A Probabilistic Doxastic Temporal Logic for Reasoning about Beliefs in
Multi-agent Systems
Karsten Martiny
1
and Ralf M¨oller
2
1
Institute for Software Systems, Hamburg University of Technology, Hamburg, Germany
2
Institute of Information Systems, University of L¨ubeck, L¨ubeck, Germany
Keywords:
Multi-agent Systems, Epistemic Logic, Doxastic Logic, Dynamic Belief Change.
Abstract:
We present Probabilistic Doxastic Temporal (PDT) Logic, a formalism to represent and reason about proba-
bilistic beliefs and their evolution in multi-agent systems. It can quantify beliefs through probability intervals
and incorporates the concepts of frequency functions and epistemic actions. We provide an appropriate se-
mantics for PDT and show how agents can update their beliefs with respect to their observations.
1 INTRODUCTION
When logically analyzing knowledge and belief in re-
alistic scenarios, an agent usually has only incomplete
and inaccurate information about the actual state of
the world, and thus considers several worlds as being
possible. As it receives new information, it has to up-
date its beliefs about possible worlds. These updates
can for example result in regarding some worlds as
impossible or judging some worlds to be more likely
than before. Thus, in addition to analyzing the set of
worlds an agent believes to be possible, it is also use-
ful to quantify these beliefs in terms of probabilities.
This provides means to specify fine-grained distinc-
tions within the range of worlds that an agent consid-
ers possible.
When multiple agents are involved in such a set-
ting, an agent may not only have varying beliefs re-
garding the facts of the actual world, but also regard-
ing the beliefs of other agents. In many scenarios, the
actions of one agent will not only depend on its belief
of ontic facts (i.e., facts of the actual world), but also
on its beliefs in some other agent’s beliefs.
To formalize reasoning about such beliefs in
multi-agent settings, we present Probabilistic Dox-
astic Temporal (PDT) Logic. PDT Logic builds
upon recent work on Annotated Probabilistic Tem-
poral (APT) Logic and provides a formalism which
enables the representation of and reasoning about dy-
namically changing quantified temporal multi-agent
beliefs through probability intervals. In this formal-
ism, analyses are intended to be carried out offline by
an external observer. In contrast to related work, PDT
Logic employs an explicit notion of time and thereby
facilitates the expression of richer temporal relations.
The remainder of this work is structured as fol-
lows: The next section presents related work about
knowledge in multi-agent systems and APT Logic.
Then, in Section 3, the syntax of PDT Logic is in-
troduced, followed by the definition of formal seman-
tics in Section 4. The evolution of multi-agent beliefs
over time is analyzed in Section 5. Finally, the paper
concludes with Section 6.
2 RELATED WORK
Approaches to formalize reasoning about knowledge
and belief date back to Hintikka’s work on epistemic
logic (Hintikka, 1962). Classical forms of epistemic
logic do not allow for a quantification of an agent’s
degree of belief in certain facts; it can only be spec-
ified whether an agent does or does not know (resp.
believe) some fact. To remove this limitation, sev-
eral approaches have been proposed to combine log-
ics of knowledge and belief with probabilistic quan-
tifications. For instance, (Fagin and Halpern, 1994)
and (van der Hoeck, 1997) define a belief operator
to quantify lower bounds on the probabilities that an
agent assigns to a formula.
To reason about dynamically changing beliefs, ex-
tensions to epistemic logics have been proposed, e.g.,
(Scherl and Levesque, 2003). In these works only
the single-agent case is considered, and therefore they
do not provide for representations of nested beliefs.
277
Martiny K. and Möller R..
A Probabilistic Doxastic Temporal Logic for Reasoning about Beliefs in Multi-agent Systems.
DOI: 10.5220/0005178802770284
In Proceedings of the International Conference on Agents and Artificial Intelligence (ICAART-2015), pages 277-284
ISBN: 978-989-758-074-1
Copyright
c
 2015 SCITEPRESS (Science and Technology Publications, Lda.)

