This paper discusses the issue of mixing di↵erent argumentation semantics in a single Dung's argumentation framework. The general notion of combination schema is defined to model a specific way of mixing argumentation semantics, and several properties that may be desirable for a combination schema are introduced at an abstract level. A specific combination schema is then evaluated in the light of such properties, showing that there are several interesting challenges both from a conceptual and a technical perspective still to be tackled.
Dung’s model of abstract argumentation focuses on evaluating the justification
of a set of arguments related by an attack relation [
With the exception of [
License Attribution 4.0 International (CC BY 4.0). the cost of evaluation errors and the computational complexity of semantics, can lead to the adoption of di↵erent semantics for di↵erent sets of arguments.
An approach to mix di↵erent argumentation semantics has been first
proposed in [
In this paper we adopt a more general perspective, by providing the following
contributions:
– We introduce the definition of combination schema as a general model of
approaches to mix argumentation semantics, and we define a number of
properties to characterize such approaches.
– We analyze the approach introduced in [
After some background in Section 2, the previous points are dealt with in Sections 3, 4 and 5, respectively. Proofs of the results are omitted due to space limitations. 2
We follow the traditional definition of argumentation framework introduced by
Dung [
Definition 1. An argumentation framework is a pair AF = (Ar , att ) in which Ar is a finite and non empty set1 of arguments and att ✓ Ar ⇥ Ar . An argument A attacks an argument B, denoted as A ! B, if (A, B) 2 att . An argument A such that for no B B ! A is called initial. An argument B such that (B, B) 2 att is called self-attacking. Given a set Args ✓ Ar , we denote as Args+ the set of arguments attacked by Args, i.e. ArgAsr+gs=+ {=A ; | .9 GBi v2enAragss,e(tBA,Arg)s 2 ✓ att }. The set Args is conflict-free i↵ Args \ Ar , the restriction of AF to Args, denoted as AF #Args , is the argumentation framework (Args, att \ (Args ⇥ Args)).
In this paper we adopt the labelling-based approach to the definition of argumentation semantics. A labelling assigns to each argument of an argumentation framework a label taken from the set {in, out, undec}, where the label in means that the argument is accepted, the label out means that the argument is rejected, and the label undec means that the status of the argument is undecided. For technical reasons, we define labellings both for argumentation frameworks and for arbitrary sets of arguments. 1 In the general definition, the set of arguments may be infinite or empty. Definition 2. Given a set of arguments Args, a labelling of Args is a total function Lab : Args ! { in, out, undec}. The set of all labellings of Args is denoted as LArgs . Given an argumentation framework AF = (Ar , att ), a labelling of AF is a labelling of Ar . The set of all labellings of AF is denoted as L(AF ). Given a labelling Lab, we write in(Lab) for {A | Lab(A) = in}, out(Lab) for {A | Lab(A) = out} and undec(Lab) for {A | Lab(A) = undec}. For a labelling Lab of Args, the restriction of Lab to a set of arguments Args0 ✓ Args, denoted as Lab#Args0 , is defined as Lab \ (Args0 ⇥ { in, out, undec}). We extend this notation to set of labellings, i.e. given a set of a labellings L ✓ LArgs , L#Args0 , {Lab#Args0 | Lab 2 L}.
A labelling-based semantics prescribes a set of labellings for each argumentation framework.
Definition 3. Given an argumentation framework AF = (Ar , att ), a labellingbased semantics S associates with AF a subset of L(AF ), denoted as LS(AF ).
Two notions of justification, i.e. skeptical and credulous, can then be introduced with respect to a semantics.
Definition 4. Given a labelling-based semantics S and an argumentation framework AF , an argument A is skeptically justified under S if 8 Lab 2 LS(AF ) Lab(A) = in; an argument A is credulously justified under S if 9 Lab 2 LS(AF ) : Lab(A) = in.
In general, a semantics encompasses a set of alternative labellings for a single argumentation framework. If a semantics S is defined in such a way that such a set is always non empty, i.e. 8 AF, LS(AF ) 6= ; , then S is said to be universally defined. Moreover, a semantics may be defined so that a unique labelling is always prescribed, i.e. for every argumentation framework AF , |LS(AF )| = 1. In this case the semantics is said to be single-status, while in the general case it is said to be multiple-status. Note that according to the previous definitions a single-status semantics is universally defined.
In the labelling-based approach, a semantics definition relies on some legality constraints relating the label of an argument to those of its attackers. Definition 5. Let Lab be a labelling of the argumentation framework (Ar , att ). An in-labelled argument is said to be legally in i↵ all its attackers are labelled out. An out-labelled argument is said to be legally out i↵ it has at least one attacker that is labelled in. An undec-labelled argument is said to be legally undec i↵ not all its attackers are labelled out and it does not have an attacker that is labelled in.
