ · · · ⊇ R n .
Definition 2.6 (n-Bisimulation). Let M = (S, ∼, V ) and M = (S , ∼ , V ) be epistemic models, and let n ∈ N. A non-empty relation R 0 ⊆ S × S is a 0-bisimulation if atoms holds for pair (s, s ) ∈ R. Then, a non-empty relation R n+1 ⊆ S × S is a (n + 1)-bisimulation if for all p ∈ P and a ∈ A: • (n + 1)-forth: for every t ∼ a s there exists t ∼ a s such that (t, t ) ∈ R n ; • (n + 1)-back: for every t ∼ a s there exists t ∼ a s such that (t, t ) ∈ R n .
Similarly to Q-bisimulations we define Q-n-bisimulations, wherein atoms is only required for p ∈ Q ⊆ P ; n-bisimilarity is denoted M s ↔ n M s , and Q-n-bisimilarity is denoted M s ↔ n Q M s .
2.3. Semantics. We continue with the semantics of our logic.
Definition 2.7 (Semantics). The interpretation of formulas in L bapal on epistemic models is defined by induction on formulas. Assume an epistemic model M = (S, ∼, V ), and s ∈ S.
Given M s and M s , if for all ϕ ∈ L bapal , M s |= ϕ iff M s |= ϕ, we write M s ≡ M s . Similarly, if this holds for all ϕ with d(ϕ) ≤ n, we write M s ≡ n M s , and if this holds for all ϕ with var (ϕ) ∈ Q ⊆ P , we write M s ≡ Q M s .
Note that the languages of APAL and BAPAL are the same, but that their semantics are different. The only difference is the interpretation of 2ϕ: in APAL, this quantifies over the 2-free fragment [BBvD + 08], so that, given the eliminability of public announcements from that fragment [Pla89], this amounts to quantifying over formulas of epistemic logic:

Semantic results
We continue with basic semantic results for the logic. They will be used in various of the later sections. Various well-known results for any dynamic epistemic logic with propositional quantification generalize straightforwardly to BAPAL.
3.1. Bisimulation invariance. We start with the bisimulation invariance of BAPAL. This is shown as for APAL. Proof. We prove that: for all ϕ ∈ L bapal , and for all M s , N s : if M s ↔ N s , then M s |= ϕ iff N s |= ϕ; from which the required follows by restricting the scope of ϕ to the consequent of the implication. The proof is by induction on the structure of ϕ, where the 2-depth D(ϕ) takes lexicographic precedence over formula structure (i.e., ψ 1 is less complex than ψ 2 , if D(ψ 1 ) < D(ψ 2 ), or if D(ψ 1 ) = D(ψ 2 ) and ψ 1 is a subformula of ψ 2 ). The non-standard inductive cases are [ϕ]ψ and 2ψ. In either case we only show one direction of the equivalence in the conclusion; the other direction is similar.
The latter is by definition equal to: M s |= ϕ implies M ϕ s |= ψ. Let us now assume M s |= ϕ. First, from M s ↔ N s and M s |= ϕ, it follows by induction that N s |= ϕ. Second, this not only holds for s but for any t in the domain of M and t in the domain of N : from M t ↔ N t and M t |= ϕ, it follows by induction that N t |= ϕ.
This allows us to show that M ϕ s ↔ N ϕ s , namely, given R : M s ↔ N s , by the relation R : M ϕ s ↔ N ϕ s such that for all t, t : (t, t ) ∈ R iff ((t, t ) ∈ R and M t |= ϕ). We now show that the relation R is a bisimulation. The clause atoms is obvious. For forth, assuming some (s, s ) ∈ R , let s ∼ a t in M ϕ . Let t be such that (t, t ) ∈ R and s ∼ a t . From (t, t ) ∈ R and M t |= ϕ, we get with induction that N t |= ϕ. Therefore, (t, t ) ∈ R . As s ∼ a t persists in N ϕ , the state t satisfies the requirements for forth. The clause back is shown similarly.
