Indepth combinatorial analysis of admissible sets for abstract argumentation

In this paper we investigate, from a graph theoretical point of view, the notion of acceptability in Dung semantics for abstract argumentation frameworks. We advance the state of the art by introducing and analyzing combinatorial structures exploited for taming, in particular cases, the exponential blowout of acceptance algorithms. We conclude the paper by a series of observations allowing to deepen the intuition with respect to the practical use of Dung acceptance based semantics.


Introduction
Dung's Theory of Formal Argumentation [13] is founded on admissible sets in argumentation frameworks. An argumentation framework is a digraph with vertices interpreted as (abstract) arguments and directed edges as attacks between arguments. A set S of arguments in an argumentation framework is an admissible set if it is conflict-free -that is, there is no attack between members of Sand, for any attack from an argument a outside S on an argument b in S, there is an argument in S attacking a. In order to make a distinction among acceptable and unacceptable arguments in a given argumentation framework, Dung defined the admissibility-based semantics, using different families of admissible sets (referred as extensions) and representing sets of collectively accepted arguments. Most decision problems concerning the acceptability of a specified argument in a given argumentation framework (also referred as reasoning tasks) was later proved to be NP-complete, P 2complete, or P 2 -complete: [9,12,16], see also the survey [18]. In order to speed-up the exponential combinatorial search to solve them, we make in this paper a graph-theoretical in-depth study of the family of admissible sets in Dung's Argumentation Frameworks.
The two main contributions of this paper are as follows: 1. In Section 3 we prove that if a set of arguments A in an argumentation framework D induces an argumentation framework having exactly one preferred extension (maximal admissible set), then there is also a unique maximal admissible set in D contained in A (Theorem 1). The above hypothesis holds trivially for A = S, where S is a conflict-free set in the argumentation framework D. The unique maximal admissible set in D contained in S, denoted S adm , is called the admissibility basis of the conflictfree set S. Ideal extension, [14], or eager extension, [6], are admissibility basis of the intersection of all preferred extensions, respectively of all semi-stables extensions (Theorem 2, Remark 1). We show (in Theorem 3) that the admissibility basis of a given conflict-free set can be found in polynomial-time (with respect to the number of arguments in the argumentation framework containing this conflict-free set). Using this, we develop an efficient algorithm for skeptical acceptance of a given argument in a bipartite argumentation framework (Lemma 1, Proposition 1, Theorem 4). 2. In Section 4 we consider a specific family of conflict-free sets, out-and-out conflict-free sets. A conflict-free set S is out-and-out if any argument outside S is either attacked by an argument in S or is attacked by an argument attacked by S. We give a linear algorithm to construct such a conflict-free set in a given argumentation framework (Theorem 5 and its proof) and show that if the argumentation framework is transitive (that is, for any distinct arguments a, b, c, if a attacks b and b attacks c, then a attacks c) then any out-and-out conflict-free set is a stable extension (Theorem 6) and, as a consequence, any preferred extension is a stable extension. The same result can be extended to what we call near-transitive argumentation framework (Theorem 7), that is argumentation framework in which for any intransitive triple a, b, c (three distinct arguments a, b, c with a attacking b, b attacking c, and a not attacking c), if c attacks an argument d, then the set {a}∪{d} is not conflict-free. We show that in general argumentation frameworks any preferred extension is contained in an out-and-out conflict-free set (Theorem 8).
We call such an out-and-out conflict-free set sound and deduce that the ideal extension is the admissibility basis of the intersection of all sound out-and-out conflict free sets (Proposition 3).
In Section 5 we further our analysis by a set of observations meant to deepen practical insights when employing admissibility based semantics in applications. We discuss the asymmetry in the strength of attacks when they are used to defeat or to attack a conflictfree set in an argumentation framework. Also, we introduce a decomposition of the set of arguments outside of a conflict free set S into different sets of arguments determined by the number of attacked and attacking arguments from S. This fine-grained decomposition can be used as a dynamic labeling of the arguments in the search algorithms for deciding the acceptability of a given argument, and also to define a score to distinguish between different extensions of the same type. We conclude the paper with Section 6.

