In abstract argumentation, the directionality principle conveys the intuition that, for an unattacked set, the choice of arguments that are part of an extension should only depend on the restriction of the framework to that set. Furthermore, having made such a choice, one should be able to select arguments from the rest of the framework so as to get an extension. In this paper we show how this idea can be generalized and used for formulating SCC-recursiveness as a stronger version of directionality. We argue that such properties characterize the information that is needed for computing the extensions of an argumentation semantics. We provide a formal approach for describing and comparing directionality-like properties. Our model provides a clear distinction between SCC-recursive semantics that use defense information and those that do not use it.
This work lies in the general setting of abstract argumentation frameworks,
as proposed by Dung [
Non-interference and directionality were proposed as desirable properties of
argumentation semantics, conveying the idea that for some sets of arguments
the selection of arguments for an extension should not depend on the rest of the
framework [
SCC-recursiveness was introduced in [
In this paper we introduce strongly connected sets (SCS's) as a generalization of SCC's and we provide a stronger characterization of SCC-recursiveness using these sets. Furthermore, we propose SCC-directionality as a natural re nement of directionality.
We also introduce general directionality as a very broad formal approach for describing the behavior of argumentation semantics with respect to local computation of extensions and the information that needs to be available from the rest of the framework.
Section 2 introduces the argumentation concepts we are going to use, with references to the literature. We generalize SCC-recursiveness and introduce SCCdirectionality, together with its properties, in Section 3. The general directionality is presented in Section 4, with an example that outlines the added value of our approach. The paper ends with conclusions and ideas for future research in Section 5. 2
In this section we aim to provide a minimal argumentation background, covering three aspects that are relevant to our work: argumentation semantics, SCCrecursiveness and properties of semantics. We start with a formal de nition of argumentation frameworks and the related terminology.
De nition 1. An argumentation framework is a pair F = (A; R), where A is a set of arguments and R A A is a binary attack relation on A. Whenever (a; b) 2 R we say that argument a attacks argument b and we write this as a ! b. We say that a set of arguments S A defends an argument a i S attacks all arguments b that attack a. We extend the notion of attack to also refer to sets of arguments, as follows: (1) (2) a ! S , 9b(b 2 S ^ a ! b) S ! a , 9b(b 2 S ^ b ! a)
S ! T , 9ab(a 2 S ^ b 2 T ^ a ! b) S
For a given argumentation framework F = (A; R) and a set of arguments A, the restriction of F to S is the argumentation framework
F#S , (S; R \ (S
S))
For example, the argumentation framework from Fig. 1 is given by A = fa; b; c; dg and R = f(a; b); (b; c); (c; d); (d; c)g. The restriction of F to the set fa; b; cg is F#fa;b;cg= (fa; b; cg; f(a; b); (b; c)g). 2.1
In the argumentation literature, semantics refer to approaches (algorithmic,
constraint-based or otherwise) for choosing sets of arguments (extensions) that
are acceptable. We rst introduce the argumentation semantics de ned in [
De nition 2. Let F = (A; R) be an argumentation framework and let E be a set of arguments.
{ E is con ict-free i there is no attack between arguments from E. The set of all con ict-free sets of F is denoted by ECF (F ). { E is admissible i E is con ict-free and E defends all the arguments it contains. The set of all admissible sets of F is denoted by EAS (F ). { E is a complete extension of F i E is admissible and E contains all the arguments it defends. The set of all complete extensions of F is denoted by ECO(F ). { E is a stable extension of F i E is con ict-free and E attacks all arguments that are not in E. The set of all stable extensions of F is denoted by EST (F ). { E is a preferred extension of F i E is a maximal (with respect to set inclusion) admissible set of F . The set of all preferred extensions of F is denoted by EPR(F ). { E is the grounded extension of F i E is the (unique) minimal complete extension (with respect to set inclusion). The (singleton) set of grounded extensions is denoted by EGR(F ).
The name of the sets (or extensions) presented in De nition 2 corresponds to the name of the argumentation semantics. Furthermore, whenever the extensions of a particular semantics are denoted by ESem(F ), it means that Sem is used as an abbreviation of the corresponding semantics. For example CO stands for the complete semantics.
