7.5 Conclusion . . . 166
Résumé du chapitre
Dans ce chapitre, nous prouvons que le problème de vérification de modèle pour toutes les logiques présentées dans la thèse—DEL-PAO dans le chapitre 2, DEL-PAOSdans le chapitre 4, DEL-PAOC dans le chapitre 5
et DEL-PAO-PP dans le chapitre 6—est en PSPACE. Nous le faisons en
fournissant un algorithme pour un langage dans lequel toutes les for- mules de toutes les précédentes logiques peuvent être traduites. Ce lan- gage comprend des programmes exécutés publiquement, utilisés pour encoder les programmes de DEL-PAO-PP, ainsi que des “programmes mentaux” qui ne modifient pas l’état d’information, ce qui nous permet d’encoder des programmes de DEL-PAO.
Pour couvrir tous les langages, nous fournissons une sémantique s’appuyant sur un état d’information et une valuation (comme dans le chapitre 6). Toutefois, les programmes publics de DEL-PAO-PP modifient l’état d’information et diffèrent donc des programmes présentés dans DEL-PAO, DEL-PAOS et DEL-PAOC. Nous avons également vu dans le
chapitre 2 et dans le chapitre 5 comment réduire les opérateurs épisté- miques et stratégiques à ces programmes. Par conséquent, il sera utile d’inclure dans notre langage à la fois les programmes publics et un nou- vel opérateur Kπ, où π sera un “programme mental”, c’est-à-dire, qui ne
modifiera pas l’état d’information. Ces opérateurs ont été introduits par
in order to include properly all presented logics: the Kleene star has been added and translations from every language, along with proofs of correctness, have been added. The model checking algorithms remain unchanged (apart from the star that was in- cluded, following [Charrier et al., 2016b]).
exemple dans [van Benthem et al., 2006; Charrier and Schwarzentru- ber, 2015; Herzig et al., 2015]. Intuitivement, [π!]ϕ est lu “ϕ sera vrai après la mise à jour des états local et d’information actuels par π”, tan- dis que Kπϕest lu “ϕ sera vrai après la mise à jour de l’état local actuel
par π (en gardant l’état d’information courant constant)”. Donc, étant donnés un état local w et un état d’information U, l’opérateur de pro- gramme public [π!] met à jour à la fois U et w, alors que l’opérateur de programme mental Kπ garde U constant et met seulement à jour w. Ce
dernier peut être vu comme traversant l’espace des possibilités épisté- miques actuelles.
Nous appelons cette logique DL-PA-PMP : une logique dynamique des affectations propositionnelles avec des programmes publics et mentaux.
7.1
DL-PA-PMP: DL-PA with public and mental pro-
grams
To show that the problem is in PSPACE, we adapt the alternating al-
gorithm in [Charrier and Schwarzentruber, 2015], originally designed for a variant of a dynamic logic with propositional assignments, pub- lic announcements and arbitrary public announcements. Here we con- sider another, novel variant without arbitrary public announcements but with public programs: DL-PA-PMP.
7.1.1 Language ofDL-PA-PMP
Remember that Prop is a countable non-empty set of propositional vari- ables and and Agt is a finite non-empty set of agents.
The set of visibility operators OBS is defined like in DEL-PAO: OBS = {Si :i ∈ Agt} ∪ {JS }.
We also include the set of control operators CTRL from DEL-PAOC: CTRL = {Ci :i ∈ Agt},
so that atoms are about both visibility and control:
ATM = {σ p : σ ∈ (OBS ∪ CTRL)∗, p ∈ Prop}.
Then the syntax of DL-PA-PMP is defined by the following grammar: πp::=α←> | α←⊥ | (πp;πp) | (πpt πp) |ϕ?
πm::=α←> | α←⊥ | (πm;πm) | (πmt πm) |πm∗ | ϕ?
where α ranges over ATM .
Unlike in previous logics, here assignments are noted α←> and α←⊥to highlight that they are different from our previous +α and −α. Indeed, even if we include joint visibility operators JS , we will see that their semantics do not take introspective consequences and introspec- tively valid atoms into account.
The other boolean operators abbreviate as usual, in the standard way. The set of atoms appearing in the formula ϕ and in the program π, noted ATM (ϕ) and ATM (π), are also defined like in DEL-PAO for boolean operators. Moreover:
ATM ([πp!]ϕ) = ATM (πp) ∪ ATM (ϕ)
ATM (Kπmϕ) = ATM (πm) ∪ ATM (ϕ)
The language of DL-PA-PMP is slightly different from the one pre- sented in [Charrier and Schwarzentruber, 2015]; here, we:
• consider general programs [πp!]ϕinstead of only public announce-
ments;
• drop arbitrary public announcements; • include the Kleene star in mental programs.