Background
In this section we present the basic concepts used for defining classical semantics in abstract argumentation frameworks introduced by Dung in 1995, [13].
Definition 1 [13] An argumentation framework is a digraph D = (ARG(D), ATT(D)), where the vertices in ARG(D) are called arguments and, if (a, b) ∈ ATT(D) is a directed edge, then (a, b) is called an attack and referred as: argument a attacks argument b, argument b is attacked by argument a, and a is an attacker of b. The set of attackers of an argument a ∈ ARG(D) is denoted by a − , and the set of arguments attacked by a is denoted by a + . If S ⊆ ARG(D) then S − = ∪ a∈S a − and S + = ∪ a∈S a + . The set of arguments S defends an argument a if a − ⊆ S + (i.e., any a's attacker is attacked by an argument in S). The set of arguments defended by a set of arguments S ⊆ ARG(D) is denoted by F (S). The map F , assigning to each set of arguments, S, the set F (S) of arguments defended by S, is called the characteristic function.
A conflict-free set in D is a set S ⊆ ARG(D) such that S + ∩ S = ∅. An admissible set is a conflict-free set such that S − ⊆ S + .
An admissible set S is called a complete extension if F (S) = S. A preferred extension is a maximal (w.r.t. set inclusion) complete extension. A grounded extension is a minimal (w.r.t. set inclusion) complete extension. A stable extension is a conflict-free set S such that ARG(D) − S = S + . An argument a ∈ ARG(D) is credulously admissible acceptable in D if there is an admissible set S such that a ∈ S.
For σ ∈ {complete, grounded, preferred, stable}, an argument a ∈ ARG(D) is credulously σ -acceptable in D if there is a σ extension S such that a ∈ S. An argument a ∈ ARG(D) is skeptically acceptable w.r.t. preferred semantics in D if it belongs to each preferred extension.

Definition 2 (Other admissibility based semantics)
Let D be an argumentation framework.
- [14] The ideal extension of D is the unique maximal (w.r.t. set-inclusion) admissible set contained in every preferred extension in D. - [5] A semi-stable extension in D is a complete extension S with the property that S ∪ S + is maximal w.r.t. set-inclusion (if T is a complete extension in D such that S ∪ S + ⊆ T ∪ T + , then S ∪ S + = T ∪ T + ). - [6] The eager extension of D is the unique maximal (w.r.t. set-inclusion) admissible set contained in every semi-stable extension in D.
Throughout this paper we consider only finite argumentation frameworks (that is, argumentation frameworks D with ARG(D) finite) without self-attacking arguments (that is, there is no argument a ∈ ARG(D) with (a, a) ∈ ATT(D)). Also if D is an argumentation framework and A ⊆ ARG(D), then we denote by D[A] the argumentation framework (induced in D by A) with ARG(D[A]) = A and ATT(D[A]) = A × A ∩ ATT(D). We prefer this notation although most authors use D ↓ A .

Admissibility basis
Let S ⊂ ARG(D) be an admissible set in the argumentation framework D. It is not difficult to see that S remains an admissible set in the argumentation framework D[A], for any set of arguments A such that S ⊆ A ⊆ ARG(D). However, if S is a maximal admissible set in D contained in A (i.e., if T ⊆ A is an admissible set in D and S ⊆ T , then S = T ) it is possible that S is not a preferred extension in D [A]. Nevertheless, if D[A] has exactly one preferred extension, then there is also a unique maximal admissible set in D contained in A, as the following theorem shows.

Theorem 1 (Unique maximal admissible set contained in a set inducing an argumentation framework with exactly one preferred extension)
Let A ⊂ ARG(D) be a set of arguments in the argumentation framework D such that D[A] has exactly one preferred extension. Then, there is A adm ⊆ A, an admissible set in D that contains any admissible set in D contained in A.
Proof Let A be the set of all admissible sets S in D satisfying S ⊆ A. A is non-empty since ∅ ∈ A. Let A adm = ∪ S∈A S. We show that A adm is an admissible set in D.
Clearly, A adm ⊆ A. Since each S ∈ A is an admissible set in D, we have S − ⊆ S + . Hence (A adm ) − = ∪ S∈A S − ⊆ ∪ S∈A S + = (A adm ) + . It remains to prove that A adm is a conflict-free set.
. It follows that each S ∈ A is a subset of P . Therefore A adm ⊆ P , and therefore it is a conflict-free set.
Prominent examples of sets A satisfying the hypothesis of the above theorem are those with the property that there is no even circuit in D[A] (by a result of [16]) and conflictfree sets. The case of conflict-free sets deserves a particular investigation as the rest of this section shows.

Theorem 2 (Admissible basis of a conflict-free set)
Let S ⊂ ARG(D) be an arbitrary conflict-free set in the argumentation framework D. There is an admissible set S adm ⊆ S that contains any admissible set contained in S. Moreover, if S satisfies F (S) ⊆ S, then S adm is a complete extension.
Proof Since S is a conflict-free set in D, we have ATT(D[S]) = ∅, S is the unique preferred extension in D[S], and Theorem 1 applies. For the second assertion, we prove that if F (S) ⊆ S then F (S adm ) = S adm . Indeed, since S adm is an admissible set, we have S adm ⊆ F (S adm ). Let a ∈ F (S adm ). From S adm ⊆ S, we have F (S adm ) ⊆ F (S). By hypothesis F (S) ⊆ S, hence a ∈ S. It follows that S adm ∪ {a} ⊆ S is a conflict-free set and because a ∈ F (S adm ) and S adm is an admissible set we obtain that S adm ∪ {a} is an admissible set contained in S. Therefore S adm ∪ {a} ⊆ S adm , that is a ∈ S adm . Hence F (S adm ) ⊆ S adm .
By the proof of Theorem 2, S adm is unique and will be referred to as the admissible basis of the conflict-free set S. Its significance is revealed by the following remark (see also [17]).

