=Paper=
{{Paper
|id=Vol-3086/paper4
|storemode=property
|title=On Weak Constrained Argumentation Frameworks
|pdfUrl=https://ceur-ws.org/Vol-3086/paper4.pdf
|volume=Vol-3086
|authors=Gianvincenzo Alfano,Sergio Greco,Francesco Parisi,Irina Trubitsyna
|dblpUrl=https://dblp.org/rec/conf/aiia/AlfanoGPT21
}}
==On Weak Constrained Argumentation Frameworks==
<pdf width="1500px">https://ceur-ws.org/Vol-3086/paper4.pdf</pdf>
<pre>
On Weak Constrained Argumentation Frameworks
Gianvincenzo Alfano, Sergio Greco, Francesco Parisi and Irina Trubitsyna
Department of Informatics, Modeling, Electronics and System Engineering (DIMES),
University of Calabria, Rende, Italy


                                      Abstract
                                      The success and popularity of Dung’s abstract Argumentation Framework (AF) is also due to its simplicity
                                      and expressiveness. Integrity constraints help to express domain knowledge in a compact and natural way,
                                      thus keeping easy the modeling task even for problems that otherwise would be hard to encode within an
                                      AF. Constraints can be expressed in the so-called Constrained Argumentation Framework (CAF). Although
                                      constraints in CAF allow restricting the set of feasible solutions, they can not be used to find “optimal”
                                      solutions to problems defined through CAFs. In this paper we present Weak constrained AFs (WAFs) that
                                      enhance CAFs with weak constraints, that express some optimal conditions. We discuss the complexity
                                      of WAFs under several well-known argumentation semantics, showing that weak constraints increase the
                                      expressive power of AFs and CAFs.

                                      Keywords
                                      Abstract Argumentation, Weak Constraints, Complexity.




1. Introduction
Despite the expressive power and generality of Dung’s abstract Argumentation Framework
(AF) [1], in some cases it is difficult to accurately model domain knowledge by an AF in a
natural and easy-to-understand way. For this reason, AF has been extended by the introduction
of further constructs, such as preferences [2, 3] and integrity constraints [4, 5], to achieve more
comprehensive, natural, and compact ways of representing useful relationships among arguments.
In particular, enhancing AFs with constraints allows us to naturally and compactly express domain
conditions that need to be taken into account to filter out unfeasible solutions, as illustrated in
what follows.
Example 1. Assume three people 𝑥, 𝑦, 𝑧 wish to attend a theatre event, but only two seats are
available. We could try to model this situation by an AF Λ with arguments 𝑥, 𝑦, 𝑧 (resp., 𝑥        ¯ , 𝑦¯, 𝑧¯),
each stating that 𝑥, 𝑦, 𝑧 attends (resp., does not attend) the event. The direct graph encoding Λ
is shown in Figure 1(a), where double arrows are used to represent mutually attacks between
arguments. Specifically, argument 𝑥 (resp., 𝑦, 𝑧) attacks and is attacked by its opposite argument
modeling the fact that only one of them can be accepted. Moreover, argument 𝑥            ¯ (resp., 𝑦¯, 𝑧¯) is
attacked by the other two arguments 𝑦¯ and 𝑧¯ (resp., 𝑥  ¯ and 𝑧¯; 𝑥
                                                                   ¯ and 𝑦¯) since 𝑥 (resp., 𝑦, 𝑧) can be
accepted only if one of the two arguments 𝑦¯ and 𝑧¯ (resp., 𝑥  ¯ and 𝑧¯; 𝑥
                                                                         ¯ and 𝑦¯) is accepted. Thus, the
set of attacks between every pair in {𝑥¯ , 𝑦¯, 𝑧¯} models the fact that at most one argument among 𝑥         ¯,
5𝑡ℎ Workshop on Advances In Argumentation In Artificial Intelligence (AI3 2021), December 1-3, 2021, Virtual
" 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)
                                    © 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
 CEUR
 Workshop
 Proceedings
               http://ceur-ws.org
               ISSN 1613-0073
                                    CEUR Workshop Proceedings (CEUR-WS.org)
                      x                     y
                                                      x                     y
                            x̄         ȳ
                                                            x̄         ȳ
                                 z̄                         z̄         w̄

                                                      z                     w
                                 z
                                 (a)                             (b)
Figure 1: (a) AF Λ of Example 1; (b) AF Λ′ of Example 4.