Multi-agent extensions to these approaches can be
found for example in (van Ditmarsch et al., 2007). A
common limitation of these works is that they are only
able to reason about step-by-step changes and there-
fore explicit reasoning about time is difficult in these
frameworks. (Renne et al., 2009) alleviates these lim-
itations by combining Dynamic Epistemic Logic (van
Ditmarsch et al., 2007) with temporal modalities.
(Shakarian et al., 2011) introduce APT Logic, a
framework to represent probabilistic temporal evolu-
tions of worlds in threads. APT Logic assigns prior
probabilities to every thread and uses these probabil-
ities to determine probabilities of events occurring in
specific threads. To represent temporal relationships
between events, APT Logic introduces the concept
of frequency functions. We utilize the approach of
APT Logic to create a doxastic multi-agent frame-
work that can explicitly reason about temporal re-
lationships through the adoption of frequency func-
tions.
3 PDT LOGIC PROGRAMS:
SYNTAX
In this section, we start with defining the syntax of
PDT Logic programs, and then give a definition of
the formal semantics in the next section.
We assume the existence of a function-free first
order logic language with finite sets of constant sym-
bols L
cons
and predicate symbols L
pred
, and an infinite
set of variable symbols L
var
. Every predicate symbol
p ∈ L
pred
has an arity. A term is any member of the
set L
cons
∪ L
var
. A term is called a ground term if it
is a member of L
cons
. If t
1
,..,t
k
are (ground) terms,
and p is a predicate symbol in L
pred
with arity n, then
p(t
1
,...,t
k
) with k ∈ {0, ..., n} is a (ground) atom. If a
is a (ground) atom, then a and ¬a are (ground) liter-
als. The set of all ground literals is denoted by L
lit
.
As usual, B denotes the Herbrand Base of L.
Time is modeled in discrete steps and we assume
that all agents reason about an arbitrarily large, but
fixed size window of time. The set of time points is
given by τ = {1, ...,t
max
}. The set of agents is denoted
by A. The number of agents (|A|) is denoted by n. To
describe what agents observe, we define observation
atoms as follows:
Definition 1 (Observation Atoms). For any group of
agents G ⊆ A and ground literal l ∈ L
lit
, Obs
G
(l) is
an observation atom. The set of all observation atoms
is denoted by L
obs
.
Intuitively, the meaning of a statement of the form
Obs
G
(l) is that all agents in the group G observe that
the fact l holds. We assume that the agents in G not
only observe that l holds, but that each agent in G is
also aware that all other agents in G make the same
observation.
Definition 2 (Formulae). Atoms and observation
atoms are formulae. If F and G are formulae, then
so are F ∧ G, F ∨ G, and ¬F. A formula is ground if
all atoms of the formula are ground.
To describe observations at a specific time, we fur-
thermore define time-stamped observation atoms:
Definition 3 (Time-stamped Observation Atoms). If
Obs
G
(l) ∈ L
obs
is an observation atom, and t ∈ τ is a
time point, then [Obs
G
(l) : t] is a time-stamped obser-
vation atom.
To express temporal relationships, we define tem-
poral rules following the approach of APT rules from
(Shakarian et al., 2011).
Definition 4 (Temporal Rules). Let F,G be formulae,
∆t a time interval, and fr a name for a so-called fre-
quency function (as defined below in Definition 11).
Then r
fr
∆t
(F,G) is called a temporal rule.
The meaning of such an expression is “F is fol-
lowed by G in ∆t time units w.r.t. fr”.
Now, we can define the belief operator B
ℓ,u
i,t
′
to ex-
press agents’ beliefs. Intuitively, B
ℓ,u
i,t
′
(·) means that at
time t
′
, agent i believes that some fact (·) is true with a
probability p ∈ [ℓ,u]. We call the probability interval
[ℓ, u] the quantification of agent i’s belief. We use F
t
to denote that formula F holds at time t.
Definition 5 (Belief Formulae). Let i be an agent, t
′
a time point, and [ℓ,u] ⊆ [0,1]. Then, belief formulae
are inductively defined as follows:
1. If F is a formula and t is a time point, then B
ℓ,u
i,t
′
(F
t
)
is a belief formula.
2. If r
fr
∆t
(F,G) is a temporal rule, then B
ℓ,u
i,t
′
(r
fr
∆t
(F,G))
is a belief formula.
3. If F and G are belief formulae, then so are
B
ℓ,u
i,t
′
(F), F ∧ G, F ∨ G, and ¬F.
4 SEMANTICS
In this section, we will provide a formal semantics
that captures the intuitions explained above. We start
with the introduction of an example, which we will re-
turn to repeatedly when introducing the various con-
cepts of the semantics.
Example 1 (Trains). Let Alice and Bob be two agents
living in two different cities C
A
and C
B
, respectively.
Suppose that Alice wants to take a train to visit Bob
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and has to change trains at a third city C
C
. We as-
