Necessary and Sufficient Explanations for
Argumentation-Based Conclusions
AnneMarie Borg1 and Floris Bex1,2
1

Department of Information and Computing Sciences, Utrecht University
2
Tilburg Institute for Law, Technology, and Society, Tilburg University{A.Borg, F.J.Bex}@UU.nl
Abstract

In this paper, we discuss necessary and sufficient explanations – the question whether andwhy a certain argument or claim can be accepted (or not) – for abstract and structured argumentation. Given a framework with which explanations for argumentation-based conclusions
can be derived, we study necessity and sufficiency: what (sets of) arguments are necessary or
sufficient for the (non-)acceptance of an argument or claim? We will show that necessary andsufficient explanations can be strictly smaller than minimal explanations, while still providing
all the reasons for a conclusion and we discuss their usefulness in a real-life application.

1

Introduction

In recent years, explainable AI (XAI) has received much attention, mostly directed at new techniques for explaining decisions of (subsymbolic) machine learning algorithms [19]. However, explanations traditionally also play an important role in (symbolic) knowledge-based systems [11].
Computational argumentation is one research area in symbolic AI that is frequently mentioned inrelation to XAI. For example, arguments can be used to provide reasons for or against decisions[1, 11, 16]. The focus can also be on the argumentation itself, where it is explained whether
and why a certain argument or claim can be accepted under certain semantics for computational
argumentation [8, 9, 10, 20]. It is the latter type of explanations that is the subject of this paper.
Two central concepts in computational argumentation are abstract argumentation frameworks [7]– sets of arguments and the attack relations between them and structured argumentation frameworks [3] – where arguments are constructed from a knowledge base and a set of rules and theattack relation is based on the individual elements in the arguments. The explanations frameworkthat is introduced in [6] is designed to provide explanations for the (non-)acceptance of arguments
and the claim of an argument in case of a structured setting. However, like the other existing
works on explanations for argumentation-based conclusions, the framework does not account forfindings from the social sciences on human explanations [16].
One of the important characteristics of explanations provided by humans is that they selectthe explanation from a possible infinite set of explanations [16]. In this paper we look at how to
select minimal,1 necessary and sufficient explanations for the (non-)acceptance of an argument.
To this end we will introduce variations to the basic framework from [6] that will provide necessary or sufficient explanations for both abstract and structured argumentation (i.e., ASPIC+ [18]).
Intuitively, a necessary explanation contains the arguments that one has to accept in order toaccept the considered argument and a sufficient explanation contains the arguments that, when
To appear in the proceedings of the 16th European Conference on Symbolic and Quantitative Approaches toReasoning with Uncertainty (ECSQARU’21).
1 Interpreting [16]’s simplicity as minimality.

1

accepted, guarantee the acceptance of the considered argument. We will show that such explanations exist in most cases and how these relate to the basic explanations from [6] as well as tominimal explanations as introduced in [8]. Moreover, we will discuss a real-life application fromthe Dutch National Police, where the necessary and sufficient explanations will reduce the size ofthe provided explanations in a meaningful way.
The paper is structured as follows. We start with a short overview of related work and present
the preliminaries on abstract and structured argumentation and the basic explanations from [6].
Then, in Section 4 we introduce necessary and sufficient explanations and study how these relate to
the better known minimal explanations. In Section 5 we show how a real-life application benefits
from these explanations and we conclude in Section 6.

2

Related WorkWe are interested in local explanations for computational argumentation: explanations for a specific argument or claim. We work here with the framework from [6] for several reasons. Often,
explanations are only defined for a specific semantics [8, 9] and can usually only be applied toabstract argumentation [9, 12, 20],2 while the framework from [6] can be applied on top of any
argumentation setting (structured or abstract) that results in a Dung-style argumentation framework. Furthermore, when this setting is a structured one based on a knowledge base and set ofrules (like ASPIC+ or logic-based argumentation [3]), the explanations can be further adjusted(something which is not considered at all in the literature). To the best of our knowledge, thisis the first approach to explanations for formal argumentation in which necessary and sufficient
explanations are considered and integrated into a real-life application.

3

Preliminaries

An abstract argumentation framework (AF) [7] is a pair AF = hArgs, Atti, where Args is a set ofarguments and Att ⊆ Args × Args is an attack relation on these arguments. An AF can be viewed
as a directed graph, in which the nodes represent arguments and the arrows represent attacks
between arguments.

D

C

A

B

E

F

Figure 1: Graphical representation of the AF AF 1 .
Example 1. Figure 1 represents AF 1 = hArgs1 , Att1 i where Args1 = {A, B, C, D, E, F } and Att1 ={(B, A), (C, B), (C, D), (D, C), (E, B), (E, F ), (F, E)}.
Given an AF, Dung-style semantics [7] can be applied to it, to determine what combinationsof arguments (called extensions) can collectively be accepted.
Definition 1. Let AF = hArgs, Atti be an argumentation framework, S ⊆ Args a set of arguments
and let A ∈ Args. Then: S attacks A if there is an A0 ∈ S such that (A0 , A) ∈ Att, let S+ denotethe set of arguments attacked by S; S defends A if S attacks every attacker of A; S is conflict-free
if there are no A1 , A2 ∈ S such that (A1 , A2 ) ∈ Att; and S is an admissible extension (Adm) if it
is conflict-free and it defends all of its elements.
An admissible extension that contains all the arguments that it defends is a complete extension(Cmp). The grounded extension (Grd) is the minimal (w.r.t. ⊆) complete extension; A preferred
extension (Prf) is a maximal (w.r.t. ⊆) complete extension; and A semi-stable extension (Sstb) is a
2 These explanations do not account for the sub-argument relation in structured argumentation. For example,
in structured argumentation one cannot remove specific arguments or attacks without influencing other arguments/attacks.