𝑦¯ and 𝑧¯ can be accepted and then, as a consequence, at least two arguments among 𝑥, 𝑦 and 𝑧 can
be accepted.                                                                                   □
   The meaning of an AF is given in terms of argumentation semantics, which intuitively tell us
the sets of arguments (called extensions) that can collectively be accepted. The preferred (and
stable) extensions of AF Λ of Example 1 are 𝐸1 = {𝑥, 𝑦, 𝑧¯}, 𝐸2 = {𝑥, 𝑦¯, 𝑧}, 𝐸3 = {𝑥    ¯ , 𝑦, 𝑧},
and 𝐸4 = {𝑥, 𝑦, 𝑧}. However, 𝐸4 is not feasible as only two seats are available and thus only
two people could attend the event. To overcome such a situation, and thus providing a natural and
compact way for expressing such kind of conditions, the use of constraints has been proposed
[4, 6, 5, 7].
Example 2. Continuing from Example 1, the constraint 𝜅 = 𝑥 ∧ 𝑦 ∧ 𝑧 ⇒ f can be used. It states
that the propositional formula 𝑥 ∧ 𝑦 ∧ 𝑧 must be false. That is, feasible solutions must satisfy the
condition that the 3 arguments 𝑥, 𝑦, and 𝑧 are not jointly accepted, i.e., the three people cannot
attend the event together. The effect of using constraint 𝜅 is that 𝐸4 is discarded from the set of
solutions of our problem.                                                                         □
   We call an AF augmented with constraints a Constrained AF (CAF). Although constraints in a
CAF allow restricting the set of feasible solutions, they do not help in finding “best” or preferable
solutions. Considering our running example, the three people may agree on the fact that “𝑥 and
𝑦 should preferably attend the event whenever there are only two seats available”. To express
this kind of conditions, we introduce weak constraints, that is, constraints that are required to be
satisfied if possible. Syntactically, they have the same form of (strong) constraints except that the
implication symbol → is used (instead of ⇒). Intuitively, these constraints can be used to find
“optimal” solutions to a problem defined by means of an AF or a CAF. A CAF with the addition
of weak constraints is said to be a Weak constrained Argumentation Framework (WAF).
Example 3. Consider a WAF obtained by adding to the CAF of Example 2 the weak constraint
t → 𝑥 ∧ 𝑦, stating that it is desirable that 𝑥 and 𝑦 attend the event together. Herein, t denotes the
truth value true. Then, extension 𝐸1 = {𝑥, 𝑦, 𝑧¯} is selected as the “best” one.                   □
  The use of strong and weak constraints substantially reduces the effort needed to figure out
how to define an AF that models a given problem. In fact, as said earlier, constraints facilitate to
express knowledge in a more compact and easy to understand way. For instance, the problem
presented in Example 1, has been represented through an AF which expresses the condition that
“at most one argument among 𝑥     ¯ , 𝑦¯, 𝑧¯ can be accepted” and then, as a consequence, at least two
arguments 𝑥, 𝑦, 𝑧 can be accepted. However, this condition is not easy to be generalized if we
have more than three people. Suppose there is a fourth guy, 𝑤, who wish to attend the event, and
there are again only two available seats. After adding the arguments 𝑤 and 𝑤          ¯ to Λ of Figure 1(a),
we cannot use the same reasoning as in Example 1 to model the fact that two of the four people
attend the event. In fact, having the attacks between every pair in {𝑥                  ¯ } does not model
                                                                            ¯ , 𝑦¯, 𝑧¯, 𝑤
this situation (it models that at least three of the four people attend the event). Remarkably, using
strong and weak constraints allow for using a common reasoning pattern to generalize to this
more complex situation, even starting from an AF having a simpler structure.
Example 4. Consider a WAF consisting of AF Λ′ of Figure 1(b) and the sets of strong and weak
constraints; 𝒞 = {𝜅, 𝑥 ∧ 𝑦 ∧ 𝑤 ⇒ f, 𝑥 ∧ 𝑧 ∧ 𝑤 ⇒ f, 𝑦 ∧ 𝑧 ∧ 𝑤 ⇒ f}; and 𝒲 = {t → 𝑥, t →
𝑦, t → 𝑧, t → 𝑤}. The strong constraints in 𝒞 (that includes 𝜅 of Example 2) filter out from
the (16 preferred) extensions of Λ′ the solutions where more than two people attend the event,
whereas the weak constraints maximize the set (or number) of people attending the event.   □

   In this paper, we present WAFs along with two criteria for interpreting weak constraints, under
any argumentation semantics 𝒮: maximal-set (ms𝒮) and maximum-cardinality (mc𝒮) according
to which the best/optimal 𝒮-extensions are those satisfying a maximal set, or a maximum number,
of weak constraints respectively. Then, we discuss the complexity of credulous and skeptical
reasoning for WAFs, showing that the introduction of weak constraints typically increases the
complexity of at least one step in the polynomial hierarchy w.r.t. AF.


2. Preliminaries
We briefly review Dung’s framework and discuss Constrained Argumentation Frameworks.

2.1. Argumentation Frameworks
An abstract Argumentation Framework (AF) is a pair ⟨𝒜, ℛ⟩, where 𝒜 is a set of arguments and
ℛ ⊆ 𝒜 × 𝒜 is a set of attacks. If (𝑎, 𝑏) ∈ ℛ then we say that 𝑎 attacks 𝑏. We can think of an AF
as a directed graph whose nodes represent arguments and edges represent attacks.
   Given an AF Λ = ⟨𝒜, ℛ⟩ and a set 𝑆 ⊆ 𝒜 of arguments, an argument 𝑎 ∈ 𝒜 is said to be
i) defeated w.r.t. 𝑆 iff ∃𝑏 ∈ 𝑆 such that (𝑏, 𝑎) ∈ ℛ, and ii) acceptable w.r.t. 𝑆 iff for every
argument 𝑏 ∈ 𝒜 with (𝑏, 𝑎) ∈ ℛ, there is 𝑐 ∈ 𝑆 such that (𝑐, 𝑏) ∈ ℛ. The sets of defeated and
acceptable arguments w.r.t. 𝑆 are defined as follows (where Λ is understood):

     • 𝐷𝑒𝑓 (𝑆) = {𝑎 ∈ 𝒜 | ∃𝑏 ∈ 𝑆 . (𝑏, 𝑎) ∈ ℛ};
     • 𝐴𝑐𝑐(𝑆) = {𝑎 ∈ 𝒜 | ∀𝑏 ∈ 𝒜 . (𝑏, 𝑎) ̸∈ ℛ ∨ 𝑏 ∈ 𝐷𝑒𝑓 (𝑆)}.

  Given an AF ⟨𝒜, ℛ⟩, a set 𝑆 ⊆ 𝒜 of arguments is said to be: conflict-free iff 𝑆 ∩ 𝐷𝑒𝑓 (𝑆) = ∅;
∙ admissible iff it is conflict-free and 𝑆 ⊆ 𝐴𝑐𝑐(𝑆).
   Different argumentation semantics have been proposed to characterize collectively acceptable
sets of arguments, called extensions [1, 8]. Every extension is an admissible set satisfying
additional conditions. Specifically, the complete, preferred, stable, semi-stable, and grounded
extensions of an AF are defined as follows.
  Given an AF ⟨𝒜, ℛ⟩, a set 𝑆 ⊆ 𝒜 is an extension called:

    • complete (co) iff it is an admissible set and 𝑆 = 𝐴𝑐𝑐(𝑆);
    • preferred (pr) iff it is a maximal (w.r.t. ⊆) complete extension;
    • stable (st) iff it is a total preferred extension, i.e. a preferred extension such that 𝑆 ∪
      𝐷𝑒𝑓 (𝑆) = 𝒜;
    • semi-stable (sst) iff it is a preferred extension such that 𝑆 ∪ 𝐷𝑒𝑓 (𝑆) is maximal;
    • grounded (gr) iff it is the smallest (w.r.t. ⊆) complete extension.

It is well-known that the set of complete extensions forms a complete semilattice with respect to
set inclusion. Arguments occurring in an extension are said to be accepted, whereas arguments
attacked by accepted arguments are said to be rejected; remaining arguments are said to be
undecided (w.r.t. the considered extension).

