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{{Paper
|id=Vol-3915/paper-9
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|title=On Counterfactual and Semifactual Explanations in Abstract Argumentation (Short paper)
|pdfUrl=https://ceur-ws.org/Vol-3915/Paper-9.pdf
|volume=Vol-3915
|authors=Gianvincenzo Alfano,Sergio Greco,Francesco Parisi,Irina Trubitsyna
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==On Counterfactual and Semifactual Explanations in Abstract Argumentation (Short paper)==
https://ceur-ws.org/Vol-3915/Paper-9.pdf
On Counterfactual and Semifactual Explanations in
Abstract Argumentation
(Discussion Paper)
Gianvincenzo Alfano1,*,† , Sergio Greco1,† , Francesco Parisi1,† and Irina Trubitsyna1,†
1
Department of Informatics, Modeling, Electronics and System Engineering (DIMES),
University of Calabria, Rende, Italy
Abstract
Explainable Artificial Intelligence and Formal Argumentation have received significant attention in recent years.
Argumentation frameworks are useful for representing knowledge and reasoning on it. Counterfactual and
semifactual explanations are interpretability techniques that provide insights into the outcome of a model by
generating alternative hypothetical instances. While there has been important work on counterfactual and
semifactual explanations for Machine Learning (ML) models, less attention has been devoted to these kinds
of problems in argumentation. In this paper, we discuss counterfactual and semifactual reasoning in abstract
Argumentation Framework recently proposed in [1].
Keywords
Formal Argumentation Theory, Explainable AI, Counterfactual and Semifactual Reasoning.
1. Introduction
In the last decades, Formal Argumentation has become an important research field in the area of
knowledge representation and reasoning [2]. Argumentation has potential applications in several
contexts, including e.g. modeling dialogues, negotiation [3, 4], and persuasion [5]. Dung’s Argumenta-
tion Framework (AF) is a simple yet powerful formalism for modeling disputes between two or more
agents [6]. 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 status is entirely determined
by the attack relation. An AF can be seen as a directed graph, whose nodes represent arguments and
edges represent attacks. Several argumentation semantics—e.g. grounded (gr), complete (co), stable (st),
preferred (pr), and semi-stable (sst) [6, 7]—have been defined for AF, leading to the characterization of
𝜎-extensions, that intuitively consist of the sets of arguments that can be collectively accepted under
semantics 𝜎 ∈ {gr, co, st, pr, sst}.
Example 1. Consider the AF Λ in Figure 1, describing tasting menus proposed by a chef. Intuitively,
(s)he proposes to have either fish, meat, or pasta and to drink either white wine or red wine.
However, if serving meat or pasta then white wine is not paired with. AF Λ has four stable extensions
(that are also preferred and semi-stable extensions) representing alternative menus: 𝐸1 ={fish, white},
𝐸2 ={fish, red}, 𝐸3 ={meat, red}, and 𝐸4 ={pasta, red}. □
AIxIA 2024 Discussion Papers - 23rd International Conference of the Italian Association for Artificial Intelligence, Bolzano, Italy,
November 25–28, 2024
*
Corresponding author.
†
These authors contributed equally.
$ g.alfano@dimes.unical.it (G. Alfano); 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://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); 0000-0003-2966-3484 (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
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
pasta
fish white red
meat
Figure 1: AF Λ of Example 1.
Argumentation semantics can be also defined in terms of labelling [8]. Intuitively, a 𝜎-labelling for
an AF is a total function ℒ assigning to each argument the label in if its status is accepted, out if its
status is rejected, and und if its status is undecided under semantics 𝜎. For instance, the 𝜎-labellings
for AF Λ of Example 1, with 𝜎 ∈ {st, pr, sst}, are as follows:
ℒ1 = {in(fish), out(meat), out(pasta), in(white), out(red)},
ℒ2 = {in(fish), out(meat), out(pasta), out(white), in(red)},
ℒ3 = {out(fish), in(meat), out(pasta), out(white), in(red)},
ℒ4 = {out(fish), out(meat), in(pasta), out(white) , in(red)},
where ℒ𝑖 corresponds to extension 𝐸𝑖 , with 𝑖 ∈ [1..4], respectively.
Integrating explanations in argumentation-based reasoners is important for enhancing argumentation
and persuasion capabilities of software agents [9, 10, 11, 12]. For this reasons, several researchers
explored how to deal with explanations in formal argumentation. Counterfactual and semifactual
explanations are types of interpretability techniques that provide insights into the outcome of a model
by generating hypothetical instances, known as counterfactuals and semifactual, respectively [13, 14].
On one hand, a counterfactual explanation reveals what should have been different in an instance to
obtain a diverse outcome [15]—minimum changes w.r.t. the given instance are usually considered [16].
On the other hand, a semifactual explanation provides a maximally-changed instance yielding the same
outcome of that considered [17].
While there has been interesting work on counterfactual and semifactual explanations for ML models,
e.g. [18, 19, 20, 21, 22, 23], less attention has been devoted to these problems in argumentation.