Third, having shown that M ϕ s ↔ N ϕ s , and also using that M s |= [ϕ]ψ and M s |= ϕ implies M ϕ s |= ψ, we now use the induction hypothesis for ψ on pair of models M ϕ s , N ϕ s , and thus obtain that N ϕ s |= ψ as required. Winding up, we now have shown that (M s |= ϕ implies M ϕ s |= ψ) is equivalent to (N s |= ϕ implies N ϕ s |= ψ), i.e., N s |= [ϕ]ψ, as required.
The property shown in the case announcement of the above proof is often used in the continuation and therefore highlighted in a corollary.
Corollary 3.2. Let ϕ ∈ L bapal such that M s |= ϕ. Then M s ↔ N s implies M ϕ s ↔ N ϕ s . The next lemma may look obvious but is actually rather special: it holds for BAPAL but not, for example, for APAL, where the quantifiers are over formulas of arbitrarily large modal depth. Lemma 3.3 plays a role in Section 4 on expressivity. Proof. We show the above by proving the following statement: We prove this by refining the complexity measure used in the previous proposition: we now, additionally, give modal depth d(ϕ) lexicographic precedence over quantifier depth D(ϕ) and D(ψ 1 ) = D(ψ 2 ) and ψ 1 is a subformula of ψ 2 ). For clarity we give the -essentially different-case K a ϕ and also the -essentially the same-cases [ϕ]ψ and 2ψ. The latter two apply to any n ∈ N and do not require the induction over n. We let these cases therefore precede the case K a ϕ.
In this case of the proof we need to use induction on subformulas Case 2ψ. Let M s ↔ n N s and M s |= 2ψ. As d(ψ) = d(2ψ), we will now use that D(ψ) < D(2ψ). This is therefore similar again to the same case in the previous Lemma 3.1. By definition, M s |= 2ψ is equal to: for all ϕ 0 ∈ L pl , M s |= [ϕ 0 ]ψ, i.e., for all ϕ 0 ∈ L pl , M s |= ϕ 0 implies M ϕ 0 s |= ψ. It is now crucial to note that, as ϕ 0 is Boolean, not only D(ϕ) = 0 but also d(ϕ 0 ) = 0. We therefore obtain by induction, as in the previous case [ϕ]ψ of this proof: for all ϕ 0 ∈ L pl , N s |= ϕ 0 implies N ϕ 0 s |= ψ, i.e., N s |= 2ψ.
Case K a ψ. Given are M s ↔ n+1 N s and M s |= K a ϕ. Let now t ∼ a s . From M s ↔ n+1 N s , t ∼ a s , and n − back follows that there is a t ∼ a s such that M t ↔ n N t . From M s |= K a ϕ and s ∼ a t follows that M t |= ϕ. As d(ϕ) = d(K a ϕ) − 1 ≤ n, we can apply the induction hypothesis for n and conclude that N t |= ϕ. As t was arbitrary, N s |= K a ϕ.
The interest of the above proof is the precedence of modal depth over quantifier depth, and of quantifier depth over subformula complexity. Essential in the proof is that in the case 2ψ, for any [ϕ 0 ]ψ witnessing that, not only D(ϕ 0 ) = 0 but also d(ϕ 0 ) = 0. Without d(ϕ 0 ) = 0 the inductive hypothesis would not have applied. In contrast, the APAL quantifier is over formulas of arbitrary finite modal depth, also exceeding the modal depth of the initial given formula, which rules out use of induction.
Both for APAL and BAPAL restricted bisimilarity does not imply restricted modal equivalence: The failure of this property is indirectly used in Proposition 4.1 in the expressivity Section 4, later, for Q = {p} (and for models with those same names).

H. van Ditmarsch and T. French
Vol. 18:1 3.2. Arbitrary announcement normal form and necessity form. We continue with results for the arbitrary announcement normal form and for the necessity form. The former are important to show decidability of BAPAL, and the latter to show that the axiomatization is complete.
Lemma 3.4. Every formula of L bapal is semantically equivalent to a formula in arbitrary announcement normal form.
