=Paper=
{{Paper
|id=Vol-1212/paper3
|storemode=property
|title=Probabilistic Argumentation with Epistemic Extensions
|pdfUrl=https://ceur-ws.org/Vol-1212/DARe-14-paper-3.pdf
|volume=Vol-1212
|dblpUrl=https://dblp.org/rec/conf/ecai/HunterT14a
}}
==Probabilistic Argumentation with Epistemic Extensions==
https://ceur-ws.org/Vol-1212/DARe-14-paper-3.pdf
Probabilistic Argumentation with Epistemic Extensions
Anthony Hunter1 and Matthias Thimm2
1
Department of Computer Science, University College London, United Kingdom
2
Institute for Web Science and Technologies (WeST), University of Koblenz-Landau, Germany
Abstract. Abstract argumentation offers an appealing way of representing and
evaluating arguments and counterarguments. This approach can be enhanced by
a probability assignment to each argument. There are various interpretations that
can be ascribed to this assignment. In this paper, we regard the assignment as
denoting the belief that an agent has that an argument is justifiable, i. e., that both
the premises of the argument and the derivation of the claim of the argument
from its premises are valid. This leads to the notion of an epistemic extension
which is the subset of the arguments in the graph that are believed to some degree
(which we defined as the arguments that have a probability assignment greater
than 0.5). We consider various constraints on the probability assignment. Some
constraints correspond to standard notions of extensions, such as grounded or
stable extensions, and some constraints give us new kinds of extensions.
1 Introduction
Abstract argumentation, as proposed by Dung [1], provides a simple and appealing
representation in the form of a directed graph. Each node denotes an argument and
each arc denotes one argument attacking another. Abstract argumentation also provides
a set of options for determining which arguments can be accepted together (i. e. an
extension), and which arguments can be rejected. Recently there has been interest in
augmenting abstract argumentation with a probabilistic assignment to each argument
[2–5]. Once we introduce a probability assignment to each argument, we have extra
information about the argumentation, and this means we can refine the evaluation of an
argument graph. In most of these proposals (i. e. [2–4]), the emphasis is on what should
the structure of the graph be. So the probability of the argument denotes the degree to
which the argument should be in the graph.
In this paper, we take a different approach. We regard the assignment as denoting
the belief that an agent has that an argument is justifiable, i. e., that both the premises
of the argument and the derivation of the claim of the argument from its premises are
valid. So for a probability function P , and an argument A, P (A) > 0.5 denotes that
the argument is believed (to the degree given by P (A)), P (A) < 0.5 denotes that the
argument is disbelieved (to the degree given by P (A)), and P (A) = 0.5 denotes that
the argument is neither believed nor disbelieved. This approach leads to the notion of an
epistemic extension: This is the subset of the arguments in the graph that are believed
to some degree (i. e. the arguments such that P (A) > 0.5). Since this is a very general
idea, our aim in this paper is to consider various properties (i. e. constraints) that hold for
classes of probability functions, and for the resulting epistemic extensions. We structure
our presentation on two views as follows:
Standard view on using probability of arguments. In this view, we provide properties
for the probability function that ensure that the epistemic extensions coincide with
Dung’s definitions for extensions. Key properties include coherence (if A attacks
B, then P (A) ≤ 1 − P (B)) and foundation (if A has no attackers, then P (A) = 1).
The advantage of using a probability function instead of Dung’s definitions is that
we can also specify the degree to which each argument is believed.
Non-standard view on using probability of arguments. In this view, we consider al-
ternative properties for the probability function. This means that the resulting epis-
temic extensions may not coincide with Dung’s definitions for extensions.
The framework that we present in this paper is appealing theoretically as it provides
further insights into semantics for abstract argumentation, and it offers a finer-grained
representation of uncertainty in arguments. Perhaps more importantly, our framework
for probability functions is appealing practically because we can now model how au-
diences judge argumentation. Consider for example how a member of the audience of
a debate hears arguments and counterarguments, but is unable (or does not want) to
express arguments. Here it is natural to consider how that member of the audience con-
siders which arguments she believes, thereby constructing an epistemic extension. More
generally, if we want to make computational models of argument where we can capture
persuasion, we need to take account of the belief that participants or audiences have
in the individual arguments that are posited. We see the non-standard view as being
particularly important in addressing this need.