Example 5. Let Λ = ⟨𝒜, ℛ⟩ be an AF where 𝒜 = {𝑎, 𝑏, 𝑐} and ℛ = {(𝑎, 𝑏), (𝑏, 𝑎), (𝑏, 𝑐), (𝑐, 𝑐)}.
AF Λ has three complete extensions: 𝐸1 = ∅, 𝐸2 = {𝑎}, 𝐸3 = {𝑏}. The set of preferred ex-
tensions is {𝐸2 , 𝐸3 }, whereas the set of stable (and semi-stable) extensions is {𝐸3 }, and the
grounded extension is 𝐸1 .                                                                    □

2.2. Constrained Argumentation Frameworks
Constrained Argumentation Framework (CAF) has been introduced in [4] and further investigated
in [5, 9]. The constrained argumentation frameworks in [6] and [7] are particular cases of those
in [5] as the set of constraints is restricted to atomic formulae only.
   Given a set of propositional symbols 𝑆, ℒ𝑆 denotes the propositional language defined in the
usual inductive way from 𝑆 using the built-in constants f, u, and t denoting the truth values
false, undef (undefined), and true, and the connectives ∧, ∨, ¬, ⇒ and ⇔.
   A general form of Constrained Argumentation Framework (CAF) has been considered in [4, 5].
A CAF is a triple Ω = ⟨𝒜, ℛ, 𝒞⟩ where ⟨𝒜, ℛ⟩ is an AF and 𝒞 is a set of (general) propositional
formulae built from ℒ𝒜 .
   Given an AF ⟨𝒜, ℛ⟩ and a set 𝑆 ⊆ 𝒜, the truth value of an argument 𝑎 ∈ 𝒜 w.r.t. 𝑆 is denoted
by 𝜗𝑆 (𝑎), or simply 𝜗(𝑎) whenever 𝑆 is understood, and 𝜗(𝑎) is 𝑖) true if 𝑎 ∈ 𝑆; 𝑖𝑖) false if
∃𝑏 ∈ 𝑆 such that (𝑏, 𝑎) ∈ ℛ; or 𝑖𝑖𝑖) undef otherwise.
   It is important to note that, regarding the operator ⇒ there is no convergence on its semantics in
CAFs (see Section 5 for a discussion). Thus, in the following we consider a simpler yet sufficiently
general form of constraints than that in [4, 5] (e.g. we do not deal with constraints with multiple
implications) and the classical interpretation of Lukasiewicz’s logic for the implication operator.
   Let ℒ′𝒜 be the propositional language defined from 𝒜 and the connectives ∧, ∨, ¬, where 𝒜 is
a set of arguments.

Definition 1. A (strong) constraint is a formula of one of the following forms: (𝑖) 𝜙 ⇒ 𝑣, or (𝑖𝑖)
𝑣 ⇒ 𝜙, where 𝜙 is a propositional formula in ℒ′𝒜 and 𝑣 ∈ {f, u, t}.
   Checking whether a constraint is satisfied (under the Lukasiewicz’logic) is equivalent to check
whether the truth value of the head is greater than or equal to the truth value of the body. For
formulae defining constraints we believe that Lucasiewicz interpretation is more appropriate as,
for instance, it allows to distinguish 𝜙 ⇒ f from 𝜙 ⇒ u, and avoids problems existing in other
interpretations [10].

Example 6. The constraint 𝑥 ∧ 𝑦 ∧ 𝑧 ⇒ f states that at least one of the arguments 𝑥, 𝑦 and 𝑧
must be false, whereas 𝑥 ∧ 𝑦 ∧ 𝑧 ⇒ u states that 𝑥, 𝑦 and 𝑧 cannot be all true.           □

 Clearly, constraints of the forms f ⇒ 𝜙 and 𝜙 ⇒ t are useless because always satisfied.
 In the following, we assume that 𝒞 is a set of satisfiable constraints built from ℒ′𝒜 and every
CAF is a triple Ω = ⟨𝒜, ℛ, 𝒞⟩.

Definition 2. Given a CAF Ω = ⟨𝒜, ℛ, 𝒞⟩ (where 𝒞 contains constraints built from ℒ′𝒜 ), a set of
arguments 𝑆 ⊆ 𝒜 is a complete (resp., grounded, preferred, stable, semi-stable) extension for Ω
if 𝑆 is a complete (resp., grounded, preferred, stable, semi-stable) extension for ⟨𝒜, ℛ⟩ and 𝒞 is
satisfied by 𝑆 (denoted as 𝑆 |= 𝒞).


3. Weak Constrained AFs
In this section, we introduce a generalization of CAFs where weak constraints are also considered.
Differently from the strong constraints discussed in the previous section, weak constraints are
propositional formulae that should be satisfied if possible.
   Weak constraints (also called relaxed constraints in some contexts) have been considered in
several research areas, including Mathematical Programming with Equilibrium Constraints [11],
Answer Set Programming [12, 13], and for modelling and solving optimization problems [14, 15].
In particular, concerning the field of Answer Set Programming, weak constraints have been
implemented in DLV [16] and clingo [17]; moreover, learning them from data is also possible [18].
   In our setting, weak constraints are logical formulae of the form 𝜙 → 𝑣 (or, equivalently,
𝑣 → 𝜙), where 𝜙 is a propositional formula built using the symbols of a given set 𝒜 and the
connectives ∧, ∨ and ¬. Herein, → denotes the logical implication connective. Observe that we
use the symbol → (instead of ⇒) to have different syntaxes for weak and strong constraints.

Definition 3. A Weak constrained Argumentation Framework (WAF) is a tuple Γ = ⟨𝒜, ℛ, 𝒞, 𝒲⟩,
where ⟨𝒜, ℛ, 𝒞⟩ is a CAF and 𝒲 is a set of weak constraints built from ℒ′𝒜 .

   The semantics of a WAF is defined by considering two possible criteria for selecting the
preferable extensions w.r.t. weak constraints—only weak constraints are considered when
selecting the preferable extensions since strong constraints must be all satisfied. The two criteria
considered for assessing to which extent an extension satisfies a set of weak constraints are:

   (i) maximal set criterion, considering as preferable (or “best”) extensions the ones that satisfy
       a maximal set of weak constraints, and
  (ii) maximum-cardinality criterion, considering as preferable (or “optimal”) extensions the
       ones that satisfy a maximal number of weak constraints.
  Clearly, the selection of preferable extensions make sense only for semantics admitting multiple
extensions, that is, complete, preferred, stable, and semi-stable semantics. Thus, in the following,
whenever we consider a generic semantics 𝒮, we refer to 𝒮 ∈ {co, pr, st, sst}.
  In the next subsections, we formally define the meaning of a WAF under the maximal-set and
maximum-cardinality semantics and provide two examples.

