The Logic of Public Announcements, Common Knowledge, and Private Suspicions

Abstract

This paper presents a logical system in which various group-level epistemic actions are incorporated into the object language. That is, we consider the standard modeling of knowledge among a set of agents by multi-modal Kripke structures. One might want to consider actions that take place, such as announcements to groups privately, announcements with suspicious outsiders, etc. In our system, such actions correspond to additional modalities in the object language. That is, we do not add machinery on top of models (as in Fagin et al., Reasoning about knowledge. MIT, Cambridge, 1995), but we reify aspects of the machinery in the logical language. Special cases of our logic have been considered in Plaza (Logics of public communications. In: Proceedings of the 4th international symposium on methodologies for intelligent systems, Charlotte, 1989), Gerbrandy (Dynamic epistemic logic. In: Moss LS, et al (eds) Logic, language, and information, vol 2. CSLI Publications, Stanford University, 1999a; Bisimulations on planet Kripke. Ph.D. dissertation, University of Amsterdam, 1999b), and Gerbrandy and Groeneveld (J Logic Lang Inf 6:147–169, 1997). The latter group of papers introduce a language in which one can faithfully represent all of the reasoning in examples such as the Muddy Children scenario. In that paper we find operators for updating worlds via announcements to groups of agents who are isolated from all others. We advance this by considering many more actions, and by using a more general semantics. Our logic contains the infinitary operators used in the standard modeling of common knowledge. We present a sound and complete logical system for the logic, and we study its expressive power.

Appendix: The Lexicographic Path Order

In this appendix, we give the details on the lexicographic path ordering (LPO), both in general and in connection with $\mathcal{L}([\alpha ])$ and $\mathcal{L}([\alpha ],\square ^{{\ast}})$.

Fix some many-sorted signature $\Sigma$ of terms. In order to define the LPO < on the $\Sigma$-terms, we must first specify a well-order < on the set of function symbols of $\Sigma$. The LPO determined by such choices is the smallest relation < such that:

(LPO1) :

If (t ₁, …, t _n ) < (s ₁, …, s _n ) in the lexicographic ordering on n-tuples, and if t _j  < f(s ₁, …, s _n ) for 1 ≤ j ≤ n, then f(t ₁, …, t _n ) < f(s ₁, …, s _n ).

(LPO2) :

If t ≤ s _i for some i, then t < f(s ₁, …, s _n ).

(LPO3) :

If g < f and t _i  < f(s ₁, …, s _n ) for all i ≤ m, then g(t ₁, …, t _m ) < f(s ₁, …, s _n ).

Here is how this is applied in this paper. We shall take two sorts: sentences and actions. Our signature contains the usual sentence-forming operators p (for p ∈ AtSen) $\neg$, ∧, and □  _A for all $A \in \mathcal{A}$. Here each p is 0-ary, $\neg$ and □  _A are unary, and ∧ is binary. We also have an operator app taking actions and sentences to sentences. We think of app(ψ, α) as merely a variation on [α]ψ. (The order of arguments to app is significant.) We further have a binary operator ∘ on actions. (This is a departure from the treatment of this paper, since we used ∘ as a metalinguistic abbreviation instead of as a formal symbol. It will be convenient to make this change because this leads to a smoother treatment of the Composition Axiom.) Finally, for each finite Kripke frame K over $\mathcal{L}([\alpha ])$ and each 1 ≤ i ≤ | K | , we have a symbol F _K ⁱ taking | K | sentences and returning an action.

Each sentence $\varphi$ has a formal version $\overline{\varphi }$ in this signature, and each action α also has a formal version $\overline{\alpha }$. These are defined by the recursion which is obvious except for the clauses

$$\displaystyle{\begin{array}{lcl} \overline{[\alpha ]\varphi }&\quad = \quad &app(\overline{\varphi },\overline{\alpha }) \\ \overline{\alpha } &\quad = \quad &\mathsf{F}_{K}^{i}(\overline{\mbox{ PRE}(k_{1})},\ldots,\overline{\mbox{ PRE}(k_{n})})\end{array} }$$

Here $\alpha =\langle K,k_{i},\mbox{ PRE}\rangle$ with K = { k ₁, …, k _n } in some specified order. However, outside of the proof of Proposition 2 we shall not explicitly mention the formal versions at all, since they are harder to read than the standard notation.

We must also first fix a wellfounded relation < on the function symbols. We set app to be greater than all other function symbols. In all other cases, distinct function symbols are unordered.