sume that train T
1
connects C
A
and C
C
, and train T
2
connects C
C
and C
B
. Both trains usually require 2
time units for their trip, but they might be running late
and arrive one time unit later than scheduled. Alice
requires one time unit to change trains at city C
C
. If
T
1
runs on time, she has a direct connection to T
2
,
otherwise she has to wait for two time units until the
next train T
2
leaves at city C
C
. If a train is running
late, she can call Bob to let him know. These calls
can be modeled as shared observations between Al-
ice and Bob. For instance, if Alice wants to tell Bob
that train T
1
is running late (i.e., T
1
does not arrive
at C
C
at the expected time), this can be modeled as
Obs
AB
(¬at(T
1
,C
C
)) at the expected arrival time.
4.1 Possible Worlds
Ontic facts and according observations form worlds.
A world w consists of a set of ground atoms and a
set of observation atoms, i.e., w ∈ 2
B
× 2
L
obs
. With a
slight abuse of notation, we use a ∈ w and Obs
G
(l) ∈
w to denote that an atom a (resp. observation atom
Obs
G
(l)) holds in world w. Since agents can only ob-
serve facts that actually hold in the respective world,
we can define consistency of worlds w.r.t. the set of
observations:
Definition 6 (World Consistency). A world w is con-
sistent, iff for every observation atom Obs
G
(l) ∈ w,
the observed fact holds, i.e., x ∈ w if l is a positive
literal x, x 6∈ w if l is a negative literal ¬x.
The set of all possible worlds is denoted by W.
For the following discussion we assume a manual suc-
cinct specification of possible worlds depending on
the respective domain. Especially, we assume in the
following discussion that W does not contain any in-
consistent worlds according to Definition 6.
Example 2 (Trains Continued). For Example 1, we
have ground terms A, B,C
A
,C
B
,C
C
,T
1
, and T
2
, rep-
resenting Alice, Bob, three cities, and two trains.
Furthermore, we have atoms on(x, y) indicating that
person y is on train x, and at(y, z) indicating that
train y is at city z. Finally, we have observa-
tion atoms of the kind Obs
G
(at(y,z)), indicating
that the agents in G observe that train y is at sta-
tion z. Thus, a possible world can for example
be w
1
= {at(T
1
,C
A
),on(T
1
,A), Obs
A
(at(T
1
,A))}, in-
dicating that train T
1
is at city C
A
and A has boarded
that train.
We define satisfaction of a ground formula F by a
world w, in the usual way (Lloyd, 1987):
Definition 7 (Satisfaction of Ground Formulae). Let
F,F
′
,F
′′
be ground formulae and w a world. Then, F
is satisfied by w (denoted w |= F)
- If F = a for some ground atom a, then a ∈ w.
- If F = ¬F
′
, then w 6|= F
′
.
- If F = F
′
∧ F
′′
, then w |= F
′
and w |= F
′′
.
- If F = F
′
∨ F
′′
, then w |= F
′
or w |= F
′′
.
4.2 Threads
We use the definition of threads from (Shakarian
et al., 2011) (equivalent to the concept of runs in (Fa-
gin et al., 1995)):
Definition 8. A thread is a mapping Th : τ → W
Thus, a thread is a sequence of worlds and Th(i)
identifies the actual world at time i according to thread
Th. The set of all possible threads is denoted by T .
Again, we refrain from using T as the set of all pos-
sible sequences constructible from τ and W, and in-
stead assume that any meaningful problem specifica-
tion gives information about possible temporal evolu-
tions of the system. For notational convenience, we
assume that there is an additional prior world Th(0)
for every thread.
Example 3 (Trains Continued). The description from
Example 1 yields the set of possible threads T de-
picted in Figure 1.
4.3 Kripke Structures
With the definition of threads, we can use a slightly
modified version of Kripke structures (Kripke, 1963).
As usual, we define a Kripke structure as a tuple
hW,K
1
,..., K
n
i, with the set of possible worlds W and
binary relations K
i
on W for every agent i ∈ A. Intu-
itively, (w, w
′
) ∈ K
i
specifies that in world w, agent i
considers w
′
as a possible world.
We initialize the Kripke structure such that the set
of possible worlds contains exactly the worlds that oc-
cur at time t = 1 in some thread Th
′
:
∀Th ∈ T : K
i
(Th(0)) :=
[
Th
′
∈T
{Th
′
(1)}, i = 1,..., n
With the evolution of time, each agent can eliminate
the worlds that do not comply with its respective ob-
servations. Through the elimination of worlds, an
agent will also reduce the set of threads it considers
possible. We assume that agents have perfect recall
and therefore will not consider some thread possible
again if it was considered impossible at one point.
Thus, K
i
is updated w.r.t. the agent’s respective obser-
vations, such that it considers all threads possible that
both comply with its current observations and were
considered possible at the previous time point:
AProbabilisticDoxasticTemporalLogicforReasoningaboutBeliefsinMulti-agentSystems
279