3.1. Maximal-Set Semantics
A WAF using the maximal-set criterion is defined as follows.

Definition 4 (Maximal-Set Semantics). Given a WAF Γ = ⟨𝒜, ℛ, 𝒞, 𝒲⟩, an 𝒮-extension 𝐸 for
⟨𝒜, ℛ, 𝒞⟩ is a maximal-set 𝒮-extension (ms𝒮-extension) for Γ if, let 𝒲𝐸 ⊆ 𝒲 be the set of weak
constraints that are satisfied by 𝐸 (that is, 𝐸 |= 𝒲𝐸 ) , there is no 𝒮-extension 𝐹 for ⟨𝒜, ℛ, 𝒞⟩
and 𝒲𝐹 ⊆ 𝒲 such that 𝐹 |= 𝒲𝐹 and 𝒲𝐸 ⊂ 𝒲𝐹 .

  Given a semantics 𝒮, ms𝒮 denotes the maximal-set version of 𝒮 (e.g. msco denotes the ms
complete semantics).

Example 7. Consider the WAF Γ = ⟨𝒜, ℛ, 𝒞, 𝒲⟩ with 𝒜 = {𝑎, 𝑏, 𝑐, 𝑑}, ℛ = {(𝑎, 𝑏), (𝑏, 𝑎),
(𝑐, 𝑑), (𝑑, 𝑐)}, 𝒞 = ∅ and 𝒲 = {𝑤1 = 𝑐 → f, 𝑤2 = 𝑎 ∨ ¬𝑎 → u} stating that 𝑐 should
preferably be false (𝑤1 ) and 𝑎 should preferably be undefined (𝑤2 ). ⟨𝒜, ℛ, 𝒞⟩ has 9 complete
extensions: 𝐸0 = {}, 𝐸1 = {𝑎}, 𝐸2 = {𝑏}, 𝐸3 = {𝑐}, 𝐸4 = {𝑑}, 𝐸5 = {𝑎, 𝑐}, 𝐸6 = {𝑎, 𝑑},
𝐸7 = {𝑏, 𝑐} and 𝐸8 = {𝑏, 𝑑}.
   In particular, 𝐸0 is the grounded extension, whereas 𝐸5 , 𝐸6 , 𝐸7 , 𝐸8 are preferred, stable, and
semi-stable extensions of ⟨𝒜, ℛ, 𝒞⟩. These are also extensions of AF ⟨𝒜, ℛ⟩, since 𝒞 = ∅.
   Regarding the satisfaction of weak constraints, we have that 𝐸0 |= {𝑤2 }, 𝐸4 |= {𝑤1 , 𝑤2 },
𝐸6 |= {𝑤1 }, and 𝐸8 |= {𝑤1 }, whereas the other complete extensions do not satisfy any constraint.
Therefore, the maximal-set preferred (stable, semi-stable) extensions are 𝐸6 and 𝐸8 , whereas
there is only one maximal-set complete extension, which is 𝐸4 .                                    □

3.2. Maximum-Cardinality Semantics
Maximum-cardinality semantics for WAFs prescribes as preferable extensions those satisfying
the highest number of weak constraints. This is similar to the semantics of weak constraints in
DLV [16] where, in addition, each constraint has assigned a weight.

Definition 5 (Maximum-Cardinality Semantics). Given a WAF Γ = ⟨𝒜, ℛ, 𝒞, 𝒲⟩, an 𝒮-
extension 𝐸 for ⟨𝒜, ℛ, 𝒞⟩ is a maximum-cardinality 𝒮-extension (mc𝒮-extension) for Γ if, let
𝒲𝐸 ⊆ 𝒲 be the set of weak constraints in 𝒲 that are satisfied by 𝐸, there is no 𝒮-extension 𝐹
for ⟨𝒜, ℛ, 𝒞⟩ and 𝒲𝐹 ⊆ 𝒲 such that 𝐹 |= 𝒲𝐹 and |𝒲𝐸 | < |𝒲𝐹 |.

Example 8. Consider the WAF Γ = ⟨𝒜, ℛ, 𝒞, 𝒲⟩ with 𝒜 = {𝑎, 𝑏, 𝑐}, ℛ = {(𝑎, 𝑏), (𝑏, 𝑎),
(𝑏, 𝑐), (𝑐, 𝑐)}, 𝒞 = ∅ and 𝒲 = {𝑤1 = t → 𝑎, 𝑤2 = t → 𝑏, 𝑤3 = 𝑐 → f} stating that it is
desirable that 𝑎 is true, 𝑏 is true, and 𝑐 is false.
   ⟨𝒜, ℛ, 𝒞⟩ has three complete extensions: 𝐸1 = {}, 𝐸2 = {𝑎}, and 𝐸3 = {𝑏}. Herein, 𝐸2
and 𝐸3 are the preferred extensions of ⟨𝒜, ℛ, 𝒞⟩, whereas the unique stable (and semi-stable)
extension is 𝐸3 . Regarding the satisfactions of weak constraints we have that 𝐸1 |= 𝒲0 = ∅,
𝐸2 |= 𝒲1 = {𝑤1 }, and 𝐸3 |= 𝒲3 = {𝑤2 , 𝑤3 }. Therefore, the only maximum-cardinality
preferred extension of Γ is 𝐸3 (as |𝒲3 | = 2 > |𝒲1 | = 1 > |𝒲0 | = 0). Note that, according to the
maximal-set semantics, both 𝐸2 and 𝐸3 are maximal-set preferred extensions. Regarding the
stable (and semi-stable) semantics, as there is only one extension, 𝐸3 is both a maximal-set and a
maximum-cardinality extension.                                                                  □