We now introduce the definitions of labellings corresponding to traditional conflict-free, admissible and complete semantics.
Definition 6. Let AF = (Ar , att ) be an argumentation framework. A conflictfree labelling is a labelling Lab where every in-labelled argument does not have an attacker that is in-labelled, and every out-labelled argument is legally out. Definition 7. Let AF = (Ar , att ) be an argumentation framework. An admissible labelling is a labelling Lab where every in-labelled argument is legally in and every out-labelled argument is legally out.
Definition 8. A complete labelling is a labelling where every in-labelled argument is legally in, every out-labelled argument is legally out and every undeclabelled argument is legally undec.
On this basis, the labelling-based definitions of several argumentation
semantics can be introduced. The semantics are defined by referring to the commitment
relation between labellings [
Definition 9. Let Lab1 and Lab2 be two labellings. We say that Lab2 is more or equally committed than Lab1 (Lab1 v Lab2) i↵ in(Lab1) ✓ in(Lab2) and out(Lab1) ✓ out(Lab2).
Definition 10. Let AF = (Ar , att ) be an argumentation framework. A stable labelling of AF is a complete labelling without undec-labelled arguments. The grounded labelling of AF is the minimal (w.r.t. v) labelling among all complete labellings. A preferred labelling of AF is a maximal (w.r.t. v) labelling among all complete labellings. The ideal labelling of AF is the maximal (under v) complete2 labelling Lab that is less or equally committed than each preferred labelling of AF (i.e. for each preferred labelling LabP it holds that Lab v LabP ).
The uniqueness of the grounded and the ideal labelling has been proved
in [
Argumentation semantics can be compared w.r.t. the tendency of making more or less committed choices about argument justification. In particular, since argument justification has been introduced according to a skeptical and credulous perspective, two corresponding skepticism relations can be defined between semantics.
S Definition 11. Given two labelling-based semantics S1 and S2, S1 SK S2 i↵ for any AF where LS1 (AF ) 6= ; and LS2 (AF ) 6= ; , it holds that 8 Lab2 2 LS2 (AF ) 9 Lab1 2 LS1 (AF ) such that Lab1 v Lab2.
Intuitively, according to the skeptical viewpoint on argument justification,
S1 SSK S2 indicates that S1 is not more committed (or, equivalently, not less
skeptical) than S2, since every labelling in LS2 (AF ) has a more skeptical
counterpart in LS1 (AF ), while LS1 (AF ) can include additional unrelated labellings
2 Literally, the original definition refers to an admissible labelling rather than a
complete labelling. However, the definition adopted here is equivalent to the original one,
since it can be shown that the ideal labelling is a complete labelling [
Definition 12. Given two labelling-based semantics S1 and S2, S1 SCRS2 i↵ for any AF where LS1 (AF ) 6= ; and LS2 (AF ) 6= ; , it holds that 8 Lab1 2
According to the credulous viewpoint on argument justification, S1 indicates that S1 is not more committed (or, equivalently, not less skeptical) than S2, since every labelling in LS1 (AF ) has a more committed counterpart in LS2 (AF ), and LS2 (AF ) can include additional unrelated labellings that can potentially lead to more committed choices w.r.t. the credulous justification of SCRS2 arguments.
The Hasse diagrams of the semantics considered in this paper according to the skeptical and the credulous perspective are shown in Figure 1. Basically, arcs connect pairs of comparable semantics, and lower semantics are less committed than higher ones.
With abuse of notation, we extend the previous notions to sets of labellings, i.e. given two sets of labellings L1 and L2 of L(AF ): – L1 – L1 that Lab1 v Lab2 that Lab1 v Lab2
S SK L2 i↵ L1 6= ; , L2 6= ; , and 8 Lab2 2 LS2 (AF ) 9 Lab1 2 LS1 (AF ) such S
CRL2 i↵ L1 6= ; , L2 6= ; , and 8 Lab1 2 LS1 (AF ) 9 Lab2 2 LS2 (AF ) such 3
The notion of mixing models an argumentation framework partitioned into a set of subframeworks each managed by an associated semantics. Definition 13. A mixing m is a tuple (AF, P, SEM, S), where AF is an agumentation framework (Ar , att ), P = {P1, . . . , Pn} is a partition3 of Ar , SEM is a set of semantics, and S : P ! SEM is a total function associating a semantics to each element of the partition such that 8 S 2 SEM, 9 Pi 2 P such that S(Pi) = S. Given a mixing m, the elements in m are denoted as AF m, Pm, SEMm, Sm, respectively. Moreover, the set of all possible mixings is denoted as M IX.
