No Finite Model Property for Logics of Quantified Announcements

Hans van Ditmarsch Open University of the Netherlands
Heerlen, The Netherlands [email protected] University of Western Australia
Perth, AustraliaUniversity of Bergen
Bergen, Norway Tim French University of Western Australia
Perth, Australia [email protected] University of Bergen
Bergen, Norway Rustam Galimullin University of Bergen
Bergen, Norway [email protected]

Abstract

Quantification over public announcements shifts the perspective from reasoning strictly about the results of a particular announcement to reasoning about the existence of an announcement that achieves some certain epistemic goal. Depending on the type of the quantification, we get different formalisms, the most known of which are arbitrary public announcement logic ( $\mathit{APAL}$ ), group announcement logic ( $\mathit{GAL}$ ), and coalition announcement logic ( $\mathit{CAL}$ ). It has been an open question whether the logics have the finite model property, and in the paper we answer the question negatively. We also discuss how this result is connected to other open questions in the field.

1 Introduction

One of the most well-known ways to introduce communication into the multi-agent epistemic logic ( $\mathit{EL}$ ) [19] is by extending the logic with public announcements, which, being dynamic operators, model the situation of all agents publicly and simultaneously receiving the same piece of information. Epistemic logic with such operators is called public announcement logic ( $\mathit{PAL}$ ) [22], and it extends $\mathit{EL}$ with constructs $[\varphi]\psi$ that mean ‘after public announcement of $\varphi$ , $\psi$ ’.

A natural way to generalise $\mathit{PAL}$ , with epistemic planning [6] flavour to it, is to consider quantification over public announcements¹¹1See a recent survey [7] for an overview.. Such an extension allows us to reason about the existence of an announcement, or a sequence thereof, that reaches certain epistemic goal. There are several ways to quantify over public announcements, and we call the resulting logics quantified public announcement logics ( $\mathit{QPAL}$ ). In the paper, we consider the Big Three of $\mathit{QPAL}$ ’s, namely arbitrary public announcement logic ( $\mathit{APAL}$ ), group announcement logic ( $\mathit{GAL}$ ), and coalition announcement logic ( $\mathit{CAL}$ ).

$\mathit{APAL}$ [5] extends $\mathit{PAL}$ with constructs $\Box\varphi$ that are read as ‘after any public announcement, $\varphi$ is true’. While $\mathit{APAL}$ modalities do not take into account who announces a formula or whether the formula can be truthfully announced by any of the agents in a system, $\mathit{GAL}$ constructs $[G]\varphi$ restrict the quantification to the formulas that agents actually know [2]. Thus, $[G]\varphi$ is read as ‘after any joint truthful announcement by agents from group $G$ , $\varphi$ is true’. ‘Truthful’ here denotes the fact that the agents know the formulas they announce. Finally, $\mathit{CAL}$ is somewhat similar to $\mathit{GAL}$ with a crucial difference that agents outside of the selected group can also make a simultaneous announcement [3, 13]. $\mathit{CAL}$ extends $\mathit{PAL}$ with constructs $[\!\langle G\rangle\!]\varphi$ that are read as ‘whatever agents from $G$ announce, there is a simultaneous announcement by the agents outside of $G$ , such that $\varphi$ is true after the joint announcement’. Given that the modalities of $\mathit{CAL}$ were inspired by coalition logic [21], it is not surprising that they are game-theoretic in nature, and, in particular, express the property of $\beta$ -effectivity²²2The dual $\langle\![G]\!\rangle\varphi$ expresses $\alpha$ -effectivity..

One of the pressing open questions of $\mathit{QPAL}$ is whether they have the finite model property (FMP).

FMP: A logic has the FMP iff every formula of the logic that is true in some model is also true in a finite model.

It is a standard result that $\mathit{EL}$ has the FMP [15]. As the reader may have already guessed, after having read the title of the paper, we show that $\mathit{APAL}$ , $\mathit{GAL}$ , and $\mathit{CAL}$ do not have the FMP.

The result is important for a couple of reasons. First, it tells us something about the expressivity of $\mathit{QPAL}$ ’s. In particular, all of $\mathit{APAL}$ , $\mathit{GAL}$ , and $\mathit{CAL}$ , are so expressive that they can force infinite models, i.e. there are formulas of the languages that can only be true on infinite structures.

Second, the lack of FMP sheds light on a connected open problem of finding finitary axiomatisations of $\mathit{QPAL}$ ’s. It is known [23] that

Finitary axiomatisation and FMP imply decidability. (*)

We also know that $\mathit{APAL}$ , $\mathit{GAL}$ , and $\mathit{CAL}$ are all undecidable [4]. The corresponding proof is quite complex, but ultimately it presents for each logic a formula over a parameterised set of tiles, such that the formula is satisfiable if and only if the set of tiles can tile the plane. The construction of the tiling is not explicit, in the sense that there is no one-to-one correspondence of worlds to unique points on the plane. Rather, given a model that satisfies the formula, a series of finite tilings is created in such a way that it guarantees the existence of an infinite tiling. It is not clear how to extract an argument against FMP from the undecidability proof, or whether it is possible. In any way, our approach presented in Section 3 is simpler and more elegant.