For the framework from Fig. 1, the semantics of De nition 2 give the following extensions:
ECF (F ) = f?; fag; fa; cg; fa; dg; fbg; fb; dg; fcg; fdgg EAS (F ) = f?; fag; fa; cg; fa; dg; fdgg ECO(F ) = ffag; fa; cg; fa; dgg EST (F ) = ffa; cg; fa; dgg EPR(F ) = ffa; cg; fa; dgg EGR(F ) = ffagg a b c
d A (3)
Several other semantics have been proposed in the literature, such as ideal
[
The general idea behind SCC-recursiveness is to compute semantics taking
advantage of the decomposition of the argumentation framework along its strongly
connected components (SCC's). We will only provide here the minimal de
nitions required for introducing the concept. For more details and the rationale
behind the idea, the reader may consult [
De nition 3. Let F = (A; R) be an argumentation framework. We de ne the path equivalence relation P EF A A as follows: { (a; a) 2 P EF for all arguments a 2 A { for any two arguments a and b, (a; b) 2 P EF i there is a path in R from a to b and a path from b to a P EF so de ned is an equivalence relation and its equivalence classes are called strongly connected components (SCC's). We will use SCCSF to refer to the set of strongly connected components of F . We will denote the strongly connected component that contains an argument a with SCCF (a).
Given an argumentation framework F and an extension E, we can partition the elements of any strongly connected component S into three di erent classes, with respect to how the extension E interacts with them from outside S. These classes are introduced in De nition 4.
De nition 4. Let F = (A; R) be an argumentation framework, E A a set of arguments and S 2 SCCSF a strongly connected component of F . The elements of S can be partitioned into three sets with respect to E: 1. DF (S; E) = fa 2 S j (E n S) ! ag { the set of arguments attacked by E from outside S 2. UF (S; E) = fa 2 S j (E n S) 6! a ^ 8b(b 2 A n S ^ b ! a ) (E n S) ! b)g { the set of arguments not attacked by E from outside S and defended by E against all attackers that are not in S. 3. PF (S; E) = S n (DF (S; E) [ UF (S; E)) { arguments that are neither attacked by E from outside of S, nor defended by E against attacks coming from outside S.
With the use of these sets, SCC-recursive semantics can be de ned. We will use U PF (S; E) as an abbreviation for UF (S; E) [ PF (S; E), which is the same as S n DF (S; E). Note that in the formulas from De nition 4 we have used ) for logical implication so that there is no confusion with the attack relation, denoted by !.
De nition 5. An argumentation semantics Sem is said to be SCC-recursive i , for any argumentation framework F = (A; R), ESem(F ) = GF (F; A), where the generic recursive function GF is de ned as follows: for any argumentation framework F = (A; R) and any two sets of arguments E; C A it holds that E 2 GF (F; C) i { in case jSCCSF j = 1, E 2 BF Sem(F; C) { otherwise, for all strongly connected components S 2 SCCSF , we have (E \ S) 2 GF (F#UPF (S;E); C \ UF (S; E)) where BF Sem is a base function that depends on Sem.
The idea conveyed in De nition 5 is that the arguments chosen from a strongly connected component S as elements of an extension E are selected based on what has already been selected from other SCC's, more precisely by taking into account which arguments are defended (UF (S; E)) or at least not defeated (U PF (S; E)) by the arguments selected for E from components that attack S.
It is shown in [
We can use the de nition of SCC-recursiveness in two ways for computing extensions. We can consider all possible sets of arguments and check whether the property speci ed in the de nition holds or we can compute extensions iteratively, starting with unattacked SCC's and considering components one by one.
E \ S1 = fag 2 CF 2(F#UPF (S1;E); A \ UF (S1; E))
= CF 2(F#S1 ; S1) = ffagg E \ S2 = ? 2 CF 2(F#UPF (S2;E); A \ UF (S2; E))
= CF 2(F#?; ?) = f?g E \ S3 = fcg 2 CF 2(F#UPF (S3;E); A \ UF (S3; E))
= CF 2(F#S3 ; S3) = ffcg; fdgg
So the requirement from De nition 5 is satis ed.