2complete extension for which S ∪ S+ is ⊆-maximal. Sem(AF) denotes the set of all the extensionsof AF under the semantics Sem ∈ {Adm, Cmp, Grd, Prf, Sstb}.
In what follows we will consider an argument accepted if it is part of at least one extension
and non-accepted if it is not part of at least one extension.3
Definition 2. Where AF = hArgs, Atti is an AF, Sem a semantics such that Sem(AF) 6= ∅, it issaid that A ∈ Args is:S• accepted if A ∈ Sem(AF): there is some Sem-extension that contains A;
T• non-accepted if A ∈/ Sem(AF): A is not part of at least one Sem-extension.
Example 2. For AF 1 we have that Grd(AF 1 ) = {∅} and there are four preferred and semi-stable
extensions: {A, C, E}, {A, C, F }, {A, D, E} and {B, D, F }. Therefore, all arguments from Args1
are accepted and non-accepted for Sem ∈ {Prf, Sstb}.

3.1

ASPIC+For our discussion on explanations for structured settings we take ASPIC+ [18] which allows
for two types of premises – axioms that cannot be questioned and ordinary premises that canbe questioned – and two types of rules – strict rules that cannot be questioned and defeasible
ones. We choose ASPIC+ since it allows to vary the form of the explanations in many ways (seeSection 3.2 and [6]).
An ASPIC+ setting starts from an argumentation system (AS = hL, R, ni), which contains
a logical language L closed under negation (¬), a set of rules R = Rs ∪ Rd consisting of strict(Rs ) and defeasible (Rd ) rules and a naming convention n for defeasible rules. Arguments areconstructed in an argumentation setting from a knowledge base K ⊆ L which consists of two
disjoint subsets K = Kp ∪ Kn : the set of axioms (Kn ) and the set of ordinary premises (Kp ).
Definition 3. An argument A on the basis of a knowledge base K in an argumentation system
hL, R, ni is:
1. φ if φ ∈ K, where Prem(A) = Sub(A) = {φ}, Conc(A) = φ, Rules(A) = ∅ and TopRule(A) =undefined;
2. A1 , . . . , An
ψ, where ∈ {→, ⇒}, if A1 , . . . , An are arguments such that there exists arule Conc(A1 ), . . . , Conc(An )ψ in Rs if= → and in Rd if= ⇒.
Prem(A) = Prem(A1 )∪. . .∪Prem(An ); Conc(A) = ψ; Sub(A) = Sub(A1 )∪. . .∪Sub(An )∪{A};
Rules(A) = Rules(A1 )∪. . .∪Rules(An )∪{Conc(A1 ), . . . , Conc(An )ψ}; DefRules(A) = {r ∈Rd | r ∈ Rules(A)}; and TopRule(A) = Conc(A1 ), . . . , Conc(An )ψ.
The above S
notation can be generalized to sets. For example, where S is a set of arguments
Prem(S) = {Prem(A) | A ∈ S} and Conc(S) = {Conc(A) | A ∈ S}.
Attacks on an argument are based on the rules and premises applied in the construction ofthat argument.
Definition 4. Let A and B be two arguments, we denote ψ = −φ if ψ = ¬φ or φ = ¬ψ. A
attacks an argument B iff A undercuts, rebuts or undermines B:• A undercuts B (on B 0 ) iff Conc(A) = −n(r) for some B 0 ∈ Sub(B) such that B 0 ’s top rule r
is defeasible, it denies a rule;• A rebuts B (on B 0 ) iff Conc(A) = −φ for some B 0 ∈ Sub(B) of the form B100 , . . . , Bn00 ⇒ φ, itdenies a conclusion;
3 In [6], four acceptance strategies are considered, the other two are not relevant for our study on necessity andsufficiency and are therefore not introduced here.

3• A undermines B (on φ) iff Conc(A) = −φ for some φ ∈ Prem(B) \ Kn , it denies a premise.
Argumentation theories and their corresponding Dung-style argumentation frameworks can
now be defined.
Definition 5. An argumentation theory is a pair AT = hAS, Ki, where AS is an argumentation
system and K is a knowledge base.
From an argumentation theory AT the corresponding AF can be derived such that AF(AT) =hArgs, Atti, where Args is the set of arguments constructed from AT and (A, B) ∈ Att iff A, B ∈ Args
and A attacks B as defined in Definition 4.
Example 3. Let AS1 = hL1 , R1 , ni where the rules in R1 are such that, with K1 = Kn1 = {r, s, t, v}
the following arguments can be derived:4d

A : s, t ⇒1 ud

D : v ⇒4 ¬qd

B : p, ¬q ⇒2 ¬n(d1 )dE : r, t ⇒5 ¬pd

C : r, s ⇒3 qd

F : v ⇒6 p.

The graphical representation of the corresponding argumentation framework AF(AT1 ) with AT1 =hAS1 , K1 i is the graph from Figure 1.
Dung-style semantics (Definition 1) can be applied in the same way as they are applied toabstract argumentation frameworks (recall Example 2). In addition to (non-)acceptance of arguments, in a structured setting we can also consider (non-)acceptance of formulas:
Definition 6. Let AF(AT) = hArgs, Atti be an AF, based on AT, let Sem be a semantics suchthat Sem(AF) 6= ∅ and let φ ∈ L. Then φ is:S• accepted : if φ ∈ Concs(Sem(AF(AT))), that is: there is some argument with conclusion φthat is accepted;T• non-accepted : if φ ∈/ Concs(Sem(AF(AT))), that is: there is some Sem-extension without
an argument with conclusion φ.
Example 4. As was the case for arguments (recall Example 2), all formulas in {p, ¬p, q, ¬q, r, s, t, u, v}
are accepted and {p, ¬p, q, ¬q, u} are also non-accepted.

3.2

Basic explanations

In [6] four types of explanations for abstract and structured argumentation were introduced. These
explanations are defined in terms of two functions: D, which determines the arguments that arein the explanation and F, which determines what elements of these arguments the explanation
presents. For the basic explanations in this paper, we instantiate D with the following functions,let A ∈ Args and E ∈ Prf(AF) for some AF AF = hArgs, Atti:5• Defending(A) = {B ∈ Args | B defends A} denotes the set of arguments that defend A andDefending(A, E) = Defending(A) ∩ E denotes the set of arguments that defend A in E.
• NoDefAgainst(A, E) = {B ∈ Args | B attacks A and E does not defend Aagainst B} denotes
the set of all attackers of A that are not defended by E.
The explanations are defined for arguments and formulas.
4 We ignore the arguments based on the elements from K , since these neither attack nor are attacked by any
1argument.
5 We write that B ∈ Args defends A ∈ Args if it attacks an attacker of A or it defends an argument that defends
A.