Proof. We give the proof by defining a truth preserving transformation δ from L bapal to L aanf . This is defined with the following recursion: We have to show that the translation is truth preserving and that the translation procedure terminates. The truth preservation is obvious for the clauses in rows 1 and 2 and for δ( The translation clauses in rows 3 to 5 employ the valid equivalences These are well-known from PAL [Pla89,vDvdHK08] (and for further reference we note that they are listed as axioms of BAPAL in Section 5).
We proceed by showing termination. For the clauses in rows 1, 2, and 6 we observe that each occurrence of δ on the right-hand side of an equation binds a formula that is less complex than the formula bound by δ on the left-hand side of the equation (and in δ(p) = p, δ has disappeared on the right-hand side). For the clauses in rows 3 to 5 we refer to complexity measures in PAL [vDvdHK08,BvD15]. Taking (e.g.) the measure [vDvdHK08, Def. 7.21], [vDvdHK08,Lemma 7.22] shows that the complexity of the right-hand side of each equivalence above is either equal to or lower than the complexity of the left-hand side of the equivalence. Therefore, δ will always return a formula in L aanf .
Proof. This is easily shown by induction on the structure of necessity forms and using that In the following lemma we show that a necessity form of arbitrary shape ψ( ) can be transformed into a necessity form of unique shape ψ 1 → [ψ 2 ] . With respect to instantiations ψ(θ) of such necessity forms this is a validity preserving transformation in both directions. Note that it is not a truth preserving transformation. This result will be used in Section 5 to show that two versions of the axiomatization with different derivation rules are both complete (the translation is not only validity preserving but also derivability preserving).
We can now observe that: • The translation τ terminates.
Here we use again that formulas of shape [ϕ][ψ]η are at least as complex as [ϕ ∧ [ϕ]ψ]η. We should also observe that in any clause of shape τ (x → y) = τ (z → w), y is at least as complex as w, and that in any claus of shape τ ([x]u) = τ (z → w), [x]u is at least as complex as w. The complexity the antecedent z of the implication does not matter. • τ (ψ(θ)) is a neccessity form of shape ψ 1 → [ψ 2 ]θ, as required.
There are four clauses in which τ does not appear on the right-hand side, and in all those cases the right-hand side has the required shape. • For all θ ∈ L bapal , |= ψ(θ) iff |= τ (ψ(θ)), as required.
This holds because all clauses are validity preserving in both directions. Apart from propositional tautologies this can be justified by one or more of the following observations: In epistemic logic, |= K a ψ implies |= ψ, and |= ψ implies |= K a ψ. The validity of these in BAPAL is also obvious. This is used in the, maybe surprising, case τ (K a ψ( )).
The PAL validities listed in the above Lemma 3.4, there justifying truth preservation, are also frequently used here; in addition (as this might otherwise be oblique) we use the validity . This is used in various cases including τ ([ϕ](ψ 1 → ψ 2 ( )).
In epistemic logic, |= ϕ → K a ψ iff |=K a ϕ → ψ. Let us prove this. We note that |= ϕ → K a ψ implies |=K a ϕ →K a K a ψ, which, asK a K a ϕ is equivalent in epistemic logic to K a ϕ, implies |=K a ϕ → K a ψ. From that and |= K a ψ → ψ we obtain |=K a ϕ → ψ. For the other direction, |=K a ϕ → ψ implies |= K aKa ϕ → K a ψ, which as K aKa ϕ is equivalent in epistemic logic toK a ϕ, implies |=K a ϕ → K a ψ. From that and |= ϕ →K a ϕ we obtain |= ϕ → K a ψ. This is used in various cases including τ (ϕ → K a ψ( )).
They play no auxiliary role as tools in later sections, but they serve to compare BAPAL to other logics with quantification over information change, where such properties sometimes hold and sometimes not.
• In order to prove the validity of CR we have to show that: for all ϕ 0 , ψ 0 ∈ L pl and for all M with non-empty denotation of ϕ 0 and of ψ 0 there are ϕ 0 , ψ 0 ∈ L pl such that M ϕ 0 ϕ 0 ↔ M ψ 0 ψ 0 . The obvious choice is ϕ 0 = ψ 0 and ψ 0 = ϕ 0 , as The proof of the validity 4 is more direct than in [BBvD + 08], where it is used that χ ψ ϕ is equivalent to χ ψ ϕ. Formula χ ψ ϕ is not equivalent to χ ∧ ψ ϕ for all χ, ψ ∈ L bapal .