De nition 6. The CF 2 semantics [
BF CF 2(F; C) = EMCF (F ) where EMCF (F ) stands for the set of maximal (with respect to set inclusion) con ict-free sets, also known as naive extensions.
For the framework from Fig. 1 we have EMCF (F ) = ffa; bg; fa; cg; fb; dgg. Let us also discuss the extensions of CF 2. The strongly connected components of F are S1 = fag, S2 = fbg and S3 = fc; dg. The extensions are ECF2(F ) = ffa; cg; fa; dgg. Let us see that E = fa; cg is indeed a CF 2 extension. We will use CF 2(F; C) instead of GF (F; C) for the recursive function. So we want to show that E 2 CF 2(F; A). We have: (4) (5)
A principle-based evaluation of argumentation semantics was proposed in [
De nition 7. An argumentation semantics Sem is universally de ned i , for any argumentation framework F , ESem(F ) 6= ?.
In words, universally de ned argumentation semantics provide at least one
extension for any argumentation framework. Of the semantics we have presented
here, only ST is not universally de ned [
De nition 8. Let F = (A; R) be an argumentation framework. A set of arguments S A is isolated i there are no attacks between S and the rest of the framework:
R \ ((S (A n S)) [ ((A n S)
S)) = ? The set of all isolated sets of F is denoted by IS(F ).
An argumentation semantics Sem is said to satisfy the non-interference principle i , for any argumentation framework F = (A; R) and every isolated set S 2 IS(F ) it holds that:
AE Sem(F; S) = ESem(F#S ) where AE Sem(F; S) , fE \ S j E 2 ESem(F )g
The intuition behind non-interference is that the elements chosen for an extension E from an isolated set S can be computed locally in the restricted framework F #S by using the same argumentation semantics. Furthermore, for any such locally computed extension, it should be possible to select additional arguments from the rest of the framework so as to get an extension of F . Again, of the semantics we have formally introduced, only ST fails to satisfy noninterference.
De nition 9. Let F = (A; R) be an argumentation framework. A set of arguments S is unattacked i S is not attacked by any argument that is not in S:
A n S 6! S The set of all unattacked sets of F is denoted by U S(F ).
An argumentation semantics Sem satis es the directionality principle i , for any argumentation framework F = (A; R) and any unattacked set S 2 U S(F ), it holds that:
AE Sem(F; S) = ESem(F#S )
Note that directionality and non-interference impose the same constraint,
the only di erence is the sets for which the relation must hold. Furthermore,
directionality implies non-interference [
If we think of SCC-recursiveness as a property as well, we have already
mentioned that all classical semantics satisfy it. Furthermore, CF 2 satis es it by
de nition. On the other hand, it is shown in [
In our work, we were inspired by the similarities that exist between noninterference, directionality and SCC-recursiveness with respect to what they describe as reasonable for the intersection of an extension with a set of arguments. 3
We have seen that non-interference and directionality require that the intersection of an extension with an isolated (or unattacked) set S can be computed as an extension of the restricted framework F #S . In this section we wish to de ne a new type of sets, one that includes unattacked and isolated sets but also generalizes strongly connected components, then formulate a principle about the intersection of extensions with such sets.
De nition 10. Let F = (A; R) be an argumentation framework and let S A be a set of arguments. S is a strongly connected set i , for all arguments a 2 A, if a indirectly attacks S and S indirectly attacks a, then a is in S. We will use SCS(F ) to refer to all strongly connected sets of F .
SCS(F ) = fS
A j 8a(a !
S ^ S ! a ) a 2 S)g (10) where !
stands for a path of attacks (one or more).
Proposition 1. Let F = (A; R) be an argumentation framework. Every strongly connected set S 2 SCS(F ) can be written as the union of zero or more strongly connected components of F . Proof. Given an argument a 2 S, for all arguments b 2 SCSF (a) we have that a ! b and b ! a, which leads to b 2 S and, thus, SCCF (a) S. It follows that S = Sa2S SCCF (a). tu
Next, we introduce several notations and concepts that will help provide another characterization for strongly connected sets, one that gives a better intuition about the actual property of SCC's that is preserved for strongly connected sets.
