=Paper=
{{Paper
|id=Vol-3871/paper7
|storemode=property
|title=On Explainable Acceptance in Probabilistic and Incomplete Abstract Argumentation Frameworks
|pdfUrl=https://ceur-ws.org/Vol-3871/paper7.pdf
|volume=Vol-3871
|authors=Gianvincenzo Alfano,Marco Calautti,Sergio Greco,Francesco Parisi,Irina Trubitsyna
|dblpUrl=https://dblp.org/rec/conf/ai3/AlfanoCGPT24
}}
==On Explainable Acceptance in Probabilistic and Incomplete Abstract Argumentation Frameworks==
https://ceur-ws.org/Vol-3871/paper7.pdf
On Explainable Acceptance in Probabilistic and
Incomplete Abstract Argumentation Frameworks
(Discussion Paper)
Gianvincenzo Alfano1,† , Marco Calautti2,† , Sergio Greco1,† , Francesco Parisi1,† and
Irina Trubitsyna1,∗,†
1
Department of Informatics, Modeling, Electronics and System Engineering (DIMES), University of Calabria, Rende, Italy
2
Department of Computer Science (DI), University of Milan, Milan, Italy
Abstract
Dung’s Argumentation Framework (AF) has been extended in several directions, including the possibility of
representing uncertainty about the existence of arguments and attacks. In this regard, two main proposals have
been introduced in the literature: Probabilistic Argumentation Framework (PrAF) and Incomplete Argumentation
Framework (iAF). PrAF is an extension of AF with probability theory, thus representing quantified uncertainty.
In contrast, iAF represents unquantified uncertainty, that is it can be seen as a special case where we only know
that some elements (arguments or attacks) are uncertain. We discuss the problem of computing the probability
that a given argument is accepted in PrAF, which is based on the concept of probabilistic explanation for any
given (probabilistic) extension [1]. Our approach can be extended to iAF, as it can be viewed as a special case of
PrAF where uncertain elements have associated a probability equal to 1/2.
Keywords
Formal Argumentation Theory, Explainable AI, Probabilistic Argumentation Framework.
1. Introduction
The abstract Argumentation Framework (AF) is a simple, yet powerful formalism for modeling disputes
between two or more agents [2]. An AF consists of a set of arguments and a binary attack relation
over the set of arguments that specifies the interactions between arguments: intuitively, if argument a
attacks argument b, then b is acceptable only if a is not. Hence, arguments are abstract entities whose
role is entirely determined by the interactions specified by the attack relation.
Recently, there has been an increasing interest in extending argumentation frameworks to manage
uncertain information. This has been carried out by either considering quantified uncertainty about
the existence of arguments and attacks, thus combining formal argumentation with probability theory,
or considering unquantified uncertainty by explicitly denoting the elements (arguments and attacks)
which are uncertain. In fact, Probabilistic Argumentation [3] can be viewed as part of the several
proposals that have been made in the last decades for extending reasoning tasks in AI frameworks
with probabilities. These include for instance Probabilistic SAT (PSAT) [4], Probabilistic Logic [5],
Probabilistic Logic Programming [6], and Probabilistic Databases [7].
One of the most popular approaches based on probability theory for modeling the uncertainty
is the so called constellations approach [8, 9, 10, 11, 12], where alternative scenarios, called possible
8th Workshop on Advances in Argumentation in Artificial Intelligence, co-located with the 23rd International Conference of the
Italian Association for Artificial Intelligence (AIxIA 2024), 25-28 November, 2024, Bolzano, Italy
∗
Corresponding author.
†
These authors contributed equally.
$ g.alfano@dimes.unical.it (G. Alfano); marco.calautti@unimi.it (M. Calautti); greco@dimes.unical.it (S. Greco);
fparisi@dimes.unical.it (F. Parisi); i.trubitsyna@dimes.unical.it (I. Trubitsyna)
https://gianvincenzoalfano.net/ (G. Alfano); https://www.unimi.it/it/ugov/person/marco-scalautti (M. Calautti);
https://people.dimes.unical.it/sergiogreco/ (S. Greco); http://wwwinfo.deis.unical.it/~parisi/ (F. Parisi);
https://sites.google.com/dimes.unical.it/itrubitsyna/home (I. Trubitsyna)
0000-0002-7280-4759 (G. Alfano); 0000-0003-0921-4040 (M. Calautti); 0000-0003-2966-3484 (S. Greco); 0000-0001-9977-1355
(F. Parisi); 0000-0002-9031-0672 (I. Trubitsyna)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
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Workshop ISSN 1613-0073
Proceedings
fish meat white red
0.6 0.8
Figure 1: Probabilistic argumentation framework Δ of Example 1.
w1 w2
fish meat white red fish meat red
w3 w4
meat white red meat red
Figure 2: Possible worlds of the probabilistic argumentation framework Δ of Example 1.
worlds, are associated with probabilities. In particular, in a Probabilistic Argumentation Framework
(PrAF) [12, 13, 14, 15, 16, 17, 18] a probability distribution function (PDF) on the set of possible worlds
is entailed by the probabilities that are associated with arguments and attacks.