The proof of the validity CR for BAPAL is easier than for other quantified epistemic logics such as APAL, where the proof needs to 'close the diamond at the bottom', which just as in the case of MK needs declaring the values of all atoms, i.e., we must choose ϕ 0 = ψ 0 = δ s (ϕ). For such a proof see e.g. [vDFH21, Lemma 3.10].
Note that not all logics with quantification over announcements satisfy all the properties of the quantifier shown in this subsection. For example, the logic known as APAL+ (quantifying over announcements of so-called positive formulas, corresponding to the universal fragment in first-order logic) does not satisfy the validity 2ϕ → 22ϕ [vDFH21]. 3.4. Boolean closure. We will now define the novel notion of Boolean closure, and prove some lemmas for it. These are used when proving the soundness of the axiomatization of the logic BAPAL, later. As P is countably infinite, and as the Booleans on P can be enumerated,P is also countably infinite. Given an epistemic model, then for each atom and for each Boolean there are also infinitely many atoms with the same value on the Boolean closure of that model. E.g., p has the same value as p p∧p (i.e., the atom q corresponding to the Boolean p ∧ p), and the same value as p p∧p∧p (the atom q corresponding to Boolean p ∧ p ∧ p), etc. We proceed with some other properties of the Boolean closure, in the form of lemmas.
Lemma 3.9. On a Boolean closed modelM , for all Booleans, including Booleans of atoms inP \ P , there is an atom with the same value.
Proof. The proof is by induction on the structure of a Boolean ϕ ∈ L pl (P ).
If ϕ is an atom, it is obvious. If ϕ = ¬ψ, by induction we may assume that there is an atom p ∈P such thaẗ V  Proof. LetM s and ψ ∈ L bapal be given. By the semantic definition of 2ψ, we have thaẗ M s |= 2ψ iffM s |= [ϕ 0 ]ψ for all ϕ 0 ∈ L pl (P ). Assuming the latter, it follows thatM s |= [p]ψ for all p ∈P because atoms are Booleans.
The next lemma involves a translation tr : L bapal (P ) → L bapal (P ) defined as tr(p ϕ 0 ) = ϕ 0 for p ϕ 0 ∈P \ P and all other clauses trivial. The property shown in ( * ) of the above proof is significant enough to be mentioned as a corollary. For ψ ∈ L bapal (P ) we have that tr(ψ) = ψ, so that: Corollary 3.13. Let ψ ∈ L bapal (P ) and model M s be given. ThenM s |= ψ iff M s |= ψ.
The following result will be needed to show the soundness of a rule in the axiomatization of BAPAL. As a similar result from the literature ([BBvD + 08, Prop. 3.7]) was later shown false, we give the proof in full detail. We recall that an atom q is fresh with respect to ϕ, if ϕ does not contain an occurrence of q.
Proof. The proof is by induction on the structure of possibility forms. Note that in the formulation of the lemma the formula is declared prior to the model. Therefore, induction hypotheses for a subformula apply to model restrictions and states in those restrictions.
For all cases, the direction from right to left is the direct application of the semantics of 2, for the dual modality 3, and where we use that any p ∈P \ P must be such that p = q ϕ 0 for some ϕ 0 ∈ L pl (P ): LetM s |= ψ{ p ϕ} be given. From that, with Lemma 3.11, we get M s |= tr(ψ{ p ϕ}). As p ∈P \ P , tr(p) = ϕ 0 for some ϕ 0 ∈ L pl (P ), and therefore M s |= tr(ψ{ p ϕ}) equals M s |= ψ{ ϕ 0 ϕ}. By the (dual) semantics of 2, from that we obtain M s |= ψ{3ϕ}.
We continue with the direction from left to right.