Another reason why we cannot get the lack of the FMP for free from the undecidability is that (*) requires the axiomatisations at hand to be finitary. To the best of our knowledge, none of $\mathit{APAL}$ , $\mathit{GAL}$ , and $\mathit{CAL}$ have a known finitary axiomatisation, although the first two are recursively axiomatisable, and providing any axiomatisation of $\mathit{CAL}$ is an open problem. However, (*) cannot be relaxed to requiring only recursive axiomatisations instead of finitary ones [23, 17].

On the whole, the relation between the FMP, finitary axiomatisations, and decidability is not trivial, and (*) can be satisfied in various ways. For example, $\mathit{EL}$ has all three properties, while there are modal logics that are decidable and recursively axiomatisable [20, 9], or finitely axiomatisable, decidable, but lack the FMP [10, 11, 8], or not finitely axiomatisable, undecidable, but have the FMP [12, 18, 16], or have none of the three properties [16], and so on.

Due to the undecidability of $\mathit{QPAL}$ ’s, up until now there was a hope that if the logics have the FMP, then we will be able to conclude they are not finitely axiomatisable. We show that, in fact, the logics do not have the FMP and the problem of the existence of finitary axiomatisations remains.

In what follows, we formally introduce $\mathit{APAL}$ , $\mathit{GAL}$ , $\mathit{CAL}$ , and some technical notions in Section 2. In Section 3 we prove that $\mathit{APAL}$ does not have the FMP. The strategy of the proof is such that we, first, present a formula that is true in an infinite model, and then claim that the formula cannot be true in any finite model. The results for $\mathit{GAL}$ and $\mathit{CAL}$ follow as a corollary. We conclude and discuss further research in Section 4.

2 Quantified public announcement logics

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). In what follows, $G\subseteq A$ , and we denote $A\setminus G$ as $\overline{G}$ .

2.1 Syntax

We start with defining the logical language and some crucial syntactic notions.

Definition 1 (Language).

The language of quantified public announcement logic is defined as follows, where $a\in A$ and $p\in P$ .

{\mathcal{L}}(A,P)\ \ni\ \varphi::=p\ |\ \neg\varphi\ |\ (\varphi\wedge\varphi)\ |\ K_{\_}a\varphi\ |\ [\varphi]\varphi\ |\ Q\varphi

$\dashv$

Other propositional connectives are defined by abbreviation, and, unless ambiguity results, we often omit parentheses occurring in formulas. We also often omit one or both of the parameters $A$ and $P$ in ${\mathcal{L}}(A,P)$ , and write ${\mathcal{L}}(P)$ or ${\mathcal{L}}$ . Formulas are denoted $\varphi,\psi$ , possibly primed as in $\varphi^{\prime},\varphi^{\prime\prime},\dots,\psi^{\prime},\dots$ . Depending on which form the quantifier $Q$ takes — $\Box$ , $[G]$ , or $[\!\langle G\rangle\!]$ , where $G\subseteq A$ — we distinguish ${\mathcal{L}}_{\mathit{apal}}$ , ${\mathcal{L}}_{\mathit{gal}}$ , and ${\mathcal{L}}_{\mathit{cal}}$ , respectively. We also distinguish the language ${\mathcal{L}}_{\mathit{el}}$ of epistemic logic (without the constructs $[\varphi]\varphi$ and $Q\varphi$ ).

For $K_{\_}a\varphi$ read ‘agent $a$ knows $\varphi$ ’. For $[\varphi]\psi$ , read ‘after public announcement of $\varphi$ , $\psi$ ’. For $\Box\varphi$ , read ‘after any announcement, $\varphi$ (is true)’. For $[G]\varphi$ , read ‘after any joint announcement by agents from $G$ , $\varphi$ is true’. And for $[\!\langle G\rangle\!]\varphi$ read ‘for each announcement by agents from $G$ , there is a counter-announcement by the remaining agents, such that $\varphi$ is true after the joint simultaneous announcement’. The dual modalities are defined by abbreviation: $\widehat{K}_{\_}a\varphi:=\neg K_{\_}a\neg\varphi$ , $\langle\varphi\rangle\psi:=\neg[\varphi]\neg\psi$ , $\Diamond\varphi:=\neg\Box\neg\varphi$ , $\langle G\rangle\varphi:=\lnot[G]\lnot\varphi$ , and $\langle\![G]\!\rangle\varphi:=\lnot[\!\langle G\rangle\!]\lnot\varphi$ .

The set of propositional variables that occur in a given formula $\varphi$ is denoted $\mathit{var}(\varphi)$ (where one that does not occur in $\varphi$ is called a fresh variable), its modal depth $d(\varphi)$ is the maximum nesting of $K_{\_}a$ modalities, and its quantifier depth $D(\varphi)$ is the maximum nesting of $Q\in\{\Box,[G],[\!\langle G\rangle\!]\}$ modalities. These notions are inductively defined as follows.

•

$\mathit{var}(p)=\{p\}$ , $\mathit{var}(\neg\varphi)=\mathit{var}(K_{\_}a\varphi)=\mathit{var}(Q\varphi)=\mathit{var}(\varphi)$ , $\mathit{var}(\varphi\wedge\psi)=\mathit{var}([\varphi]\psi)=\mathit{var}(\varphi)\cup\mathit{var}(\psi)$ ;
•

$D(p)=0$ , $D(\neg\varphi)=D(K_{\_}a\varphi)=D(\varphi)$ , $D(\varphi\wedge\psi)=D([\varphi]\psi)=\max\{D(\varphi),D(\psi)\}$ , $D(Q\varphi)=D(\varphi)+1$ ;
•

$d(p)=0$ , $d(\neg\varphi)=d(Q\varphi)=d(\varphi)$ , $d(\varphi\wedge\psi)=\max\{d(\varphi),d(\psi)\}$ , $d([\varphi]\psi)=d(\varphi)+d(\psi)$ , $d(K_{\_}a\varphi)=d(\varphi)+1$ .