This implies that the model checking procedure will have to consider general publicly executed programs and the Kleene star. The latter is however included in an extended version of [Charrier and Schwarzen- truber, 2015] that can be found in [Charrier et al., 2016b].
7.1.2 Semantics of DL-PA-PMP
Formulas are interpreted on pointed models hU, wi (see Section 6.2): public programs will modify the public information state while men- tal programs will not. The latter will allow us to simulate DEL-PAO programs within the framework of DL-PA-PMP.
As we have mentioned, assignments do not take into account intro- spective consequences even with JS operators, so that a finite number of atoms is modified; it is the translation of other logics formulas into DL-PA-PMPthat will deal with consequences. We define:
w+α = w ∪ {α} w−α = w \ {α} and
U +α = {w+α : w ∈ U } U −α = {w−α : w ∈ U }
Then the truth conditions of DL-PA-PMP are as follows:
U, w |= α iff α ∈ w
U, w |= ¬ϕ iff not (U, w |= ϕ)
U, w |= ϕ ∧ ϕ0 iff U, w |= ϕ and U, w |= ϕ0
U, w |= [π!]ϕ iff U0, w0|= ϕfor every hU0, w0isuch that hU, wiPπhU0, w0i
U, w |= Kπϕ iff U, w0|= ϕfor every w0 ∈ U such that hU, wiMπhU, w0i
where Pπ is the public relation on pointed models is defined by:
hU, wiPα←>hU0, w0i iff U0 =U +αand w0 =w+α
hU, wiPα←⊥hU0, w0i iff U0 =U −αand w0 =w−α
hU, wiPπ;π0hU0, w0i iff hU, wi(Pπ◦ Pπ0)hU0, w0i
hU, wiPπtπ0hU0, w0i iff hU, wi(Pπ∪ Pπ0)hU0, w0i
hU, wiPχ?hU0, w0i iff U, w |= χ, w0=w, and U0= {u ∈ U : U, u |= χ}
and Mπ is the mental relation on pointed models that is defined by:
hU, wiMα←>hU0, w0i iff U0 =U and w0=w+α
hU, wiMα←⊥hU0, w0i iff U0 =U and w0=w−α
hU, wiMπ;π0hU0, w0i iff hU, wi(Mπ◦ Mπ0)hU0, w0i
hU, wiMπtπ0hU0, w0i iff hU, wi(Mπ∪ Mπ0)hU0, w0i
hU, wiMπ∗hU0, w0i iff hU, wi(
[
k∈N0
(Mπ)k)hU0, w0i
hU, wiMχ?hU0, w0i iff U0 =U, w0 =wand U, w |= χ
Observe that mental programs indeed do not change the public in- formation state: when hU, wiMπhU0, w0i then U0 =U. However, the ex-
ecution of mental programs may exit the information state. To see this, take as an example U = {w ∈ 2ATM :p /∈ w}, i.e., the set of valuations
where p is false. Mp←> relates the pointed model hU, ∅i to the model
hU, {p}i, but {p} /∈ U.
As we will translate DEL-PAO-PP’s epistemic operators, that depend on the information state, into mental programs, we impose that truth condition for Kπ requires w0 ∈ U: valuations outside the U will not be
taken into account.
Note also that unlike in all other semantics presented in this thesis, introspectively valid atoms can be removed by the public or mental ex- ecution of α←⊥. Like introspective consequences, the translation into DL-PA-PMPwill have to deal with them.
7.2
Translation into DL-PA-PMP
In this section, we show how to translate formulas of the previously presented logics into the language of DL-PA-PMP. We prove that the translation is correct on models we are interested in: the formula is satisfied in a given (introspective) model of its original framework if and only if it is satisfied in the corresponding (finite) model of DL-PA- PMPthat we will use for model checking.
We define the restriction of a set of valuations to a set of atoms A as: U |A= {u ∩ A : u ∈ U }.
Note that U|Ais a finite set of finite valuations whenever A is finite.
To avoid confusion, we indicate to which semantics we refer to by subscripting the satisfaction relation |= by the name of the logic. When we mention translation functions, we always refer to the ones defined at the beginning of the current subsection.
7.2.1 From DEL-PAO
Take a DEL-PAO formula ϕ. We define the following procedure of trans- lation of ϕ into DL-PA-PMP.