4. Complexity of Credulous and Skeptical Acceptance
In this section we discuss the complexity of the credulous and skeptical acceptance problems. We
assume the reader is familiar with the main complexity classes [19], which are briefly recalled in
what follows.
   Classes Σ𝑃𝑘 , Π𝑃𝑘 and Δ𝑃𝑘 , with 𝑘 ≥ 0 are defined as follows: Σ𝑃0 = Π𝑃0 = Δ𝑃0 = 𝑃 ;
                                          𝑃               𝑃
Σ𝑃1 = 𝑁𝑃 and Π𝑃1 = 𝑐𝑜𝑁𝑃 ; Δ𝑃𝑘 = 𝑃 Σ𝑘−1 , Σ𝑃𝑘 = 𝑁𝑃 Σ𝑘−1 , and Π𝑃𝑘 = 𝑐𝑜Σ𝑃𝑘 , ∀𝑘 > 0. Thus, 𝑃 𝐶
(resp., 𝑁𝑃 𝐶 ) denotes the class of problems that can be solved in polynomial time using an oracle
in the class 𝐶 by a deterministic (resp., non-deterministic) Turing machine. The class Δ𝑃𝑘 [𝑙𝑜𝑔 𝑛]
denotes the subclass of Δ𝑃𝑘 containing the problems that can be solved in polynomial time by
a deterministic Turing machine by performing a number of calls bounded by 𝑂(𝑙𝑜𝑔 𝑛) to an
oracle in the class Σ𝑃𝑘−1 . It is known that: Σ𝑃𝑘 ⊂ Δ𝑃𝑘+1 [𝑙𝑜𝑔 𝑛] ⊂ Δ𝑃𝑘+1 ⊂ Σ𝑃𝑘+1 ⊆ 𝑃 𝑆𝑃𝐴𝐶𝐸
and Π𝑃𝑘 ⊂ Δ𝑃𝑘+1 [𝑙𝑜𝑔 𝑛] ⊂ Δ𝑃𝑘+1 ⊂ Π𝑃𝑘+1 ⊆ 𝑃 𝑆𝑃𝐴𝐶𝐸.
   We now recall the definition of credulous and skeptical acceptance problems. Given an AF-
based framework Λ (e.g., AF, CAF, WAF), an argument 𝑎, and an argumentation semantics
𝒮 ∈ {co, pr, st, sst, gr},

    • the credulous acceptance problem, denoted as 𝐶𝐴𝒮 , is the problem of deciding whether
      argument 𝑎 is credulously accepted, that is, deciding whether 𝑎 belongs to at least an
      𝒮-extension of Λ.
    • the skeptical acceptance problem, denoted as 𝑆𝐴𝒮 , is the problem of deciding whether
      argument 𝑎 is skeptically accepted, that is, deciding whether 𝑎 belongs to every 𝒮-extension
      of Λ.

For AFs, the complexity of the credulous and skeptical acceptance problems has been investigated
in [1] for the grounded semantics (as grounded semantics admits exactly one extension that can
be computed in polynomial time, these problems become identical and polynomial), in [20] for
the stable semantics, in [20, 21] for the preferred semantics, and in [22, 23] for the semi-stable
semantics. The results for AFs are summarized in the second and third column of Table 1.
   As for the complexity of CAFs, it is important to note that, given a CAF Ω = ⟨𝒜, ℛ, 𝒞⟩, if
we consider the corresponding AF Λ = ⟨𝒜, ℛ⟩, then the set of complete extensions of Λ that
satisfy 𝒞 does not always form a complete meet-semilattice. This means that the constraints break
the lattice by marking as unfeasible some extensions. As a result, the complexity of credulous
and skeptical acceptance problems in CAF is slightly different from that of AF as 𝑆𝐴co becomes
NP-complete and 𝐶𝐴pr is shown to be in Σ𝑝2 (and still NP-hard) [9].
   Concerning WAFs, given an 𝒮-extension, checking satisfaction of a maximal-set of weak
constraints means ensuring that no any other 𝒮-extension is better according to that criterion.
Table 1
Complexity of credulous (𝐶𝐴𝒮 ) and skeptical (𝑆𝐴𝒮 ) acceptance problems under complete (co),
stable (st), preferred (pr), and semi-stable (sst) semantics.



                       AF                                           WAF
  𝒮         𝐶𝐴𝒮               𝑆𝐴𝒮                𝐶𝐴ms𝒮            𝑆𝐴ms𝒮           𝐶𝐴mc𝒮 /𝑆𝐴mc𝒮
                                              𝑃
  co    NP-complete             P            Σ2 -complete     Π𝑃
                                                               2 -complete
                                                                                𝑃
                                                                               Δ2 [log 𝑛]-complete
  st    NP-complete      coNP-complete       Σ𝑃
                                              2 -complete     Π𝑃
                                                               2 -complete     Δ𝑃
                                                                                2 [log 𝑛]-complete
  pr    NP-complete       Π𝑃
                           2 -complete      Σ𝑃            𝑃
                                             2 -hard, in Σ3   Π𝑃
                                                               3 -complete     Δ𝑃
                                                                                3 [log 𝑛]-complete
 sst    Σ𝑃
         2 -complete      Π𝑃
                           2 -complete
                                              𝑃
                                             Σ3 -complete     Π𝑃
                                                               3 -complete     Δ𝑃
                                                                                3 [log 𝑛]-complete




This is an additional source of complexity that makes, in most cases, credulous and skeptical
reasoning in WAFs one level higher in the polynomial-time hierarchy than AFs.
Theorem 1. For any WAF ⟨𝒜, ℛ, 𝒞, 𝒲⟩, the problem
    • 𝐶𝐴ms𝒮 is: (𝑖) Σ𝑃2 -complete for any semantics 𝒮 ∈ {co, st}, (𝑖𝑖) Σ𝑃2 -hard and in Σ𝑃3 for
      𝒮 = pr, and (𝑖𝑖𝑖) Σ𝑃3 -complete for 𝒮 = sst.
    • 𝑆𝐴ms𝒮 is: (𝑖) Π𝑃2 -complete for 𝒮 ∈ {co, st}, and (𝑖𝑖) Π𝑃3 -complete for 𝒮 ∈ {pr, sst}.
   It turns out that, under standard complexity assumptions, computing credulous and skeptical
acceptance in WAFs under maximum-cardinality semantics is easier than using maximal-set
semantics. Roughly speaking, this follows from the fact that a binary search strategy can be used
for deciding whether the cardinality of a set of constraints satisfied by an 𝒮-extension containing
a given argument is maximum.
Theorem 2. For any WAF ⟨𝒜, ℛ, 𝒞, 𝒲⟩ with |𝒲| = 𝑛, the problems 𝐶𝐴mc𝒮 and 𝑆𝐴mc𝒮 are:
    - Δ𝑃2 [log 𝑛]-complete for any semantics 𝒮 ∈ {co, st},
    - Δ𝑃3 [log 𝑛]-complete for 𝒮 ∈ {pr, sst}.
   The results for WAFs are summarized in the last three columns of Table 1.
   Restricted forms of WAFs, that is, Linearly ordered WAFs where constraints are linearly
ordered, for which maximal-set and maximum-cardinality semantics coincide, are investigated
in [9]: 𝐶𝐴𝒮 and 𝑆𝐴𝒮 are Δ𝑃2 -complete for the complete and stable semantics and Δ𝑃3 -complete
for the preferred and semistable semantics. It is also shown that, for the case of WAFs where
constraints are expressed by negative constraints (i.e., denials constraints whose body is a
conjunction of literals), the complexity of credulous acceptance for the preferred semantics
decreases to Σ𝑃2 -complete.