4

Definition 7. Let AF = hArgs, Atti be an AF and suppose that A ∈ Args [resp. φ ∈ L] is accepted
w.r.t. Sem. Then:
SemAcc(A) = {Defending(A, E) | E ∈ Sem(AF) and A ∈ E}.
SemAcc(φ) = {F(Defending(A, E)) | E ∈ Sem(AF) such that A ∈ E andConc(A) = φ} .
An acceptance explanation, for an argument or formula, contains all the arguments that defend
the argument (for that formula) in an extension. If it is an explanation for a formula, the function
F can be applied to it.
Definition 8. Let AF = hArgs, Atti be an AF and suppose that A ∈ Args [resp. φ ∈ L] is
non-accepted w.r.t. Sem. Then:[
SemNotAcc(A) =NoDefAgainst(A, E).
E∈Sem(AF ) and A∈E/

SemNotAcc(φ) =[[A∈Args and Conc(A)=φE∈Sem(AF ) and A∈E/

F(NoDefAgainst(A, E)).

A non-acceptance explanation contains all the arguments that attack the argument [resp. anargument for the formula] and to which no defense exists in some Sem-extension. For a formula F
can be applied again.
The function F can be instantiated in different ways. We recall here some of the variations
introduced in [6].
• F = id, where id(S) = S. Then explanations are sets of arguments.
• F = Prem. Then explanations only contain the premises of arguments (i.e., knowledge baseelements).
• F = AntTop, where AntTop(A) = hTopRule(A), Ant(TopRule(A))i. Then explanations contain the last applied rule and its antecedents.
• F = SubConc, where SubConc(A) = {Conc(B) | B ∈ Sub(A), Conc(B) ∈/ K ∪ {Conc(A)}}.
Then the explanation contains the sub-conclusions that were derived in the construction of
the argument.
Example 5. Consider the AF AF(AT1 ) from Examples 1 and 3. We have that:• PrfAcc(A) ∈ {{C}, {E}, {C, E}} and PrfAcc(B) = {D, F };• PrfNotAcc(A) = {B, D, F } and PrfNotAcc(B) = {C, E}.
If we take F = Prem, then: PrfAcc(u) ∈ {{r, s}, {r, t}, {r, s, t}}, PrfAcc(¬n(d1 )) = PrfNotAcc(u) ={v} and PrfNotAcc(¬n(d1 )) = {r, s, t}.
A conclusion derived from an argumentation system can have many causes and therefore many
explanations. When humans derive the same conclusion and are asked to explain that conclusion
they are able to select the explanation from all the possible explanations. In the social sciencesa large amount of possible selection criteria that humans might apply have been investigated,see [16] for an overview. In this paper we focus on necessity and sufficiency.

4

Necessity and Sufficiency

Necessity and sufficiency in the context of philosophy and cognitive science are discussed in, forexample, [14, 15, 21]. Intuitively, an event Γ is sufficient for ∆ if no other causes are required for∆ to happen, while Γ is necessary for ∆, if in order for ∆ to happen, Γ has to happen as well.
5In the context of logical implication (denoted by →), one could model sufficiency by Γ → ∆ andnecessity by ∆ → Γ [13].
In the next sections we formulate these logical notions in our argumentation setting. We willassume that the arguments on which the explanation for an argument A is based are relevant for
A: B ∈ Args [resp. S ⊆ Args] is relevant for A if B (in)directly attacks or defends A (i.e., there isa path from B to A) and does not attack itself [resp. for each C ∈ S, C is relevant for A].

4.1

Necessity and Sufficiency for AcceptanceIn the context of argumentation, a set of accepted arguments is sufficient if it guarantees, independent of the status of other arguments, that the considered argument is accepted, while an accepted
argument is necessary if it is impossible to accept the considered argument without it.
Definition 9. Let AF = hArgs, Atti be an AF and let A ∈ Args be accepted (w.r.t. some Sem).
Then:• S ⊆ Args is sufficient for the acceptance of A if S is relevant for A, S is conflict-free and S
defends A against all its attackers;• B ∈ Args is necessary for the acceptance of A if B is relevant for A and if B ∈/ E for someE ∈ Adm(AF), then A ∈/ E.
We denote by Suff(A) = {S ⊆ Args | S is sufficient for the acceptance of A} the set of all sufficient
sets of arguments for the acceptance of A and by Nec(A) = {B ∈ Args | B is necessary for theacceptance of A} the set of all necessary arguments for the acceptance of A.
Example 6. In AF 1 both {C} and {E} are sufficient for the acceptance of A but neither isnecessary, while for B, {D, F } is sufficient and D and F are necessary.
Necessary and sufficient explanations are now defined by replacing Defending in the basic
explanations from Section 3.2 with Nec resp. Suff.
Definition 10. Let AF = hArgs, Atti be an AF and let A ∈ Args [resp. φ ∈ L] be accepted. Thensufficient explanations are defined by:• Acc(A) ∈ Suff(A);S• Acc(φ) ∈ {F(Suff(A)) | A ∈ Args and Conc(A) = φ}.
Necessary explanations are defined by:• Acc(A) = Nec(A);T• Acc(φ) = {F(Suff(A)) | A ∈ Args and Conc(A) = φ}.
Example 7. For AF(AT1 ) we have that sufficient explanations are Acc(A) ∈ {{C}, {E}, {C, E},{C, F }, {D, E}}, Acc(B) = {D, F }, Acc(u) ∈ {{r, s}, {r, t}, {r, s, t}} and Acc(¬n(d1 )) = {v}.
Moreover, necessary acceptance explanations are Acc(A) = ∅, Acc(B) = {D, F }, Acc(u) = {r}and Acc(¬n(d1 )) = {v}.
Next we show that the sets in Suff(A) are admissible and contain all the needed arguments.
Additionally, we look at conditions under which Suff and Nec are empty, as well as the relation between Suff and Nec. These last results provide the motivation for the necessary formula acceptance
explanation.6
Proposition 1. Let AF = hArgs, Atti be an AF and let A ∈ Args be accepted w.r.t. some Sem ∈{Adm, Cmp, Grd, Prf, Sstb}. Then:
1. For all S ∈ Suff(A), {S, S ∪ {A}} ⊆ Adm(AF);6 Full proofs of our results can be found in Appendix A.