t
1 2
3
4
5 6
7
8
9
10
Th
1
on(T
1
, A)
at(T
1
,C
A
)
on(T
1
, A)
at(T
1
,C
C
)
on(T
2
, A)
at(T
2
,C
C
)
on(T
2
, A)
at(T
2
,C
B
)
2
Th
2
on(T
1
, A)
at(T
1
,C
A
)
on(T
1
, A)
at(T
1
,C
C
)
on(T
2
, A)
at(T
2
,C
C
)
on(T
2
, A)
at(T
2
,C
B
)
Obs
A
(¬at(T 2,C
B
))
2
Th
3
on(T
1
, A)
at(T
1
,C
A
)
on(T
1
, A)
at(T
1
,C
C
)
on(T
2
, A)
at(T
2
,C
C
)
on(T
2
, A)
at(T
2
,C
B
)
Obs
AB
(¬at(T
2
,C
B
))
1
Th
4
on(T
1
, A)
at(T
1
,C
A
)
on(T
1
, A)
at(T
1
,C
C
)
on(T
2
, A)
at(T
2
,C
C
)
on(T
2
, A)
at(T
2
,C
B
)
Obs
A
(¬at(T
1
,C
C
))
1
Th
5
on(T
1
, A)
at(T
1
,C
A
)
on(T
1
, A)
at(T
1
,C
C
)
on(T
2
, A)
at(T
2
,C
C
)
on(T
2
, A)
at(T
2
,C
B
)
Obs
AB
(¬at(T
1
,C
C
))
1 2
Th
6
on(T
1
, A)
at(T
1
,C
A
)
on(T
1
, A)
at(T
1
,C
C
)
on(T
2
, A)
at(T
2
,C
C
)
on(T
2
, A)
at(T
2
,C
B
)
Obs
A
(¬at(T
1
,C
C
))
Obs
A
(¬at(T
1
,C
B
))
1 2
Th
7
on(T
1
, A)
at(T
1
,C
A
)
on(T
1
, A)
at(T
1
,C
C
)
on(T
2
, A)
at(T
2
,C
C
)
on(T
2
, A)
at(T
2
,C
B
)
Obs
AB
(¬at(T
1
,C
C
))
Obs
A
(¬at(T
1
,C
B
))
1 2
Th
8
on(T
1
, A)
at(T
1
,C
A
)
on(T
1
, A)
at(T
1
,C
C
)
on(T
2
, A)
at(T
2
,C
C
)
on(T
2
, A)
at(T
2
,C
B
)
Obs
AB
(¬at(T
1
,C
B
))
Obs
A
(¬at(T
1
,C
C
))
1 2
Th
9
on(T
1
, A)
at(T
1
,C
A
)
on(T
1
, A)
at(T
1
,C
C
)
on(T
2
, A)
at(T
2
,C
C
)
on(T
2
, A)
at(T
2
,C
B
)
Obs
AB
(¬at(T
1
,C
B
))
Obs
AB
(¬at(T
1
,C
C
))
Figure 1: Visualization of the possible threads Th
k
from Example 1: at(T
i
,C
j
) denotes that train T
i
is currently at city C
j
,
on(T
i
,A) that Alice is currently on train T
i
, and Obs
AG
(¬at(T
i
,C
j
)) denotes a call from Alice to inform Bob that train T
i
is
currently not at cityC
j
. For the sake of simplicity, facts irrelevant to the analysis (such as on(T
i
,A) for time points 2 and 5) are
omitted from the presentation. Note that if a train is running late (the respective threads are marked with according circles),
there are always two possible threads: one where only A observes this and one where both share the observation. For an easier
distinction, we have marked the according group of an observation with boldface indices.
K
i
(Th(t)) := {Th
′
(t) : (Th
′
(t − 1) ∈ K
i
(Th(t − 1))∧
{Obs
G
(l)∈Th(t): i∈ G}={Obs
G
(l)∈Th
′
(t): i∈G})}
(1)
The next lemmata describe key properties of K
i
following immediately from the above definitions.
Lemma 1. K
i
defines an equivalence relation over
the possible worlds K
i
(Th(t)) at time t.
Lemma 2. The set of threads Th
′
considered possible
w.r.t. K
i
is narrowing to a smaller and smaller subset
over time, i.e., {Th
′
: Th
′
(t) ∈ K
i
(Th(t))} ⊆ {Th
′
:
Th
′
(t − 1) ∈ K
i
(Th(t − 1))} for all Th ∈ T and t ∈ τ.
Example 4 (Trains Continued). From Figure 1, we
obtain that at time 1, the only possible world is
{{at(T
1
,C
A
),on(T
1
,A)}}, which is contained in all
possible threads. Thus, K
i
(Th
j
(1)) contains exactly
this world for all agents i and threads j. Conse-
quently, both agents consider all threads as possible
at time 1.
Now, assume that time evolves for two steps
and the actual thread is Th
4
(i.e., train T
1
is
running late, but A does not inform B about
this). Both agents will update their possibility
relations accordingly, yielding K
1
(Th
4
(3)) =
{{Obs
A
(¬at(T
1
,C
C
))}} and K
2
(Th
4
(3)) =
{{at(T
1
,C
C
),on(T
1
,A)}, {Obs
A
(¬at(T
1
,C
C
))}},
i.e., A knows that T1 is not on time, while B is
unaware of this.
4.4 Subjective Posterior Temporal
Probabilistic Interpretations
Each agent has probabilistic beliefs about the ex-
pected evolution of the world over time. This is ex-
pressed through subjective temporal probabilistic in-
terpretations:
Definition 9 (Subjective Posterior Probabilistic Tem-
poral Interpretation). Given a set of possible threads
T , some thread Th
′
∈ T , a time point t and an
agent i, I
Th
′
i,t
: T → [0,1] specifies the subjective
posterior probabilistic temporal interpretation from
agent i’s point of view at time t in thread Th
′
, i.e.,
a probability distribution over all possible threads:
∑
Th∈T
I
Th
′
i,t
(Th) = 1. We call Th
′
the point of view
(pov) thread of interpretation I
Th
′
i,t
.
The prior probabilities of each agent for all threads
are then given by I
Th
′
i,0
(Th). Since all threads are in-
distinguishable a priori, there is only a single prior
distribution for each agent (i.e., ∀Th,Th
′
,Th
′′
∈ T :
I
Th
′
i,0
(Th) = I
Th
′′
i,0
(Th)). Furthermore, in order to be
able to reason about nested beliefs (as discussed be-
low), we assume that the prior probability assess-
ments of all agents are commonly known (i.e., all
agents know how all other agents assess the prior
probabilities of each thread). This in turn requires
that all agents have exactly the same prior probabil-
ity assessment over all possible threads: if two agents
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have different, but commonly known prior probabil-
ity assessments, we essentially have an instance of
Aumann’s well-known problem of “agreeing to dis-
agree” (Aumann, 1976). Intuitively, if differing priors
are commonly known, it is common knowledge that
(at least) one of the agents is at fault and should revise
its probability assessments. As a result, we have only
one prior probability distribution which is the same
from all viewpoints, denoted by I . Note that I di-
rectly corresponds to the concept of temporal proba-
bilistic interpretations in (Shakarian et al., 2011).
Example 5 (Trains Continued). A meaningful inter-
pretation is
I =