Remark 1 (Ideal and Eager Semantics)
Let D be an argumentation framework.
- [14] If S is the conflict free set obtained by intersecting all preferred extensions of D, then its admissible basis, S adm , is the ideal extension of D. - [6] If S is the conflict free set obtained by intersecting all complete extensions T of D with the property that T ∪ T + are maximal with respect to set-inclusion (semi-stable extensions), then its admissible basis, S adm , is the eager extension of D.
By Theorem 2, we can define a function S → S adm , which associates to each conflict free set S its admissible basis S adm . Interesting enough, this function can be evaluated in polynomial time, as the following theorem shows.
Proof If the current conflict-free set, S adm , is an admissible set, then the algorithm returns it. Otherwise, the set (S adm ) − − (S adm ) + , of attackers of S adm that are not counterattacked by S adm , is not empty. The arguments of S adm attacked by this set can not be members of any admissible set contained in S adm . Hence they can be deleted from S adm . At each while iteration, at least one argument is deleted from the current conflict-free set, hence the algorithm terminates.
We note that, when the uniqueness of the maximal (w.r.t. inclusion) admissible set contained in S is assured, an equivalent (fixed point) form of the above algorithm is given in [17]. Theorems 2 and 3 can be combined to decide in polynomial time the acceptability of an argument in a bipartite argumentation framework. An argumentation framework D is bipartite if ARG(D) is the disjoint union of two conflict-free sets L and R, ARG(D) = L∪R. An example is given in Fig. 1, where L = {a 1 , a 2 } and R = {b 1 , b 2 , . . . , b n }. It is not difficult to see that, in this case, we have L adm = L and R adm = ∅.
Parts of the following lemma are well-known, but some of them are new. We prove all of them for the sake of completeness. ii) By i), D has a stable extension S . Since S is an admissible set, it follows that S ∪ S is a stable extension in D. iii) Let S be a preferred extension in the bipartite argumentation framework D. In the above proof of ii) S is a subset of the stable extension S ∪ S . The only possibility is that S = ∅, that is ARG(D ) is empty, and, therefore, S is a stable extension. iv) Since S is a conflict-free set and S L and S R are subsets of S, it follows that these are also conflict-free sets. Since S is an admissible set, we have It follows that S R and S L are admissible sets. Conversely, since S L and S R are admissible sets, we have By hypothesis S is a conflict-free set, therefore it is an admissible set in D. v) Let S be a stable extension in D (by i), S exists). By iv), S L = S ∩ L and S R = S ∩ R are admissible sets contained in L, respectively R,

Bipartite Credulous Acceptance
Proof Suppose that a ∈ L, that is, S = L (the case a ∈ R is similarly). Then, by Lemma 1 iv), there is an admissible set T such that a ∈ T if and only if a ∈ T L . By Theorem 2, this happens if and only if a ∈ L adm .
We note that, essentially, the above algorithm has been previously also given in [16] (see also [15]).

Theorem 4 (Admissible bases and bipartite skeptical evaluation) The following algorithm is correct:
Proof Suppose that a ∈ L, that is, S = L and T = R (the case a ∈ R is similarly).
By the above theorem, if a ∈ L adm then a is not credulously accepted, and the answer is NO. Suppose a ∈ L adm and there is a preferred extension U such that a ∈ U . Then, by Lemma 1 iii), U is a stable extension and, therefore a − ∩ U = ∅. By Lemma 1 iv), we have a − ∩ U R = ∅, and the algorithm returns NO, since a − ∩ U R ⊆ a − ∩ R adm , by Theorem 2. Hence if a ∈ L adm and a − ∩ R adm = ∅, then every preferred extension contains a and, therefore, the algorithm returns YES.
Please note that the above algorithm improves that given in [7] (see also, [15]), since it avoids testing if there is an argument in a − that is credulously accepted, by the use of the admissible bases. For example, testing if a 1 is skeptical accepted in the bipartite argumentation framework in Fig. 1, needs only to compute L adm = {a 1 , a 2 } and, since R adm = ∅, the answer is positive. On the other hand, the algorithm in [7] requires O(n) credulously acceptance testing of the arguments b i ∈ a − 1 . It is well-known that the acceptability problems on bipartite argumentation problems can be solved in polynomial time (see, e.g., [18]) but the speed-up of a factor of |ARG(D)| of the worst time complexity is important.