6

2. Suff(A) = ∅ iff there is no B ∈ Args such that (B, A) ∈ Att.
3. Nec(A) = ∅ iff there is no B ∈ Args such that (B, A) ∈ Att orT
4. Nec(A) ⊆ Suff(A).

T

Suff(A) = ∅.

The next proposition relates the introduced notions of necessity and sufficiency with Defending
and therefore with the basic explanations from Section 3.2.
Proposition 2. Let AF = hArgs, Atti be an AF and let A ∈ Args be accepted w.r.t. Sem ∈{Adm, Cmp, Grd, Prf, Stb}. Then:• for all E ∈ Sem(AF) such that A ∈ E, Defending(A, E) ∈ Suff(A);T• E∈Sem(AF ) and A∈E Defending(A, E) = Nec(A).

4.2

Necessity and Sufficiency for Non-Acceptance

When looking at the non-acceptance of an argument A, the acceptance of any of its direct attackers
is a sufficient explanation. However, other arguments (e.g., some of the indirect attackers) might
be sufficient as well. An argument is necessary for the non-acceptance of A, when it is relevantand A is accepted in the argumentation framework without it. In what follows we will assume
that (A, A) ∈/ Att, since otherwise A itself is the reason for its non-acceptance.
We need the following definition for our notion of sufficiency.
Definition 11. Let AF = hArgs, Atti be an AF and let A, B ∈ Args such that A indirectly attacks
B, via C1 , . . . , Cn ∈ Args, i.e., (A, C1 ), (C1 , C2 ), . . . , (Cn , B) ∈ Att. It is said that the attack fromA on B is uncontested if there is no D ∈ Args such that (D, C2i ) ∈ Att for i ∈ {1, . . . , n2 }. It iscontested otherwise, in which case it is said that the attack from A is contested in C2i .
The need for the above definition is illustrated in the next example:
Example 8. In AF 1 , the indirect attacks from D and F on A are contested: the attack from D iscontested in B, since (E, B) ∈ Att and the attack from F is also contested in B since (C, B) ∈ Att.
It is therefore possible that A and D or F are part of the same extension (recall Example 2).
For the definition of necessity for non-acceptance we define subframeworks, which are needed
because an argument might be non-accepted since it is attacked by an accepted or by anothernon-accepted argument.7
Definition 12. Let AF = hArgs, Atti be an AF and let A ∈ Args. Then AF ↓A = hArgs\{A}, Att∩
(Args \ {A} × Args \ {A})i denotes the AF based on AF but without A.
Since indirect attacks might be sufficient for not accepting an argument, but they also might
be contested, the definition of sufficiency for non-acceptance is defined inductively.
Definition 13. Let AF = hArgs, Atti be an AF and let A ∈ Args be non-accepted (w.r.t. Sem).
Then:• S ⊆ Args is sufficient for the non-acceptance of A if S is relevant for A and there is a B ∈ Ssuch that:– (B, A) ∈ Att; or– B indirectly attacks A and that attack is uncontested; or– B indirectly attacks A and for every argument C in which the attack from B on A
is contested and every D ∈ Args such that (D, C) ∈ Att, there is an S0 ⊆ S that is
sufficient for the non-acceptance of D.
7 In terms of labeling semantics (see e.g., [2]) an argument is non-accepted if it is out (i.e., attacked by an inargument) or undecided.

7• B ∈ Args is necessary for the non-acceptance of A if B is relevant for A and A is accepted
w.r.t. Sem in AF ↓B .
We denote by SuffNot(A) = {S ⊆ Args | S is sufficient for the non-acceptanceof A} the set of
all sufficient sets of arguments for the non-acceptance of A and by NecNot(A) = {B ∈ Args |B is necessary for the non-acceptance of A} the set of all necessary arguments for the non-acceptance of A.
Example 9. For AF 1 from Example 1 we have that B is both necessary and sufficient for the
non-acceptance of A. Moreover, while D and F are neither sufficient for the non-acceptance ofA, {D, F } is. For the non-acceptance of B we have that C and E are sufficient, but neither isnecessary.
Definition 14. Let AF = hArgs, Atti be an AF and let A ∈ Args [resp. φ ∈ L] be non-accepted.
Then sufficient explanations are defined by:• NotAcc(A) ∈ SuffNot(A);S• NotAcc(φ) ∈ {F(SuffNot(A)) | A ∈ Args and Conc(A) = φ};Necessary explanations are defined by:• NotAcc(A) = NecNot(A);T• NotAcc(φ) = {F(SuffNot(A)) | A ∈ Args and Conc(A) = φ}.
Example 10. For AF 1 we have, for sufficiency NotAcc(A) ∈ {{B}, {D, F }, {B, D, F }}, NotAcc(B) ∈{{C}, {E}, {C, E}}, NotAcc(u) = {v} and NotAcc(¬n(d1 )) ∈ {{r, s}, {r, t}, {r, s, t}} and for necessity NotAcc(B) = ∅ and NotAcc(u) = {v}.
The next propositions are the non-acceptance counterparts of Propositions 1 and 2. First somebasic properties of sufficiency and necessity for non-acceptance.
Proposition 3. Let AF = hArgs, Atti be an AF and let A ∈ Args be non-accepted w.r.t. Sem ∈{Adm, Cmp, Grd, Prf, Sstb}. Then: SuffNot(A) 6= ∅; and NecNot(A) = ∅ implies that there are atleast two direct attackers of A.
Now we show how NoDefAgainst (and hence the basic explanations from Section 3.2) is related
to our notions of sufficiency and necessity for non-acceptance.
Proposition 4. Let AF = hArgs, Atti be an AF and let A ∈ Args be an argument that is not
accepted w.r.t. Sem ∈ {Cmp, Grd, Prf, Sstb}. Then:• for all E ∈ Sem(AF) such that A ∈/ E, NoDefAgainst(A, E) ∈ SuffNot(A);T• NecNot(A) ⊆ E∈Sem(AF ) and A∈E/ NoDefAgainst(A, E).

