DEL-GENERATED ETL MODELS 35 of informational events However, it is a common assumption of the framework that,

given some consistency constraints, the agents’ informational states can change in any way. On the other hand, ETL describes the processes that govern the interactions between agents, but without giving a good account of how agents’ informational states are affected by informational events. The results that such a combination between the two different frameworks produces are elegantly presented in Hoshi 2010, (27), pp. 414-415: (i) “[u]sing DEL product update to produce ETL models provides a “bridge” between the two logical paradigms allowing us to investigate their precise relationship”; (ii) “[t]he merged framework generalizes models in DEL. We can then reinterpret the formal language of DEL over this class of generalized models and investigate new logical systems”; and (iii) “[m]odels in the merged framework (. . . ) are powerful tools for studying concrete scenarios of intelligent interactions.” A good philosophical introduction to these issues can be found in Hoshi 2010 (27), section 1. Some reasons to the effect that ETL does not have the last word on intelligent interaction are presented in Halpern and Fagin 1989 (23), see especially Theorem 5.1. Wang 2010 (57), section 9.3.2, has a brief but informed presentation of the advantages of reasoning with protocols in DEL vs. reasoning with protocols in ETL. Let E be the class of all pointed models, E = {(E, e) ∶E an event model and e∈ D(E)}. A history is a finite sequence of event models from E. Let E∗ _{be the set}

of all histories built from elements of E. Let σ denote an element of E∗ _{(so σ}

is a sequence of pointed event models). I write σn for the initial segment of σ of length n (n ≤ len(σ)) and write σ(n) for the nth component of σ. For example, if

σ= (E1, e1)(E2, e2)(E3, e3), then len(σ) =3, σ(2)= (E2, e2) and σ2= (E1, e1)(E2, e2).

Definition 15 (DEL protocol). A DEL protocol is a set P ⊆ E∗ _{closed under}

finite initial segments.

Let P tcl(E)be the class of all DEL protocols, then

Definition 16 (State-dependent DEL protocol). Let M be an epistemic model.

A state-dependent DEL protocol on M is any function p∶D(M) →P tcl(E).

If all states in a model get assigned the same protocol, then that model has a uniform DEL protocol. The reason for distinguishing between state-dependent and uniform protocols is because we want to be able to tell apart models in which the protocol is common knowledge (in which it is uniform) from model in which the agents may have uncertainties regarding which protocol is running (state-dependent DEL protocols). For the rest of this section we will be focusing on state-dependent DEL protocols.

Definition 17 (p-generated model). Given an epistemic model M = ⟨S, Ri, V⟩

and p, a state-dependent DEL protocol on M, the p-generated model at level n is

Mn,p= ⟨_Sn,p_{, R}n,p

(1) S0,p=_{S, R}0,p i =Ri, V0,p=V; (2) sσ∈Sn+1,p _iff(₁)_s∈_S;(₂)_len(_σ) =_n+_1;(₃)_sσ n∈Sn,p;(4)σ∈p(s);(5)Mn,p, sσ⊧ pre(σ(n+1)); (3) For each sσ, tσ′∈_Sn+1,p∶_sσRn+1,p i tσ ′ _{iff sσ} nRn,ptσn′ and σ(n+1)→i σ ′ (n+1);

(4) For all q∈P ROP ∶Vn+1,p(_q) = {_sσ ∈_Sn+1,p∶_s∈_V(_q)}_.

Now, the way to get from this construction to an ETL model is described in the following definition.

Definition 18 (DEL-generated ETL model). Given an epistemic model M =

⟨S, Ri, V⟩and a state-dependent DEL protocol onM,p, and ETL model,F orest(M, p) =

⟨H, R′

i, V′⟩ is defined as follows:

(1) H= {h∶ there is a s∈S, σ∈ ⋃sinSp(s) with h=sσ∈S

len(σ),p}_;

(2) for all h, h′ ∈ _{H with h} = _{sσ and h}′ = _tσ′_{, hR}′

ih

′ _{iff len}(_σ) = _len(_σ′) _and

sσRlen(σ),p

i tσ

′_;

(3) for every q∈P ROP and h=sσ∈H, h∈V′(_q) _{iff h}∈_Vlen(σ),p(_q)_.

van Benthem et al. 2009 (51) prove that for an arbitrary epistemic model and state-dependent DEL protocol, F orest(M, p) is indeed an ETL model.

There are a few logics that could be interpreted on these models and the two most obvious represent extensions of PAL and DEL, i.e. TPAL and TDEL. Interpreting public announcement logic and dynamic epistemic logic on these models is not trivial. One significant difference when adding a protocol is that reduction axioms from the dynamic logic to its static counterpart are no longer available. However, since I said that I do not want to talk about syntax in this thesis, I will not explore this issue further. Instead I will define a few operators on these models that will be helpful later in the thesis. These definitions are taken from Hoshi 2009 (26):

Let X be a set of pointed event models, such that if (E, e) ∈X then (E, e′) ∈_X,

for all e′∈_D(_E)_{. The following operators could be defined in TDEL(X):}

H, h⊧N EXT ϕ iff ∃e∈X∶he∈H and H, he⊧ϕ

H, h⊧F U T U REϕ iff ∃σ∈X∗∶

hσ∈HandH, hσ⊧ϕ

H, h⊧BEF OREϕ iff ∃e∈X,∃h′∶

h=h′

e and H, h′⊧

ϕ

The intended interpretation of these operators is as follows: “some event can happen such that afterwards ϕ holds”, “some sequence of events can happen such that afterwards ϕ holds”, and respectively “event e has happened and before it happened ϕwas holding”. Of course, duals could be defined in the usual way.

CHAPTER 3