Procedure 7.1.
1. eliminate all epistemic operators Ki and CK from ϕ using Proce-
dure 2.2 on page 61; call the resulting formula ϕ0;
2. translate ϕ0into the DL-PA-PMP formula trfml
ATM (ϕ0)(ϕ0)according to
the following definition:
trfmlA (α) = α trfml A (¬ϕ) = ¬trfmlA (ϕ) trfml A (ϕ1∧ ϕ2) = trfmlA (ϕ1) ∧ trfmlA (ϕ2) trfmlA ([π]ϕ) = Ktrprg A (π)tr fml A (ϕ) where trprg A (π)is defined as: trprgA (+α) = β1←>; . . . ; βm←> trprgA (−α) = (
fail if α valid in INTR
β0 1←⊥; . . . ; βp0←⊥ otherwise trprgA (π1;π2) = trprgA (π1); trprgA (π2) trprgA (π1t π2) = trAprg(π1) t trprgA (π2) trprgA (π∗) = trprgA (π)∗ trprgA (χ?) = trfml A (χ)?
with
{β1, . . . , βm} = α⇒∩ A
{β10, . . . , βp0} = α⇐∩ A
Remember that α⇐and α⇒ are the introspective causes and conse-
quences of α. We note trDEL-PAO(ϕ)the formula obtained after applying
Procedure 7.1 to ϕ.
Lemma 7.1. Let π be a DEL-PAO program and A ⊆ ATM a set of atoms. Let w be a DL-PA-PMP valuation such that w ∈ INTR. Then for every w0
such that hINTR|A, w ∩ AiMtrprgA (π)hINTR|A, w0i, we have w0∈ INTR|A.
Proof. Observe that w ∈ INTR implies that w ∩ A ∈ INTR|A. We prove
the property by induction on the form of π.
• π = +α. In this case, we have trprgA (π) = β1←>; . . . ; βm←> with
{β1, . . . , βm} = α⇒∩ A. Hence by the truth conditions of DL-PA-
PMP, w0= (w ∩ A) ∪ (α⇒∩ A) = (w ∪ α⇒) ∩A. Since w ∈ INTR, we have seen in DEL-PAO that w ∪ α⇒ ∈ INTR, hence (w ∪ α⇒) ∩A ∈
INTR|A.
• π = −α.If α is valid in INTR, trprgA (−α) = fail and the property is trivially valid. Otherwise, the proof is similar to the case π = +α. • The proofs for π = π1;π2 and π = π1 t π2 are straightforward
since their translation is homomorphic and the associated relation Mtrprg A (π)is a composition or union of Mtr prg A (π1)and Mtr prg A (π2). • π = χ?. Then trprgA (π) = trfml
A (χ)?. Like when assigning to false
an introspectively valid atom, the proof is obvious if the program fails. Otherwise, the truth condition indicates that w0 = w ∩ A,
hence w0 trivially belongs to INTR| A.
In all cases we have w0 ∈ INTR| A.
Lemma 7.1 implies that on specific states, translations of DEL-PAO programs cannot exit the public information state. This helps us prove the next equivalences.
Proposition 7.1. Let A ⊆ ATM be a set of atoms and w ∈ INTR an introspective valuation. Then we have:
INTR|A, w ∩ A |=DL-PA-PMP KtrprgA (π1;π2)ϕ ↔ KtrprgA (π1)KtrprgA (π2)ϕ
INTR|A, w ∩ A |=DL-PA-PMP KtrprgA (π1tπ2)ϕ ↔ KtrprgA (π1)ϕ ∧ KtrprgA (π2)ϕ
INTR|A, w ∩ A |=DL-PA-PMP KtrprgA (π∗)ϕ ↔ K
trprgA (π22|ATM (π)|)ϕ
Proof. All these equivalences are plainly valid in DEL-PAO (see Proposi- tion 2.11 on page 50) and most are equivalences of dynamic logics. They still apply in DL-PA-PMP on certain models because, first, the transla- tion of these program operators is homomorphic, and second, because of Lemma 7.1 that ensures that on this kind of model, translations of pro- grams for the set A do not exit the public information state. This is es- pecially important for the sequence: KtrprgA (π1;π2)ϕ ↔ KtrprgA (π1)KtrprgA (π2)ϕ
is not valid in the general case.