In the proof of the following Lemma 3.15 (and only in this proof), we do not only have to keep track explicitly of the parameter set of atoms for which the logical language is defined, but similarly of the set of atoms for which a model is defined, and where this set of atoms may contain P instead of being contained in it. We therefore resort to let M s (Q) denote that model M is defined for variables Q (and where s is the designated state). This also implies that, given some M s (P ), its Boolean closure isM s (P ). We also let BAPAL(Q) mean 'BAPAL for set of atoms Q'. Outside this proof, the parameter set of atoms remains P .
Proof. Let ϕ be BAPAL(P ) valid. We first show that for any Q with P ⊆ Q: (3.1) It is more intuitive to show the contrapositive, where for notational convencience we replaced ϕ by ¬ϕ: If ϕ ∈ L bapal (P ) is BAPAL(Q) satisfiable, then ϕ is BAPAL(P ) satisfiable. Case ϕ = ϕ ϕ . Suppose M s (Q) |= ϕ ϕ . Then M s (Q) |= ϕ and M ϕ s (Q) |= ϕ . By induction it follows that M s (P ) |= ϕ and M ϕ s (P ) |= ϕ (as usual, note that the inductive hypothesis applies to any model N t (Q), not merely to M s (Q); it therefore applies to N t = M ϕ s (Q)). Therefore M s (P ) |= ϕ ϕ .
Then there is ψ 0 ∈ L pl (Q) such that M s (Q) |= ψ 0 ϕ , and therefore M s (Q) |= ψ 0 and M ψ 0 s (Q) |= ϕ . For any atom q ∈ var (ψ 0 ) such that q ∈ Q \ P , choose a fresh atom p ∈ P (fresh with respect to ψ 0 and ϕ , and with respect to prior choices of such atoms in var (ψ 0 )), and transform M into N = (S, ∼, V ) with V (p ) = V (q) and V (q) = V (p ). Let ψ 0 ∈ L pl (P ) be the result of all such substitutions.  s (P ) |= ϕ follows N s (P ) |= 3ϕ . Now observe that N s (P ) and M s (P ) only differ in variables not occurring in 3ϕ , so that N s (P ) |= 3ϕ iff M s (P ) |= 3ϕ (where similarly to above we take into account occurrences of 3 in ϕ ). Therefore, M s (P ) |= 3ϕ . This shows (3.3). We can now quickly close the argument. Let ϕ be satisfiable for Q. Then there is M s (Q) such that M s (Q) |= ϕ. Using (3.3), M s (P ) |= ϕ. Therefore ϕ is satisfiable for P . This shows (3.2). Therefore, we have now shown, for arbitrary ϕ, (3.1): if ϕ is valid for P , and P ⊆ Q, then ϕ is valid for Q. We now apply (3.1) for Q =P .
Clearly P ⊂P . So, if ϕ is BAPAL(P ) valid, then ϕ is BAPAL(P ) valid. For any validity ϕ in a logic BAPAL(P ), p ∈ P , and fresh q ∈ P , ϕ[q /p ] is also a BAPAL(P ) validity. Therefore, given that ϕ is a validity of BAPAL(P ) and a (obviously) fresh q ∈P \ P , also ϕ[q/p] is a validity of BAPAL(P ). Here is it important to observe that this is a validity for the class of epistemic models for variablesP , and that this is not a validity for the class of Boolean closures of epistemic models for variables P . But of course, the latter is contained in the former: a Boolean closed model for P is, after all, a model forP . Therefore, alsö M s (P ) |= ϕ[q/p], as required.

Expressivity
Given logical languages L and L , and a class of models in which L and L are both interpreted (employing a satisfaction relation |= resp. |= ), we say that L is at least as expressive as L , if for every formula ϕ ∈ L there is a formula ϕ ∈ L such that for all models M and states s ∈ D(M ), M s |= ϕ iff M s |= ϕ. If L is not at least as expressive as L and L is not at least as expressive as L, then L is incomparable to L . If L is at least as expressive as L , and L is at least as expressive as L, then L is as expressive as L . If L is at least as expressive as L but L is not at least as expressive as L, then L is (strictly) more expressive than L . The combination of a language with a semantics given a class of models determines a logic. In this work we only consider model class S5. We abbreviate "given logic L determined by language L, model class S5 and satisfaction relation |=, and logic L determined by language L , model class S5 and satisfaction relation |= , L is at least as expressive as L ," by "L is at least as expressive as L ," and similarly for other expressivity terminology. Note that in this work the language L bapal is the same for BAPAL and APAL, whereas the semantics of the quantifier are different in BAPAL and APAL. We also consider language fragments, namely L pal and L el .