Out-and-out conflict free sets
In this section we consider a specific family of conflict-free sets and use it to obtain results on the existence of stable extensions in special argumentation frameworks. A conflict-free set S in an argumentation framework D is called an out-and-out conflict-free set if any argument in ARG(D) − (S ∪ S + ) is attacked by an argument in S + . This corresponds to quasi-kernels in Graph Theory terminology (introduced by [10], see [11]) and appears also in the Theory of Voting in the study of uncovered sets (see, e.g., [3]).

Definition 3 (Out-and-out conflict-free set) Let D be an argumentation framework.
A conflict-free set S in D is called an out-and-out conflict-free set if Example. In the argumentation framework D in

Theorem 5 (Out-and-out conflict-free sets in AFs)
In every argumentation framework, D, there is an out-and-out conflict-free set.
Proof Let {a 1 , a 2 , . . . , a n } be an arbitrary ordering of the arguments in ARG(D), and let us consider the following algorithm

Construction of an out-and-out conflict-free set
Clearly, the set S constructed by the algorithm is non-empty: in the first scan at least a 1 is added to S, and in the second scan at least the last argument added to S in the first scan, remains in S.
Also, S is a conflict-free set: in the first scan, the attacks (a i , a j ) ∈ ATT(D) with a i , a j ∈ S and i < j are cleared, and in the second scan the attacks (a i , a j ) ∈ ATT(D) with a i , a j ∈ S and i > j are avoided.
Clearly, at the end of the algorithm, we have ARG(D) = S∪ Discard. The arguments in Discard are those attacked by the returned set S, that is S + , and those attacked by the arguments added to S in first scan and deleted from S in the second scan. Therefore, these arguments belongs to (S + ) + , and the theorem holds.
For the argumentation framework D in Fig. 2, taking {a 1 , . . . , a 6 } = {b, c, d, e, f, a}, the set S constructed by the above algorithm in the first scan is {b, d, f, a}, and after the second scan, the algorithm returns {a, f, d}. Also, taking {a 1 , . . . , a 6 } = {a, b, c, d, e, f }, we obtain after the first scan {a, c, e}, which is not modified by the second scan. We already verified that these are out-and-out conflict-free sets.
Out-and-out conflict-free sets can be used to obtain admissibility results in particular argumentation frameworks.
An It follows that S ∪ S + = ARG(D), that is, S is a stable extension. ii) Since D is a transitive argumentation framework, by i), D has a stable extension S .
Since S is an admissible set, no argument in ARG(D ) attacks S, therefore we obtain that S ∪ S is a conflict-free set in D attacking all arguments outside it, that is, it is stable extension in D. iii) Let S be a preferred extension in the transitive argumentation framework D. Note that D , as defined in ii), is also a transitive argumentation framework. In the above proof of ii) S is a subset of the stable extension S ∪ S . Since S is a maximal admissible set, it follows that S = ∅, that is, ARG(D ) is empty. Hence, S is a stable extension in D.
The above proof can be adapted to obtain similar results for a class of argumentation frameworks containing the transitive ones.
An intransitive triple in the argumentation framework D is a set of three distinct arguments {a, b, c} ⊆ ARG(D), such that Proof i) By Theorem 5, there is in D an out-and-out conflict-free set S such that S ∪ . Because D is a near-transitive argumentation framework, it follows that the set {a} ∪ {d} ⊆ S is not conflict-free, a contradiction. It follows that U 1 = ∅, that is S is an admissible set. Moreover, it follows that: (*) there is no attack between an argument in S and an argument in U . Let c ∈ U , and as above, the intransitive triple {a, b, c} implied by the definition of U . There is no attack from c against an argument in U , that is U is a conflict-free set. Otherwise, if there is d ∈ U such that (c, d) ∈ ATT(D), then (since D is a neartransitive argumentation framework) the set {a}∪{d} is not conflict-free, contradicting the above remark (*) . It follows that S ∪ U is a conflict-free set in D, hence a stable extension, since ARG(D) − (S ∪ U) = S + = (S ∪ U) + . ii) Similar to the proof of Theorem 6 ii). iii) Similar to the proof of Theorem 6 iii), with the observation that if D is a near-transitive argumentation framework, then D is also a near-transitive argumentation framework.
Le us note here the algorithmic importance of Theorem 6 ii) and Theorem 7 ii). Clearly, verifying if the argumentation framework D is transitive or near-transitive can be done in polynomial time (e.g., testing the conditions in their definitions for each triple of arguments). Also, by the proofs of Theorem 6 i) and Theorem 7 i), the stable extension containing the admissible set S in the (arbitrary) argumentation framework D can be constructed in polynomial time with the algorithm in the proof of Theorem 5. Hence, the stable extension containing the admissible set S can be constructed in polynomial time whenever the (polynomial) test whether D is (near)transitive is positive.