In this paper, we discuss counterfactual and semifactual reasoning in AF [1]. Analogously to coun-
terfactual explanations in ML that reveal what should have been minimally different in an instance
to obtain a different outcome, our counterfactuals tell what should have been minimally different in a
solution, i.e. a 𝜎-labeling with a given acceptance status for a goal argument, to obtain an alternative
solution where the goal has a different status.
Example 2. Continuing with Example 1, assume that the chef suggests the menu ℒ3 = {out(fish),
in(meat), out(pasta), out(white), in(red)} and the customer replies that (s)he likes everything
except meat (as (s)he is vegetarian). Therefore, the chef looks for the closest menus not containing meat,
that are ℒ2 = {in(fish), out(meat), out(pasta), out(white), in(red)} and ℒ4 = {out(fish),
out(meat), in(pasta), out(white), in(red)}. In this context, we say that ℒ2 and ℒ4 are counterfac-
tuals for ℒ3 w.r.t. the goal argument meat. □
Given a 𝜎-labelling ℒ of an AF Λ, and a goal argument 𝑔, a counterfactual of ℒ w.r.t. 𝑔 is a closest
𝜎-labelling ℒ′ of Λ that changes the acceptance status of 𝑔. Hence, counterfactuals explain how to
minimally change a solution to avoid a given acceptance status of a goal argument.
In contrast, semifactuals give the maximal changes to the considered solution in order to keep the
status of a goal argument. That is, a semifactual of ℒ w.r.t. goal 𝑔 is a farthest 𝜎-labelling ℒ′ of Λ that
keeps the acceptance status of argument 𝑔.
Example 3. Continuing with Example 1, suppose now that a customer has tasted menu ℒ3 =
{out(fish), in(meat), out(pasta), out(white), in(red)}, and asks to try completely new flavors
while still maintaining the previous choice of wine as (s)he liked it a lot. Here the chef is interested in
the farthest menus containing red wine. These menus are ℒ2 = {in(fish), out(meat), out(pasta),
out(white), in(red)} and ℒ4 = {out(fish), out(meat), in(pasta), out(white), in(red)}. We
say that the labellings ℒ2 and ℒ4 are semifactuals for the labelling ℒ3 w.r.t. the goal argument red. □
2. Counterfactual and Semifactual Reasoning
Intuitively, a counterfactual of a given 𝜎-labelling w.r.t. a given goal argument 𝑔 is a minimum-distance 𝜎-
labelling altering the acceptance status of 𝑔. More in detail, let ⟨A, R⟩ be an AF, 𝜎 ∈ {gr, co, st, pr, sst}
a semantics, 𝑔 ∈ A a goal argument, and ℒ a 𝜎-labelling for ⟨A, R⟩. Then, a labelling ℒ′ ∈ 𝜎(⟨A, R⟩) is
a counterfactual of ℒ w.r.t. 𝑔 if:
(𝑖) ℒ(𝑔) ̸= ℒ′ (𝑔), and
(𝑖𝑖) there exists no ℒ′′ ∈ 𝜎(⟨A, R⟩) such that ℒ(𝑔) ̸= ℒ′′ (𝑔) and 𝛿(ℒ, ℒ′′ ) < 𝛿(ℒ, ℒ′ ).
We use 𝒞ℱ 𝜎 (𝑔, ℒ) to denote the set of counterfactuals of ℒ w.r.t. 𝑔.
Example 4. Continuing with Example 2, under stable semantics, for the labelling ℒ3 = {out(fish),
in(meat), out(pasta), out(white), in(red)}, we have that ℒ2 = {in(fish), out(meat), out(pasta),
out(white), in(red)} and ℒ4 = {out(fish), out(meat), in(pasta), out(white), in(red)} are its
only counterfactuals w.r.t. argument meat, as their distance, 𝛿(ℒ3 , ℒ2 ) = 𝛿(ℒ3 , ℒ4 ) = 2, is min-
imal. The other labelling ℒ1 = {in(fish), out(meat), out(pasta), in(white), out(red)}, such
that ℒ3 (meat) ̸= ℒ1 (meat) is not at minimum distance as 𝛿(ℒ3 , ℒ1 ) = 4 > 𝛿(ℒ3 , ℒ2 ). Therefore,
𝒞ℱ st (meat, ℒ3 ) = {ℒ2 , ℒ4 }. □
The concept of semifactual is, in a sense, symmetrical and complementary to that of a counterfactual.
Indeed, let ⟨A, R⟩ be an AF, 𝜎 ∈ {gr, co, st, pr, sst} a semantics, 𝑔 ∈ A a goal argument, and ℒ a
𝜎-labelling for ⟨A, R⟩. Then, ℒ′ ∈ 𝜎(⟨A, R⟩) is a semifactual of ℒ w.r.t. 𝑔 if:
(𝑖) ℒ(𝑔) = ℒ′ (𝑔), and
(𝑖𝑖) there exists no ℒ′′ ∈ 𝜎(⟨A, R⟩) such that ℒ(𝑔) = ℒ′′ (𝑔) and 𝛿(ℒ, ℒ′′ ) > 𝛿(ℒ, ℒ′ ).