2.2 Structures

We consider the following structures and structural notions in this work.

Definition 2 (Model).

An (epistemic) model $M=(S,\sim,V)$ consists of a non-empty domain $S$ (or ${\mathcal{D}}(M)$ ) of states (or ‘worlds’), an accessibility function $\sim:A\rightarrow{\mathcal{P}}(S\times S)$ , where each $\sim_{\_}a$ is an equivalence relation, and a valuation $V:P\rightarrow{\mathcal{P}}(S)$ , where each $V(p)$ represents the set of states where $p$ is true. For $s\in S$ , a pair $(M,s)$ , for which we write $M_{\_}s$ , is a pointed (epistemic) model. $\dashv$

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, $\sim^{M}_{\_}a$ the accessibility relation for an agent $a$ , and $V^{M}$ its valuation.

Definition 3 (Bisimulation).

Let $M=(S,\sim,V)$ and $M^{\prime}=(S^{\prime},\sim^{\prime},V^{\prime})$ be epistemic models. A non-empty relation $\mathfrak{R}\subseteq S\times S^{\prime}$ is a bisimulation if for every $(s,s^{\prime})\in\mathfrak{R}$ , $p\in P$ , and $a\in A$ the conditions atoms, forth and back hold.

•

atoms: $s\in V(p)$ iff $s^{\prime}\in V^{\prime}(p)$ .
•

forth: for every $t\sim_{\_}{a}s$ there exists $t^{\prime}\sim^{\prime}_{\_}{a}s^{\prime}$ such that $(t,t^{\prime})\in\mathfrak{R}$ .
•

back: for every $t^{\prime}\sim^{\prime}_{\_}{a}s^{\prime}$ there exists $t\sim_{\_}{a}s$ such that $(t,t^{\prime})\in\mathfrak{R}$ .

If there exists a bisimulation $\mathfrak{R}$ between $M$ and $M^{\prime}$ such that $(s,s^{\prime})\in\mathfrak{R}$ , then $M_{\_}{s}$ and $M^{\prime}_{\_}{s^{\prime}}$ are bisimilar, notation $M_{\_}{s}{\raisebox{1.29167pt}[0.0pt][0.0pt]{$\medspace\underline{\!\leftrightarrow\!}\medspace$}}M^{\prime}_{\_}{s^{\prime}}$ (or $\mathfrak{R}:M_{\_}{s}{\raisebox{1.29167pt}[0.0pt][0.0pt]{$\medspace\underline{\!\leftrightarrow\!}\medspace$}}M^{\prime}_{\_}{s^{\prime}}$ , to be explicit about the bisimulation).

Let $Q\subseteq P$ . A relation $\mathfrak{R}$ between $M$ and $M^{\prime}$ satisfying atoms for all $p\in Q$ , and forth and back, is a $Q$ -bisimulation (a bisimulation restricted to $Q$ ). The notation for $Q$ -restricted bisimilarity is ${\raisebox{1.29167pt}[0.0pt][0.0pt]{$\medspace\underline{\!\leftrightarrow\!}\medspace$}}_{\_}Q$ . $\dashv$

The notion of $n$ -bisimulation, for $n\in\mathbb{N}$ , is given by defining relations $\mathfrak{R}^{0}\supseteq\dots\supseteq\mathfrak{R}^{n}$ .

Definition 4 ( $n$ -Bisimulation).

Let $M=(S,\sim,V)$ and $M^{\prime}=(S^{\prime},\sim^{\prime},V^{\prime})$ be epistemic models, and let $n\in\mathbb{N}$ . A non-empty relation $\mathfrak{R}^{0}\subseteq S\times S^{\prime}$ is a $0$ -bisimulation if atoms holds for pair $(s,s^{\prime})\in\mathfrak{R}$ . Then, a non-empty relation $\mathfrak{R}^{n+1}\subseteq S\times S^{\prime}$ is a $(n+1)$ -bisimulation if for all $p\in P$ and $a\in A$ :

•

$(n+1)$ -forth: for every $t\sim_{\_}{a}s$ there exists $t^{\prime}\sim^{\prime}_{\_}{a}s^{\prime}$ such that $(t,t^{\prime})\in\mathfrak{R}^{n}$ ;
•

$(n+1)$ -back: for every $t^{\prime}\sim^{\prime}_{\_}as^{\prime}$ there exists $t\sim_{\_}{a}s$ such that $(t,t^{\prime})\in\mathfrak{R}^{n}$ .

