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BM S with L∗EDA

This section is about embedding the language L∗

BM S in the language L∗EDA.

Thus, we consider theextendedlanguages that we have briefly introduced in chapter 3 and 4: the extension ofLBM S with common knowledge operators,

and the extension ofLEDAwith common knowledge operators both on states

and actions. The use of the word “embedding” should not be taken too literally, since both languages are interpreted in different types of structures. For this reason, we prefer to speak about asimulation ofL∗BM S usingL∗EDA.

For such a simulation to be possible, we will need a way to make both languages comparable. For this purpose, we will use a construction similar to the one we have seen in section 2.4; in fact, the construction in this section will be a rather straightforward extension of construction 2.9.

We first introduce this construction and the result that comes along with it. The shortcoming of this result is that it does not make any essential use

of action formulas, besides using action labels to refer to individual actions. However, we designed LEDA and L∗EDA to give us expressivity in describing

epistemic action structure and the domain of applicability of actions. We put this expressive power to use in the second half of the section, where we present a translation of L∗

BM S into L∗EDA, which works modulo the con-

struction presented in the first half. This translation will map formulas of L∗

BM S to state formulas of L∗EDA, and epistemic actions overL∗BM S to action

formulas ofL∗

EDA. Formulas of L∗EDA will be equivalent to their translation

(along the construction), while epistemic actions will be characterized by their translation up to epistemic bisimilarity, in a sense to be made precise in due course.

While the first half of the section does not depend on the fact that we are considering the extended languages in an essential way, the second part will make crucial use of the common knowledge operatorsL∗

EDA has on actions.

For the remainder of this section, fix a finite set of epistemic actions

Act over L∗

EDA. Without loss of generality, we assume these actions to be

mutuallydisjoint.

Construction 5.13 Let a state model M= (S, V) be given. In prepara- tion of the following, for any given state model N, let CN

Act :=

S

α∈ActCαN.

Let furthermoreAActbe the union of all action frames underlying epistemic

actions inAct. Then we define:

M0 :=M Mn+1 :=Mn⊗_CMn Act AAct Mω := [ n<ω Mn Cω := [ n<ω CMn Act .ω :=idCω

In the last clause,idCω denotes the identity map onCω. To obtain a labelling

of the actions in A with actions labels, we identify the set of labels L of L∗

EDA with the epistemic actions in Act, and for any such epistemic action

Note that construction 5.13 is an extended version of construction 2.9 in chapter 2.4. We inductively construct an update model using the product update mechanism given by the semantics of L∗

BM S: the action frame is

just given by taking the union of all action frames underlying the epistemic actions. The state model is given by repeated applications of the product update. We then collect the access conditions that were generated during the construction, and turn them into an access map. We finally label each action in the action frame of the resulting update model with anepistemic action in Act; namely, V(a) = α iff α = (A,PRE, a). It is easy to see that

this is a total, surjective map. Note that we have chosen our language L∗

EDA depending on the set of epistemic actions Act. As a result, we have

L∗BM S ⊆ L∗EDA.

The construction induces a product update model (Mω,(AAct, V), .ω),

which we call, for a given state model M,the product update model corre- sponding toM, referring to it withMM

⊗ . We now have the following:

Theorem 5.14 Let M be a state model. LetMM

⊗ = (Mω,(AAct, V), .ω)

be the product update model corresponding toM. Then M, s|=L∗ BM S φ iffM M ⊗ , s|=L∗ EDA φ for all φ∈ L∗ BM S. Proof: By construction. tu

As pointed out above, this result does not use the expressive power of L∗

EDA on actions in any sense. In effect, we have just turned part of the

syntax of L∗

BM S into semantic objects for L∗EDA, and adapted our set of

labels ofL∗

EDA to the set of epistemic actions at hand.

We now want to describe the epistemic actions, using a translation of formulas ofL∗

BM S intoL∗EDA. We define, by double recursion, a function T r,

which maps each formulaφ∈ L∗

BM S to a state formulaT r(φ)∈ L∗EDA, which

will be a state formula, and which furthermore maps each epistemic action α over L∗

EDA, to a translationT r(α), which will be an action formula. For

the formulas, we define: 1. T r(p) =p

2. T r(¬φ) =¬T r(φ)

3. T r(φ∧ψ) =T r(φ)∧T r(ψ) 4. T r(Kiφ) =KiT r(φ)

5. T r(K∗

Grφ) =KGr∗ T r(φ)

6. T r([α]φ) = [T r(α)]T r(φ)

We now need to define T r(α), for an epistemic action α = (A,PRE, a).

A model world pair is a pair consisting of an epistemic model M, and a distinguished world inM. We can now code the preconditions as proposition letters. For this purpose, let a total injective map I from the range of

PRE to P, the set of proposition letters, be given. Define for all a ∈ A,

V(a) = I(PRE(a)). This gives us a model world pair (A, V, a). A well–

known fact in modal logic tells us that any finite model world pair can be characterized by a formula of the standard modal language with common knowledge operators. Let δ be the modal definition of (A, V, a). This is a formula, built up from proposition letters p, boolean operators, knowledge and common knowledge operators. Since δ contains proposition letters, it is not the formula we want. However, we obtain the desired translation as follows: consider any proposition letterpoccurring inδ. By injectivity ofI, we find a unique formulaφ∈ L∗

BM S, such thatI(φ) =p. Replacepuniformly

inδ with↑T r(φ), where we have T r(φ by the first part of the definition of T r above. Do this for all proposition letters occurring in δ, and call the resulting formulaδ0_{. This is the translation we want, i.e. we put}

T r(α) =δ0

for each epistemic action α. Note that this is an action formula of the languageL∗

EDA.

To state the theorem accompanying the translationT r, we need a further piece of notation. Let M= ((S, V),(C, VC), .) and N= ((S, V),(D, VD), .0)

be update models. Letc∈ C and d∈D. We write c≈d, if there exists a relationX ⊆C×D, such that cXd, where X satisfies the back and forth clauses of the definition of an epistemic bisimulation, and mXn implies D(m) = D(n). So c ≈d expresses that c and dshare the same epistemic pattern up to bisimilarity, and that furthermore c and d have the same domain.

Theorem 5.15

Let a state model M be given, and let MM

⊗ = (Mω,(AAct, V), .ω) be the

update model corresponding to M. Let N= (Mω,(B, V0), .) be an update

model. 1. Lets∈S and φ∈ L∗ BM S. Then M, s|=L∗ BM S φiffM M ⊗, s|=L∗ EDA T r(φ)

2. Letα= (A,PRE, a), andb∈B. Then

a≈biff N, b|=L∗

EDA T r(α)

Proof: By double induction. tu Thus we see that, along the construction of associated product update models,L∗

EDAis a conservative extension ofL∗BM S. The key to the preceding

theorem lies the possibility of describing both epistemic uncertainty patterns and preconditions of actions by means of operators ofL∗

EDA. It can thus be

seen as a verification of the claim that in L∗

EDA, mechanisms of describing

actions replace the syntactic use of epistemic actions.