4.3

Necessity, Sufficiency and Minimality

In this paper we have introduced necessity and sufficiency to reduce the size of an explanation.
More common in the literature is to place a minimality condition on the explanation [8, 9]. In
this section we show that our notions of necessity and sufficiency result in explanations that do
not contain more arguments than the minimal explanations from [8]. To this end we introduce,for ∈ {⊆, ≤}:• MinDefending (A, E) = {S ∈ Defending(A, E) | @S0 ∈ Defending(A, E) such that S0  S}
denotes the -minimal Defending(A, E) sets.8
8 In view of the result in [6], these sets correspond to the minimal ( = ≤ and where S ≤ S0 denotes |S| ≤ |S0 |)and compact ( = ⊆) explanations from [8].

8• MinNotDefAgainst (A, E) = {S ∈ NoDefAgainst(A, E) | there is no S0 ∈ NoDefAgainst(A, E)
such that S0  S}, denotes the set with all -minimal NoDefAgainst(A, E) sets.
• MinSuff  (A) = {S ∈ Suff(A) | @S0 ∈ Suff(A) such that S0  S}, denotes the set of all
-minimally sufficient sets for the acceptance of A.
• MinSuffNot (A) = {S ∈ SuffNot(A) | @S0 ∈ SuffNot(A) such that S0  S}, denotes the set ofall -minimally SuffNot sets for the non-acceptance of A.
The next example shows that sufficient explanations can be smaller than minimal basic explanations.
Example 11. Let AF 2 = hArgs2 , Att2 i, shown in Figure 2. Here we have that Prf(AF 2 ) ={{A, B}, {C, D}} and that PrfAcc(B) = {A, B}, PrfAcc(D) = {C, D}, PrfNotAcc(B) = {C, D}
and PrfNotAcc(D) = {A, B}. These are the explanations for B and D, whether as defined in Section 3.2 or with Defending [resp. NoDefAgainst] replaced by MinDefending [resp. MinNotDefAgainst].

A

C

B

D

Figure 2: Graphical representation of AF 2 .
When looking at minimally sufficient sets instead, we have that Acc(B) = {A} and NotAcc(D) ∈{{A}, {B}}. To see that these explanations are still meaningful, note that A defends B against
all of its attackers and as soon as A is accepted under complete semantics, B will be accepted
as well. Thus, the minimally sufficient explanations for B and D are ≤- and ⊆-smaller than theminimal basic explanations for B and D, but still meaningful.
That minimally sufficient explanations can be smaller than minimal explanations is formalized
in the next propositions.
Proposition 5. Let AF = hArgs, Atti be an AF, let A ∈ Args be accepted w.r.t. Sem ∈ {Adm,
Cmp, Grd, Prf, Sstb} and let ∈ {⊆, ≤}. Then:• for all E ∈ Sem(AF) and all S ∈ MinDefending (A, E) there is some S0 ∈ MinSuff  (A) suchthat S0  S;
• for all E ∈ Adm(AF) and all S ∈ MinSuff  (A) also S ∈ MinDefending (A, E);• for all E ∈ Sem(AF) and all S ∈ MinDefending (A, E), Nec(A) ⊆ S.
Proposition 6. Let AF = hArgs, Atti be an AF and let A ∈ Args be not accepted w.r.t. Sem ∈{Cmp, Grd, Prf, Sstb}. Then:• for all E ∈ Sem(AF) and all S ∈ MinNotDefAgainst (A, E), there is some S0 ∈ MinSuffNot (A)such that S0  S;
• for all E ∈ Sem(AF) and all S ∈ MinNotDefAgainst (A, E), NecNot(A) ⊆ S.

5

Applying Necessity and Sufficiency

At the Dutch National Police several argumentation-based applications have been implemented [4].
These applications are aimed at assisting the police at working through high volume tasks, leaving
more time for tasks that require human attention. In this section we illustrate how necessity andsufficiency can be applied in the online trade fraud application from [17].
Consider the following language L3 : the complainant delivered (cd ), the counterparty delivered(cpd ); the received product seems fake (fake); a package is expected (pex ); the complainant waited
9

before filing the complaint (wait); the received packages is indeed fake (recfake); the delivery may
still arrive (deco); it is a case of fraud (f ); and their negations. Based on Dutch Criminal Law(i.e., Article 326) we can derive the following arguments:
A1 : cpd

A2 : ¬cpd

A7 : wait

A8 : ¬wait

B2 : A2 , A6 ⇒ ¬deco
C1 : A9 , B1 ⇒ f
C4 : A9 , B4 ⇒ ¬f

A3 : fakeA4 : ¬fake

A5 : pex

A9 : cd

A10 : ¬cd

B1 : A1 , A3 ⇒ recfake

B3 : A2 , A5 , A7 ⇒ ¬deco

B4 : A5 , A8 ⇒ deco

C2 : A2 , A9 , B2 ⇒ f
C5 : A4 , A9 ⇒ ¬f

A6 : ¬pex

C3 : A2 , A9 , B3 ⇒ f
C6 : A10 ⇒ ¬f.

The above arguments are only a small subset of the possible arguments in the actual application,
yet this framework already results in 30 preferred and semi-stable extensions. We can therefore
not provide a detailed formal analysis. However, we can already show the usefulness of necessary
and sufficient explanations.
The necessary explanation for the acceptance of f is cd, while for the acceptance of ¬f thenecessary explanation is empty. The reason for this is that, by Article 326, the complainant musthave delivered (e.g., sent the goods or money) before it is a case of fraud but ¬f can be accepted
for a variety of reasons. In the basic explanations it is not possible to derive this explanation, yet
it can be the sole reason for not accepting f . Moreover, minimal sufficient explanations for theacceptance of ¬f when cd and F = Prem are {cd, pex, ¬wait} and {cd, ¬fake}, these are both ⊂and <-smaller than any basic explanation for the acceptance of ¬f , while still providing the main
reasons for the acceptance of ¬f .
Therefore, with necessary and sufficient explanations, we can provide compact explanations
that only contain the core reasons for a conclusion, something which is not possible with the(minimal) explanations from the basic framework.