This paper builds on previous work [5, 6] but extends it in several directions. In
particular, the contributions of this paper are as follows:
1. We investigate the notion of epistemic extensions and show their usefulness with
respect to classical extensions (Section 3).
2. We adopt and significantly extend properties for standard epistemic extensions
from [5, 6] and show that these probabilistic concepts coincide with their corre-
sponding concepts from abstract argumentation (Section 4).
3. We introduce non-standard epistemic extensions and a corresponding set of prop-
erties as a means to extend the standard view and provide a complete picture of the
relationships between our different probabilistic properties (Section 5).
4. We apply the notion of epistemic extensions to the problem of completing partial
probability assignments and show the feasibility of this approach (Section 6).
Proofs of technical results can be found in an extended technical report [7].
2 Preliminaries
An abstract argumentation framework AF is a tuple AF = (Arg, →) where Arg is a
set of arguments and → is a relation →⊆ Arg × Arg. For two arguments A, B ∈ Arg
the relation A → B means that argument A attacks argument B. For A ∈ Arg define
AttAF (A) = {B | B → A}.
Semantics are given to abstract argumentation frameworks by means of extensions
[1] or labellings [8]. In this work, we use the latter. A labelling L is a function L :
Arg → {in, out, undec} that assigns to each argument A ∈ Arg either the value in,
meaning that the argument is accepted, out, meaning that the argument is not accepted,
or undec, meaning that the status of the argument is undecided. Let in(L) = {A |
L(A) = in} and out(L) resp. undec(L) be defined analogously. The set in(L) for
a labelling L is also called extension. A labelling L is called conflict-free if for no
A, B ∈ in(L) we have that A → B.
Arguably, the most important property of a semantics is its admissibility. A labelling
L is called admissible if and only if for all arguments A ∈ Arg
1. if L(A) = out then there is B ∈ Arg with L(B) = in and B → A, and
2. if L(A) = in then L(B) = out for all B ∈ Arg with B → A,
and it is called complete if, additionally, it satisfies
3. if L(A) = undec then there is no B ∈ Arg with B → A and L(B) = in and there
is a B 0 ∈ Arg with B 0 → A and L(B 0 ) 6= out.
The intuition behind admissibility is that an argument can only be accepted if there are
no attackers that are accepted and if an argument is not accepted then there has to be
some reasonable grounds. The idea behind the completeness property is that the status
of an argument is only undec if it cannot be classified as in or out. Different types of
classical semantics can be phrased by imposing further constraints. Let AF = (Arg, →)
be an abstract argumentation framework and L : Arg → {in, out, undec} a complete
labelling. Then
– L is grounded if and only if in(L) is minimal,
– L is preferred if and only if in(L) is maximal,
– L is stable if and only if undec(L) = ∅, and
– L is semi-stable if and only if undec(L) is minimal.
All statements on minimality/maximality are meant to be with respect to set inclusion.
Note that a grounded labelling is uniquely determined and always exists [1].