5. Related Work
The use of constraints in AFs has been firstly proposed in [4] and then further investigated in
[6, 5, 7]. As mentioned in Section 2.2, there is no convergence on the semantics of operator ⇒
for constraints in CAFs. The semantics proposed in [4] relies on classical 2-valued logic for
the evaluation of constraints by making use of the concept of completion of extensions, which
converts undefined truth values to negated truth values. More formally, for any set of arguments
𝑆 ⊆ 𝒜, the completion of 𝑆 is 𝑆 ^ = 𝑆 ∪ {¬𝑎 | 𝑎 ∈ 𝒜 ∖ 𝑆}. Then, 𝑆 satisfies a set of constraints
                  ^
𝒞 if and only if 𝑆 is a (2-valued) model of 𝒞. A drawback of this semantics is that in checking
whether an extension 𝐸 satisfies a set of constraints it does not distinguish between false and
undefined arguments, e.g., a constraint of the form 𝑎 ∧ ¬𝑎 ⇒ f is always satisfied, even when
𝜗𝐸 (𝑎) = undef (e.g. assuming an extension 𝐸 = ∅ stating that 𝑎 is undefined, 𝐸       ^ = {¬𝑎} and
then the 2-valued semantics is applied w.r.t. 𝐸^ ).
   The semantics proposed in [5] assumes the standard 3-valued Kleene’s semantics for the
connectives ∧, ∨ and ¬, whereas for ⇒ it assumes the Slupecki’s interpretation which is defined
as follows: 𝜗(𝜙 ⇒ 𝜓) is true if 𝜗(𝜙) ∈ {false, undef}, otherwise it is 𝜗(𝜓). Thus, while a
constraint of the form t ⇒ 𝑎 ∨ ¬𝑎 is useless according to [4] (since it is always satisfied), in
the Arieli’s 3-valued semantics this constraint indicates that argument 𝑎 cannot have a neutral,
undefined, status. The use of 3-valued semantics allows us to distinguish between different
conditions on arguments. For instance, the constraint t ⇒ ¬𝑎 means that 𝑎 should be rejected,
while the constraint 𝑎 ⇒ f is a somewhat weaker demand: 𝑎 should not be accepted, and so
its status may be undecided. A drawback of Arieli’s semantics, due to the assumption of the
Slupecki’s logic for interpreting the implication operator, is that it does not distinguish two
constraints of the form 𝜙 ⇒ f and 𝜙 ⇒ u, though it distinguishes two constraints of the
form t ⇒ 𝜙 and u ⇒ 𝜙. In order to avoid the above-mentioned problems, we considered the
interpretation of Lukasiewicz’s logic for the implication operator.
   Besides being related to the proposals for CAFs in [4, 5], our work is also related to the
approach in [24] that provides a method for generating non-empty conflict-free extensions
for CAFs. Constraints have been also used in the context of dynamic AFs to refer to the
enforcement of some change [25]. In this context, extension enforcement aims at modifying
an AF to ensure that a given set of arguments becomes (part of) an extension for the chosen
semantics [26, 27, 28, 29, 30]. This is different from our approach where integrity constrains are
used to discard unfeasible solutions (extensions), without enforcing that a new set of arguments
becomes an extension.
   Weak constraints allow for selecting “best” or “optimal” extensions satisfying some conditions
on arguments, if possible. This can be viewed as expressing a kind of preference over the set
of extensions. Dung’s framework has been extended in several ways for allowing preferences
over arguments [31, 32, 33, 34]. In particular, preferences relying to so-called critical attacks, i.e.,
attacks from a less preferred argument to a more preferred one, can be handled by removing or
invalidating such attacks or by “inverting” them [35]. Such kind of preferences can be encoded
into AFs, possible through reductions relying on additional (meta)-arguments and attacks [36].
Thus these preferences do not increase expressiveness of AFs from a computational standpoint.
   Preferences can be also expressed in value-based AFs [37, 38], where each argument is associ-
ated with a numeric value, and a set of possible orders (preferences) among the values is defined.
In [39] weights are associated with attacks, and new semantics extending the classical ones on
the basis of a given numerical threshold are proposed. [40] extends [39] by considering other ag-
gregation functions over weights apart from sum. Except for weighted solutions under grounded
semantics (that prescribes more than one weighted solution), the complexity of credulous and
skeptical reasoning in the above-considered AF-based frameworks is lower than that of WAFs,
which suggests that WAFs are more expressive and can be used to model those frameworks (we
plan to formally investigate these connections in future work).