Similarly to $Q$ -bisimulations we define $Q$ - $n$ -bisimulations, wherein atoms is only required for $p\in Q\subseteq P$ ; $n$ -bisimilarity is denoted $M_{\_}s{\raisebox{1.29167pt}[0.0pt][0.0pt]{$\medspace\underline{\!\leftrightarrow\!}\medspace$}}^{n}M^{\prime}_{\_}{s^{\prime}}$ , and $Q$ - $n$ -bisimilarity is denoted $M_{\_}s{\raisebox{1.29167pt}[0.0pt][0.0pt]{$\medspace\underline{\!\leftrightarrow\!}\medspace$}}^{n}_{\_}QM^{\prime}_{\_}{s^{\prime}}$ . $\dashv$

2.3 Semantics

We continue with the semantics of our logic. Let ${\mathcal{L}}_{\mathit{el}}^{G}:=\{\bigwedge_{\_}{i\in G}K_{\_}i\varphi_{\_}i\mid\varphi_{\_}i\in{\mathcal{L}}_{\mathit{el}}\}$ be the set of all announcement by group $G$ .

Definition 5 (Semantics).

The interpretation of formulas in ${\mathcal{L}}_{\mathit{apal}}\cup{\mathcal{L}}_{\mathit{gal}}\cup{\mathcal{L}}_{\mathit{cal}}$ on epistemic models is defined by induction on formulas.

Assume an epistemic model $M=(S,\sim,V)$ , and $s\in S$ .

\begin{array}[]{lcl}M_{\_}s\models p&\mbox{iff}&s\in V(p)\\ M_{\_}s\models\neg\varphi&\mbox{iff}&M_{\_}s\not\models\varphi\\ M_{\_}s\models\varphi\wedge\psi&\mbox{iff}&M_{\_}s\models\varphi\text{ and }M_{\_}s\models\psi\\ M_{\_}s\models K_{\_}a\varphi&\mbox{iff}&\mbox{for all }t\in S:s\sim_{\_}at\text{ implies }M_{\_}t\models\varphi\\ M_{\_}s\models[\varphi]\psi&\mbox{iff}&M_{\_}s\models\varphi\text{ implies }M^{\varphi}_{\_}s\models\psi\\ M_{\_}s\models\Box\psi&\mbox{iff}&\mbox{for all }\varphi\in{\mathcal{L}}_{\mathit{el}}:M_{\_}s\models[\varphi]\psi\\ M_{\_}s\models[G]\psi&\mbox{iff}&\mbox{for all }\varphi_{\_}G\in{\mathcal{L}}_{\mathit{el}}^{G}:M_{\_}s\models[\varphi_{\_}G]\psi\\ M_{\_}s\models[\!\langle G\rangle\!]\psi&\mbox{iff}&\mbox{for all }\varphi_{\_}G\in{\mathcal{L}}_{\mathit{el}}^{G}\mbox{ there is }\chi_{\_}{\overline{G}}\in{\mathcal{L}}_{\mathit{el}}^{\overline{G}}:M_{\_}s\models\varphi_{\_}G\rightarrow\langle\varphi_{\_}G\land\chi_{\_}{\overline{G}}\rangle\psi\end{array}

where $[\![\varphi]\!]_{\_}M:=\{s\in S\mid M_{\_}s\models\varphi\}$ ; and where epistemic model $M^{\varphi}=(S^{\prime},\sim^{\prime},V^{\prime})$ is such that: $S^{\prime}=[\![\varphi]\!]_{\_}M$ , ${\sim^{\prime}_{\_}a}={\sim_{\_}a}\cap([\![\varphi]\!]_{\_}M\times[\![\varphi]\!]_{\_}M)$ , and $V^{\prime}(p):=V(p)\cap[\![\varphi]\!]_{\_}M$ . For $(M^{\varphi})^{\psi}$ we may write $M^{\varphi\psi}$ . Formula $\varphi$ is valid on model $M$ , notation $M\models\varphi$ , if for all $s\in S$ , $M_{\_}s\models\varphi$ . Formula $\varphi$ is valid, notation $\models\varphi$ , if for all $M$ , $M\models\varphi$ . We call $\varphi$ satisfiable if there is $M_{\_}s$ such that $M_{\_}s\models\varphi$ . $\dashv$

Observe that the quantification in the definition of semantics is restricted to the quantifier-free fragment. Moreover, given the eliminability of public announcements from that fragment [22], this amounts to quantifying over formulas of epistemic logic.

For clarity, we also give the semantics of the diamond versions of public and quantified announcements.

\begin{array}[]{lcl}M_{\_}s\models\langle\varphi\rangle\psi&\mbox{iff}&M_{\_}s\models\varphi\text{ and }M^{\varphi}_{\_}s\models\psi\\ M_{\_}s\models\Diamond\psi&\mbox{iff}&\mbox{there is }\varphi\in{\mathcal{L}}_{\mathit{el}}:M_{\_}s\models\langle\varphi\rangle\psi\\ M_{\_}s\models[G]\psi&\mbox{iff}&\mbox{there is }\varphi_{\_}G\in{\mathcal{L}}_{\mathit{el}}^{G}:M_{\_}s\models\langle\varphi_{\_}G\rangle\psi\\ M_{\_}s\models\langle\![G]\!\rangle\psi&\mbox{iff}&\mbox{there is }\varphi_{\_}G\in{\mathcal{L}}_{\mathit{el}}^{G}\mbox{ such that for all }\chi_{\_}{\overline{G}}\in{\mathcal{L}}_{\mathit{el}}^{\overline{G}}:M_{\_}s\models\varphi_{\_}G\land[\varphi_{\_}G\land\chi_{\_}{\overline{G}}]\psi\end{array}

Definition 6 (Modal Equivalence).