3 Epistemic extensions
We now go beyond classical three-valued semantics of abstract argumentation and turn
to probabilistic interpretations of the status of arguments. Let 2X denote the power set
PX . A probability function P Arg
of a set on some finite set X is a function P : 2X → [0, 1]
with X⊆X P (X) = 1. Let P : 2 → [0, 1] be a probability function on Arg. We
P
abbreviate P (A) = A∈E⊆Arg P (E). This means that the probability of an argument
is the sum of the probabilities of all sets of arguments that contain that argument. The
following definition is a generalization of the notion of epistemic extensions given in
[6]. For an argument A, it is labelled in when it is believed to some degree (which we
identify as P (A) > 0.5), it is labelled out when it is disbelieved to some degree (which
we identify as P (A) < 0.5), and it is labelled undec when it is neither believed nor
disbelieved (which we identify as P (A) = 0.5). More specifically, let AF = (Arg, →)
be an abstract argumentation framework and P : 2Arg → [0, 1] a probability function on
Arg. The labelling LP : Arg → {in, out, undec} defined via the following constraints
is called the epistemic labelling of P :
– LP (A) = in iff P (A) > 0.5
– LP (A) = out iff P (A) < 0.5
– LP (A) = undec iff P (A) = 0.5
The epistemic extension of P is the set of arguments that are labelled in by the epistemic
labelling, i. e. X is an epistemic extension iff X = in(LP ). Furthermore, we say that
a labelling L and a probability function P are congruent, denoted by L ∼ P , if for
all A ∈ Arg we have L(A) = in ⇔ P (A) = 1, L(A) = out ⇔ P (A) = 0, and
L(A) = undec ⇔ P (A) = 0.5. Note, that if L ∼ P then L = LP , i. e., if a labelling L
and a probability function P are congruent then L is also the epistemic labelling of P .
An epistemic labelling can be used to give either a standard semantics (as we will
investigate in Section 4) or a non-standard semantics (see Section 5).
To further illustrate the epistemic extensions, consider the graph given in Figure 1.
Here, we may believe that, say, A is valid and that B and C are not valid. In which case,
with this extra epistemic information about the arguments, we can resolve the conflict
and so take the set {A} as the “epistemic” extension. In contrast, there is only one
admissible set which is the empty set. So by having this extra epistemic information, we
get a more informed extension (in the sense that it has harnessed the extra information
in a sensible way).
B = Bob will go to the party A = Ann will go to the C = Chris will go to the
and this means that Chris party and this means that party and this means that
will not go to the party Bob will not go to the party Ann will not go to the party
Fig. 1. Example of three arguments in a simple cycle.
In general, we want epistemic extensions to allow us to better model the audience of
argumentation. Consider, for example, when a member of the audience of a TV debate
listens to the debate at home, she can produce the abstract argument graph based on
the arguments and counterarguments exchanged. Then she can identify a probability
function to represent the belief she has in each of the arguments. So she may disbelieve
some of the arguments based on what she knows about the topic. Furthermore, she may
disbelieve some of the arguments that are unattacked. As an extreme, she is at liberty
of completely disbelieving all of the arguments (so to assign probability 0 to all of
them). If we want to model audiences, where the audience either does not want to or is
unable to add counterarguments to an argument graph being constructed in some form
of argumentation, we need to take the beliefs of the audience into account, and we need
to consider which arguments they believe or disbelieve.
4 Standard epistemic extensions
We now consider some constraints on the probability function which may take different
aspects of the structure of the argument graph into account. We will show how these
constraints are consistent with Dung’s notions of admissibility.
For the remainder of this paper let AF = (Arg, →) be an abstract argumentation
framework and P : 2Arg → [0, 1]. Consider the following properties (note that COH is
from [6] and JUS is from [5]):
COH P is coherent wrt. AF if for every A, B ∈ Arg, if A → B then P (A) ≤ 1−P (B).
SFOU P is semi-founded wrt. AF if P (A) ≥ 0.5 for every A ∈ Arg with AttAF (A) =
∅.
FOU P is founded wrt. AF if P (A) = 1 for every AP ∈ Arg with AttAF (A) = ∅.
SOPT P is semi-optimistic wrt. AF if P (A) ≥ 1 − B∈AttAF (A) P (B) for every A ∈
Arg with AttAF (A) 6= ∅. P
OPT P is optimistic wrt. AF if P (A) ≥ 1 − B∈AttAF (A) P (B) for every A ∈ Arg.
JUS P is justifiable wrt. AF if P is coherent and optimistic.
TER P is ternary wrt. AF if P (A) ∈ {0, 0.5, 1} for every A ∈ Arg.
COH ensures that if argument A attacks argument B, then the degree to which A is
believed caps the degree to which B can be believed; SFOU ensures that if an argument
is not attacked, then the argument is not disbelieved (i. e. P (A) ≥ 0.5); FOU ensures
that if an argument is not attacked, then the argument is believed without doubt (i. e.