Theorem 8 (Out-and-out conflict-free set extending a preferred extension) Let P be a preferred extension in the argumentation framework D. There is an out-and-out conflict-free set S ⊆ ARG(D) such that P ⊆ S.
Proof Let U = ARG(D) − (P ∪ P + ). If U = ∅, then P is a stable extension. Since any stable extension is an out-and-out conflict-free set, we can take S = P .
If U = ∅, let D = D[U ] be the argumentation framework induced by U in D. By Theorem 5, there is T , an out-and-out conflict-free set in D . Since there are no attacks between the arguments in P and U , it follows that S = P ∪ T is a conflict-free set. It is easy to see that S ∪ S + ∪ S ++ = ARG(D).
Note that, for the out-and-out conflict-free set S in the above theorem, we have S adm = P .

Proposition 2 (Maximal out-and-out conflict-free sets) Let S be a maximal (w.r.t set-inclusion) out-and-out conflict-free set S ⊆ ARG(D).
Then, for any argument a ∈ ARG(D) − S, either a ∈ S + or a ∈ S − − S + . . (a, b) ∈ ATT(D)}, and U 2 = U − U 1 . If we prove that U 1 = ∅, the theorem holds.
Suppose that U 1 = ∅, and let D = D[U 1 ] be the argumentation framework induced by U 1 in D. By Theorem 5, there is T , an out-and-out conflict-free set in D . Since there are no attacks between the arguments in S and U 1 , it follows that S = S ∪ T is a conflict-free set. By the construction of T , S is an out-and-out conflict set, a contradiction to the hypothesis that S is a maximal out-and-out conflict-free set.

Definition 4 (Sound out-and-out conflict-free set) Let D be an argumentation framework.
A maximal (w.r.t set-inclusion) out-and-out conflict-free set S ⊆ ARG(D) is called a sound conflict-free set if S adm is a preferred extension of D. The set of all sound conflict-free sets of D is denoted by S D .
Let P D the set of all preferred extensions in the argumentation framework D. By Theorem 8, for every P ∈ P there is S P ∈ S D such that P ⊆ S P . It follows that ∩ P ∈P D P ⊆ ∩ P ∈P D S P . By the Definition 4, the set {S P | P ∈ P D } is exactly S D . Hence, the following inclusion holds In fact, we have equality here. 1 Indeed, suppose that there is x ∈ ∩ S∈S D S − ∩ P ∈P D P . It follows that there is P 0 ∈ P such that x ∈ P 0 and P 0 is not a stable extension (since a stable extension is a sound conflict-free set). Since P 0 is a preferred extension, there is y ∈ ARG(D) − (P 0 ∪ P + 0 ) such that (y, x) ∈ ATT(D). In the argumentation framework D[ARG(D) − (P 0 ∪ P + 0 )] we construct an out-and-out conflict free S 0 by applying the algorithm in the proof of Theorem 5 with any ordering of its arguments starting with y. Clearly, in the first scan of this algorithm x is discarded from S 0 . It follows that P 0 ∪ S 0 is a sound out-and-out conflict free set not containing x, contradicting its choice (x ∈ ∩ S∈S D S).
Hence P ∈P D P = S∈S D S, and using Remark 1, we obtain the following result.

Further observations
In this section we conclude our analysis by two types of observations aimed at deepening the intuitive understanding of how such admissibility notions can play out in practice.
To this end we will: 1. Discuss our assumption that the argumentation frameworks considered have no selfattacks and, also, show that we can not change the admissibility acceptance of an argument a by adding (missing) attacks issued from the attackers of a. These two simple remarks (new to our knowledge) could be very useful from an algorithmic point of view. 2. Decompose the set of arguments outside of a conflict free set S into different sets of arguments determined by the number of attacked and attacking arguments from S. This fine-grained decomposition can be used in two ways in practice. First, as a dynamic labeling of the arguments (and, consequently, as a topological searching space by using appropriate heuristics) and, second, in order to define a score to distinguish between different extensions of the same type.