Proposition 7.2. Let V ∈ INTR be an introspective valuation and ϕ a DEL-PAOformula. Then we have:
V |=DEL-PAO ϕif and only if
INTR|RATM (ϕ), V ∩ RATM (ϕ) |=DL-PA-PMP trDEL-PAO(ϕ)
Proof. We follow the translation procedure and use properties of DEL- PAOand DL-PA-PMP programs to prove the equivalence.
• First of all, remove epistemic operators from ϕ, and write the re- sulting formula ϕ0; by Proposition 2.17 on page 59, it is equiva-
lent to ϕ since V is introspective. We have seen that RATM (ϕ) = ATM (ϕ0). Call this set A. Translate the resulting formula into
DL-PA-PMP; we obtain trfml
A (ϕ
0).
• In DEL-PAO, apply validities (Red;), (Redt), (Red∗) and (Red?) of
Proposition 2.11 on page 50 to ϕ0. We obtain a formula ϕ00 equiva-
lent to ϕ0 and with the same atoms, but with only boolean opera-
tors and assignment programs.
In DL-PA-PMP, apply validities of Proposition 7.1 to trfml
A (ϕ
0), with-
out “breaking” sequences of assignments that originate from the same DEL-PAO assignment. (For example, suppose we translate [+JSp]Sip, we will obtain KJS p←>;Sip←>Sip, with JS p←>; Sip←>
both originating from +JS p.) We obtain a new formula that is equivalent to trfml
A (ϕ0), and actually identical to the translation of
ϕ00(noted trfml
A (ϕ
00)) since, first, translations of program operators ;,
t,∗and ? are homomorphic and second, because the equivalences that we apply are identical to the ones of Proposition 2.11.
Therefore we only need to prove that
V |=DEL-PAOϕ00if and only if INTR|A, V ∩ A |=DL-PA-PMPtrfmlA (ϕ 00
). We do it by induction on the form of ϕ00(remember that it only contains
• ϕ00is boolean. In this case, trfml
A (ϕ00) =ϕ00. Then V |=DEL-PAO ϕ00
is equivalent to V ∩ A |=DEL-PAO ϕ00by Proposition 2.9 on page 38,
which is equivalent to U, V ∩ A |=DL-PA-PMPϕ00for any public infor-
mation state U since the truth conditions for boolean formulas are identical in DEL-PAO and in DL-PA-PMP and only depend on the local state. In particular, INTR|A, V ∩ A |=DL-PA-PMP ϕ00.
• ϕ00 = [+α]ψ.Then we have trfml A (ϕ 00) =K β1←>;...;βm←>trfmlA (ψ)with {β1, . . . , βm} = α⇒∩ A. Hence: INTR|A, V ∩ A |=DL-PA-PMPKβ1←>;...;βm←>trfmlA (ψ) ⇔ INTR|A, (V ∩ A) ∪ (α⇒∩ A) |=DL-PA-PMPtrfmlA (ψ) ⇔ INTR|A, (V ∪ α⇒) ∩A |=DL-PA-PMP trfmlA (ψ)
⇔ V ∪ α⇒|=DEL-PAO ψ (by induction hypothesis) ⇔ V |=DEL-PAO [+α]ψ.
• ϕ00 = [−α]ψ.The proof is obvious if α is introspectively valid (as it translates to fail) and similar to the case ϕ00= [+α]ψotherwise.
Therefore the translation is correct.
This settles the case of DEL-PAO. The properties of other logics are slightly different due to their respective operators and semantics, but we will see that the method is the same.
7.2.2 FromDEL-PAOS
Observe that DEL-PAOSis a fragment of DEL-PAOC: neither of them con-
sider the operator of joint visibility JS or common knowledge, and their semantics are strictly identical for boolean, epistemic and dynamic op- erators; DEL-PAOC simply further includes the strategic operator 3J.
Therefore we do not detail any procedure for DEL-PAOS, as it is com-
pletely identical to the one for DEL-PAOC, without the reduction of strate- gic operators at the beginning.
7.2.3 FromDEL-PAOC
Take a DEL-PAOC formula ϕ. We define the following procedure of trans- lation of ϕ into DL-PA-PMP.
Procedure 7.2.
1. eliminate all epistemic operators Ki and strategic operators 3J
from ϕ using Procedure 5.1 on page 121; call the resulting formula ϕ0;
2. translate ϕ0into the DL-PA-PMP formula trfml(ϕ0)according to the following definition: trfml(α) = α trfml(¬ϕ) = ¬trfml(ϕ) trfml(ϕ1∧ ϕ2) = trfml(ϕ1) ∧ trfml(ϕ2) trfml([π]ϕ) = Ktrprg(π)trfml(ϕ)
where trprg(π)is defined as:
trprg(+α) = α←> trprg(−α) =
(
fail if α valid in INTR
α←⊥ otherwise trprg(π 1;π2) = trprg(π1); trprg(π2) trprg(π 1t π2) = trprg(π1) t trprg(π2) trprg(π∗) = trprg(π)∗ trprg(χ?) = trfml(χ)?