6

Conclusion

We have discussed how the explanations from the basic framework introduced in [6] can be adjusted
to account for findings from the social sciences on necessary and sufficient explanations [14, 15, 21].
To this end we have introduced necessary and sufficient sets of arguments for the (non-)acceptanceof an argument and formula and integrated these into the explanations definition. The result is ameaningful reduction in the size of an explanation, which almost always exists. Moreover, we haveshown that our necessary and sufficient explanations can be smaller than the minimal explanations
from [8] and reduce the explanations to the core reasons of (not) accepting a conclusion in a real-life
application.
To the best of our knowledge this is the first investigation into necessary and sufficient setsfor (non-)acceptance of arguments, especially in the context of integrating findings from the social
sciences (e.g., [12]) into (explanations for) argumentation-based conclusions and into a real-life
application. In future work we plan to investigate how to integrate further findings, such as
contrastiveness and other selection mechanisms.
Acknowledgements. This research has been partly funded by the Dutch Ministry of Justiceand the Dutch National Police.

References[1] Katie Atkinson, Pietro Baroni, Massimiliano Giacomin, Anthony Hunter, Henry Prakken,
Chris Reed, Guillermo Simari, Matthias Thimm, and Serena Villata. Towards Artificial
Argumentation. AI magazine, 38(3):25–36, 2017.
[2] Pietro Baroni, Martin Caminada, and Massimilliano Giacomin. Abstract argumentation
frameworks and their semantics. In Pietro Baroni, Dov Gabay, Massimiliano Giacomin,
10and Leon van der Torre, editors, Handbook of Formal Argumentation, pages 159–236. College
Publications, 2018.
[3] Philippe Besnard, Alejandro Garcia, Anthony Hunter, Sanjay Modgil, Henry Prakken,
Guillermo Simari, and Francesca Toni. Introduction to structured argumentation. Argument& Computation, 5(1):1–4, 2014.
[4] Floris Bex, Bas Testerink, and Joeri Peters. AI for online criminal complaints: From naturaldialogues to structured scenarios. In Workshop proceedings of Artificial Intelligence for Justiceat ECAI 2016, pages 22–29, 2016.
[5] Guido Boella, Souhila Kaci, and Leendert van der Torre. Dynamics in argumentation withsingle extensions: Abstraction principles and the grounded extension. In Claudio Sossai
and Gaetano Chemello, editors, Proceedings of the 10th European Conference on Symbolicand Quantitative Approaches to Reasoning with Uncertainty (ECSQARU’09), volume 5590
of Lecture Notes in Computer Science, pages 107–118. Springer, 2009.
[6] AnneMarie Borg and Floris Bex. A basic framework for explanations in argumentation. IEEE
Intelligent Systems, 36(2):25–35, 2021.
[7] Phan Minh Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence, 77(2):321–357,
1995.
[8] Xiuyi Fan and Francesca Toni. On computing explanations in argumentation. In Blai Bonetand Sven Koenig, editors, Proceedings of the 29th AAAI Conference on Artificial Intelligence(AAAI’15), pages 1496–1502. AAAI Press, 2015.
[9] Xiuyi Fan and Francesca Toni. On explanations for non-acceptable arguments. In Elizabeth
Black, Sanjay Modgil, and Nir Oren, editors, Proceedings of the 3rd International Workshop
on Theory and Applications of Formal Argumentation, (TAFA’15), LNCS 9524, pages 112–
127. Springer, 2015.
[10] Alejandro Garcı́a, Carlos Chesñevar, Nicolás Rotstein, and Guillermo Simari. Formalizing
dialectical explanation support for argument-based reasoning in knowledge-based systems.
Expert Systems with Applications, 40(8):3233–3247, 2013.
[11] Carmen Lacave and Francisco J Diez. A review of explanation methods for heuristic expert
systems. The Knowledge Engineering Review, 19(2):133–146, 2004.
[12] Beishui Liao and Leendert van der Torre. Explanation semantics for abstract argumentation. In Henry Prakken, Stefano Bistarelli, Francesco Santini, and Carlo Taticchi, editors, Proceedings of the 8th International Conference on Computational Models of Argument(COMMA’20), volume 326 of Frontiers in Artificial Intelligence and Applications, pages 271–
282. IOS Press, 2020.
[13] Fangzhen Lin. On strongest necessary and weakest sufficient conditions. Artificial Intelligence,
128(1):143–159, 2001.
[14] Peter Lipton. Contrastive explanation. Royal Institute of Philosophy Supplement, 27:247–266,
1990.
[15] Tania Lombrozo. Causal-explanatory pluralism: How intentions, functions, and mechanisms
influence causal ascriptions. Cognitive psychology, 61(4):303–332, 2010.
[16] Tim Miller. Explanation in artificial intelligence: Insights from the social sciences. Artificial
Intelligence, 267:1–38, 2019.

11

[17] Daphne Odekerken, AnneMarie Borg, and Floris Bex. Estimating stability for efficientargument-based inquiry. In Henry Prakken, Stefano Bistarelli, Francesco Santini, and Carlo
Taticchi, editors, Proceedings of the 8th International Conference on Computational Models of
Argument (COMMA’20), volume 326 of Frontiers in Artificial Intelligence and Applications,pages 307–318. IOS Press, 2020.
[18] Henry Prakken. An abstract framework for argumentation with structured arguments. Argument & Computation, 1(2):93–124, 2010.
[19] Wojciech Samek, Thomas Wiegand, and Klaus-Robert Müller. Explainable artificial intelligence: Understanding, visualizing and interpreting deep learning models. arXiv preprint
arXiv:1708.08296, 2017.
[20] Zeynep Saribatur, Johannes Wallner, and Stefan Woltran. Explaining non-acceptability inabstract argumentation. In Proceedings of the 24th European Conference on Artificial Intelligence (ECAI’20), volume 325 of Frontiers in Artificial Intelligence and Applications, pages
881–888. IOS Press, 2020.
[21] James Woodward. Sensitive and insensitive causation. Philosophical Review, 115(1):1–50,
2006.