Self-Attacks and Reinforced Attackers
By the Definition 1 of an argumentation framework, it is possible to have self-attacking arguments. An argument a ∈ ARG(D) is a self-attacking argument if (a, a) ∈ ATT(D). Clearly, no self-attacking argument can be member of a conflict-free set, therefore no selfattacking argument is acceptable in an argumentation framework. However, a self-attacking argument can have a decisive influence in the non-acceptance of some other arguments.
The following proposition shows that we can restrict to argumentation frameworks without self-attacking arguments without any loss of the generality. Conversely, let S ⊆ ARG(D) be an admissible set in D . If a ∈ S, then a 2 ∈ S − in D . Since a 1 is the only attacker of a 2 in D , and a 1 ∈ S, it follows that S does not defend a against the attack (a 1 , a), a contradiction. Therefore, S ⊆ ARG(D) − {a}. As above, we obtain that S is an admissible set in D. 2 The strength of Dung's collective view of arguments acceptance is illustrated by the following proposition which surprisingly shows that the admissibility based acceptability of a given argument is not influenced by adding attacks issued from the attackers of the specified argument, against other arguments. Proof We specify the argumentation framework in the notations x + (x − , S + , S − ) by writing

Proposition 5 (Attackers reinforcement is worthless) Let
Suppose that there is an admissible set S ⊆ ARG(D) with a ∈ S (S is conflict free in D and S − D ⊆ S + D ). Since ARG(D) = ARG(D ), we have S ⊆ ARG(D ). Since a ∈ S and the attacks added to D to obtain D are from the arguments in a − D ⊆ ARG(D) − S to other arguments, it follows that S is a conflict-free set in D . Also, Conversely, suppose that there is an admissible set S ⊆ ARG(D ) with a ∈ S (S is conflict free in D and S − D ⊆ S + D ). Since ARG(D ) = ARG(D), we have S ⊆ ARG(D). Since ATT(D) ⊆ ATT(D ) and S is a conflict-free set in D , it follows that S is a conflictfree set in D. Since a ∈ S and in D only attacks from a − D are added, it follows that

Dynamic labelling
An equivalent way to express Dung's extension-based semantics is using argument labeling as proposed by [4] (originally introduced in [24]). The idea underlying the labeling-based approach is to assign to each argument a label from the set {I, O, U }. The label I (i.e., In) means the argument is accepted, the label O (i.e., Out) means the argument is rejected, and the label U (i.e., Undecided) means one abstains from an opinion on whether the argument is accepted or rejected.
We consider here a related approach, more appropriate for the combinatorial search of σ -extensions containing a specified argument in a given argumentation framework.

Definition 5 (Hashing ARG(D) by a conflict-free set)
Let D = (ARG(D), ATT(D)) be an argumentation framework and S ⊆ ARG(D) be a conflict-free set in D. S decomposes ARG(D) into disjoint (possible empty) sets where, S i,j = {a ∈ ARG(D) − S | |a − ∩ S| = i and |a + ∩ S| = j } is the set of arguments of D outside S attacked by i arguments from S and attacking j arguments in S.
In order to capture Dung's extension-based semantics, we need a coarser decomposition of the argument set of an argumentation framework, as follows. It is not difficult to verify (using the Definition 6) that the following important property holds.
Remark 2 Let S and S be conflict-free sets such that S ⊆ S . Then, S − S ⊆ Neutral(S) and Neutral(S ) ⊆ Neutral(S).
The argumentation semantics in Definition 1 can be equivalently characterized using our decomposition. Some parts of the the following proposition are obvious, some parts are already established (e.g. Proposition 3.2 in [2]) but they are interesting from a combinatorial search point of view. Proof i) and v) are obvious. ii) is Caminada's characterization of a complete labeling [4]. iv) follows from Dung's [13] characterization of grounded extension as the unique least fixed point of the characteristic function. We prove only iii).