0.7 0.02 0.09 0.02 0.09 0.01 0.02 0.02 0.03

,
which assigns the highest probability to Th
1
(no train
running late), lower probabilities to the threads where
one train is running late and A informs B (Th
3
and
Th
5
), even lower probabilities to the events that either
both trains are running late and A informs B (Th
7
,
Th
8
, and Th
9
) or that one train is running late and A
does not inform B (Th
2
and Th
4
), and lowest proba-
bility to the thread where both trains are running late
and A does not inform B (Th
6
).
Even though we only have a single prior proba-
bility distribution over the set of possible threads, it
is still necessary to distinguish the viewpoints of dif-
ferent agents in different threads, as the definition of
interpretation updates shows:
Definition 10 (Interpretation Update). Let i be an
agent, t a time point, and Th
′
a pov thread. Then,
if the system is actually in thread Th
′
at time t, agent
i’s probabilistic interpretation over the set of possible
threads is given by the update rule:
I
Th
′
i,t
(Th) =
(
1
α
Th
′
it
· I
Th
′
i,t− 1
(Th) if Th(t) ∈ K
i
(Th
′
(t))
0 if Th(t) 6∈ K
i
(Th
′
(t))
(2)
with
1
α
Th
′
it
being a normalization factor:
α
Th
′
it
=
∑
Th∈T ,Th(t)∈K
i
(Th
′
(t))
I
Th
′
i,t− 1
(Th) (3)
The invocation of K
i
in the update rule yields ob-
vious ramifications about the evolution of interpreta-
tions, as stated in the following lemma:
Lemma 3. The subjective temporal probabilistic in-
terpretation I
Th
′
i,t
of an agent i assigns nonzero prob-
abilities exactly to the set of threads that i still
considers possible at time t, i.e., I
Th
′
i,t
(Th) > 0 ⇔
K
i
(Th(t),Th
′
(t))
Essentially, the update rule assigns all impossible
threads a probability of zero and scales the probabili-
ties of the remainingthreads such that they are propor-
tional to the probabilities of the previous time point.
Example 6 (Trains Continued). Applying the update
rule from (2) to the situation described in Example 4,
with I as given in Example 5, yields the updated in-
terpretation for A:
I
Th
4
A,3
=

0 0 0 0.4 0 0.2 0 0.4 0

(4)
i.e., A considers exactly those threads possible, where
the train is running late and she does not inform B
(threads Th
4
, Th
6
, and Th
8
). Due to the lack of any
new information, B can only eliminate the situations
where A does inform him about being late, and thus
B’s interpretation is updated to:
I
Th
4
B,3
≈

0.82 0.02 0.10 0.02 0 0.02 0 0.02 0

. (5)
4.5 Frequency Functions
To represent temporal relationships within threads,
we utilize the concept of frequency functions as in-
troduced in (Shakarian et al., 2011). Frequency func-
tions enable us to represent temporal relations be-
tween the occurrence of specific events and are de-
fined axiomatically as follows:
Definition 11 (Frequency Functions). (Shakarian
et al., 2011) Let Th be a thread, F and G be ground
formulae, and ∆t > 0 be an integer. A frequency func-
tion fr maps quadruples of the form (Th, F,G,∆t) to
[0, 1] such that the following axioms hold:
(FF1) If G is a tautology, then fr(Th,F,G,∆t) = 1.
(FF2) If F is a tautology and G is a contradiction, then
fr(Th, F,G,∆t) = 0.
(FF3) If F is a contradiction, fr(Th,F,G, ∆t) = 1.
(FF4) If G is not a tautology, and either F or ¬G
is not a tautology, and F is not a contradiction,
then there exist threads Th
1
, Th
2
∈ T such that
fr(Th
1
,F,G,∆t) = 0 and fr(Th
2
,F,G,∆t) = 1.
To illustrate the concept of frequency functions,
we present the point and existential frequency func-
tions from (Shakarian et al., 2011):
The point frequency function pfr expresses how
frequently some event F is followed by another event
G in exactly ∆t time units:
pfr(Th, F,G, ∆t)=
|{t : Th(t) |= F ∧ Th(t + ∆t) |= G}|
|{t : (t ≤ t
max
− ∆t) ∧ Th(t) |= F}|
(6)
The existential frequency function efr expresses
how frequently some event F is followed by another
event G within the next ∆t time units:
efr(Th, F,G, ∆t) = (7)
efn(Th,F,G, ∆, 0,t
max
)
fn(Th, F,∆t) + efn(Th, F,G,∆t,t
max
− ∆t,t
max
)
,
fn(Th,F,∆t) := |{t : (t ≤ t
max
− ∆t) ∧ Th(t) |= F}|,
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281