Theorem 1 (Kamin and Levy 1980; Dershowitz 1982).

Let < be an LPO on $\Sigma$ -terms.

• 1. < is transitive.

• 2. < has the subterm property : if t is a proper subterm of u, then t < u.

• 3. < is monotonic (it has the replacement property ): if y < x _i for some i, then

  $$\displaystyle{f(x_{1},\ldots,y,\ldots x_{n}) < f(x_{1},\ldots,x_{i},\ldots x_{n}).}$$

• 4. < is wellfounded.

• 5. Consider a term rewriting system every rule of which of the form $l \leadsto r$ with r < l. Then the system is terminating : there are no infinite sequences of rewritings.

Proof.

Here is a sketch for part (1): We check by induction on the construction of the least relation < that if s < t, then for all u such that t < u, s < u. For this, we use induction on the term u. We omit the details. Further, (2) follows easily from (1) and (LPO2), and (3) from (LPO1), (1) and (2). Moreover, (5) follows easily from (4) and (3), since the latter implies that any replacement according to the rewrite system results in a smaller term in the order < .

Here is a proof of of the wellfoundedness property (4), taken from on Buchholz (1995). (We generalized it slightly from the one-sorted to the many-sorted setting and from the assumption that < is a finite linear order on $\Sigma$ to the assumption that < is any wellfounded order.)

Let W be the set of terms t such that the order < is wellfounded below t. W is then itself wellfounded under < . So for all n, W ⁿ is wellfounded under the induced lexicographic order. We prove by induction on the given wellfounded relation on function symbols of $\Sigma$ that for all n-ary f, $f[W^{n}] \subseteq W$. So assume that for g < f, say with arity m, $g[W^{m}] \subseteq W$. We check this for f by using induction on W ⁿ. Fix $\vec{s} \in W^{n}$, and assume that whenever $\vec{u} <\vec{ s}$ in W ⁿ, that $f(\vec{u}) \in W$. We prove that $f(\vec{s}) \in W$ by checking that for all t such that $t < f(\vec{s})$, t ∈ W. And this is done by induction on the structure of t. If $t = f(\vec{u}) < f(\vec{s})$ via (LPO1), then $\vec{u} <\vec{ s}$ lexicographically, and each $u_{i} < f(\vec{s})$. This last point implies that $\vec{u} \in W^{n}$ by induction hypothesis on t, so t ∈ W by induction hypothesis on W ⁿ. If t ≤ s _i so that $t < f(\vec{s})$ via (LPO2), then t ∈ W by definition of W. And if $t = g(u_{1},\ldots,u_{m}) < f(\vec{s})$ via (LPO3), then g < f and each $u_{i} < f(\vec{s})$. By induction hypothesis on t, each u _i  ∈ W. So by induction hypothesis on f, $g(\vec{u}) \in W$.

Now that we know that each f takes tuples in W ⁿ to elements of W, it follows by induction on terms that all terms belong to W.

For more on the LPO, its generalizations and extensions, see the surveys Dershowitz (1982) and Plaisted (1985).

Proposition 2.

Consider the LPO < on $\mathcal{L}([\alpha ],\square ^{{\ast}})$ defined above.

• 1. If $\alpha \,\stackrel{\ast}{\rightarrow} \beta$ , then PRE(β) < α.

• 2. If $\alpha \,\stackrel{\ast}{\rightarrow} \beta$ , then $[\beta ]\psi < [\alpha ]\square _{\mathcal{C}}^{{\ast}}\psi$ .

• 3. $\mbox{ PRE}(\alpha ) \rightarrow p < [\alpha ]p$ .

• 4. $\mbox{ PRE}(\alpha ) \rightarrow \neg [\alpha ]\psi < [\alpha ]\neg \psi$ .

• 5. [α]ψ ∧ [α]χ < [α](ψ ∧χ).

• 6. $\mbox{ PRE}(\alpha ) \rightarrow \bigwedge \{\square _{A}[\beta ]\psi:\alpha \, \stackrel{A}{\rightarrow} \beta \} < [\alpha ]\square _{A}\psi$ .

• 7. $[\alpha \circ \beta ]\varphi < [\alpha ][\beta ]\psi$ .

In particular, for all rules $\varphi \leadsto \psi$ of the rewriting system $\mathcal{R}^{{\ast}}$ , $\psi <\varphi$ .

Proof.