iii) Let D = D[Neutral(S)] the argumentation framework induced by Neutral(S) in D.
If S is a preferred extension in D then it is a complete extension and there is no complete extension S such that S ⊂ S (S = S ). Suppose that T = ∅ is an admissible set in D . Then S ∪T is an admissible set in D and (by Remark 2) Neutral(S ∪T ) ⊆ Neutral(S). Since S is a complete extension, by (ii), we have a − ∩Neutral(S ∪T ) = ∅, for every a ∈ Neutral(S ∪T ). Hence, again by (ii), S ∪ T is a complete extension, contradicting the maximality of S.
Conversely, suppose that the empty set is the only admissible set in D . If there is a complete extension S such that S − S = ∅, then (by Remark 2) S − S ⊆ Neutral(S) and Attacker(S − S) ∩ Neutral(S) = ∅, that is S − S is a non-empty admissible set in D , a contradiction.
Our intention is to use the decomposition in Definition 6 as a dynamic labeling of the arguments (having as parameter the conflict-free set): each argument of the argumentation framework has one of the following labels: S, A(ttacker), N(eutral) and D(iscard). Since the conflict-free sets containing S can be obtained only by adding arguments from Neutral(S), deciding the acceptability of a given argument can be done as follows.
Let us denote by S the set of all conflict-free sets in D containing a given argument a 0 . We can decide the (credulously) acceptability of a 0 by considering search algorithms that operate on the state space S = (S I , S * , succ), where S I = {a 0 } ∈ S is the initial state, S * ⊆ S is the set of goal states (that are conflict-free sets corresponding the various admissibility based semantics, as characterized in Theorem 6) and succ is a successor function that maps each state S ∈ S (i.e., a conflict-free set in D containing a 0 ) to a finite (possible empty) set of successor states: succ(S) = {S ∪ {a}|a ∈ Neutral(S)}. Note that for preferred semantics, a goal state can be recognized only by solving a set of search problems of the same type.
S can be organized as a state space topology [21] by defining heuristic functions, h : S → R + 0 ∪{∞}, in order to speed up the search for a goal state or to add fast pruning criteria that allow some states to be discarded.
For example, an obvious heuristic, h 1 , estimates the "degree of inadmissibility" of any conflict-free set S ∈ S by the number of arguments attacking S and not counterattacked, i.e., h 1 (S) = |Attacker(S)|. Clearly, h 1 (S) = 0 if and only if S is an admissible set in D. Assuming that states with lower h 1 values are on a path to the closest goal state, the successor state of an arbitrary (non goal) state S will be S ∪ {x S }, selected in a greedy best first search (GBFS) manner. Obviously, if h 1 (S) > 0 but Neutral(S) = ∅, then S is a dead-end: there is no way to extend S to goal state (no admissible set extends S).
The decomposition in the Definition 5 can be also used to define the score of a conflictfree set S with respect to an argument a ∈ ARG(D)−S. If a ∈ S i,j then Score(S, a) = i−j . Equivalently, we have the following definition. Note that a non-negative score, Score(S, a) ≥ 0, shows that the conflict-free set S counter-attacks every attack from a. This is not the same as graded defense (or neutrality) introduced in [20], but shares the motivation and significance, excellently discussed in their paper.
Examples.  [19]; D 4 is depicted in Fig. 3). D n has 2 n stable extensions: S 0 = {a 0 , a n+1 , . . . , a 2n } and S A = A ∪ {a n+1 , . . . , a 2n } − {a n+i |a i ∈ A}, for each A ⊆ {a 1 , . . . , a n }, A = ∅. Note that each argument is credulously preferred (stable) accepted in D n . However, despite of the exponential number of preferred extension, no argument is skeptically accepted, since for each argument a there is a preferred extension not containing a. The same conclusion holds for the ideal or eager extension (which are empty). Let us analyze the score of each such stable extensions with respect to the outside arguments. It follows that S {a 1 ,...,a n } = {a 1 , . . . , a n } is the only stable extension having non-negative scores with respect to the arguments outside it and maximizing the sum of these scores.  Fig. 4). Clearly, S = {a 0 , . . . , a n } is the grounded extension in D n . Since ARG(D n ) − S = {b}, the score of S against b is Score(S, b) = |b − ∩ S| − |b + ∩ S| = 1 − n < 0. On the other hand, S 0 = {a 0 } and S i = {a 0 , a i } are admissible sets with non-negative scores with respect to arguments outside them: Score(S 0 , b) = 1 and Score(S 0 , a i ) = 0, for all i ≥ 1; Score(S i , b) = 0 and Score(S i , a j ) = 0, for all j ≥ 1 and j = i. It follows that S 0 is an admissible set containing and defending the argument a 0 and its score against any other argument is non-negative (which is not the case for the grounded extension). This can be more convincing for the acceptance of the argument a 0 in some applications (of course, for non logical argumentation frameworks). Similarly, S i can be used as a better justification for the acceptability of a i .