Example 1. Consider a PrAF Δ = ⟨{fish, meat, white, red}, {(fish, meat), (meat, fish), (meat,
white), (white, red), (red, white)}, {fish/0.6, white/ 0.8}⟩, whose corresponding graph is shown
in Figure 1, where nodes and edges represent arguments and attacks, respectively, and probabilities
different from 1 are specified nearby them. For the sake of brevity, we do not specify the probabilities of
certain elements in Δ (all the elements different from fish and white have probability 1). Intuitively,
Δ describes what a person is going to have for lunch as follows. They will have either fish or meat,
and will drink either white wine or red wine. However, if they will have meat, then they will not
drink white wine. Furthermore, the probability that fish is available is 0.6, whereas the probability
that white wine is available is 0.8. 2
Intuitively, PrAF is a combination of two powerful approaches to reasoning and decision making:
probabilistic reasoning and abstract argumentation. Probabilities are assigned to arguments and attacks
to indicate their degree of uncertainty. One of the benefits of probabilistic abstract argumentation is its
ability to handle quantified uncertainty in the analysis. In fact, PrAF can help to model and analyze
situations where there is uncertainty by capturing both the relationships between arguments and the
uncertainty degrees of arguments and attacks.
Several argumentation semantics—e.g. grounded (gr), complete (co), preferred (pr), stable (st), and
semi-stable (sst)—have been defined for AFs, leading to the characterization of σ-extensions, which
intuitively consist of the sets of arguments that can be collectively accepted under semantics σ. Consider
for instance the deterministic version of the PrAF in Example 1, obtained by assuming that all arguments
are certain (i.e. they have probability 1). Considering the preferred semantics, the pr-extensions are
E1 = {fish, white}, E2 = {fish, red}, and E3 = {meat, red}.
The semantics of a PrAF is given by considering all possible worlds (i.e. AFs) obtained by removing
consistent subsets of the probabilistic elements. Here, for consistent subset we mean any subset of
probabilistic elements (arguments and attacks) whose deletion from the initial framework results in an
AF (for instance we cannot delete an argument without also deleting the attacks towards or from that
argument). Every possible world has associated a probability value derived from the probabilities of the
elements that have been kept or removed. Moreover, every possible world admits a set of σ-extensions.
The probability of a possible world w is computed by multiplying the probabilities of the elements
occurring in w and the complement to 1 of the probabilities of the elements not occurring in w.
Example 2. Continuing with Example 1, the possible worlds of Δ are shown in Figure 2. The probability
of a possible world wi is obtained by multiplying the probabilities P (a) of each argument a occurring
in wi and the probabilities (1 − P (b)) of every argument b not occurring in wi . Since P (fish) =
0.6, P (white) = 0.8, and P (meet) = P (red) = 1, the probabilities of w1 , w2 , w3 , and w4 are
0.6 · 1 · 0, 8 · 1 = 0.48, 0.6 · 1 · 0.2 · 1 = 0.12, 0.4 · 1 · 0.8 · 1 = 0.32, and 0.4 · 1 · 0.2 · 1 = 0.08. Since
w1 coincides with the deterministic version of Δ, its pr-extensions are E1 , E2 , and E3 given earlier.
The pr-extensions of w2 are E2 and E3 , while w3 and w4 admit only E3 as their preferred extension.
2
2. Explanation-based Probabilistic Acceptance
Interesting problems recently investigated in the context of probabilistic argumentation are probabilistic
credulous acceptance (PrCA) and probabilistic skeptical acceptance (PrSA) [19, 15]. In particular, given a
PrAF Δ whose set of arguments is A, a goal argument g ∈ A and a semantics σ, PrCA is the problem of
computing the probability P rCAσΔ (g) that the goal g is credulously accepted, that is, there is a possible
world w of Δ such that g belongs to a σ-extension of w. Moreover, PrSA is the problem of computing
the probability P rSAσΔ (g) that the goal g is skeptically accepted, that is, g is credulously accepted and
belongs to all σ-extensions of w.
However, the answer to these problems does not reflect our intuition of probability that a goal
argument is accepted under a given semantics. For instance, considering the PrAF Δ of Figure 1, the
probability that meat is credulously accepted under preferred semantics is 1, whereas the probability that
pr
meat is skeptically accepted under preferred semantics is 0.4. However, the fact that P rCAΔ (meat) =
1 does not mean that the person in our example will surely have meat in any scenario (i.e. possible
world). In fact, even if meat belongs to at least one preferred extension of every world of Δ, we expect
that the probability of acceptance of meat should be lower than 1. Indeed, in any possible world, the
presence of multiple extensions is an additional source of uncertainty that should be taken into account.
