1 Introduction
An Abstract Argumentation Framework (AF) [dung] is represented by a pair consisting of a set of arguments and a binary relation of attack defined between some of them. Given a framework, it is possible to examine the question on which set(s) of arguments can be accepted, hence collectively surviving the conflict defined by . Answering this question corresponds to define an argumentation semantics. The key idea behind extensionbased semantics is to identify some sets of arguments (called extensions) that survive the conflict “together”. A very simple example of AF is , where two arguments and attack each other. In this case, each of the two positions represented by either or can be intuitively valid, since no additional information is provided on which of the two attacks prevails. However, having weights on attacks results in such additional information, which can be fruitfully exploited in this direction. For instance, in case the attack is stronger than (or preferred to) , taking the position defined by may result in a better choice for an intelligent agent, since it can be regarded as more reliable or relevant on the framework.
In a recent work, Xu and Cayrol represent an AF by a binary matrix and they give a characterization for stable, admissible and complete extensions by analysing subblocks of this matrix [xu15]. Also, they present the reduced matrix w.r.t. conflictfree subsets, by which the determination of extensions becomes more efficient, and that allows to determine grounded and preferred extensions.
Our aim is to extend the above mentioned results to Weighted Argumentation Frameworks (WAFs) by adopting the paradigm introduced in [DBLP:conf/cilc/BistarelliPS10, DBLP:conf/flairs/BistarelliRS16] for the semiringbased version of classical semantics. In particular, (i) we characterize conflictfree, admissible, stable and complete extensions by analysing subblocks of a nonbinary matrix representing a given WAF, (ii) we show how to reduce this matrix to another one of smaller size that allows to more efficiently determine extensions, and (iii) we provide two algorithms that allow to build incrementally grounded and preferred extensions.
This paper is organized as follows: we first recall the basic definitions on AFs and on WAFs, then we give characterizations for weighted extensions by analysing the matrix associated with the given WAF. Finally, we present the matrix reductions of WAFs based on contraction and division of WAFs, and we provide methods for incrementally building grounded and preferred extensions.
2 Weighted Argumentation Frameworks
In this section, we recollect the main definitions at the basis of AFs [dung], and introduce csemirings for dealing with attackweights. We then rephrase some of the classical definitions, with the purpose to parametrise them with the notion of weighted attack and csemiring. Last, we give definitions about the matrix representation for AFs.
2.1 Abstract Argumentation Frameworks
In his pioneering work [dung], Dung proposed Abstract Frameworks for Argumentation, where (as shown in Figure 1) an argument is an abstract entity whose role is solely determined by its relations to other arguments:
Definition 1.
An Abstract Argumentation Framework (AF) is a pair of a set of arguments and a binary relation on , called attack relation. , (or ) means that attacks ( is asymmetric).
Let be an AF and . denotes the set of arguments attacked by (a set attacks a set if exist and with ). denotes the set of arguments attacking . denotes the set of arguments which are not attacked (also called initial arguments of ).
An argumentation semantics is the formal definition of a method ruling the argument evaluation process. In the extensionbased approach, a semantics definition specifies how to derive from an AF a set of extensions, where an extension of an AF is simply a subset of . In Definition 2 we define conflictfree sets:
Definition 2 (Conflictfree).
A set is conflictfree iff no two arguments and in exist such that attacks .
All the following semantics rely (explicitly or implicitly) upon the concept of defence:
Definition 3 (Defence [dung]).
An argument is defended by a set (or defends ) iff for any argument , if then s.t., .
Definition 4 (Extensionbased semantics).

A conflictfree set is admissible iff each argument in is defended by .

An admissible extension is a complete extension iff each argument that is defended by is in .

A preferred extension is a maximal (w.r.t. set inclusion) admissible subset of .

A grounded extension is a minimal (w.r.t. set inclusion) complete subset of .

A conflictfree set is a stable extension iff for each argument which is not in , there exists an argument in that attacks it.
2.2 Csemirings
Csemirings are commutative ( is commutative) and idempotent semirings (i.e., is idempotent), where defines a partial order . The obtained structure can be shown to be a complete lattice.
Definition 5 (csemirings).
A commutative semiring is a tuple such that is a set, , and are binary operators making the triples and commutative monoids (semigroups with identity), satisfying i) (distributivity), and ii) (annihilator). If , the semiring is said to be absorptive.
Wellknown instances of csemirings are:

^{1}^{1}1Boolean csemirings can be used to model crisp problems and classical Argumentation [dung].,