Given $M_{\_}s$ and $M^{\prime}_{\_}{s^{\prime}}$ , if for all $\varphi\in{\mathcal{L}}$ , $M_{\_}s\models\varphi$ iff $M^{\prime}_{\_}{s^{\prime}}\models\varphi$ , we write $M_{\_}{s}\equiv M^{\prime}_{\_}{s^{\prime}}$ . Similarly, if this holds for all $\varphi$ with $d(\varphi)\leq n$ , we write $M_{\_}{s}\equiv^{n}M^{\prime}_{\_}{s^{\prime}}$ , and if this holds for all $\varphi$ with $\mathit{var}(\varphi)\in Q\subseteq P$ , we write $M_{\_}s\equiv_{\_}QM^{\prime}_{\_}{s^{\prime}}$ . $\dashv$

It is a standard standard model-theoretic result for ${\mathcal{L}}_{\mathit{el}}$ that bisimulation between models implies their modal equivalence [15]. We can extend the result to ${\mathcal{L}}_{\mathit{apal}}$ , ${\mathcal{L}}_{\mathit{gal}}$ , and ${\mathcal{L}}_{\mathit{cal}}$ .

Theorem 7.

Let $M_{\_}s$ and $M^{\prime}_{\_}{s^{\prime}}$ be models. Then $M_{\_}s{\raisebox{1.29167pt}[0.0pt][0.0pt]{$\medspace\underline{\!\leftrightarrow\!}\medspace$}}M^{\prime}_{\_}{s^{\prime}}$ implies $M_{\_}{s}\equiv M^{\prime}_{\_}{s^{\prime}}$ . $\dashv$

Proof.

By an induction on the structure of a formula. The proof for the case of public announcements can be found, for example, in [8]. The cases of quantified announcements follow by the induction hypothesis in the presence of an appropriate ordering of subformulas such that the $D$ -depth takes precedence over $d$ -depth.

Another well-known result is that for ${\mathcal{L}}_{\mathit{el}}$ (and hence for ${\mathcal{L}}_{\mathit{pal}}$ given the translation from ${\mathcal{L}}_{\mathit{pal}}$ into ${\mathcal{L}}_{\mathit{el}}$ [22]), $M_{\_}s{\raisebox{1.29167pt}[0.0pt][0.0pt]{$\medspace\underline{\!\leftrightarrow\!}\medspace$}}^{n}M^{\prime}_{\_}{s^{\prime}}$ implies $M_{\_}{s}\equiv^{n}M^{\prime}_{\_}{s^{\prime}}$ [15]. Observe that this is not the case for any of the $\mathit{QPAL}$ ’s because the quantification is over formulas of arbitrary finite modal depth and thus may exceed the given $n$ . Finally, due to the fact that the quantification in the presented languages is implicit, it is also not the case for $\mathit{QPAL}$ ’s that $M_{\_}s{\raisebox{1.29167pt}[0.0pt][0.0pt]{$\medspace\underline{\!\leftrightarrow\!}\medspace$}}_{\_}QM^{\prime}_{\_}{s^{\prime}}$ implies $M_{\_}s\equiv_{\_}QM^{\prime}_{\_}{s^{\prime}}$ .

Example 8.

In order to highlight the differences between different types of quantification, let us consider models $M_{\_}{s_{\_}0}$ , $M_{\_}{s_{\_}0}^{\psi}$ , and $M_{\_}{s_{\_}0}^{\psi_{\_}a}$ in Figure 1.

Figure 1: Models, from left to right,

M_{\_}{s_{\_}0}

M_{\_}{s_{\_}0}^{\psi}

, and

M_{\_}{s_{\_}0}^{\psi_{\_}a}

. The names of the worlds indicate which atoms are true there.

There is a formula $\psi$ that can be announced in $M_{\_}{s_{\_}0}$ , and such that $\varphi:=\widehat{K}_{\_}aK_{\_}bp\land\widehat{K}_{\_}a\widehat{K}_{\_}b\lnot p$ will hold after the announcement. Indeed, let $\psi:=\lnot p\rightarrow\lnot q$ . The result of updating $M_{\_}{s_{\_}0}$ with $\psi$ is presented in the figure and the reader can verify that $M_{\_}{s_{\_}0}^{\psi}\models\varphi$ . This means that $M_{\_}{s_{\_}0}\models\langle\psi\rangle\varphi$ , and hence $M_{\_}{s_{\_}0}\models\Diamond\varphi$ . Observe that $M_{\_}{s_{\_}0}\not\models\langle\{a\}\rangle\varphi$ , since, according to the semantics, each announcement by agent $a$ should be prefixed with $K_{\_}a$ . This implies that in order to remove $s_{\_}1$ , we also have to remove all $a$ -reachable states, in particular $t_{\_}1$ . On the other hand, $M_{\_}{s_{\_}0}\models\langle\{a\}\rangle(K_{\_}bq\land\lnot K_{\_}aq)$ . Indeed, let $\psi_{\_}a:=K_{\_}ap$ . Such an announcement results in model $M_{\_}{s_{\_}0}^{\psi_{\_}a}$ in which $K_{\_}bq\land\lnot K_{\_}aq$ is true. Finally, $M_{\_}{s_{\_}0}\not\models\langle\![\{a\}]\!\rangle(K_{\_}bq\land\lnot K_{\_}aq)$ , as any announcement by agent $a$ that results in a model with worlds $s_{\_}0$ and $t_{\_}0$ can be countered by agent $b$ with a simultaneous announcement $K_{\_}bq$ . Such a joint announcement results in a singleton model with the only world $s_{\_}0$ . $\dashv$

3 APAL, GAL, and CAL do not have the finite model property

In this section, we show that none of the $\mathit{QPAL}$ ’s have the FMP. We do this by proving the result for $\mathit{APAL}$ first, and then state the corresponding results for $\mathit{GAL}$ and $\mathit{CAL}$ as a corollary.