A

Proofs

Before we turn to the proofs, we introduce some useful notation:• SemWith(A) = {E ∈ Sem(AF) | A ∈ E} denotes the set of all Sem-extensions that containthe argument A.
• SemWithout(A) = {E ∈ Sem(AF) | A ∈/ E} denotes the set of all Sem-extensions that do notcontain the argument A.
Proposition 1. Let AF = hArgs, Atti be an AF and let A ∈ Args be accepted w.r.t. some Sem ∈{Adm, Cmp, Grd, Prf, Sstb}. Then:
1. For all S ∈ Suff(A), {S, S ∪ {A}} ⊆ Adm(AF);
2. Suff(A) = ∅ iff there is no B ∈ Args such that (B, A) ∈ Att.
3. Nec(A) = ∅ iff there is no B ∈ Args such that (B, A) ∈ Att orT
4. Nec(A) ⊆ Suff(A).

T

Suff(A) = ∅.

Proof. Let AF = hArgs, Atti be an AF and let A ∈ Args be accepted w.r.t. some Sem.
1. Let S ∈ Suff(A). Note that, by definition, S is conflict-free. If there is some B ∈ S such that
(B, A) ∈ Att then there is some C ∈ S that defends against this attack (i.e., (C, B) ∈ Att),a contradiction. If, there is some B ∈ S such that (A, B) ∈ Att, then A indirectly attacks
itself. Since there is no D ∈ S such that (D, A) ∈ Att it follows that B is not defended
against the attack from A. A contradiction with the definition of S that it defends A against
all attackers. Hence S ∪ {A} is conflict-free.
Now suppose that there is some B ∈ Args and some C ∈ S such that (B, C) ∈ Att. Since C(in)directly defends A, B indirectly attacks A. By definition of a sufficient set of arguments
S defends A against B. It follows that there is some D ∈ S such that (D, B) ∈ Att. Hence S
defends A and all its own elements against any attacker. Therefore {S, S∪{A}} ⊆ Adm(AF).
2. Suppose that Suff(A) = ∅, then there is no S ⊆ Args such that S is relevant for A anddefends A against all its arguments. Since A is accepted by assumption, it follows that A is
not attacked at all. Now suppose that there is no B ∈ Args such that (B, A) ∈ Att. Then
there is no S ⊆ Args that is relevant for A and hence Suff(A) = ∅.
12