,

,

,

.
Csemirings provide a structure that reveals to be suitable for Weighted Argumentation Frameworks. In fact, values in can be used as weights for relations, while the operators and allow to define an ordering among weights.
2.3 Weighted AFs
The following definition reshapes the notion of Weighted Argumentation Framework into semiringbased WAF, called :
Definition 6 (Semiringbased WAF).
A semiringbased WAF () is a quadruple , where is a csemiring , is a set of arguments, the attack binaryrelation on , and is a binary function. Given and , then means that attacks with a weight . Moreover, we require that iff .
In Figure 2, we provide an example of a WAF describing the defined by , , with , and (i.e., the weighted semiring).
Therefore, each attack is associated with a semiring value that represents the “strength” of an attack between two arguments. We can consider the weights in Figure 2 as supports to the associated attack, as similarly suggested in [DBLP:journals/ai/DunneHMPW11]. A semiring value equal to the top element of the csemiring (e.g., for the weighted semiring) represents a noattack relation between two arguments. On the other side, the bottom element, i.e., (e.g., for the weighted semiring), represents the strongest attack possible. In the following, we will use to indicate the operator of the csemiring on a set of values:
Definition 7 (Attacks to/from sets of arguments).
Let be a . A set of arguments attacks a set of arguments and the weight of such attack is , if
For example, looking at Figure 2, we have that , , and .
Definition 8 (defence [DBLP:conf/flairs/BistarelliRS16]).
Given a , , defends iff such that , we have that .
A set defends an argument from , if the of all attack weights from to is worse^{2}^{2}2When considering the partial order of a generic semiring, we use “worse” or “better” because “greater” or “lesser” would be misleading: in the weighted semiring, , i.e., lesser means better. (w.r.t. ) than the of the attacks from to . For example, the set in Figure 2 defends from because , i.e., ().
Definition 9 (conflictfree).
Given a , a subset of arguments is conflictfree if .
Definition 10 (admissible).
Given a , a conflictfree set is admissible iff the arguments in are defended by from the arguments in .
Definition 11 (complete).
A admissible extension is also a complete extension iff each argument such that is admissible belongs to , i.e., .
Definition 12 (preferred and grounded).
A preferred extension is a maximal (w.r.t. set inclusion) admissible subset of . The least (w.r.t. set inclusion) complete extension is the grounded extension.
Definition 13 (stable).
Given , a admissible set is also a stable extension iff such that .
3 The Matrix Representation for WAFs
Given an AF , we can obtain a matrix representing by using Definition 4 in [xu15]. We extend this definition to represent WAFs through matrices.
Definition 14.
Let be a WAF with . The matrix of corresponding to the permutation of , denoted by , is a matrix of order , its elements being determined by the following rules: iff and ; iff .
Example 1.
Given as in Figure 3. The matrices of corresponding to the permutations and are
4 Characterizing extensions of a WAF
In this section, we mainly focus on the characterization of various extensions in the matrix representing a WAF.
4.1 Characterizing the wconflictfree subsets
The basic requirement for extensions is conflictfreeness. So, we will discuss the matrix condition which insures that a subset of a WAF is conflictfree.
Definition 15.
Let be a WAF with and . The subblock
of is called the cfsubblock of , and denoted by for short. We use this subblock to find conflictfree subsets of arguments.
Claim 1.
Given with , is conflictfree iff all the elements in the cfsubblock are .
4.2 Characterizing the wadmissible subsets
From Definition 10, we know that arguments belonging to a admissible subset are defended from the arguments in .
Definition 16.
Let be a WAF with , and . The subblock
of is called the ssubblock of , and denoted by for short. The subblock of
is called the subblock^{3}^{3}3In [xu15], is denoted as and it is called the asubblock. of , and denoted by .
Theorem 1.
Given with , a conflictfree subset is admissible iff , , where
refers to the column vector
of the ssubblock and refers to the column vector of the subblock .Example 3.
Let’s consider the conflictfree subsets and (see Figure 3). We have
and
, the weight associated to the column vector of is while the weight associated to the row vector of is . Since , is not admissible in according to Theorem 1.
However, from
and
, we know that the weight associated to the column vector of is while the weight associated to the row vector of is . Since , we claim that is admissible in by Theorem 1.
4.3 Characterizing the wstable extensions
We can say whether a admissible subset is also a stable extension by checking if all arguments in are attacked by arguments in . On this purpose, we can use the already defined matrix .
Theorem 2.
Given with , a admissible subset is a stable extension iff each column vector of the ssubblock of contains only elements different from , where is a permutation of .
4.4 Characterizing the wcomplete extensions
From the definition of complete extension, it comes that in addition of considering relations between arguments all inside and between arguments in and those outside , we also need to take into account attacks thoroughly outside . We give the following definition and theorem.
Definition 17.
Let be a WAF with , and .The subblock
of is called the csubblock of , and denoted by for short.
Theorem 3.
Given with , a admissible subset is complete iff

if some column vector of the ssubblock contains only elements, then its corresponding column vector of the csubblock contains some element different from and