To better grasp the issue behind the probability of credulous acceptance, consider the following
AF (where all elements are certain): Λ = ⟨{fish, meat}, {(fish, meat), (meat, fish)}⟩ saying that
fish and meat are mutually exclusive. Again, the probability that a person will have meat is 1, under
probabilistic credulous acceptance, when considering the preferred semantics, whereas we believe
that the expected answer should be 0.5. Moreover, if we consider AF w1 of Example 2 (that can be
obtained from Λ by adding arguments white and red and attacks (white, red), (red, white) and
(meat, white)) we expect that the probability of having meat does not change.
With the aim of providing more intuitive answers for probabilistic acceptance, a new problem called
Probabilistic Acceptance (denoted as PrA, or PrA[σ] when considering a given semantics σ) has beed
investigated [1, 20], i.e. given a PrAF Δ and a goal argument g, compute the probability that g is accepted
under semantics σ ∈ {gr, co, pr, st, sst}. In this framework, acceptance still relies on σ-extensions
but, differently from credulous acceptance, we get rid of the assumption that no uncertainty exists at
the level of the extensions of a world (i.e. AF). In more detail, PrA[σ] implicitly assumes that a PDF
over the set of σ-extensions of any AF (and thus of any possible world of PrAF Δ) is defined. Thus, a
concrete instance of PrA is obtained after defining such a PDF. This can be carried out by exploiting
the concept of explanation for an extension.
In general, in abstract argumentation an explanation for an extension E can be viewed as a (possibly
minimal) subset S ⊆ E such that, by assuming that the elements in S are acceptable, it turns out that
all elements in E \ S are “univocally” determined as acceptable (w.r.t. the underlying semantics). For
instance, considering AF w1 of Example 2, for the preferred extension E = {meat, red}, the set S1 =
{meat} is an explanation for E, whereas the set S2 = {red} is not. In our perspective, explanations
are sequences of “choices” to be made to justify how an extension is obtained and they provide a tool
to assign probabilities to extensions. Integrating explanations in argumentation systems is important
for enhancing the argumentation and persuasion capabilities of software agents [21, 22, 23]. For this
reasons, several researchers have explored how to deal with explanations in formal argumentation
[24, 25, 26].
An instantiation of PrA[σ] where the PDF over the set of σ-extensions of a world relies on the concept
of explanation is called Explanation-based Probabilistic Acceptance problem, and denoted by PrEA (and
PrEA[σ] for a specific semantics σ). Intuitively, an explanation for an σ-extension E is a sequence of
arguments occurring in E that “justify” E. Every explanation is associated with a probability entailed
by the possible choices that can be made when building it. These choices must be consistent with an
ordering entailed by the strongly connected components of the given AF, and they are used to guide the
construction of an extension. The sum of the probabilities of the explanations for an extension E gives
the probability of E. Thus, we still assign to each possible world w of Δ a probability as in the standard
way, but in addition propose to distinguish among extensions of a given world w by associating with
them a probability based on explanations.
Example 3. Continuing with Example 1, take for instance the possible world w1 having probability
0.48. As shown in Example 2, w1 has three pr-extensions, namely E1 , E2 and E3 . As shown in [1], in
this case, for each extension there is only one explanation. In particular, X1 = ⟨fish, white⟩ is the
explanation for E1 . The intuition of explanation X1 is that, considering that the AF consists of two
strongly connected components, we first choose fish (with probability 1/2 as we can only choose
between fish and meat) in the first component and determine that meat cannot belong to the extension;
then we choose white (with probability 1/2 as we can only choose between white and red) in the
second component, obtaining that X1 has probability 1/2 · 1/2 = 1/4. Analogously, X2 = ⟨fish, red⟩
is the only explanation for E2 with probability 1/2 · 1/2 = 1/4. Considering explanation X3 = ⟨meat⟩
for extension E3 , we have that we first choose meat with probability 1/2 as it belongs to the first
component, and we can only choose between fish and meat. Next, since we determine that fish and
white cannot belong to the extension, whereas red does, the probability of X3 turns out to be 1/2.
Since the probabilities of X1 , X2 and X3 are 1/4, 1/4 and 1/2, respectively, the probabilities associated
with E1 , E2 and E3 in the world w1 are 1/4, 1/4 and 1/2, respectively. Moreover, since E1 is not an
extension of any other possible world, the probability of E1 in Δ is 1/4 · 0.48 = 0.12. It turns out that
the answer to PrEA[pr] for meat is 0.70, while that for fish is 0.30. 2
The definition of Explanation-based Probabilistic Acceptance has been also carried out to another
argumentation framework extending AF that has received an increasing attention in the last years and
is tightly related to PrAF, that is, to incomplete AF (iAF) [27, 28]. This follows from the fact that iAF can
be viewed as a special case of PrAF where uncertain elements have associated a probability equal to 1/2.
Acknowledgements
We acknowledge the support from PNRR MUR project PE0000013-FAIR and PE0000014-SERICS, project
Tech4You ECS0000009, and MUR project PRIN 2022 EPICA (CUP H53D2300 3660006).
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