6. Conclusions and Future Work
We have introduced a general argumentation framework where weak constraints can be easily
expressed. These constraints impact on the complexity of credulous and skeptical reasoning: it
turns out that they generally increase the expressivity of AFs and CAFs. WAFs allow us to model
optimization problems such as for instance Min Coloring and Maximum Satisfiability, where
some kind of preferences (e.g. use the minimum number of colors) are expressed on solutions.
This is not possible for AFs whose expressivity is lower than that of WAFs (AFs can capture
simpler problems such as k-coloring and SAT).
   We envisage implementations of the proposed WAF semantics by exploiting ASP-based sys-
tems and analogies with logic programs with weak constraints [12, 13] (the relationship between
the semantics of some frameworks extending AF and that of logic programs has been recently
investigated in [41]). For WAFs, DLV system [16] could be used for computing maximum-
cardinality stable semantics.
   Although we considered ground constraints, the framework can be easily extended for general
formulae with variables, whose ground version is a propositional set. For instance, the strong and
weak constraints in Example 4 could be written by using only one strong constraint 𝑋 ∧ 𝑌 ∧
𝑍 ∧ (𝑋 ̸= 𝑌 ) ∧ (𝑋 ̸= 𝑍) ∧ (𝑌 ̸= 𝑍) ⇒ f and only one weak constraint t → 𝑋, where 𝑋,
𝑌 and 𝑍 are variables whose domain is the set of arguments. Future work will be also devoted
to considering more general forms of constraints, not only using variables ranging on the sets
of arguments, but also constraints allowing to express conditions on aggregates (e.g., at least 𝑛
arguments from a given set 𝑆 should be accepted/rejected).
   Finally, given the inherent nature of argumentation and the typical high computational complex-
ity of most of the reasoning tasks [42, 43, 44, 45], there have been several recent efforts toward
the investigation of incremental techniques that use AF solutions (e.g. extensions, skeptical ac-
ceptance) at time 𝑡 to recompute updated solutions at time 𝑡 + 1 after that an update (e.g. adding/
removing an attack) is performed [46, 47, 48, 49, 50, 51, 25]. These approaches have been
extended to argumentation frameworks more general than AFs [52, 53, 54, 55]. Following this
line of research, we plan to investigate incremental techniques for recomputing WAF semantics
after performing updates consisting of changes to the AF component or to the sets of strong and
weak constraints.