We show that BAPAL is more expressive than EL and that BAPAL is not as least as expressive as two other logics with quantification over announcements: APAL, and group announcement logic (GAL) [ÅBvDS10]. It is not known whether APAL (or GAL) is at least as expressive as BAPAL or not. We conjecture that it is not. To prove that, one would somehow have to show that the BAPAL-2 in a given formula 2ϕ can be 'simulated' by an APAL-2 that is properly entrenched in preconditions and postconditions relative to ϕ, thus providing an embedding of L bapal into L apal . This seems quite hard.
In the models depicted below we use the following visual conventions: the names of states are replaced by the sets of atoms true in those states; the accessibility relations for the two agents a, b are reflexively and symmetrically (and transitively) closed, in other words, 20:16

H. van Ditmarsch and T. French
Vol. 18:1 they partition the domain into equivalence classes; and the actual state (the designated world) is underlined.
Proposition 4.1. BAPAL is more expressive than EL.
Proof. To prove that BAPAL is more expressive than EL we first observe that L el ⊆ L bapal (and that on that restriction they have the same semantics), so that BAPAL is at least as expressive as EL, and we then observe that the (standard) proof that EL is not at least as expressive as APAL [BBvD + 08] can also be used to show that EL is not at least as expressive as BAPAL.
We recall the proof in [BBvD + 08], wherein the formula 3(K a p ∧ ¬K b K a p) is shown not to be equivalent to an epistemic logical formula ψ as follows. There must be a propositional variable q not occurring in ψ. Two models that are bisimilar except for q will either make ψ true in both or false in both. On the other hand, 3(K a p ∧ ¬K b K a p) may be true in one and false in the other, as it quantifies over variable q as well. This quantification is implicit, as q ∈ var (3(K a p ∧ ¬K b K a p). We can therefore easily make q (K a p ∧ ¬K b K a p) true in one and false in the other, as shown below for M s and M s .
As the announcement q witnessing the diamond is a Boolean, this also proves the case for BAPAL. Proof. Consider (again, but to other usage) L apal formula 3(K a p∧¬K b K a p). Let us suppose that there exists an equivalent L bapal formula ψ. Given the modal depth d(ψ) of ψ, consider two models N t , O t , with a difference between them further away from the root than d(ψ), Formally, N t and O t can be defined as follows. Model N t has domain Z, equivalence classes for relation ∼ a consisting of pairs {2i, 2i + 1} for i ∈ Z and for relation ∼ b consisting of pairs {2i − 1, 2i} for i ∈ Z, and with V (p) = i∈Z {4i − 1, 4i}. The actual state t is state 0. Note that M s ↔ N t , where M s is the model that was used in the previous proposition. Model O t is as model N t (and with t = 0), except that the domain is restricted to the range i ≤ 4j, where j is the least positive integer for which d(ψ) < 4j (so, on the left model O is infinite, on the right it ends in two p worlds). As the argument in the proof is abstract (ψ is hypothetical) and only needs j to be in excess of d(ψ) 4 (we only need to refer to formulas of modal depth larger than d(ψ)), the schematic visualization of these models below, wherein we have abstracted from the names of states, suffices in the proof. We use Lemma 3.3 that n-bisimilarity implies n-logical equivalence: from N t ↔ d(ψ) O t it follows that N t ≡ d(ψ) O t and thus, as the formula ψ itself has depth d(ψ), that N t |= ψ iff O t |= ψ ( †). On the other hand, N t |= 3(K a p ∧ ¬K b K a p) (obviously, consider the bisimilar M s ) whereas O t |= 3(K a p ∧ ¬K b K a p). To prove the latter we observe that any finite subset of the model O can be distinguished from its complement by a formula in the logic (by 'distinguished' we mean that the formula is true in all the states of that subset and false in all other states of the domain of that model), where we use that any state can be distinguished from all others by its distance to the rightmost terminal state 1 , that is distinguished by K a p. In particular, there must therefore be a distinguishing formula ϕ of a three-state subset of O such that O ϕ is as depicted. As O ϕ t |= K a p ∧ ¬K b K a p, we get that O t |= ϕ (K a p ∧ ¬K b K a p), and thus O t |= 3(K a p ∧ ¬K b K a p). This is a contradiction with ( †). Therefore, no such ψ ∈ L bapal exists.