Definition 9 (Finite Model Property).

A logic has the finite model property if every satisfiable formula is satisfied in a finite model. $\dashv$

Thie idea of the proof is to show that there is a formula, fmp, that expresses the property that:

There is some subset of worlds in the model that can be partitioned into sets $X$ and $Y$ , such that for every element of $x\in X$ , there is some announcement, $\psi^{x}$ , that preserves $x$ and no states from $Y$ , but there is no announcement that preserves only the states from $Y$ , and none of the states from $X$ .

As the announcements in $\mathit{QPAL}$ ’s are closed under negations and conjunctions, if $X$ were finite up to bisimulation, then the announcement $\bigwedge_{\_}{x\in X}\lnot\psi^{x}$ , would be adequate to preserve only the states of $Y$ and none of the states from $X$ . Therefore, if it can be shown that such a formula is satisfiable, then it would follow that $\mathit{QPAL}$ ’s do not have the finite model property. To show the satisfiability for such a formula, we need a means to identify $X$ and $Y$ states. An epistemic formula will not do, as any formula that characterises $X$ , will have a negation that characterises $Y$ , and announcing the negation would violate the property we need. However, rather than distinguishing between two partitions using a modal property, we can distinguish them using a second order property that the $\mathit{APAL}{}$ quantifier does not range over. We know [4] that $\mathit{APAL}$ can express such properties; particularly it can specify whether or not two states are $n$ -bisimilar for all $n$ . It is this property we use to define the necessary partition of worlds.

The proof will be via construction, where we will present a $\mathit{APAL}$ formula, show that it has an infinite model, and then show that it is impossible that a finite model exists.

Theorem 10.

$\mathit{APAL}$ does not have the finite model property. $\dashv$

Proof.

We will give the proof by construction. Consider the following formula fmp.

$\displaystyle\mathbf{root}$	$\displaystyle=$	$\displaystyle\Box(\widehat{K}_{\_}a(\lnot x\land K_{\_}b\lnot x)\rightarrow K_{\_}a(\lnot x\rightarrow K_{\_}b\lnot x))$
$\displaystyle\mathbf{stem}$	$\displaystyle=$	$\displaystyle\Diamond(\widehat{K}_{\_}a(\lnot x\land K_{\_}b\lnot x)\land\widehat{K}_{\_}a(\lnot x\land\widehat{K}_{\_}bx))$
$\displaystyle\mathbf{tier}$	$\displaystyle=$	$\displaystyle K_{\_}b(x\land\widehat{K}_{\_}a\lnot x\land K_{\_}a(\lnot x\rightarrow\widehat{K}_{\_}bx))$
fmp	$\displaystyle=$	$\displaystyle\bigwedge\left(\begin{array}[]{l}\mathbf{tier}\land\widehat{K}_{\_}b\mathbf{root}\land\widehat{K}_{\_}b\mathbf{stem}\\ K_{\_}b(\mathbf{stem}\rightarrow\Diamond(\mathbf{tier}\land K_{\_}b\mathbf{stem}))\\ K_{\_}b(\mathbf{root}\rightarrow\Box(\mathbf{tier}\rightarrow\widehat{K}_{\_}b\mathbf{stem}))\end{array}\right)$

The formula fmp partitions a set of $b$ -related worlds into two sets: root (the worlds where the formula $\mathbf{root}$ is true); and stem (the worlds where the formula $\mathbf{stem}$ is true). We note that $\mathbf{stem}$ is equivalent to $\lnot\mathbf{root}$ so this is a partition.

The formula $\mathbf{tier}$ sets a label $x$ , where $x$ is true at all the $b$ -related worlds (tier-0), at every $b$ -related world there is an $a$ -related world where $x$ is false (tier-1), and from each of those worlds there is a $b$ -related world where $x$ is true (tier-2). This creates a consistent labelling used to define $\mathbf{root}$ and $\mathbf{stem}$ .

The formula $\mathbf{root}$ is true at a world if there is only one $a$ -reachable tier-1 world, up to finite bisimulation, while $\mathbf{stem}$ is true if there is more than one $a$ -reachable tier-1 world.

The first line of fmp states that $\mathbf{tier}$ is true, and at least one of the $b$ -related worlds satisfies $\mathbf{stem}$ , and at least one of the $b$ -related worlds satisfies $\mathbf{root}$ . The second line of fmp states that there is some public announcement that removes all of the $\mathbf{root}$ worlds (and possibly some, but not all, of the $\mathbf{stem}$ worlds too), and the final line of fmp states that there is no public announcement that removes all of the $\mathbf{stem}$ worlds leaving a $\mathbf{root}$ world. This line needs a small caveat. Rather than just talking about removing $\mathbf{stem}$ and $\mathbf{root}$ worlds, there is the possibility that an announcement could change a $\mathbf{stem}$ world to a $\mathbf{root}$ world, or vice-versa. However each of these announcement quantifiers is guarded by the formula $\mathbf{tier}$ , and we have the property $\mathbf{root}\rightarrow\Box\mathbf{root}$ . Therefore, we do not need to consider the cases where $\mathbf{root}$ worlds are transformed into $\mathbf{stem}$ worlds.