P (A) = 1); SOPT ensures that the belief in A is bounded from below if the belief in its
attackers is not high; OPT ensures that if an argument is not attacked, then the argument
is believed without doubt (i. e. P (A) = 1) and that the belief in A is bounded from
below if the belief in its attackers is not high; JUS combines COH and OPT to provide
bounds on the belief in an argument based on the belief in its attackers and attackees;
and TER ensures that the probability assignment is a three-valued assignment.
Example 1. Consider AF = (Arg, →) depicted in Fig. 2 and the probability functions
depicted in Table 1 (note that the probability functions there are only partially defined by
giving the probabilities of arguments). The following observations can be made: 1.) P1
is semi-founded, founded, but neither coherent, optimistic, semi-optimistic, ternary, nor
justifiable, 2.) P2 is coherent and semi-optimistic, but neither semi-founded, founded,
optimistic, ternary, nor justifiable, 3.) P3 is coherent, semi-optimistic, semi-founded,
founded, optimistic, and justifiable, but not ternary, 4.) P4 is semi-founded, founded,
optimistic, and semi-optimistic, but neither coherent, justifiable, nor ternary, and 5.) P5
is coherent, semi-founded, semi-optimistic, and ternary but neither optimistic, justifi-
able, nor founded.
A1 A2 A3 A4 A5 A6
P1 0.2 0.7 0.6 0.3 0.6 1
P2 0.7 0.3 0.5 0.5 0.2 0.4
P3 0.7 0.3 0.7 0.3 0 1
P4 0.7 0.8 0.9 0.8 0.7 1
P5 0.5 0.5 0.5 0.5 0.5 0.5
Table 1. Some probability functions for Example 1
Let P be the set of all probability functions, P(AF) be the set of all probability functions
on Arg, and Pt (AF) be the set of all t-probability functions with t ∈ {COH,SFOU,FOU,
OPT,SOPT,JUS,TER}. We obtain the following relationships between the different
classes of probability functions.
A4
A1 A2 A3
A5 A6
Fig. 2. A simple argumentation framework
Proposition 1. Let AF = (Arg, →) be an abstract argumentation framework.
1. ∅ ( PJUS (AF) ( PCOH (AF) ( P(AF)
2. POPT (AF) = PSOPT (AF) ∩ PFOU (AF).
3. PFOU (AF) ( PSFOU (AF).
4. ∅ ( PTER (AF) ( P(AF).
For the proof of item 1.) of the above proposition see [5] and [6]. The remaining rela-
tionships follow directly from these definitions.
For all probability functions P such that LP is admissible in the classical sense, we
have that P assigns some degree of belief to each argument that is unattacked, thereby
P satisfies the SFOU constraint.
Proposition 2. For all probability functions P , if LP is admissible then P ∈ PSFOU (AF).
We can further constrain a probability assignment, so that the epistemic labelling straight-
forwardly captures the standard semantics (i. e. Dung’s semantics). By setting the prob-
ability function appropriately, its epistemic labelling corresponds to grounded, com-
plete, stable, preferred, or semi-stable labellings. All we require is a three-valued prob-
ability function that simulates each complete labelling function. For this, we provide
the following definition that provides the counter-part in our framework for a complete
labelling.
Definition 1. Let AF = (Arg, →) be an argumentation framework. Then a complete
probability function P ∈ P(AF) for AF is a probability function P such that for every
A ∈ Arg the following conditions hold:
1. P ∈ PTER (AF);
2. if P (A) = 1 then P (B) = 0 for all B ∈ Arg with B → A;
3. if P (B) = 0 for all B with B → A then P (A) = 1;
4. if P (A) = 0 then there is B ∈ Arg with P (B) = 1 and B → A;
5. if P (B) = 1 for some B with B → A then P (A) = 0.
Note that the above definition straightforwardly follows the definition of completeness
for classical semantics. Therefore, we have that P is a complete probability function if
and only if there is a complete labeling L and P ∼ L.
Completeness of probability functions can be characterized by the aforementioned
properties as follows.