We use 𝒮ℱ 𝜎 (𝑔, ℒ) to denote the set of semifactuals of ℒ w.r.t. 𝑔.
Example 5. Consider the stable labelling ℒ3 = {out(fish), in(meat), out(pasta),out(white),
in(red)} for the AF of Example 3. We have that ℒ2 = {in(fish), out(meat), out(pasta), out(white),
in(red)} and ℒ4 = {out(fish), out(meat), in(pasta), out(white), in(red)} are the only semi-
factuals of ℒ3 w.r.t. the argument red as there is no other st-labelling agreeing on red and having
distance greater than 𝛿(ℒ3 , ℒ2 ) = 𝛿(ℒ3 , ℒ4 ) = 2. In fact, ℒ1 = {in(fish), out(meat), out(pasta),
in(white), out(red)}, having distance 𝛿(ℒ3 , ℒ1 ) = 4, is not a semifactual for ℒ3 w.r.t. red as
ℒ1 (red) ̸= ℒ3 (red). Thus, 𝒮ℱ st (red, ℒ3 ) = {ℒ2 , ℒ4 }. □
2.1. Existence and Verification Problems
Finding a counterfactual (resp., semifactual) means looking for a minimum (resp., maximum) distance
labelling. The first problem we consider is a natural decision version of that problem.
Given as input an AF Λ = ⟨A, R⟩, a semantics 𝜎 ∈ {co, st, pr, sst}, a goal argument 𝑔 ∈ A, an
integer 𝑘 ∈ N, and a 𝜎-labelling ℒ ∈ 𝜎(Λ), CF-EX𝜎 (resp., SF-EX𝜎 ) is the problem of deciding whether
there exists a labelling ℒ′ ∈ 𝜎(Λ) s.t. ℒ(𝑔) ̸= ℒ′ (𝑔) (resp., ℒ(𝑔) = ℒ′ (𝑔)) and 𝛿(ℒ, ℒ′ ) ≤ 𝑘 (resp.,
𝛿(ℒ, ℒ′ ) ≥ 𝑘).
The complexity of the existence problem under counterfactual and semifactual reasoning (i.e., CF-EX𝜎
and SF-EX𝜎 ) has been recently proved to be 𝑖) NP-complete for 𝜎 ∈ {co, st}; and 𝑖𝑖) Σ𝑝2 -complete for
𝜎 ∈ {pr, sst} [1].
A problem related to CF-EX𝜎 and SF-EX𝜎 is that of verifying whether a given labelling ℒ′ is a
counterfactual/semifactual for ℒ and 𝑔, and thus that the distance between the two labelling is mini-
mum/maximum.
Given as input an AF Λ = ⟨A, R⟩, a semantics 𝜎 ∈ {co, st, pr, sst}, a goal argument 𝑔 ∈ A, a
𝜎-labelling ℒ ∈ 𝜎(Λ), and a labelling ℒ′ , CF-VE𝜎 (resp., SF-VE𝜎 ) is the problem of deciding whether
ℒ′ belongs to 𝒞ℱ 𝜎 (𝑔, ℒ) (resp., 𝒮ℱ 𝜎 (𝑔, ℒ)).
The problems CF-VE𝜎 and CF-EX𝜎 (resp., SF-VE𝜎 and SF-EX𝜎 ) are on the same level of the polynomial
hierarchy. In fact CF-VE𝜎 and SF-VE𝜎 are 𝑖) coNP-complete for 𝜎 ∈ {co, st}; and 𝑖𝑖) Π𝑝2 -complete for
𝜎 ∈ {pr, sst} [1].
3. Conclusions
Several researchers explored how to deal with explanations with in formal argumentation [24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. Counterfactual reasoning in AF has been firstly introduced
in [39], where considering sentences of the form “if 𝑎 were rejected, then 𝑏 would be accepted”, an AF
Λ is modified to another AF Λ′ such that (i) argument 𝑎 which is accepted in Λ is rejected in Λ′ (ii) and
the Λ′ is as close as possible to Λ.
However, none of the above-mentioned approaches deals with semifactual reasoning and most of
them manipulate the AF by adding arguments or meta-knowledge. In contrast, in our approach, focusing
on a given AF, novel definitions of counterfactual and semifactual are introduced to help understand
what should be different in a solution (not in the AF) to accommodate a user requirement concerning
a given goal. It turns out that the complexity of the considered problems is not lower than those of
corresponding classical problems in AF, and is provably higher for fundamental problems such as the
verification problem.
Although counterfactual- and semifactual-based reasoning suffers from high computational com-
plexity (as many other computational problems in argumentation [40, 41, 42, 43, 44, 45, 46, 47]), several
tools and techniques emerged in the last few years that can tackle such kinds of computational issues,
including ASP- and SAT-based solvers. This is witnessed by the several efficient approaches presented
at the ICCMA competition,1 which aims at nurturing research and development of implementations for
computational models of argumentation.
Acknowledgements
We acknowledge the support from project Tech4You (ECS0000009), and PNRR MUR projects FAIR
(PE0000013) and SERICS (PE00000014).
Declaration on Generative AI
The author(s) have not employed any Generative AI tools.
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