Observe that we do not need to keep the set of atoms of ϕ0 in this
case, as we do not have to deal with introspective causes and conse- quences in assignments. We note trDEL-PAOC(ϕ) the formula obtained
after applying Procedure 7.2 to ϕ.
Lemma 7.2. Let π be a DEL-PAOC program and A ⊆ ATM a set of atoms such that ATM (π) ⊆ ATM . Let w be a DL-PA-PMP valua- tion such that w ∈ INTR. Then for every w0 such that hINTR|
A, w ∩
AiMtrprg(π)hINTR|A, w0i, we have w0∈ INTR|A.
Proof. We prove it by induction on the form of π.
• π = +α. In this case, trprg(π) = α←>. Hence by the truth con-
ditions of DL-PA-PMP, w0 = (w ∩ A) ∪ {α} = (w ∪ {α}) ∩ A since
α ∈ A(because ATM (π) ⊆ ATM ). Since w ∈ INTR, we have seen in DEL-PAOC that w ∪ {α} ∈ INTR, hence (w ∪ {α}) ∩ A ∈ INTR|A.
• π = −α.If α is valid in INTR, trprg(−α) = fail and the property is
trivially valid. Otherwise, the proof is similar to the case π = +α. • The proofs for π = π1;π2, π = π1 t π2 and π = χ? are similar to
their cases in the proof of Lemma 7.1 as their truth conditions are identical to DEL-PAO’s.
In all cases we have w0 ∈ INTR| A.
This counterpart of Lemma 7.1 indicates that, again, the execution of mental programs cannot exit the public information state in models that interest us.
Proposition 7.3. Let A ⊆ ATM be a set of atoms and w ∈ INTR an introspective valuation. Then we have:
INTR|A, w ∩ A |=DL-PA-PMP Ktrprg(π1;π2)ϕ ↔ Ktrprg(π1)Ktrprg(π2)ϕ
INTR|A, w ∩ A |=DL-PA-PMP Ktrprg(π1tπ2)ϕ ↔ Ktrprg(π1)ϕ ∧ Ktrprg(π2)ϕ
INTR|A, w ∩ A |=DL-PA-PMP Ktrprg(π∗)ϕ ↔ K
trprg(π22|ATM (π)|)ϕ
INTR|A, w ∩ A |=DL-PA-PMP Ktrprg(χ?)ϕ ↔ trfml(χ) → ϕ
Proof. The proof is similar to Proposition 7.1 for the translation of DEL- PAOprograms. It likewise relies on Lemma 7.2.
Proposition 7.4. Let V ∈ INTR be a valuation and ϕ a DEL-PAOC for- mula. Then we have:
V |=DEL-PAOCϕif and only if
INTR|RATM (ϕ), V ∩ RATM (ϕ) |=DL-PA-PMP trDEL-PAOC(ϕ)
Remember that relevant atoms of DEL-PAOC also include control atoms that are “hidden” in strategic operators.
Proof. The proof follows the lines of Proposition 7.2 from the previous section: we first remove epistemic and strategic operators from ϕ and obtain a formula ϕ0, equivalent to ϕ by Proposition 5.1 on page 118
and such that RATM (ϕ) = ATM (ϕ0). Then we apply validities (Red ;),
(Redt), (Red∗) and (Red?) of Proposition 2.11 on page 50 to ϕ0, getting ϕ00,
and equivalences of Proposition 7.3 to its translation trfml(ϕ0), obtaining
trfml(ϕ00)like in the previous setting. We again need to prove that
V |=DEL-PAOCϕ00 if and only if INTR|A, V ∩ A |=DL-PA-PMP trfml(ϕ00),
with A = ATM (ϕ0). We again do it by induction on the form of ϕ00.
• The case when ϕ00 is boolean is the same as for Proposition 7.2 (because the translation is still homomorphic).