Th(1) Th(2) Th(3) Th(4) Th(5) Th(6) Th(7) Th(8)
F
G
F
G G
F
G
F
Figure 2: Example thread Th with τ = {1, ...,8}. This figure
shows each world that satisfies formula F or formula G.
efn(Th,F,G,∆t,t
1
,t
2
) = |{t : (t
1
<t ≤t
2
) ∧ Th(t) |= F
∧ ∃t
′
∈ [t + 1, min(t
2
,t + ∆t)] (Th(t
′
) |= G)}|
To illustrate the concept of frequency functions,
we adapt an example from (Shakarian et al., 2011):
consider the thread Th depicted in Figure 2. The
thread evolves over 8 time steps and in each of the
respective worlds, either F or G is satisfied. Suppose
that we want to determine how often F is followed by
G exactly after two time steps. This can be expressed
through a point frequency function: pfr(Th, F,G, 2) =
1
3
. If instead we want to know howoften F is followed
by G within the next two time steps, we can use an ex-
istential frequency function: efr(Th, F,G,2) =
3
3
= 1.
Note that the frequency functions are defined such
that neither of them considers the world at time 8
in the denominator, even though Th(8) |= F. This
is because there cannot be any world beyond time 8
such that G is satisfied and consequently, considering
this world would mitigate any result of the frequency
functions.
4.6 Semantics of the Belief Operator
Now, with the definitions of subjectiveposterior prob-
abilistic temporal interpretations and the introduction
of frequency functions, we can build upon the defini-
tions from (Shakarian et al., 2011) for the satisfiability
of interpretations to provide formal semantics for the
belief operators defined in Section 3:
Definition 12 (Belief in Ground Formulae). Let I
Th
′
i,t
′
be agent i’s interpretation at time t
′
in pov thread Th
′
.
Then, it holds w.r.t. this interpretation that agent i be-
lieves that some formula F holds at time t with a prob-
ability in the range [ℓ,u] (denoted by I
Th
′
i,t
′
|= B
ℓ,u
i,t
′
(F
t
))
iff
ℓ ≤
∑
Th∈T ,Th(t)|=F
I
Th
′
i,t
′
(Th) ≤ u. (8)
Definition 13 (Belief in Rules). Let F and G be
ground formulae, fr be a frequency function, and
I
Th
′
i,t
′
(Th) be agent i’s interpretation at time t
′
in pov
thread Th
′
. Then, it holds w.r.t. this interpretation that
agent i believes that some rule r
fr
∆t
(F,G) holds with
a probability in the range [ℓ, u] (denoted by I
Th
′
i,t
′
|=
B
ℓ,u
i,t
′
(r
fr
∆t
(F,G))) iff
ℓ ≤
∑
Th∈T
I
Th
′
i,t
′
(Th) · fr(Th, F,G, ∆t) ≤ u. (9)
Definition 14 (Nested Beliefs). Let i, j be agents,
B
ℓ
k
,u
k
k,t
(·) be some belief formula, and I
Th
′
i,t
′
(Th) be
agent i’s interpretation at time t
′
in pov thread Th
′
.
Then, it holds w.r.t. this interpretation that agent i be-
lieves at time t
′
that with a probability in the range
[ℓ, u] agent j has some belief B
ℓ
k
,u
k
k,t
(·) at time t (de-
noted by I
Th
′
i,t
′
|= B
ℓ,u
i,t
′
(B
ℓ
k
,u
k
k,t
(·))) iff
ℓ ≤
∑
Th∈T , I
Th
j,t
|=B
ℓ
k
,u
k
k,t
I
Th
′
i,t
′
(Th) ≤ u. (10)
Example 7 (Trains Continued). We can use a point
frequency function to express beliefs about the punc-
tuality of trains. Assume that both A and B judge the
probability of a train running late (i.e., arriving after
3 instead of 2 time units, expressed through the tem-
poral rule r
pfr
3
) as being at most 0.4. This yields the
following belief formulae
B
0,0.4
i,0
(r
pfr
3
(at(T
1
,C
A
),at(T
1
,C
C
)))
B
0,0.4
i,0
(r
pfr
3
(at(T
2
,C
C
),at(T
2
,C
B
)))
, i ∈ {A,B}.
One can easily verify that these formulae are satisfied
by the interpretation given in Example 5.
From the above definitions, we can use the belief
about some fact (·) to quantify the belief about the
negation of this fact ¬(·):
Lemma 4. I
Th
′
i,t
|= B
ℓ,u
i,t
′
(¬(·)) iff I
Th
′
i,t
|= B
ℓ
′
,u
′
i,t
(·) with
ℓ
′
= 1− u and u
′
= 1− ℓ.
5 EVOLUTION OVER TIME
In order to completely specify a problem in PDT
Logic, we introduce the concept of doxastic systems.
Definition 15 (Doxastic System). Let A be a set of
agents, T be a set of threads, A
|A|×|T |
0
be a matrix
of prior probability distributions across T for every
agent in A, and F be a set of frequency functions.
Then, we call the quadruple D = hA,T ,F ,A
|A|×|T |
0
i
a doxastic system.
Note that several of the parameters discussed be-
fore are not explicitly specified in a doxastic system:
the set of possible worlds W, the set of ground atoms
B, the set of observation atoms L
obs
, nor the set of
time points τ are explicitly specified. However, all
relevant information regarding these parameters is al-
ready contained in the specification of T .
To identify specific situations in a doxastic system
after time has passed and some observations occurred,
we furthermore define pointed doxastic systems:
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Definition 16 (Pointed Doxastic System, pds). Let
D = hA,T ,F ,A
|A|×|T |
0
i be a doxastic system and H
be a set of timestamped observation atoms such that
all observation atoms from H occur in at least one of
the worlds (implicitly) defined in T . Then we call the
pair hD,Hi a pointed doxastic system.
Intuitively, the set of timed observations specified
in a pds points to a certain situation in a doxastic sys-
tem. One could view
ˆ
t(H) = max{t : ∃[Obs
G
(l) : t] ∈
H} as the present time in a pds: the most recent obser-
vation occurred at
ˆ
t(H), all observations that actually
occurred in the past (t <
ˆ
t) are specified in H (and
thus deterministic in retrospective), and no further in-
formation about future observations t >
ˆ
t is given. In
this sense, H specifies a certain history up to
ˆ
t(H) in
a doxastic system and points to the last event of this
history.
Example 8 (Trains Continued). A doxastic system for
the train example can be specified as
D = h{A, B}, {Th
1
,..., Th
9
}, {pfr,efr}, A
0
i,
A
0
=