As a corollary of Proposition 4.2 we can very similarly show that BAPAL is not at least as expressive as GAL, as in GAL we also quantify over announcements of arbitrarily large modal depth. We then use the same models as above but with an additional agent c who has the identity accessibility relation on the model. On models where the accessibility relation for c is the identity, K c ϕ ↔ ϕ is valid for any ϕ. The language of GAL has a primitive c ϕ which stands for 'there is a formula ψ ∈ L el such that K c ψ ϕ' (see [ÅBvDS10]). On this three-agent model, K c ψ ϕ is equivalent to ψ ϕ, and as 'there is a formula ψ ∈ L el such that ψ ϕ' is equivalent to '3ϕ' in the APAL semantics, we obtain that on this model c ϕ is equivalent to 3ϕ. We now copy the above argument but with c (K a p ∧ ¬K b K a p) instead of 3(K a p ∧ ¬K b K a p). Therefore: Corollary 4.3. BAPAL is not at least as expressive as GAL.

Axiomatization
We now provide a sound and complete finitary axiomatisation for BAPAL.
With these results we can now easily demonstrate that not only bapal bapal bapal ω but also the other two axiomatizations are complete and define the same set of theorems. Below, let the name of the axiomatization stand for the set of derivable theorems. Again, we follow the same argument as in [BBvD + 08].
• bapal bapal bapal ω ⊆ bapal bapal bapal 1 : A derivation in bapal bapal bapal ω is not a finite sequence of formulas but a converse well-founded sequence of formulas, because a R2 ω rule application has an infinite number of premisses. We can transform such a bapal bapal bapal ω derivation into a bapal bapal bapal 1 derivation as follows. If it contains no R2 ω rule applications it is already a bapal bapal bapal 1 derivation. Otherwise, consider a R2 ω rule application with conclusion ψ(2ϕ). One of its infinite premisses must be ψ([p]ϕ) for a fresh atom p. Discarding all other premisses from that R2 ω rule application makes it a R2 1 rule application. Successively doing this for all R2 ω rule applications in the derivation (where we note that this is a finite number, as the derivation is converse well-founded) therefore transforms this bapal bapal bapal ω derivation into a bapal bapal bapal 1 derivation. • bapal bapal bapal 1 = bapal bapal bapal: In Lemma 5.7, using the transformation τ defined in Lemma 3.6, was shown that a derivation with R2 1 rule applications can be transformed into one with R2 rule applications. This shows bapal bapal bapal 1 ⊆ bapal bapal bapal. The other direction of the mutual inclusion, bapal bapal bapal ⊆ bapal bapal bapal 1 , is trivial, as a R2 rule application is also a R2 1 rule application, and therefore a bapal bapal bapal derivation also a bapal bapal bapal 1 derivation. • bapal bapal bapal ⊆ bapal bapal bapal ω : Here we use completeness of bapal bapal bapal ω . Let a bapal bapal bapal theorem be given.

Conclusions and further research
We proposed the logic BAPAL. It is an extension of public announcement logic. It contains a modality 2ϕ intuitively corresponding to: "after every public announcement of a Boolean formula, ϕ is true". We have shown that BAPAL is more expressive than EL and not at least as expressive as APAL, and that it has a finitary complete axiomatization. For further research we wish to report the decidability of the satisfiability problem of BAPAL and of yet another logic with quantification over announcements, called positive arbitrary public announcement logic, APAL+. The logic APAL+ has a primitive modality "after every public announcement of a positive formula, ϕ is true". The positive formulas correspond to the universal fragment in first-order logic. These are the formulas where negations do not bind modalities. It has been reported in [vDFH21].