De nition 11. Let F = (A; R) be an argumentation framework and let S be a set of arguments. We introduce the following notations:
A (11) LA1(F; S) = fT 2 SCCSF j T 6 S ^ S ! T g LLLBBA1nn(++F11;((FSF;);SS=)) f==TLL2BAS11(C(FFC;;SSSF[[SjSTinin==611LLSBA^i(i(TFF;!;SS)))S)g LA (F; S) = Sn 1 LAn(F; S)
LB (F; S) = Sn 1 LBn(F; S) These notations correspond to the following concepts: { LAn { level n look-ahead of S in F { LBn { level n look-back of S in F { LA { total look-ahead of S in F { LB { total look-back of S in F
Note that for arbitrary sets it is possible that S \ LA1(F; S) 6= ? and S \ LA (F; S) 6= ?. On the other hand, whenever S is the union of zero or more strongly connected components, S \ LA1(F; S) = ? follows from the de nition, since T 6 S and T is a strongly connected component. Similar considerations apply to the look-back.
Proposition 2. Let F = (A; R) be an argumentation framework and let S be a set of arguments. The following relations hold: A (a) S 2 IS(F ) , LA (F; S) = LB (F; S) = ? (b) S 2 U S(F ) , LB (F; S) = ? (c) S 2 SCS(F ) , LA (F; S) \ LB (F; S) = ?
The alternative characterizations from Proposition 2 follow from the corresponding de nitions. The advantage of using look-back and look-ahead consists in clustering the attackers of S and the arguments attacked by S into strongly connected components and distinguishing between the ones that are closer to S and those that are farther away.
Note that only one-way attacks are possible between a strongly connected set and a strongly connected component. On the other hand, this is not necessarily the case for two SCS's. Indeed, suppose we have four SCC's: S1, S2, S3 and S4 and the attacks between them are S1 ! S2 and S3 ! S4. Then the strongly connected sets S1 [ S4 and S2 [ S3 attack each other.
Corollary 1. Let F = (A; R) be an argumentation framework. All isolated sets of F are also unattacked and all unattacked sets of F are also strongly connected sets in F
IS(F )
U S(F )
SCS(F ) (12)
Furthermore, let us see that, for any set S, the set S [ LB (F; S) is an unattacked set (it can be shown that it is the minimal unattacked set that contains S). Similarly, S [ LA (F; S) [ LB (F; S) is the minimal isolated set that contains S.
We are now ready to prove one of the main results of this paper, the generalization of SCC-recursiveness.
Theorem 1. Let GF be the generic function of an SCC-recursive argumentation semantics. Then, for any argumentation framework F = (A; R) and for any partition of A into disjoint strongly connected sets Part = fS1; S2; : : : ; Sng, the following holds:
E 2 GF (F; C) , 8S(S 2 Part ) (E \ S) 2 GF (F#UPF (S;E); C \ UF (S; E))) (13) Proof. We know that the result holds if the partition consists of all the SCC's of F . What we need to show is that it holds for other partitions as well. We proceed by induction on the size of F . For frameworks with a single argument there will be a single SCC and a single partition into strongly connected sets, so the result holds trivially.
For the induction step, we assume that the claim is true for all frameworks that have fewer arguments than F and we prove it for F . We start with the converse. Let us see that if Part = fAg the claim trivially holds. In what follows we focus on partitions containing at least two distinct strongly connected sets. We must prove that if E \ S 2 GF (F#UPF (S;E); C \ UF (S; E)) for all S 2 Part, then E 2 GF (F; C).
Let S 2 Part be an arbitrary SCS. Then E \ S 2 GF (F #UPF (S;E); C \ UF (S; E)). We know from Proposition 1 that S is the union of one or more SCC's of F . For any such component T , we have that T \U PF (S; E) is a strongly connected set in F#UPF (S;E), so we can apply the induction hypothesis for F 0 = F #UPF (S;E) and the extension E \ S. We get (E \ S) \ (T \ U PF (S; E)) 2 GF (F 0#UPF 0 (T \UPF (S;E);E\S); C \ UF (S; E) \ UF 0 (T \ U PF (S; E); E \ S)).