• ϕ00 = [+α]ψ. This time we have trfml(ϕ00) = K
α←>trfml(ψ). Ob-
serve that α ∈ A. Hence:
INTR|A, V ∩ A |=DL-PA-PMP Kα←>trfml(ψ)
⇔ INTR|A, (V ∩ A) ∪ {α} |=DL-PA-PMP trfml(ψ)
⇔ INTR|A, (V ∪ {α}) ∩ A |=DL-PA-PMP trfml(ψ)
⇔ V ∪ {α} |=DEL-PAOCψ (by induction hypothesis)
• ϕ00= [−α]ψ.The proof is obvious if α is introspectively valid (as it translates to fail) and similar to the case ϕ00= [+α]ψotherwise.
Therefore the translation is correct.
7.2.4 From DEL-PAO-PP
In DEL-PAO-PP, we consider public programs and epistemic operators. The formers will be translated homomorphically (except, as before, for assignments) to DL-PA-PMP public programs. We have already seen in DEL-PAO how to reduce epistemic operators to programs; we reuse these in DEL-PAO-PP. As epistemic operators do not modify the public information state and as we only keep indistinguishable worlds that are in this information state, we translate them to mentally executed programs, whose semantics fit perfectly.
Take a DEL-PAO-PP formula ϕ. We define the following procedure of translation of ϕ into DL-PA-PMP.
Procedure 7.3.
1. translate ϕ into the DL-PA-PMP formula trfml
RATM (ϕ)(ϕ)according to
the following definition: trfmlA (α) = α trfml A (¬ϕ) = ¬trfmlA (ϕ) trfml A (ϕ1∧ ϕ2) = trfmlA (ϕ1) ∧ trfmlA (ϕ2) trfmlA (Kiϕ) = KtrprgA (varyIfNotSeen(i,RATM (ϕ)))tr fml A (ϕ) trfmlA (CKϕ) = Ktrprg A (varyIfNotSeen(Agt ,RATM (ϕ)))tr fml A (ϕ) trfmlA ([π!]ϕ) = [trprgA (π)!]trfml A (ϕ) where trprg A (π)is defined as: trprgA (+α) = β1←>; . . . ; βm←> trprgA (−α) = (
fail if α valid in INTR
β0 1←⊥; . . . ; βp0←⊥ otherwise trprgA (π1;π2) = trprgA (π1); trprgA (π2) trprgA (π1t π2) = trAprg(π1) t trprgA (π2) trprgA (χ?) = trfml A (χ)? with {β1, . . . , βm} = α⇒∩ A {β10, . . . , βp0} = α⇐∩ A
Observe that unlike in Procedure 7.1 for DEL-PAO and in Proce- dure 7.2 for DEL-PAOC, we use RATM (ϕ) instead of ATM (ϕ0) (with ϕ0
the formula resulting from eliminating epistemic operators from ϕ) as epistemic operators have not be eliminated yet. This is however equiv- alent, as we have seen that RATM (ϕ) = ATM (ϕ0)in previous cases. We
simulate epistemic operators with our programs varyIfNotSeen(., .) on the relevant atoms of the nested formula as it may contain other epistemic operators; then we translate these programs so that assignments be- have correctly. This translation is identical for public programs (and is the same as previous translations of DEL-PAO and DEL-PAOC programs). We do not include the star since it does not appear in varyIfNotSeen(i, A) and varyIfNotSeen(Agt, A). We note trDEL-PAO-PP(ϕ)the formula obtained
after applying Procedure 7.3 to ϕ.
We verify that epistemic operators are indeed equivalent to their mentally executed programs counterparts.
Proposition 7.5. Let U be a public information state such that U ⊆ INTR, w ∈ U a valuation, i an agent and ϕ a formula without epistemic operators. Then:
1. for every A ⊆ ATM such that RATM (Kiϕ) ⊆ A,
U, w |=DEL-PAO-PP Kiϕ if and only if U|A, w ∩ A |=DL-PA-PMP
Ktrprg
A (varyIfNotSeen(i,ATM (ϕ)))tr fml
A (ϕ);
2. for every A ⊆ ATM such that RATM (CK ϕ) ⊆ A,
U, w |=DEL-PAO-PP CKϕ if and only if U|A, w ∩ A |=DL-PA-PMP
Ktrprg
A (varyIfNotSeen(Agt ,RATM (ϕ)))tr fml
A (ϕ).
Proof. We only examine the first case as the case of common knowledge is similar.
Observe that since ϕ is boolean, RATM (ϕ) = ATM (ϕ). We have seen in the proof of Proposition 2.17 on page 59 that varyIfNotSeen(i, ATM (ϕ)) correctly simulates the epistemic operator on valuations that are intro- spective enough (which is the case of w ∩ A since RATM (Kiϕ) ⊆ A).
However, we are now also dealing with public information states.