for each column vector of the csubblock appearing in (1), which contains some element different from , there is at least one element of such that , where refers to the column vector of the ssubblock , where refers to the column vector of the ssubblock , and .
Example 5.
Given as in Figure 4. According to Definition 14, the matrix of is as follows
By Theorem 1, we have that is admissible. Note that the matrix has a column vector corresponding in to the column vector . For in , the corresponding column vector in has . Since , according to Theorem 3, we claim that is a complete extension of .
5 Matrix reduction for WAFs
Most of the time, it is convenient to reduce the size of the matrix before performing further operations on it. Below, we provide a method to contract the conflictfree subset of a matrix into a single entity, without affecting the computation of the extensions. Moreover, we show an iterative procedure for building grounded and preferred extensions.
5.1 Matrix reduction by contraction
Starting from a conflictfree subblock, we can characterize admissible, stable and complete extensions of a WAF. Contracting such a subblock, we obtain a new matrix of smaller size, but with the same semantics status as the original one.
Definition 18.
Let be the matrix of a WAF. The combination of two rows and of the matrix consists in “combining” the elements in the same position of the rows. If and are elements in the same position of the rows and respectively, their combination is given by the rule . The combination of two columns of the matrix is similar as the combination of two rows.
For a conflictfree subset , we can contract the subblock into a single entry in the matrix. This new entry will have the same status as w.r.t. the extensionbased semantics. Thus the matrix can be reduced into another matrix with order by applying the following rules: let , for each such that and ,

combine rows to the row ;

combine column to the column ;

delete row and column .
The matrix obtained in this way is called the reduced matrix w.r.t. the conflictfree subset . Also, the original WAF can be reduced into a new one with arguments by applying the following rules. Let and . For each such that and , and each such that , set and . Then,

if , combine to and set ;

if , combine to and set ;

delete and from .
Let denote the new relation and , then is a new AF called the reduced AF w.r.t. . Obviously, the reduced matrix is exactly the matrix obtained from and .
Theorem 4.
Given with , let be conflictfree and . Then is stable (resp. admissible, complete, preferred) in AF iff is stable (respectively admissible, complete, preferred) in the reduced .
5.2 Matrix reduction by division
Let be a WAF. The grounded extension of can be viewed as the union of two subsets and : consists of the initial arguments of and is the grounded extension, defended by , of the remaining subAF w.r.t. (that is , where ). On the other hand, a preferred extension coincides with an admissible extension , defended by , from which the associated remaining subAF (where ) has no nonempty admissible extension. We have the following theorem.
Theorem 5.
Let be a WAF, be a admissible extension of , and . If is a admissible (resp. stable, complete, preferred) extension, defended by , of the remaining subAF w.r.t. (), then is a admissible (resp. stable, complete, preferred) extension of .
Building wgrounded extensions
A grounded extension can be built incrementally by starting from a admissible extension. Let be the set of initial arguments of , then is a admissible extension. If has no initial arguments, then the grounded extension of is empty. Otherwise, let be the set of initial arguments of . We proceed to construct by computing the sets as follows:

;

and ;


compute ;

, , , ;

(with ), if then and ;

repeat (c) until ;

, with ;


repeat 3. until or .
This process can be done repeatedly until, for some , , where . From Theorem 5, we know that the set union between admissible extensions is a admissible extension in turn. At this point, the set is the grounded extension of . Note that the set coincides with the set of undec arguments in the labelling of where is the set of in arguments.
Example 7.
Let be a WAF as in Figure 5. We have , so we look for the sets . , so , and . Consider that implies . is the grounded extension of .
Building wpreferred extensions
A preferred extension can be built incrementally by starting from some admissible extension. Since the preferred semantics admits more extensions, different preferred extensions can be built, depending on both the initial extension and the selection of the nonempty admissible on each step of the procedure. Let be any admissible extension of and compute:

;

and ;


compute ;

, , , ;

(with ), if then and ;

repeat (c) until ;

, with ;


repeat 3. until or .
This process can be done repeatedly until, for some , , where . At this point, by Theorem 5, the set is the preferred extension of .
Example 8.
Let be a WAF as in Figure 5. Let’s consider the admissible extension of . Thus , and . Since and , is the preferred extension of .
Computational Complexity. We analysed the above described algorithms from the computational point of view. The first algorithm, which computes grounded extensions, has an overall time complexity of . The algorithm for preferred extensions reveals worse performance than the first one, with a time complexity of . This is due to the fact that an admissible extension has to be found at each execution of step 3. A more extended study of the complexity is left for future work.
6 Conclusion and Future Work
In this work, we introduce a matrix approach for studying extensions of semiringbased semantics. A WAF is represented as a matrix in which all elements correspond to weights assigned to relations among arguments. In particular, by extracting subblocks from this matrix, it is possible to check if a set of arguments is an extension for some semantics. Also, we describe an incremental procedure for building grounded and preferred extensions and we study how to reduce the number of arguments of a WAF in order to obtain a contracted matrix with the same status as the original one (w.r.t. the semantics). A possible application for this approach could be the identification of equational representation of semiringbased extensions, by using the method proposed in [DBLP:conf/ecsqaru/Gabbay11]. We plan to extend our current implementation^{4}^{4}4http://www.dmi.unipg.it/conarg [DBLP:conf/ictai/BistarelliS11, DBLP:conf/tafa/BistarelliS11] with the proposed approaches, and to test their performance on real applications. Finally, we would like to investigate whether such methodologies can be applied when considering coalitions of arguments [DBLP:journals/fuin/BistarelliS13].