3. First suppose that Nec(A) = ∅. Then there is no argument relevant for A (from which it
follows
T that there is no B ∈ Args such that (B, A) ∈ Att) or there is no B ∈ Args such thatB ∈ SemWith(A).
T Note that for each S ∈ Suff(A)T there is some E ∈ SemWith(A) suchthat S ⊆ E. Since SemWith(A) = ∅ it follows that Suff(A) = ∅ as well.
For the other direction suppose first that A is not attacked at all, then there is no argument
relevant for A from which it follows that Nec(A) = ∅. Now suppose that Suff(A) = ∅. By
assumption A is accepted and A is attacked, hence Suff(A) 6= ∅. It follows that for eachS ∈ Suff(A) and for each B ∈ S there is an S0 ∈ Suff(A) such that B ∈/ S0 and therefore also
an E ∈ Adm(AF) with B ∈/ E but A ∈ E. Therefore none of the arguments is necessary:
Nec(A) = ∅.
4. In view of the above two items,
T suppose that A is attacked by some argument. Let B ∈Nec(A) and suppose that B ∈/ Suff(A). Then there is some S ∈ Suff(A) such that B ∈/ S.
Note that S ∪ {A} ∈ Adm(AF). However, B ∈/ S, a contradiction with B ∈ Nec(A).
Proposition 2. Let AF = hArgs, Atti be an AF and let A ∈ Args be accepted w.r.t. Sem ∈{Adm, Cmp, Grd, Prf, Stb}. Then:• for all E ∈ Sem(AF) such that A ∈ E, Defending(A, E) ∈ Suff(A);T• E∈Sem(AF ) and A∈E Defending(A, E) = Nec(A).
Proof. Let AF = hArgs, Atti be an AF and let A ∈ Args be an argument that is accepted w.r.t.
Sem. Consider both items.
• Since A is accepted, there is some E ∈ Sem(AF) such that A ∈ E. Let T = Defending(A, E).
By definition, T is relevant for A (i.e., all B ∈ T (in)directly defend A and since B ∈ E,
(B, B) ∈/ Att). Now suppose that there is some C ∈ Args such that C attacks A and A is
not defended by T. By assumption A ∈ E. Hence there is a D ∈ E such that (D, C) ∈ Att.
But then D (in)directly defends A and therefore D ∈ T. Thus T defends A against all its
attackers and therefore T ∈ Suff(A, ∅).
T• Let T = E∈SemWith(A) Defending(A, E), since A is accepted, SemWith(A) 6= ∅. Suppose
there is some B ∈ T which is not necessary for the acceptance of TA. Then there is an
E ∈ SemWith(A) such that B ∈/ E. However, by definition of T, B ∈ SemWith(A). Hence
T is necessary for A. To see that T contains all the necessary arguments, assume it doesnot. Then there is some B ∈ Args such that B ∈/ T but B is necessary for the acceptance ofA. However, since B ∈/ T, there is some E ∈ SemWith(A) such that B ∈/ E, but A ∈ E. A
contradiction.
Proposition 3. Let AF = hArgs, Atti be an AF and let A ∈ Args be non-accepted w.r.t. Sem ∈{Adm, Cmp, Grd, Prf, Sstb}. Then: SuffNot(A) 6= ∅; and NecNot(A) = ∅ implies that there are atleast two direct attackers of A.
Proof. Let AF = hArgs, Atti be an AF and let A ∈ Args be an argument that is not accepted
w.r.t. Sem ∈ {Adm, Cmp, Grd, Prf, Sstb}.
• Suppose that Suff(A) = ∅. Then there is no S ⊆ Args that is relevant for A and in which
B ∈ Args (in)directly attacks A. It follows that there is no B ∈ Args such that (B, A) ∈ Att.
A contradiction with the assumption that A is non-accepted and that (A, A) ∈/ Att.
• It follows that there are B1 , . . . , Bn ∈ Args such that (B1 , A), . . . , (Bn , A) ∈ Att. Assume
that NecNot(A, ∅) = ∅ but that n = 1. Since by assumption in this section (A, A) ∈/ Att, itfollows that A is not attacked in AF ↓B1 and should therefore be accepted in any complete
extension. Hence n ≥ 2.
Proposition 4. Let AF = hArgs, Atti be an AF and let A ∈ Args be an argument that is not
accepted w.r.t. Sem ∈ {Cmp, Grd, Prf, Sstb}. Then:13• for all E ∈ Sem(AF) such that A ∈/ E, NoDefAgainst(A, E) ∈ SuffNot(A);T• NecNot(A) ⊆ E∈Sem(AF ) and A∈E/ NoDefAgainst(A, E).
In order to prove the proposition, we first need the following lemma:9
Lemma 1. Let AF = hArgs, Atti, E ∈ Sem(AF) for some Sem ∈ {Adm, Cmp, Grd, Prf, Sstb} andA ∈ Args. If there is a B ∈ E such that (B, A) ∈ Att, then E ∈ Sem(AF ↓A ).
Proof. Let AF = hArgs, Atti, E ∈ Sem(AF) for some Sem and A, B ∈ Args such that B ∈ E and(B, A) ∈ Att. Note that E is still admissible in AF ↓A since no new attacks are added.
Sem ∈ {Cmp, Prf}. Now suppose there is some C ∈ Args such that C ∈/ E but C is defended by
E in AF ↓A . If C is not attacked at all in AF ↓A , since C ∈/ E, (A, C) ∈ Att, but then E defends C in
AF, a contradiction. Hence there is some D ∈ Args such that (D, C) ∈ Att and E defends against
this attack in AF ↓A , but then E would defend C in AF as well. Again a contradiction. Hence
E is complete in AF ↓A and if E was maximally complete in AF it is still maximally complete in
AF ↓A .
Sem = Sstb. Any argument, other than A, attacked by E is still attacked by E in AF ↓A . Since
E is still complete, it follows that E is also still semi-stable.
We now turn to the proof of the proposition itself.
Proof. Let AF = hArgs, Atti be an AF and let A ∈ Args be non-accepted w.r.t. Sem ∈ {Cmp, Grd,
Prf, Sstb}. Consider both items:• By definition of NoDefAgainst, T = NoDefAgainst(A, E) is relevant for A. We show that there
is a B ∈ T such that (B, A) ∈ Att. Suppose there is no such B, then A is not attacked at
all or E defends A against all its direct attackers and therefore against all its attackers, bothare a contradiction with the completeness of E. Hence there is such a B ∈ T. From which it
follows that NoDefAgainst(A, E) ∈ SuffNot(A).
T• Let B ∈ NecNot(A) and suppose that B ∈/ E∈SemWithout(A) NoDefAgainst(A). Then there issome E ∈ SemWithout(A) such that B ∈/ NoDefAgainst(A, E). By assumption, B is relevantfor A and thus (in)directly attacks A. From which it follows that there is some C ∈ E suchthat (C, B) ∈ Att. By Lemma 1, E ∈ Sem(AF ↓B ), a contradiction with the assumption thatB ∈ NecNot(A).
Proposition 5. Let AF = hArgs, Atti be an AF, let A ∈ Args be accepted w.r.t. Sem ∈ {Adm,
Cmp, Grd, Prf, Sstb} and let ∈ {⊆, ≤}. Then:• for all E ∈ Sem(AF) and all S ∈ MinDefending (A, E) there is some S0 ∈ MinSuff  (A) suchthat S0  S;
• for all E ∈ Adm(AF) and all S ∈ MinSuff  (A) also S ∈ MinDefending (A, E);• for all E ∈ Sem(AF) and all S ∈ MinDefending (A, E), Nec(A) ⊆ S.
Proof. Let AF = hArgs, Atti be an AF and let A ∈ Args be accepted w.r.t. Sem ∈ {Adm, Cmp, Grd,
Prf, Sstb}:• Let S ∈ MinDefending (A, E), then S = Defending(A, E) for some E ∈ SemWith(A). By
Proposition 2 it follows that S ∈ Suff(A). Hence there is some S0 ∈ MinSuff  (A) such thatS0  S, for any of the considered semantics.
9 For Sem = Grd this lemma was shown in [5].

14• Let Sem = Adm and S ∈ MinSuff  (A). By Proposition 1, S ∪ {A} ∈ Adm(AF) and bydefinition of a sufficient set of arguments, S defends A against all its attackers. Therefore Defending(A,S ∪ {A}) = S. Suppose that S ∪ {A} is such that Defending(A, S ∪S{A}) ∈/ E∈Adm(AF ) MinDefending (A, E). Then there is some E ∈ AdmWith(A) such that
Defending(A, E) ≺ Defending(A, S ∪ {A}). By Proposition 2, Defending(A, E) ∈ Suff(A). A
contradiction since S ∈ MinSuff  (A) and Defending(A, E) ≺ S.
• This follows immediately from the second item in Proposition 2.
Proposition 6. Let AF = hArgs, Atti be an AF and let A ∈ Args be not accepted w.r.t. Sem ∈{Cmp, Grd, Prf, Sstb}. Then:• for all E ∈ Sem(AF) and all S ∈ MinNotDefAgainst (A, E), there is some S0 ∈ MinSuffNot (A)such that S0  S;
• for all E ∈ Sem(AF) and all S ∈ MinNotDefAgainst (A, E), NecNot(A) ⊆ S.
Proof. Let AF = hArgs, Atti be an AF and let A ∈ Args be non-accepted w.r.t. Sem ∈ {Cmp, Grd,
Prf, Sstb}:• Let S ∈ MinNotDefAgainst (A, E) for some E ∈ SemWithout(A). Note that we have S =NoDefAgainst(A, E). Hence, by Proposition 4 it follows that S ∈ SuffNot(A). Hence, there issome S0 ∈ MinSuffNot (A) such that S0  S, for any of the considered semantics.
• This follows from the second item in Proposition 4.

15