0.7 0.02 0.09 0.02 0.09 0.01 0.02 0.02 0.03
0.7 0.02 0.09 0.02 0.09 0.01 0.02 0.02 0.03

To identify the situation described in Example 4 (T
1
is running late), we can specify the following pointed
doxastic system: hD, [Obs
A
(¬at(T
1
,C
C
) : 3]i
5.1 Evolution of Probabilistic
Interpretations
In accordance with the prior probability matrix A
0
from Definition 15, we define an interpretation matrix
A
Th
′
t
to store the interpretations of all agents 1, ..., n
across all threads Th
1
,..., Th
m
given that the doxastic
system is in pov thread Th
′
at time t:
A
Th
′
t
=



I
Th
′
1,t
(Th
1
) . .. I
Th
′
1,t
(Th
m
)
.
.
.
.
.
.
.
.
.
I
Th
′
n,t
(Th
1
) . .. I
Th
′
n,t
(Th
m
)



(11)
With the definition of K
i
from (1), the update rule
from (2), and using the prior probability matrix A
0
from Definition 15, we can provide an update matrix
U
Th
′
t
to calculate the interpretation matrix for any pov
thread Th
′
at any time point t (◦ denotes the element-
wise multiplication of matrices):
A
Th
′
t
= A
Th
′
t−1
◦U
Th
′
t
, with (12)
(u
Th
′
t
)
ij
=
(
0 if Th
j
(t) 6∈ K
i
(Th
′
(t))
1
α
Th
′
it
if Th
j
(t) ∈ K
i
(Th
′
(t))
(13)
and α
Th
′
it
a normalization factor as defined in (3).
The timed observations specified in the history H
of a pds hD, Hi induce an updated set of reachability
relations K
i
(Th(t)) for every thread Th that complies
with the given observations (for threads Th
⊥
that do
not comply with the givenobservationsK
i
(Th
⊥
(t)) =
/
0). These updated reachability relations in turn yield
the updated interpretations in A
Th
′
t
. The complete
state of interpretations at any time point for every pos-
sible pov thread Th
1
,..., Th
m
can then be specified as
a block matrix, which we call the belief state (bs) of a
pds at time t:
bs(hD,Hi,t) =

A
Th
1
t
,..., A
Th
m
t

(14)
The belief state can be viewed as a specifica-
tion of conditional probabilities: the kth entry of
bs(hD,Hi,t) specifies the interpretations of all agents
across all threads at time t given that the system is in
pov thread Th
k
.
5.2 Evolution of Beliefs
In order to analyze the temporal evolution of beliefs,
we use the update rule from (12) to update belief
states. Since different possible observations yield dif-
ferent branches in the evolution of beliefs, we have
to update every thread in the belief state individually,
using the update matrices U
Th
t
as defined in (13):
bs(hD,Hi,t) = bs(hD, Hi,t − 1) ◦ (U
Th
1
t
,...,U
Th
m
t
)
(15)
Furthermore, to analyze satisfiability and valid-
ity of arbitrary finite belief expressions B
ℓ,u
i,t
′
(·) w.r.t.
a given pds hD,Hi, we define an auxiliary belief vec-
tor
~
b(·) for different beliefs B
ℓ,u
i,t
′
(·) as follows:
B
ℓ,u
i,t
′
(F
t
) : (
~
b(F
t
))
j
=
(
1 if Th
j
(t) |= F
0 if Th
j
(t) 6|= F
(16)
B
ℓ,u
i,t
′
(r
fr
∆t
(F,G)) : (
~
b(r
fr
∆t
(F,G)))
j
=fr(Th
j
,F,G,∆t)
B
ℓ,u
i,t
′
(B
ℓ
k
,u
k
k,t
(·)) : (
~
b(B
ℓ
k
,u
k
k,t
(·)))
j
=
(
1 if I
Th
j
k,t
|= B
ℓ
k
,u
k
k,t
(·)
0 if I
Th
j
k,t
6|= B
ℓ
k
,u
k
k,t
(·)
Using (11) and (16), we can determine a matrix P
t
′
with the probabilities p
Th
k
i,t
′
(·) that each agent i assigns
at time t
′
to some event (·), for all possible pov threads
Th
1
,..., Th
m
:
P
t
′
(·) =