As an example consider a finite model in Figure 2. The model does not satisfy fmp, and in particular it does not satisfy the third conjunct. Indeed, let the current world be $s_{\_}0$ that satisfies $\mathbf{root}$ . We show that $M_{\_}{s_{\_}0}\not\models\Box(\mathbf{tier}\rightarrow\widehat{K}_{\_}b\mathbf{stem})$ , or, equivalently, $M_{\_}{s_{\_}0}\models\Diamond(\mathbf{tier}\land K_{\_}b\lnot\mathbf{stem})$ . By the semantics this amounts to the fact that there is $\psi\in{\mathcal{L}}_{\mathit{el}}$ such that $\mathbf{tier}\land K_{\_}b\lnot\mathbf{stem}$ holds in the updated model. Let, for example, $\psi:=K_{\_}a(\lnot x\rightarrow K_{\_}b(x\rightarrow K_{\_}a\lnot p_{\_}4))$ . It is easy to verify that the resulting updated model, which only consists of states $s_{\_}0$ , $t_{\_}0$ , and $u_{\_}0$ , satisfies $\mathbf{tier}\land K_{\_}b\lnot\mathbf{stem}$ .

Figure 2: A model that does not satisfy formula fmp. The names of the worlds indicate which atoms are true there, e.g.

x

is true and

\overline{x}

means that

x

is false.

Now we show that fmp is a satisfiable formula. The model $M=(S,\sim,V)$ , which satisfies fmp, is built as follows:

1.

$S=\{s_{\_}0,s_{\_}1,\ldots\}\cup\{t_{\_}0,t_{\_}1,\ldots\}\cup\{u_{\_}0,u_{\_}1,\ldots\}$
2.

$\forall i\geq 0$ , $s_{\_}i\sim_{\_}as_{\_}i$ , $s_{\_}i\sim_{\_}at_{\_}{2i}$ , $t_{\_}{2i}\sim_{\_}as_{\_}i$ , $t_{\_}i\sim_{\_}at_{\_}i$ and $u_{\_}i\sim_{\_}au_{\_}i$ .
3.

$\forall i>0$ , $s_{\_}i\sim_{\_}at_{\_}{2i-1}$ , $t_{\_}{2i}\sim_{\_}at_{\_}{2i-1}$ , $t_{\_}{2i-1}\sim_{\_}as_{\_}i$ and $t_{\_}{2i-1}\sim_{\_}at_{\_}{2i}$ .
4.

$\forall i,j\geq 0,\ s_{\_}i\sim_{\_}bs_{\_}j$ , $t_{\_}i\sim_{\_}bu_{\_}i$ , $u_{\_}i\sim_{\_}bt_{\_}i$ , $t_{\_}i\sim_{\_}bt_{\_}i$ and $u_{\_}i\sim_{\_}bu_{\_}i$ .
5.

$V(x)=\{s_{\_}0,s_{\_}1,\ldots\}\cup\{u_{\_}0,u_{\_}1,\ldots\}$
6.

$\forall i\geq 0$ , $V(p_{\_}i)=\{u_{\_}k\ |\ 0<k\leqslant i\}$ .

This model is represented in Figure 3.

Figure 3: A model that does satisfy formula fmp. For each world

s_{\_}i

, where

i>0

, there is an announcement that preserves

s_{\_}i

, but not

s_{\_}0

. However, there is no announcement that preserves only

s_{\_}0

The formula $\mathbf{root}$ is true only at the state $s_{\_}0$ , and the formula $\mathbf{stem}$ is true at $s_{\_}i$ for all $i>0$ . The states $s_{\_}i$ are tier-0 states, the states $t_{\_}i$ are tier-1 states, and the states $u_{\_}i$ are tier-2 states. Therefore $M_{\_}{s_{\_}0}\models\mathbf{tier}\land\widehat{K}_{\_}b\mathbf{root}\land\widehat{K}_{\_}b\mathbf{stem}$ . In any state $s_{\_}i$ that satisfies $\mathbf{stem}$ , we can make an announcement $\psi_{\_}i:=\widehat{K}_{\_}a\widehat{K}_{\_}bp_{\_}{2i}$ that will preserve all the states $s_{\_}j,t_{\_}j,u_{\_}j$ where $0<j\leq 2i$ . Therefore $M_{\_}{s_{\_}i}\models\langle\psi_{\_}i\rangle(\mathbf{tier}\land K_{\_}b\mathbf{stem})$ , so $M_{\_}{s_{\_}0}\models K_{\_}b(\mathbf{stem}\rightarrow\Diamond(\mathbf{tier}\land K_{\_}b\mathbf{stem}))$ . Finally, consider any announcement, $\psi$ that preserves the root state $s_{\_}0$ and keeps $\mathbf{tier}$ true. Suppose that $\mathit{var}(\psi)\subseteq P_{\_}n=\{x,p_{\_}0,\ldots,p_{\_}n\}$ . The state $s_{\_}0$ is $P_{\_}n$ -bisimilar to all states $s_{\_}i$ for $i>n$ , so all these states will be preserved and continue to satisfy $\mathbf{stem}$ . Therefore $M_{\_}{s_{\_}0}\models K_{\_}b(\mathbf{root}\rightarrow\Box(\mathbf{tier}\rightarrow\widehat{K}_{\_}b\mathbf{stem}))$ as required. Since all three conjuncts are now satisfied, $M_{\_}{s_{\_}0}\models\textbf{fmp}$ .