Part (1) holds because we regard α as a term $\alpha = \mathsf{F}_{K}^{i}(\overline{\gamma _{1}},\ldots,\overline{\gamma _{n}})$, for some frame K and i. So whenever $\alpha \,\stackrel{\ast}{\rightarrow} \beta$, each PRE(β) is a proper subterm of α.

Here is the argument for part (2): We need to see that $app(\psi,\overline{\beta }) < app(\square _{\mathcal{C}}^{{\ast}}\psi,\overline{\alpha })$. Now lexicographically, $(\psi,\overline{\beta }) < (\square _{\mathcal{C}}^{{\ast}}\psi,\overline{\alpha })$. So we only need to know that $\overline{\beta } < app(\square _{\mathcal{C}}^{{\ast}}\psi,\overline{\alpha })$. Let $\alpha = \mathsf{F}_{K}^{i}(\overline{\gamma _{1}},\ldots,\overline{\gamma _{n}})$. Now according to Eq. (38.2) in section “Syntax”, $\overline{\beta }$ is $\mathsf{F}_{K}^{j}(\overline{\gamma _{1}},\ldots,\overline{\gamma _{n}})$, for the same K and γ ₁, …, γ _n but perhaps for j ≠ i. Then it is clear by (LPO2) that $\overline{\gamma _{i}} < app(\square _{\mathcal{C}}^{{\ast}}\psi,\overline{\alpha })$ for all i. So by (LPO3), $\overline{\beta } < app(\square _{\mathcal{C}}^{{\ast}}\psi,\overline{\alpha })$.

The remaining parts are similar.

A normal form in a rewriting system is a sentence which cannot be rewritten in the system. Of course, we are interested in the systems $\mathcal{R}$ and $\mathcal{R}^{{\ast}}$ from sections “A Logic for” and “A Logic for”, respectively. It follows from the wellfoundedness of < that for every $\varphi$ there is a normal form $nf(\varphi ) \leq \varphi$ obtained by rewriting $\varphi$ in some arbitrary fashion until a normal form is reached.

Lemma 3.

A sentence $\varphi \in \mathcal{L}([\alpha ])$ is a normal form of $\mathcal{R}^{{\ast}}$ iff $\varphi$ is a modal sentence (i.e., iff $\varphi$ contains no actions). Moreover, the rule $[\alpha ][\beta ]\varphi \leadsto [\alpha \circ \beta ]\varphi$ is not needed to reduce $\varphi$ to normal form. So for $\mathcal{L}([\alpha ])$ , $\mathcal{R}$ has the same normal forms as $\mathcal{R}^{{\ast}}$ .

A sentence $\varphi \in \mathcal{L}([\alpha ],\square ^{{\ast}})$ is a normal form of $\mathcal{R}^{{\ast}}$ iff $\varphi$ is built from atomic sentences using $\neg$ , ∧, □ _A , and $\square _{\mathcal{B}}^{{\ast}}$ , or if $\varphi$ is of the form $[\alpha ]\square _{\mathcal{B}}^{{\ast}}\psi$ , where α is a normal form action, and ψ is a normal form. An action α is a normal form if whenever $\alpha \,\stackrel{\ast}{\rightarrow} \beta$ , then $\mbox{ PRE}(\beta )$ is a normal form.

Proof.

It is immediate that every modal sentence is a normal form in $\mathcal{L}([\alpha ])$, that every $[\alpha ]\square _{\mathcal{C}}^{{\ast}}\varphi$ is a normal form in $\mathcal{L}([\alpha ],\square ^{{\ast}})$, and that if each PRE(β), with $\alpha \,\stackrel{\ast}{\rightarrow} \beta$, is a normal form, then α is a normal form action. Going the other way, we check that if $\varphi \in \mathcal{L}([\alpha ])$, $[\alpha ]\varphi$ is not a normal form. So we see by an easy induction that the normal forms of $\mathcal{L}([\alpha ])$ are exactly the modal sentences. We also argue by induction for $\mathcal{L}([\alpha ],\square ^{{\ast}})$, and we note that every $[\alpha ][\beta ]\varphi$ is not a normal form, using the rule $[\alpha ][\beta ]\varphi \leadsto [\alpha \circ \beta ]\varphi$.

One fine point concerning $\mathcal{R}$ and our work in section “A Logic for” is that to reduce sentences of $\mathcal{L}([\alpha ])$ to normal form we may restrict ourselves to rewriting sentences which are not subterms of actions. This simplification accounts for the differences between parallel results of sections “A Logic for” and “A Logic for”.