A
Th
1
t
·
~
b(·),...,A
Th
m
t
·
~
b(·)

(17)
The rows in P
t
′
can be seen as conditional proba-
bilities: agent i believes at time t
′
that a fact (·) is true
with probability p
Th
k
i,t
′
given that the pov thread is Th
k
.
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283

Using Definitions 12 - 14 and (17), we can provide
a definition for the satisfiability and validityof beliefs:
Definition 17 (Validity and Satisfiability of Beliefs).
Let F
B
be a belief formula as defined in Definition 5.
F
B
is satisfiable (valid) iff
• For F
B
≡ B
ℓ,u
i,t
′
(·): ℓ ≤ p
Th
k
i,t
′
(·) and u ≥ p
Th
k
i,t
′
(·) for
at least one (all) p
Th
k
i,t
′
in P
t
′
.
• For F
B
≡ ¬B
ℓ,u
i,t
′
(·): ℓ > p
Th
k
i,t
′
(·) or u < p
Th
k
i,t
′
(·) for
at least one (all) p
Th
k
i,t
′
in P
t
′
.
• For F
B
≡ F
′
B
∧ F
′′
B
: for at least one (all) p
Th
k
i,t
′
in P
t
′
both F
′
B
and F
′′
B
are satisfied.
• For F
B
≡ F
′
B
∨ F
′′
B
: F
′
B
is satisfiable (valid) or F
′′
B
is
satisfiable (valid).
To illustrate the evolution of beliefs, we finish the
example with an analysis of expected arrival times.
Example 9 (Trains Continued). From D, as specified
in Example 8, we can infer that Bob (and of course
Alice, too) can safely assume at time 1 that Alice will
arrive at time 8 at the latest (i.e., the actual thread
is one of Th
1
,..., Th
5
) with a probability in the range
[0.9, 1] because from Definition 17 we obtain that the
following belief is valid w.r.t. D for t = 1:
F
Bob,t
≡ B
0.9,1
B,t
(r
efr
8
(on(T
1
,A), (at(T
2
,C
B
)∧on(T
2
,A))).
Now, consider the previously described situation,
where T
1
is running late and A does not inform B
about it. This leads to the updated interpretations
given in (4) and (5). These updates lead to a signif-
icant divergence in the belief of the expected arrival
time: Alice’s belief exhibits a drastically reduced cer-
tainty and changes to
B
0.4,1
A,3
(r
efr
8
(on(T
1
,A), (at(T
2
,C
B
) ∧ on(T
2
,A))),
while Bob’s previous belief remains valid.
Even though Alice’s beliefs have changed signif-
icantly, she is aware that Bob maintains beliefs con-
flicting with her own, as is shown by the following
valid expression of nested beliefs: B
0.6,1
A,3
(F
Bob,3
)
Finally, consider the pointed doxastic system
hD,[Obs
AB
(¬at(T
1
,C
C
) : 3]i, i.e., the same situation
as before with the only difference that Alice now
shares her observation of the delayed train with Bob.
It immediately follows that Bob updates his beliefs
in the same way as Alice, which in turn yields an
update in Alice’s beliefs about Bob’s beliefs so that
now the following expression is valid (because 0.6 is
not a valid lower bound any longer): ¬B
0.6,1
A,3
(F
Bob,3
).
This example shows how Alice can reason about the
influence of her own actions on Bob’s belief state
and therefore she can decide on actions that improve
Bob’s utility (as he does not have to wait in vain).
6 CONCLUSION
In this paper, by extending APT Logic to dynamic
scenarios with multiple agents, we have developed a
general framework to represent and reason about the
belief change in multi-agent systems. Next to lift-
ing the single-agent case of APT Logic to multiple
agents, we have also provided a suitable semantics
to the temporal evolution of beliefs. The resulting
framework extends previous work on dynamic multi-
agent epistemic logics by enabling the quantification
of agents’ beliefs through probability intervals. An
explicit notion of temporal relationships is provided
through temporal rules building on the concept of fre-
quency functions.
PDT Logic as introduced in this work provides the
foundation for future work. While a basic decision
procedure can be obtained through a direct applica-
tion of the given semantics, we will continue to in-
vestigate optimized algorithms, using both exact and
approximate methods. With a focus on inferring con-
sistent possible threads automatically, this will give
rise to a thorough complexity analysis of the decision
problems. With efficient algorithms, we can apply
PDT Logic to realistic problems.
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