References
 [1] P. M. Dung, On the acceptability of arguments and its fundamental role in nonmonotonic
     reasoning, logic programming and n-person games, Artif. Intell. 77 (1995) 321–358.
 [2] L. Amgoud, C. Cayrol, On the acceptability of arguments in preference-based argumentation,
     in: Proc. of Conference on Uncertainty in Artificial Intelligence (UAI), 1998, pp. 1–7.
 [3] S. Modgil, H. Prakken, A general account of argumentation with preferences, Artif. Intell.
     195 (2013) 361–397.
 [4] S. Coste-Marquis, C. Devred, P. Marquis, Constrained argumentation frameworks, in: Proc.
     of International Conference on Principles of Knowledge Representation and Reasoning
     (KR), 2006, pp. 112–122.
 [5] O. Arieli, Conflict-free and conflict-tolerant semantics for constrained argumentation
     frameworks, J. Appl. Log. 13 (2015) 582–604.
 [6] O. Arieli, Towards constraints handling by conflict tolerance in abstract argumentation
     frameworks, in: Proc. of International Florida Artificial Intelligence Research Society
     Conference (FLAIRS), 2013.
 [7] O. Arieli, On the acceptance of loops in argumentation frameworks, J. Log. Comput. 26
     (2016) 1203–1234.
 [8] M. Caminada, Semi-stable semantics, in: Proc. of Computational Models of Argument
     (COMMA), 2006, pp. 121–130.
 [9] G. Alfano, S. Greco, F. Parisi, I. Trubitsyna, Argumentation frameworks with strong and
     weak constraints: Semantics and complexity, in: Proc. of AAAI Conference on Artificial
     Intelligence (AAAI), 2021, pp. 6175–6184.
[10] A. Avron, Natural 3-valued logics–characterization and proof theory, The Journal of
     Symbolic Logic 56 (1991) 276–294.
[11] A. Ramos, Two new weak constraint qualifications for mathematical programs with
     equilibrium constraints and applications, J. Optim. Theory Appl. 183 (2019) 566–591.
[12] F. Buccafurri, N. Leone, P. Rullo, Enhancing disjunctive datalog by constraints, IEEE Trans.
     Knowl. Data Eng. 12 (2000) 845–860.
[13] S. Greco, Non-determinism and weak constraints in datalog, New Gener. Comput. 16
     (1998) 373–396.
[14] W. Faber, M. Vallati, F. Cerutti, M. Giacomin, Solving set optimization problems by
     cardinality optimization with an application to argumentation, in: Proc. of European
     Conference on Artificial Intelligence (ECAI), 2016, pp. 966–973.
[15] M. Li, R. Lv, The binary algorithm for finding the best weak-constraints in contradictory
     weak-constrained optimization problems, in: Proc. of International Conference on Natural
     Computation (ICNC), 2014, pp. 475–479.
[16] M. Alviano, F. Calimeri, C. Dodaro, D. Fuscà, N. Leone, S. Perri, F. Ricca, P. Veltri,
     J. Zangari, The ASP system DLV2, in: Proc. of International Conference on Logic
     Programming and Nonmonotonic Reasoning (LPNMR), 2017, pp. 215–221.
[17] M. Gebser, R. Kaminski, B. Kaufmann, T. Schaub, Multi-shot ASP solving with clingo,
     Theory Pract. Log. Program. 19 (2019) 27–82.
[18] M. Law, A. Russo, K. Broda, Learning weak constraints in answer set programming, Theory
     Pract. Log. Program. 15 (2015) 511–525.
[19] C. H. Papadimitriou, Computational complexity., Addison-Wesley, 1994.
[20] Y. Dimopoulos, A. Torres, Graph theoretical structures in logic programs and default
     theories, Theor. Comput. Sci. 170 (1996) 209–244.
[21] P. E. Dunne, T. J. M. Bench-Capon, Coherence in finite argument systems, Artif. Intell. 141
     (2002) 187–203.
[22] P. E. Dunne, M. Caminada, Computational complexity of semi-stable semantics in abstract
     argumentation frameworks, in: Proc. of European Conference on Logics in Artificial
     Intelligence (JELIA), volume 5293, 2008, pp. 153–165.
[23] W. Dvorák, S. Woltran, Complexity of semi-stable and stage semantics in argumentation
     frameworks, Inf. Process. Lett. 110 (2010) 425–430.
[24] R. Booth, S. Kaci, T. Rienstra, L. W. N. van der Torre, A logical theory about dynamics
     in abstract argumentation, in: Proc. of International Conference Scalable Uncertainty
     Management (SUM), 2013, pp. 148–161.
[25] S. Doutre, J. Mailly, Constraints and changes: A survey of abstract argumentation dynamics,
     Argument & Computation 9 (2018) 223–248.
[26] R. Baumann, G. Brewka, Expanding argumentation frameworks: Enforcing and mono-
     tonicity results, in: Proc. of Computational Models of Argument (COMMA), 2010, pp.
     75–86.
[27] R. Baumann, What does it take to enforce an argument? minimal change in abstract
     argumentation, in: Proc. of the European Conference on Artificial Intelligence (ECAI),
     2012, pp. 127–132.
[28] S. Coste-Marquis, S. Konieczny, J. Mailly, P. Marquis, Extension enforcement in abstract
     argumentation as an optimization problem, in: Proc. of International Joint Conference on
     Artificial Intelligence (IJCAI), 2015, pp. 2876–2882.
[29] J. P. Wallner, A. Niskanen, M. Järvisalo, Complexity results and algorithms for extension
     enforcement in abstract argumentation, J. Artif. Intell. Res. 60 (2017) 1–40.
[30] A. Niskanen, J. P. Wallner, M. Järvisalo, Extension enforcement under grounded seman-
     tics in abstract argumentation, in: International Conference on Principles of Knowledge
     Representation and Reasoning (KR), 2018, pp. 178–183.
[31] L. Amgoud, C. Cayrol, A reasoning model based on the production of acceptable arguments,
     Ann. Math. Artif. Intell. 34 (2002) 197–215.
[32] S. Kaci, L. W. N. van der Torre, Preference-based argumentation: Arguments supporting
     multiple values, Int. J. Approx. Reason. 48 (2008) 730–751.
[33] S. Modgil, Reasoning about preferences in argumentation frameworks, Artif. Intell. 173
     (2009) 901–934.
[34] L. Amgoud, S. Vesic, A new approach for preference-based argumentation frameworks,
     Ann. Math. Artif. Intell. 63 (2011) 149–183.
[35] L. Amgoud, S. Vesic, Rich preference-based argumentation frameworks, Int. J. Approx.
     Reason. 55 (2014) 585–606.
[36] S. Kaci, L. W. N. van der Torre, S. Villata, Preference in abstract argumentation, in: Proc.
     of Computational Models of Argument (COMMA), 2018, pp. 405–412.
[37] T. J. M. Bench-Capon, Persuasion in practical argument using value-based argumentation
     frameworks, J. Log. Comput. 13 (2003) 429–448.
[38] P. E. Dunne, T. J. M. Bench-Capon, Complexity in value-based argument systems, in: Proc.
     of European Conference on Logics in Artificial Intelligence (JELIA), 2004, pp. 360–371.
[39] P. E. Dunne, A. Hunter, P. McBurney, S. Parsons, M. J. Wooldridge, Weighted argument
     systems: Basic definitions, algorithms, and complexity results, Artif. Intell. 175 (2011)
     457–486.
[40] S. Coste-Marquis, S. Konieczny, P. Marquis, M. A. Ouali, Weighted attacks in argumen-
     tation frameworks, in: Proc. of International Conference on Principles of Knowledge
     Representation and Reasoning (KR), 2012.
[41] G. Alfano, S. Greco, F. Parisi, I. Trubitsyna, On the semantics of abstract argumentation
     frameworks: A logic programming approach, Theory Pract. Log. Program. 20 (2020)
     703–718.
[42] G. Alfano, M. Calautti, S. Greco, F. Parisi, I. Trubitsyna, Explainable acceptance in proba-
     bilistic abstract argumentation: Complexity and approximation, in: Proc. of International
     Conference on Principles of Knowledge Representation and Reasoning (KR), 2020, pp.
     33–43.
[43] G. Alfano, S. Greco, F. Parisi, On scaling the enumeration of the preferred extensions of
     abstract argumentation frameworks, in: Proc. of ACM Symposium on Applied Computing
     (SAC), 2019, pp. 1147–1153.
[44] G. Alfano, S. Greco, F. Parisi, G. I. Simari, G. R. Simari, Incremental computation for
     structured argumentation over dynamic delp knowledge bases, Artif. Intell. 300 (2021)
     103553.
[45] G. Alfano, S. Greco, F. Parisi, I. Trubitsyna, Incomplete argumentation frameworks:
     Properties and complexity, in: Proc. of AAAI Conference on Artificial Intelligence (AAAI),
     2022, p. (to appear).
[46] S. Greco, F. Parisi, Efficient computation of deterministic extensions for dynamic abstract
     argumentation frameworks, in: Proc. of European Conference on Artificial Intelligence
     (ECAI), 2016, pp. 1668–1669.
[47] S. Greco, F. Parisi, Incremental computation of deterministic extensions for dynamic
     argumentation frameworks, in: Proc. of European Conference on Logics in Artificial
     Intelligence (JELIA), 2016, pp. 288–304.
[48] G. Alfano, S. Greco, F. Parisi, Efficient computation of extensions for dynamic abstract
     argumentation frameworks: An incremental approach, in: Proc. of International Joint
     Conference on Artificial Intelligence (IJCAI), 2017, pp. 49–55.
[49] G. Alfano, S. Greco, F. Parisi, An efficient algorithm for skeptical preferred acceptance
     in dynamic argumentation frameworks, in: Proc. of International Joint Conference on
     Artificial Intelligence (IJCAI), 2019, pp. 18–24.
[50] G. Alfano, S. Greco, Incremental skeptical preferred acceptance in dynamic argumentation
     frameworks, IEEE Intell. Syst. 36 (2021) 6–12.
[51] G. Alfano, S. Greco, F. Parisi, Incremental computation in dynamic argumentation frame-
     works, IEEE Intelligent Systems (2021). doi:10.1109/MIS.2021.3077292.
[52] G. Alfano, S. Greco, F. Parisi, Computing skeptical preferred acceptance in dynamic
     argumentation frameworks with recursive attack and support relations, in: Proc. of Compu-
     tational Models of Argument (COMMA), 2020, pp. 67–78.
[53] G. Alfano, A. Cohen, S. Gottifredi, S. Greco, F. Parisi, G. R. Simari, Dynamics in abstract
     argumentation frameworks with recursive attack and support relations, in: Proc. of European
     Conference on Artificial Intelligence (ECAI), 2020, pp. 577–584.
[54] G. Alfano, S. Greco, F. Parisi, G. I. Simari, G. R. Simari, An incremental approach
     to structured argumentation over dynamic knowledge bases, in: Proc. of International
     Conference on Principles of Knowledge Representation and Reasoning (KR), 2018, pp.
     78–87.
[55] G. Alfano, S. Greco, F. Parisi, G. I. Simari, G. R. Simari, Incremental computation of
warranted arguments in dynamic defeasible argumentation: the rule addition case, in: Proc.
of ACM Symposium on Applied Computing (SAC), 2018, pp. 911–917.

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