Since E\S U PF (S; E) and T S, we can write (E\S)\(T \U PF (S; E)) = E \ T . Furthermore, let us see that U PF 0 (T \ U PF (S; E); E \ S) = U PF (T; E). Indeed, for any argument a 2 A, we have a 2 U PF 0 (T \ U PF (S; E); E \ S) , a 2 T ^ a 2 U PF (S; E) ^ ((E \ S) n (T \ U PF (S; E))) 6! a , a 2 T ^ (E n S) 6! a ^ ((E \ S) n T ) 6! a , a 2 T ^ (E n T ) 6! a , a 2 U PF (T; E).
Similarly, we show that UF (S; E) \ UF 0 (T \ U PF (S; E); E \ S) = UF (T; E). Here, we actually need to prove the double inclusion. We start with . Let a be an argument from UF (S; E) \ UF 0 (T \ U PF (S; E); E \ S) and let b be an attacker of a, with b 2 A n T . If b 62 S, then (E n S) ! b, because a 2 UF (S; E). If b 2 S, we have either b 2 DF (S; E), in which case (E n S) ! b, or b 2 U PF (S; E) and then (E \ S) n (T \ U PF (S; E)) ! b because a 2 UF 0 (T \ U PF (S; E); E \ S). In all three cases we can infer (E n T ) ! b. Thus, we conclude that a 2 UF (T; E).
For the other inclusion, let us consider a 2 UF (T; E) and b an attacker of a. If b 2 U PF (S; E) n T . Since a 2 UF (T; E), we have (E n T ) ! b. But b 62 DF (S; E), so (EnS) 6! b, which leads to ((E\S)nT ) ! b, so a 2 UF 0 (T \U PF (S; E); E\S). If b 2 A n S we still have (E n T ) ! b, but what we need for a 2 UF (S; E) is (E n S) ! b. Suppose that this is not the case, i.e. there exists c 2 (E \ S) n T such that c ! b. Since both c and a are in S, we have S ! b and b ! S so we must have b 2 S because S is strongly connected. However, this contradicts the assumption that b 2 A n S.
Putting it all together, we got E \ T 2 GF (F#UPF (T;E); C \ UF (T; E)). And this holds for every SCS S 2 Part and every SCC T S. Since Part is a partition, this covers all possible SCC's of F so, based on the SCC-recursiveness of Sem, we can conclude that E 2 GF (F; C), as desired.
We now discuss the direct implication. So we have E 2 GF (F; C). Just as before, we consider S 2 Part and T 2 SCCSF such that T S. From SCCrecursiveness we have E \ T 2 GF (F #UPF (T;E); C \ UF (T; E)). We now use the same equalities as above in order to obtain (E \ S) \ (T \ U PF (S; E)) 2 GF (F #UF0 (T \UPF (S;E);E\S); C \ UF (S; E) \ UF 0 (T \ U PF (S; E); E \ S)). Since F 0 has fewer arguments than F , we use the induction hypothesis and we get E \ S 2 GF (F#UPF (S;E); C \ UF (S; E)). This completes our proof. tu
The very strong result covered by Theorem 1 reveals that we don't need to split a framework along its SCC's, it is enough to choose any partition into strongly connected sets in order to get the same property as that required by SCC-recursiveness. On a more practical note, this also means that whenever we put two frameworks together and only add attacks from one of them towards the other, we can compute the extensions of the union framework using the SCC-recursive approach. The theorem also has two immediate corollaries. Corollary 2. Let F = (A; R) be an argumentation framework and let GF be the generic function of an SCC-recursive argumentation semantics. For every E; C A, the following relation holds: E 2 GF (F; C) , 8S(S 2 SCS(F ) ) (E \ S) 2 GF (F#UPF (S;E); C \ UF (S; E))) (14) Corollary 3. Let F = (A; R) be an argumentation framework and let Sem be an SCC-recursive argumentation semantics whose generic function does not use its second argument. Then, for every set of arguments E, we have: E 2 ESem(F ) , 8S(S 2 SCS(F ) ) (E \ S) 2 ESem(F#UPF (S;E))) (15)
Let us see that the result in Corollary 3 is somewhat similar to the condition that characterizes both non-interference and directionality, in the sense that the same semantics is applied for a restricted framework. Based on this similarity, we formalize the idea of SCC-directionality.