Finally, we reason that fmp cannot have a finite model via a contradiction. Suppose that fmp did have some finite model $M=(S,\sim,V)$ . We know $\mathbf{root}$ is equivalent to $\lnot\mathbf{stem}$ , and $\mathbf{root}\rightarrow\Box(\mathbf{tier}\rightarrow\mathbf{root})$ . As $M_{\_}s\models\textbf{fmp}$ we have that for each $s_{\_}1,s_{\_}2,\ldots,s_{\_}k$ , where $s\sim_{\_}bs_{\_}i$ and $M_{\_}{s_{\_}i}\models\mathbf{stem}$ , there is some announcement $\psi_{\_}i$ such that $M_{\_}{s_{\_}i}^{\psi_{\_}i}\models\mathbf{tier}\land K_{\_}b\mathbf{stem}$ . Therefore, for some $t\sim_{\_}bs$ where $M_{\_}t\models\mathbf{root}$ , we have $M_{\_}t\models\lnot\psi_{\_}i$ for $i=1,\ldots,k$ . Announcing $\varphi:=\bigwedge_{\_}{i=1}^{k}(\mathbf{tier}\rightarrow\lnot\psi_{\_}i)$ at $t$ therefore preserves $t$ , and removes every $s_{\_}i\sim s$ where $M_{\_}{s_{\_}i}\not\models\mathbf{root}$ . So it follows that $M_{\_}t\models\Diamond(\mathbf{tier}\land K_{\_}b\mathbf{root})$ , contradicting the fact that $M_{\_}s\models\textbf{fmp}$ .

The proof of Theorem 10 can be extended to the cases of $\mathit{GAL}$ and $\mathit{CAL}$ . To see this, it is enough to notice that the formula fmp forces a model with a root-and-stem structure, and thus arbitrary announcements can be ‘modelled’ by announcements of the grand coalition. In particular, we need to substitute $\Box$ and $\Diamond$ in fmp with $[A]$ and $\langle A\rangle$ for $\mathit{GAL}$ , and with $[\!\langle A\rangle\!]$ and $\langle\![A]\!\rangle$ for $\mathit{CAL}$ . Finally, the model in Figure 3 will also work, since the intersection of $a$ - and $b$ -relations is the identity, and the set of all possible $\mathit{APAL}$ updates then coincides with those of $\mathit{GAL}$ and $\mathit{CAL}$ .

Corollary 11.

$\mathit{GAL}$ and $\mathit{CAL}$ do not have the finite model property. $\dashv$

4 Conclusions and further research

It has been an open question for quite a while whether quantified public announcement logics have the finite model property, and we have answered the question for $\mathit{APAL}$ , $\mathit{GAL}$ , and $\mathit{CAL}$ negatively. Not only this result is interesting in itself, it also clarifies some other properties of $\mathit{QPAL}$ ’s. In particular, from the expressivity perspective, we presented a formula that forces infinite models. Moreover, we have found the value of one of the unknowns in the expression

Finitary axiomatisation and FMP imply decidability

and thus only the problem of finding finitary axiomatisations of $\mathit{QPAL}$ ’s stands. Finally, the result trivially extends to the logics that are extensions of any of $\mathit{QPAL}$ ’s (e.g. [14]).

Interestingly, restricting the quantification in $\mathit{APAL}$ to just announcements of Boolean formulas still results in a logic with no FMP. This was shown in [8], where the authors used a somewhat simpler partition of worlds distinguishable by a formula of epistemic logic. Since in $\mathit{APAL}$ the quantifiers range over all epistemic formulas, we used a more complex second-order property of worlds being $n$ -bisimilar for arbitrary $n$ . It is unknown whether $\mathit{APAL}$ with the quantification restricted only to positive (universal) fragment of epistemic logic [9] has the FMP, but given the aforementioned results, it seems very unlikely.

Acknowledgments

We thank the anonymous reviewers of the paper. This work was carried out while Hans van Ditmarsch was affiliated to LORIA, CNRS, University of Lorraine, France.

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No Finite Model Property for Logics of Quantified Announcements

Abstract

1 Introduction

2 Quantified public announcement logics

2.1 Syntax

Definition 1 (Language).

2.2 Structures

Definition 2 (Model).

Definition 3 (Bisimulation).

Definition 4 (nn-Bisimulation).

2.3 Semantics

Definition 5 (Semantics).

Definition 6 (Modal Equivalence).

Theorem 7.

Proof.

Example 8.

3 APAL, GAL, and CAL do not have the finite model property

Definition 9 (Finite Model Property).

Theorem 10.

Proof.

Corollary 11.

4 Conclusions and further research

Acknowledgments

References

Definition 4 ( $n$ -Bisimulation).