assignment is a couple of the form either hi, p
+
i or
hi, p
−
i, where i is an agent from I and p is a variable
from P. The intuition for the former is that i sets the
variable p to true; for the latter it is that i sets p to
false. For ease of notation we write i:+p and i:−p in-
stead of hi, p
+
i and hi, p
−
i. An atomic action is a finite
set of authored assignments. Given an atomic action
δ and an agent i ∈ I, we define i’s part of δ as:
δ|
i
= {i:+p | i:+p ∈ δ} ∪ {i:−p | i:+p ∈ δ}
The set of all atomic actions is noted ∆.
Beyond the modal operators Bel
i
and Ch
i
, our
language has two dynamic modal operators h.i and
hh.ii the first of which is from Propositional Dynamic
Logic PDL. These operators have complex actions as
arguments. The formula hπiϕ reads “the action π is
executable and ϕ is true afterwards”. In contrast, for
hhπiiϕ reads “π is executed and ϕ is true afterwards”.
The latter implies the former: when π is executed then
π clearly should be executable. It is also clear that the
other way round, executability should not imply exe-
cution.
Therefore hi:+pi> reads “i is able to set p to true”,
while hhi:+pii> reads “i is going to set p to true”. The
formula hhi:+p, j:−qiiϕ expresses that the agents i and
j are going to assign the value ‘true’ to the proposi-
tional variable p and ‘false’ to q, and that afterwards
ϕ will be true; and Bel
k
hhi:+p, j:−qiiϕ expresses that
agent k believes that this is going to happen. As the
reader may have noticed, we drop the set parenthe-
ses around the atomic assignments in formulas such
as hi:+pi>, hhi:+pii> and hhi:+p, j:−qiiϕ.
Formally, we define by induction the set of actions
(programs) Prog and the set of well-formed formulas
Fml of ABC logic.
π F δ | skip | fail | π;π | if ψ then π else π
ϕ F p | > | ¬ϕ | ϕ ∧ ϕ | Bel
i
ϕ | Ch
i
ϕ | hπiϕ | hhπiiϕ
where p ranges over P, i over I and δ over ∆. Here is
an example of a complex action. Let L mean that the
light is on. Then if L then j:−L else j:+L describes
j’s action of toggling the light switch.
We use the standard abbreviations for ∨ and →.
Moreover, ⊥ abbreviates ¬> and [π]ϕ abbreviates
¬hπi¬ϕ.
The formulas of our language have a semantics is
in terms of Kripke models and model updates, as cus-
tomary in dynamic epistemic logics (van Ditmarsch
et al., 2007). We do not go into the details of the se-
mantics here (see (Herzig et al., 2013)) and instead
rely on the reader’s intuitions. Here are some exam-
ples of validities:
• hhπii> → hπi> (do implies can)
• Bel
i
ϕ → Ch
i
ϕ (realistic choice)
• Ch
i
ϕ → Bel
i
Ch
i
ϕ (positive introspection)
• ¬Ch
i
ϕ → Bel
i
¬Ch
i
ϕ (negative introspection)
• hh j:aii> → Ch
j
hh j:aii> (intentional action)
It is decidable whether a formula is true in a given
ABC model. It is also decidable whether a formula is
satisfiable in the set of ABC models.
5.1 Formalizing Trust
We now turn to a formalization of a theory of trust
in complex actions in ABC logic. The trust theory
basically extends Castelfranchi and Falcone’s.
Among the different theories of trust, the cognitive
theory of Castelfranchi and Falcone, henceforth ab-
breviated C&F, is probably most prominent (Castel-
franchi and Falcone, 1998; Falcone and Castelfranchi,
2001).
According to C&F, the trust relation involves a
truster i, a trustee j, an action a that is performed by
j and a goal ϕ of i. They defined the predicate Trust
as a goal together with a particular configuration of
beliefs of the trustee. Precisely, i trusts j to do a in
order to achieve ϕ if and only if i has the goal that ϕ
and i believes that:
1. j is capable to perform a,
2. j is willing to perform a,
3. j has the power to achieve ϕ by doing a.
C&F distinguish external from internal conditions
in trust assessment: j’s capability to perform a is an
external condition, while j’s willingness to perform
a is an internal condition (being about the trustee’s
mental state). Finally, j’s power to achieve ϕ by do-
ing a relates internal and external conditions: if j per-
forms a then ϕ will result. Observe that in the power
condition, the result is conditioned by the execution
of a; therefore the power condition is independent
from the capability condition. In particular, j may
well have the power to achieve ϕ without being capa-
ble to perform a: for example, right now I have the
power to lift a weight of 50kg, but I am not capable to
do this because there is no such weight at hand.
We follow Jones who argued that the core notion
of trust need not involve a goal of the truster (Jones
and Firozabadi, 2001; Jones, 2002) and consider a
simplified version of C&F’s definition in terms of a
truster, a trustee, an action of the trustee, and an ex-
pected outcome of that action.
Complex action expressions involve multiple
agents that occur in the action expressions. We there-
fore need not identify the trustee as a separate argu-
ment of the trust predicate. Our official definition of
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