Note however that, while both non-interference and directionality state that the local computation of an extension (its intersection with an isolated or unattacked set) should be computable independently of the rest of the framework, the core intuition behind SCC-recursiveness is quite di erent. What we need to say for strongly connected sets is that the local computation is reasonably in uenced by the computation performed for some other set of arguments. De nition 12. Let F = (A; R) be an argumentation framework. An argumentation semantics Sem satis es the SCC-directionality principle i , for any strongly connected set S 2 SCS(F ) and any T 2 AE Sem(F; LB (F; S)), we have
BE Sem(F; S; LB (F; S); T ) = ESem(F#UPF (S;T )) where, for any sets of arguments S, D, T , we de ne (16) (17)
BE Sem(F; S; D; T ) , fE \ S j E 2 ESem(F ) ^ E \ D = T g
The meaning of SCC-directionality is that, given a strongly connected set S, the arguments we can choose from S for an extension E are determined by what we have chosen in E from the total look-back of S and, furthermore, for any meaningful choice that we can make in LB (F; S), the choices from S can be computed using the same semantics, but applied to a restricted framework that accounts for the selection T .
Note that the signi cant novelty of SCC-directionality lies more in the use of
strongly connected sets and the actual formalization using BE than in the total
look-back, which is in fact equivalent to the union of all ancestor SCC's that are
mentioned in [
Proposition 3. Every universally de ned SCC-recursive semantics Sem whose generic function does not use its second argument satis es SCC-directionality. Proof. Let F = (A; R) be an argumentation framework and let S 2 SCS(F ) be an arbitrary strongly connected set. Also, let T 2 AE Sem(F; LB (F; S)). Given U 2 BE Sem(F; S; LB (F; S); T ), we have that there exists E 2 ESem(F ) such that E \ LB (F; S) = T and E \ S = U . Furthermore, from Corollary 3 we have that E \ S 2 ESem(F #UPF (S;E)). However, let us see that, in fact, U PF (S; E) = U PF (S; E \ LB (F; S)), which leads to U PF (S; E) = U PF (S; T ). Thus, we have proved that BE Sem(F; S; LB (F; S); T ) ESem(F#UPF (S;T )).
For the other inclusion, let us consider U 2 ESem(F #UPF (S;T )). We denote S1 = S [ LB (F; S) and S2 = A n S1 and take the partition Part = fS1; S2g. Both elements are strongly connected sets and, since Sem is universally de ned, we have ESem(F #UPF (S2;T [U)) 6= ? so we can choose an extension V from this set. but then T [ U [ V satis es the conditions of Theorem 1, which leads to T [ U [ V 2 ESem(F ).
The previous result covers the CF and CF 2 semantics. For the other semantics, let us consider the argumentation framework from Fig. 2 and see that SCC-directionality is violated for AS. We consider S = fc; dg. Then LB (F; S) = fa; b; eg. Since fa; cg is admissible, we have that fag 2 AE AS (F; LB (F; S)), so we can choose T = f g
a . But then EAS (F #UPF (S;T )) = f?; fcg; fdgg. However, there is no admissible set that contains d. The proof for the other semantics based on admissibility (CO, ST , GR and PR) is similar.
a b c d e
Furthermore, let us see that, for universally de ned semantics, the noninterference and directionality principles are just special cases of the newly introduced SCC-directionality, where the relation is required to hold only for isolated (respectively unattacked) sets. The proof relies on the fact that, for unattacked sets, BE Sem(F; S; LB (F; S)) = AE (F; S) and U PF (S; T ) = S. 4
In this section we introduce general directionality, based on the intuition that, for certain sets of arguments S, the intersection of an extension E with S depends on some set of arguments D in the sense that this intersection can be computed locally, using a generic function that works with information compiled from the intersection of E with D. We formalize this in Def. 13.