Proposition 3. For an argument graph AF, and for P ∈ P(AF), P is a complete prob-
ability function iff P ∈ PCOH (AF) ∩ PFOU (AF) ∩ PTER (AF).
In the same way that Caminada and Gabbay [9] showed that different semantics can be
obtained by imposing further restrictions on the choice of labelling, we can obtain the
different semantics by imposing further restrictions on the choice of complete probabil-
ity function. These constraints, as shown in the following result, involve minimizing or
maximizing particular assignments. So for instance, if the assignment of 1 to arguments
is maximized, then a preferred labelling is obtained.
Proposition 4. Let AF = (Arg, →) be an abstract argumentation framework and P ∈
P(AF). If P is a complete probability function for AF and the restriction specified in
Table 2 holds for P , then the corresponding type of epistemic labelling is obtained.
Restriction on probability function P Classical semantics
No restriction complete extensions
No arguments A such that P (A) = 0.5 stable
Maximal no. of A such that P (A) = 1 preferred
Maximal no. of A such that P (A) = 0 preferred
Maximal no. of A such that P (A) = 0.5 grounded
Minimal no. of A such that P (A) = 1 grounded
Minimal no. of A such that P (A) = 0 grounded
Minimal no. of A such that P (A) = 0.5 semi-stable
Table 2. Correspondences between probabilistic and classical semantics
For an argumentation framework AF we can identify specific probability functions in
P ∈ PJUS (AF) that are congruent with admissible labellings, grounded labellings, or
stable labellings, for AF as follows.
Proposition 5. (From [5]) Let AF = (Arg, →) be an argumentation framework.
1. For every admissible L there is P ∈ PJUS (AF) with L ∼ P .
2. Let L be the grounded labelling and let3 P = arg maxQ∈PJUS (Arg) H(Q). Then
L ∼ P.
3. Let stable labellings exist for AF and let L be a stable labelling. Then there is
P ∈ arg minQ∈PJUS (Arg) H(Q) with L ∼ P .
So Proposition 4 and Proposition 5 provide two ways to identify probability functions
that capture specific types of labellings. Each of these results show that standard notions
of classical semantics (e. g. admissibility) can be captured using probability functions.
The next result shows that using probability functions to capture labellings gives a
finer-grained formalization of classical semantics.
Proposition 6. For each complete labelling L, if there is A such that L(A) 6= undec,
then there are infinitely many probability functions P such that LP = L.
Obviously, for every probability function P , there is by definition exactly one epistemic
labelling LP . This means that using a probability function to identify which arguments
are in, undec, or out, subsumes using labels. Furthermore, the probability function cap-
tures more information about the arguments. The granularity can differentiate between
3 P
Define the entropy H(P ) of P as H(P ) = − E⊆Arg P (E) log P (E)
for example a situation where A is believed (i. e. it is in) with certainty by P (A) = 1
from a situation where A is only just believed (i. e. it is only just in) for example by
P (A) = 0.51. Similarly, we can differentiate a situation where an attack by B on A is
undoubted when P (B) = 1 and P (A) = 0 from a situation where an attack by B on A
is very much doubted when for example P (B) = 0.55 and P (A) = 0.45.
In conclusion, we have shown how axioms can be used to constrain the probability
function, and thereby constrain the epistemic labelings and the epistemic extensions.
This allows us to subsume Dung’s notions of extensions as epistemic extensions. Fur-
thermore, we get a finer-grained representation of the labelling of arguments by repre-
senting the belief in each of the arguments.
5 Non-standard epistemic extensions
Before exploring the notion of non-standard epistemic extensions, we will augment the
set of properties we introduced in the previous section with the following properties.
Let AF = (Arg, →) be an abstract argumentation framework and P : 2Arg → [0, 1].
RAT P is rational wrt. AF if for every A, B ∈ Arg, if A → B then P (A) > 0.5
implies P (B) ≤ 0.5.
NEU P is neutral wrt. AF if P (A) = 0.5 for every A ∈ Arg.
INV P is involutary wrt. AF if for every A, B ∈ Arg, if A → B, then P (A) =
1 − P (B).