Let
We can also use the scores to pass beyond of the notion of admissibility, as follows. For k = 0, 0-strict admissibility will be referred, simply, as strict admissibility. Fig. 4 The grounded extension has not the best scores Clearly, every strict admissible set S is an admissible set: if a ∈ ARG(D) − S is an attacker of S, then there is at least one argument in S attacking a, since Score(S, a) ≥ 0. Converse is not true, not every admissible set is strict admissible. A simple example is given in Fig. 5. Strict admissibility, imposing an external condition is different from strong admissibility introduced in [1], which imposes a supplementary internal condition on a Dung's admissible set.
There are particular argumentation frameworks D and a ∈ ARG(D) such that every admissible set containing a is also a strict admissible set. A prominent example is the argumentation framework in the standard proof of the polynomial time reduction of the SAT problem to the problem of deciding if a given argument in a given argumentation framework is credulously accepted [12,16].
Let F = C 1 ∧ . . . ∧ C m be an arbitrary instance of SAT, where for each i ∈ {1, . . . , m}, the clause C i is a a disjunction of literals, C i = l i 1 ∨ . . . ∨ l i k i , and a literal l is either a variable x j or its negation x j , where {x 1 , . . . , x n } is the set of Boolean variables occurring in F . An argumentation framework D F and a F ∈ ARG(D F ) can be constructed in polynomial time such that F is satisfiable if and only if a F is (credulously) admissible acceptable in D F . Firstly, the special argument a F is added to ARG(D F ). Second, for each clause C i an argument a C i is added to ARG(D F ) and an attack (a C i , a F ) is added to ATT(D F )). Finally, for each Boolean variable x j occurring in F , two arguments a x j and a x j are added to ARG(D F ), attacking each other in D F and adding all the attacks (a x j , a C i ), if x j is a Fig. 5 The argument c is not strict admissible acceptable, since the conflict-free sets S 1 = {c} and S 2 = {a, c}, containing it, are not strict admissible sets (S 1 is not admissible, S 2 is admissible, but Score(S 2 , b) = 1 − 2 = −1 < 0) Fig. 6 The argumentation framework D F associated to instance F = C 1 ∧C 2 ∧C 3 ∧C 4 , where C 1 = x 1 ∨x 2 , C 2 =x 1 ∨ x 2 ∨x 3 , C 3 = x 2 ∨x 3 , C 4 =x 2 ∨x 3 ∨x 4 literal in C i , and (a x j , a C i ), if x j is a literal in C i , for all clauses C i . Clearly, D F can be constructed in polynomial time in the size of the instance of SAT. An example of this construction is illustrated in Fig. 6.
It is not difficult to see that any admissible set in D F containing a F is of the form S = {a F } ∪ T , where T selects for each variable x j at most one of the arguments a x j or a x j such that for each clause C i , the argument a C i has at least one attacker in S. But this is equivalent to the fact that F is satisfiable (by assigning the truth value to the literals corresponding to the arguments in T ). The arguments a outside S are either of the type a x j and a x j not selected by the set T (for which Score(S, a) = 0), or of the form a C i (for each clause C i ), for which Score(S, a) ≥ 0, since a attacks a F , but it is attacked by at least one argument in T .
It follows that we proved the hardness of the problem of deciding if a specified argument in a given argumentation framework is strictly accepted, by exhibiting a polynomial time reduction from the CNF satisfiability problem SAT. Since the membership of this problem to NP is obvious, the following theorem holds. Theorem 9 (Strict admissibility acceptance is NP-complete) Deciding if a specified argument in a given argumentation framework is strictly accepted is an NP-complete problem.

Discussion
In this paper we presented a combinatorial analysis of the notion of admissibility in Dung semantics for abstract argumentation frameworks. The two main contributions were the study of admissibility basis of conflict free sets and the study of a specific family of conflict free sets -the out-and-out conflict free sets. The first contribution allowed for the development of an efficient skeptical acceptance algorithm in bipartite argumentation frameworks. The second allowed for the development of linear algorithms to compute stable extensions in transitive argumentation frameworks and the further study of admissibility in near transitive argumentation frameworks. Despite of the fact that the results concern very specific sub-classes of argumentation frameworks, since their algorithmic recognition can be made in polynomial time, they are applicable as sub-tasks in the algorithms dedicated to reasoning with general argumentation frameworks (see, e.g., Lemma 1 i), Theorem 6 ii), Theorem 7 ii)). In the same line of idea, [23] reported remarkable experimental results on checking skeptical admissibility in general argumentation frameworks using bipartite argumentation framework recognition. It is therefore not necessary to show that they correspond to interesting applications of argumentation (in order to sustain the practical applicability of these results).
We closed the paper by providing two types of observations relevant for the practical use of admissibility notions. In fact, the paper is conceived as a theoretical basis for developing an efficient argumentation solver using a dedicated combinatorial search space (as suggested in Section 5.2). Several software systems for deciding acceptability in abstract argumentation frameworks have been developed and, also, an international competition 3 for empirical evaluation and comparison of solvers appeared in 2015. In the survey [8], the existing implementation methods are classified into two categories: reduction approaches and direct approaches. The reduction approaches transform (in polynomial time, sometimes iteratively) the graph model into inputs for established solvers (SAT, CSP, or ASP) assessed for well-studied problem domains. The direct methods access the framework directly (without having the overhead of transformation and, as a result, a potential loss of structural information). In a future work (carrying on this paper) we intend to introduce a new direct approach based on a systematic and informative search method on the space of conflict-free sets of arguments associated to the input of the problem of deciding acceptability.