De nition 13. An argumentation semantics Sem is said to satisfy general directionality with signature (Sets; Det; Local; Inf o) i , for any argumentation framework F = (A; R), any set S 2 Sets(F ) and any set T 2 AE Sem(F; Det(F; S)), the following holds:
BE Sem(F; S; Det(F; S); T ) = GF Sem(F#Local(F;S); Inf o(F#Det(F;S)[Local(F;S); S; T )) (18) where: { GF Sem is a function that depends on Sem (and on the signature of the general directionality) { Sets(F ) returns the sets of arguments S for which the relation holds. For example, Sets 2 fIS; U S; SCSg { Det(F; S) returns the set of arguments that reasonably in uences the intersection of extensions with S; it must satisfy Det(F; S) \ S 6= ?. For example, Det 2 fLBn; LB ; LAn; LA ; LB1 [ LA1g { Local(F; S) gives the set that can be used for restricting the framework that is available to GF ; by convention, we use LS(F; S) = S and All(F; S) = A.
For example Local 2 fLS; LS [ LB1g { Inf o(F; S; T ) encodes the knowledge of T into information that is included in Local(F; S); based on the way we used it in the de nition, this information should be computable in the restriction of F to Local(F; S) [ Det(F; S). Furthermore, we de ne the following D(F; S; T ) = DF (S; T ), U (F; S; T ) = UF (S; T ), S(F; S; T ) = S and T (F; S; T ) = T .
Proposition 4. Any argumentation semantics Sem satis es (S; T ; Det))-directionality, for any Sets and Det. (Sets; Det; All; Proof. We can take GF Sem(F; (S; T; Det(F; S))) = BE Sem(F; S; Det(F; S); T ). Proposition 5. For universally de ned semantics, SCC-directionality is equivalent to (SCS; LB ; LS; (S; D))-directionality.
S n DF#S[LB (F;S) (S; T ).
Proof. The result follows from the following relation U PF (S; T ) = SnDF (S; T ) =
Furthermore, it is easy to see that we can get the equivalent characterization of non-interference and directionality in the general directionality setting by replacing SCS with IS, respectively U S in the signature.
We also provide the following result, without proof: Proposition 6. Any SCC-recursive argumentation semantics that is universally de ned satis es (SCS; LB ; LS; (D; U ))-directionality.
If we note that the actual computation of DF and UF only depends on the parent SCC's of S, we can re ne the results in Proposition 5 and Proposition 6 by replacing LB with LB1.
If we compare SCC-recursiveness with respect to the use of the second argument (the defense information), we can see that our model for describing directionality properties clearly shows the distinction between the two classes of SCC-recursiveness, namely the defense information U . Furthermore, the defeat information, which is somewhat hidden in the original de nition of SCCrecursiveness, is present in our model.
All the results presented so far were based on the look-back information. A simple example that uses forward information can be given for con ict-free sets: Proposition 7. CF satis es S j a ! (T n S)g.
(SCS; LA1; LS; (D0)), where D0(F; S; T ) = fa 2 Proof. Con ict-free sets are independent of the direction of the attacks. If all attacks are reversed, the meaning of forward and backward information interchanges, while the defeat information D becomes D0.
The rst contribution of this paper is the generalization of strongly connected components as strongly connected sets and their use for a stronger version of SCC-recursiveness. We have also turned the simple SCC-recursiveness (without defense information) into a directionality-like property that re nes plain directionality.
Based on the similarities and di erences between non-interference, directionality and SCC-recursiveness, we have provided in this paper a unifying model that can capture these properties and at the same time provide the means for comparing such properties with respect to various aspects.
The fact that all argumentation semantics satisfy some (trivial) form of general directionality suggests the possibility of searching for \minimal" kinds of directionality that are satis ed by each semantics. For SCC-recursiveness we have seen already that ancestor information can be replaced with just parent information.
Future work will explore the relations between various kinds of general directionality. We will also give more attention to the look-ahead information, since it might be relevant for argumentation semantics such as stage or semi-stable.
This work has been funded by the Sectoral Operational Programme Human Resources Development 2007-2013 of the Romanian Ministry of Labour, Family and Social Protection through the Financial Agreement POSDRU/88/1.5/S/61178 and by project ERRIC (Empowering Romanian Research on Intelligent Information Technologies), number 264207/FP7-REGPOT-2010-1.