MAX P is maximal wrt. AF if P (A) = 1 for every A ∈ Arg.
MIN P is minimal wrt. AF if P (A) = 0 for every A ∈ Arg.
We explain these constraints as follows: RAT ensures that if argument A attacks argu-
ment B, and A is believed (i. e. P (A) > 0.5), then B is not believed (i. e. P (B) ≤ 0.5);
NEU ensures that all arguments are neither believed nor disbelieved (i. e. P (A) = 0.5
for all arguments); INV ensures that if argument A attacks argument B, then the belief
in A is the inverse of the belief in B; MAX ensures that all arguments are completely
believed; and MIN ensures that all arguments are completely disbelieved.
Example 2. We continue Example 1, the framework AF = (Arg, →) depicted in Fig. 2,
and the probability functions depicted in Table 1. The following observations can be
made: 1.) P2 and P3 are rational but neither neutral, involutary, maximal, nor minimal,
2.) P1 and P4 are neither rational, neutral, involutary, maximal, nor minimal, and 3.) P5
is rational, neutral, and involutary but neither maximal nor minimal.
As before let Pt (AF) be the set of all t-probability functions with t ∈ {COH,SFOU,
FOU, SOPT, OPT, JUS, TER, RAT, NEU, INV, MAX, MIN}. We extend the classi-
fication from Proposition 1 as follows.
Proposition 7. Let AF = (Arg, →) be an abstract argumentation framework.
1. ∅ ( PJUS (AF) ( PCOH (AF) ( PRAT (AF) ( P(AF)
2. ∅ ( PNEU (AF) ⊆ PINV (AF) ( PCOH (AF)
3. ∅ ( PINV (AF) ( PSOPT (AF)
4. ∅ ( PMIN (AF) ( PCOH (AF)
P(AF)
PSFOU (AF)
PRAT (AF) PFOU (AF)
PSOPT (AF)
PTER (AF) PCOH (AF) POPT (AF)
PINV (AF) PMIN (AF) PJUS (AF) PMAX (AF)
PNEU (AF)
∅
Fig. 3. Classes of probability functions (a normal arrow → indicates a strict subset relation, a
dashed arrow 99K indicates a subset relation)
5. ∅ ( PMAX (AF) ( POPT (AF)
Together with Examples 1 and 2 we obtain the strict classification of classes of proba-
bility functions as depicted in Figure 3.
The RAT constraint is a weaker version of the COH constraint, and it can be used
to capture each admissible labelling as a probability function.
Proposition 8. If L is an admissible labelling, then there is a P ∈ PRAT (AF) such that
L ∼ P.
Furthermore, the epistemic labelling corresponding to each probability function that
satisfies the RAT property is conflict-free.
Proposition 9. Let AF = (Arg, →) be an abstract argumentation framework. For each
P ∈ PRAT (AF ), in(LP ) is a conflict-free set of arguments in AF.
When the argument graph has odd cycles, there is no probability function that is invo-
lutary, apart from a neutral probability function.
Proposition 10. Let AF = (Arg, →) be an abstract argumentation framework. If AF
contains an odd cycle (i. e. there is a sequence of attacks A1 → A2 → ...... → Ak
where A1 = Ak and k is an even number), and P ∈ PINV (AF) then P ∈ PNEU (AF).
Even when the graph is acyclic, it may be the case that there is no involutary proba-
bility function (apart from the neutral probability function). Consider for example an
argument graph containing three arguments A, B and C, with A attacking both B and
C, and B attacking C. For this, there is no involutary probability function (apart from
the neutral probability function). If we restrict consideration to trees, then we are guar-
anteed to have a probability function that is involutary and not neutral. But even here
there are constraints as captured in the next result.
Proposition 11. If P ∈ PINV (AF), then for all Bi , Bj ∈ AttAF (A) we have P (Bi ) =
P (Bj ).
When P ∈ PMAX (AF), the probability function does not take the structure of the graph
into account. Hence, there is an incompatibility between a probability function being
maximal and a probability function being either rational or coherent. However, there is
compatibility between a probability function being maximal and a probability function
being founded since each P ∈ PMAX (AF) is in PFOU (AF).
Proposition 12. Let AF = (Arg, →) be an abstract argumentation framework. If there
are A, B ∈ Arg such that A → B, then PRAT (AF ) ∩ PMAX (AF ) = ∅.
In conclusion, we have identified epistemic extensions that are obtained from ratio-
nal probability functions as being an appealing alternative to extensions obtained by
Dung’s definitions. Rational probability functions are more general than coherent prob-
ability functions, and allow the audience more flexibility in expressing their beliefs
whilst taking the structure of the argument graph into account. We have also considered
alternatives such as the involutary probability functions but these are over-constrained.
6 Application: Partial probability functions
Assigning a probability value to an argument can be useful for a variety of purposes
such as representing the belief that the premises of the argument are valid, or the be-
lief that the claim is valid given that the premises are valid, or the belief that both the
premises and claim are valid, or the belief that the argument should appear in the ar-
gument graph, etc. However, given an argument graph, it may be difficult for a user to
assign a value to every argument. The user might have knowledge in order to identify a
value for some arguments, but the user may be unable or unwilling to make assignments
to the remaining arguments. This means that the user can only provide a partial assign-
ment. If this is the case, then it would be desirable to have techniques to handle this
incomplete information. Ideally, we would like to identify an appropriate assignment
for all the arguments based on the assignment to the subset of arguments.
More specifically, a partial function π : Arg → [0, 1] on Arg is called a partial
probability assignment. A probability function P ∈ P(Arg) is π-compliant if for every
A ∈ dom π we have π(A) = P (A). Let P π (AF) ⊆ P(AF) be the set of all π-compliant
probability functions. The question that arises is that given an abstract argumentation
framework AF = (Arg, →) and a partial probability assignment π, how do we deter-
mine P ∈ P(Arg) that is most compatible with both AF and π, i. e., which P ∈ P(Arg)
do we select as a meaningful representative? This question has also been addressed in
similar ways for partial probabilistic information without argumentation, cf. e. g. [10].
There, the principle of maximum entropy has been used to complete incomplete proba-
bilistic information in probabilistic logics. As a first step, we investigate the properties
of the sets of probability functions which are defined by our different axioms. An im-
portant requirement for applying maximum entropy approaches is that the probability
function with maximum entropy is uniquely determined. A sufficient property to en-
sure this, is that the set under consideration is both convex and closed.4 More generally,
4
A set X is convex if for x1 , x2 ∈ X it also holds that δx1 +(1−δ)x2 ∈ X for every δ ∈ [0, 1];
a set X is closed if for every converging sequence x1 , x2 , . . . with xi ∈ X (i ∈ N) we have
that limi→∞ xi ∈ X
maximizing any strictly convex function over a convex set has a unique solution and
also most interesting distance measures fall into this category. So if we are to identify
probability functions that complete the missing assignments, knowing that for specific
sets of probability functions (i. e. those that satisfy specific axioms) that they are convex
and closed, means that we may find an appropriate probability function.
Proposition 13. Let AF = (Arg, →) be an abstract argumentation framework.
1. The sets P(AF), PCOH (AF), PTER (AF), PNEU (AF), PINV (AF), PSFOU (AF), PFOU (AF),
POPT (AF), PSOPT (AF), PJUS (AF), PMIN (AF), PMAX (AF) are convex and closed.
2. The set PRAT (AF) is not convex but closed.
The above proposition suggests that most of our properties are suitable for convex op-
timization problems such as maximum entropy estimation. Moreover, the same is true
for the notion of partial probability assignments.
Proposition 14. For every partial probability assignment π the set P π (AF) is convex
and closed.
Let t be any one of our properties which lead to a convex and closed set of probabil-
ity functions (or any combination of those). If it is the case that there is at least one
π-compliant P in Pt (AF) then (thanks to the convexity properties) we have that the in-
tersection of P π (AF) and Pt (AF) is convex and closed as well, cf. [10]. In that case, we
can select the probability function with maximal entropy within this intersection (which
is uniquely defined). As for the rationale of this decision, several results from probabil-
ity reasoning, as for example discussed in [10], could be harnessed. We continue with
A B A B
A B
C C
(a) (b) (c)
Fig. 4. Argumentation frameworks for Example 3
some examples to illustrate the definitions and to investigate some of our concerns in
dealing with partial assignments.
Example 3. For the argumentation framework depicted in Figure 4(a) consider π1 with
π1 (A) = 1. Obviously, the most reasonable choice for a π1 -compliant P ∈ P(AF)
would be P (A) = 1 and P (B) = 0 (by obeying the property of involution). Further-
more, for π2 (B) = 0.3 we would have P (A) = 0.7 and P (B) = 0.3 following the
same rationale.
For the argumentation framework depicted in Figure 4(b) consider π3 with π3 (C) =
0.4. A possible choice for P would be P (A) = 0.6, P (B) = 0.4, and P (C) = 0.4
(having thus a maximally committed function that is coherent). But note that the set
P π (AF) ∩ PCOH (AF) does contain more than this single probability function. Further-
more, for π4 with π4 (B) = 0.7 and π4 (C) = 0.6 one would only guess P (A) ≤ 0.3 but
due to the “inconsistency” of π4 (violating the coherence condition), what is the best
choice?
For the argumentation framework depicted in Figure 4(c) consider π5 with π5 (A) =
0.4 and the following four selections P1 , P2 , P3 , P4 ∈ P(AF):
P1 (A) = 0.4 P2 (A) = 0.4 P3 (A) = 0.4 P4 (A) = 0.4
P1 (B) = 0.6 P2 (B) = 0.4 P3 (B) = 0.5 P4 (B) = 0.2
P1 (C) = 0.4 P2 (C) = 0.6 P3 (C) = 0.5 P4 (C) = 0.3
All of the above probability functions are π5 -compliant and coherent. Function P4 is
not maximally committed and as such is perhaps not a good choice. Both P1 and P2 are
“extreme points of view” and model some kind of probabilistic stable semantics. The
function P3 is as unbiased as possible but still “reasonable” as it models probabilistic
grounded semantics. Note that P3 is also the probability function with maximal entropy
in P π (AF) ∩ PCOH (AF).
Given P π (AF) and Pt (AF), we can either select P ∈ P π (AF) that is “as close as
possible to” Pt (AF) or P ∈ Pt (AF) that is “as close as possible to” P π (AF). In future
work, we will investigate definitions for “as close as possible to”, and we will explore
the pros and cons of each of these alternatives for selecting P .
7 Discussion
In this paper, we have investigated the use of a probability function to represent belief
in an argument. We use this to identify an epistemic labelling, and thereby an epistemic
extension. We have considered various constraints on the probability function leading
to two views on using the probability functions, namely the standard view, and the non-
standard view. We applied our classification of properties of probabilistic argumentation
to the problem of completing partial probability assignments. A first investigation leads
us to believe that maximizing entropy within probability functions of a specific type
gives appropriate results for this problem. In future work, we will investigate this issue
in more depth.
The work in this paper contrasts with other research on introducing a probability as-
signment to arguments such as [3]. There, a probability distribution over the subgraphs
of the argument graph is introduced, and this can then be used to give a probability
assignment for a set of arguments being an admissible set or extension of the argument
graph. They assume independence between arguments which in general is not appro-
priate. To address this shortcoming, the set of spanning subgraphs can be used as a
sample space, thereby obviating the need for an independence assumption between ar-
guments [4]. This probability distribution over subgraphs is used to give a probability
assignment to extensions. For a semantics X (such as grounded, preferred, or stable),
the probability that a set of arguments Γ is an extension according to semantics X,
denoted PX (Γ ), is the sum of the probability assigned to the subgraphs for which Γ
is an extension according to semantics X. The uncertainty that is being handled in [3,
4] is about the structure of the graph, and it is therefore a different kind of uncertainty
model to the one being addressed by this paper. Similarly, the work [11] extends abstract
argumentation by allowing the attack relation to be a fuzzy relation.
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