In general there are many possible ways to combine the semantics, each corresponding to a specific combination schema as in the following definition. Definition 14. A combination schema cs is a function which, given a mixing m 2 M IX, returns an element of 2L(AF m), i.e. a set of labellings of the argumentation framework associated to m.
In a sense, the notion of combination schema generalizes the notion of semantics. Whereas a semantics identifies a way to assign a set of labellings to an argumentation framework, a combination schema identifies a way to assign a set of labellings to a mixing, on the basis of the argumentation semantics associated to the elements of the partition.
In the following, several properties of combination schemas will be introduced to characterize di↵erent proposals for combining argumentation semantics. These properties are not meant to be mandatory, but they may allow one to know whether a combination schema is suitable to a specific application context.
In general, a property of a combination schema cs requires the set of labellings cs(m) (with m possibly subject to specific constraints) to satisfy a set of conditions, where these conditions may depend in turn on the characteristics of the input mixing m. As it will be shown in the following, some properties do not hold in general, but they do if the set of semantics in the input mixing is restricted and/or the partition of the arguments in the input mixing satisfies specific constraints.
As to partitions, the relevant constraints can be expressed by the notion of
partition selector of [
Definition 15. A partition selector F is a function receiving as input an argumentation framework AF = (Ar , att ) and returning a set of partitions of Ar .
In this paper three restrictions on partitions are worth considering. The first one restricts the partitions to those including one set only, i.e. the whole set of arguments of the argumentation framework.
Definition 16. For any argumentation framework AF = (Ar , att ), FWH(AF ) , {{Ar }}.
The other two restrictions, as in [
Pi \ Pj = ; Definition 17. Given an argumentation framework AF = (Ar , att ), the set of strongly connected components of AF , denoted as SCCSAF , consists of the equivalence classes of arguments induced by the binary relation of path-equivalence, i.e. the relation ⇢ (A, B) defined over Ar ⇥ Ar such that ⇢ (A, B) holds if and only if A = B or there are directed paths from A to B and from B to A in AF .
In particular, we consider the partition coinciding with the strongly connected components SCCSAF , identified by the partition selector FSCC, and the partitions where each element is the union of some strongly connected components, identified by the partition selector F[ SCC.
Definition 18. For any argumentation framework AF = (Ar , att ), FSCC(AF ) , {SCCSAF }, F[ SCC(AF ) , {{P1, . . . , Pn} | {P1, . . . , Pn} is a partition of Ar and 8 i 2 { 1, . . . n} ((S 2 SCCSAF ^ Pi \ S 6= ; ) ! S ✓ Pi}).
Summing up, as to property restrictions we say that: – a property is satisfied by a combination schema cs under a given set of semantics SSEM if it holds for any mixing m such that SEMm ✓ SSEM; – a property is satisfied by a combination schema cs w.r.t. a partition selector F if it holds for any mixing m such that Pm 2 F (AF m).
In the following the properties considered in this paper are introduced. 3.1
First, an expected requirement for any reasonable combination schema is that it should actually depend on the semantics assigned to the partition elements of the input mixing. This requirement can however be enforced at di↵erent levels of strictness.
At the lowest level, if a semantics S is applied to the whole argumentation framework (i.e. the partition includes just one set coinciding with the set of arguments Ar ), then one can require the labellings returned by the combination schema to coincide with those returned by S.
M IX, if Pm =
A more demanding requirement is to extend this property to all cases where a unique semantics S is assigned to the partition elements.
Definition 20. A combination schema cs is adherent to a unique semantics if 8 m 2 M IX, if SEMm = {S} then cs(m) = LS(AF m).
It is immediate to see that a combination schema is reasonable i↵ it is adherent to a unique semantics w.r.t. FWH.
One can view the property of adherence to a unique semantics as a conjunction of two partial properties, namely top-down and bottom-up adherence to a unique semantics. Definition 21. A combination schema cs is top-down adherent to a unique semantics if 8 m 2 M IX, if SEMm = {S} then LS(AF m) ✓ cs(m). Definition 22. A combination schema cs is bottom-up adherent to a unique semantics if 8 m 2 M IX, if SEMm = {S} then cs(m) ✓ LS(AF m).
In a sense, top-down adherence requires the application of the combination schema to be a complete procedure w.r.t. the semantics involved, i.e. the obtained labellings include those prescribed by the semantics, while bottom-up adherence corresponds to correctness of the procedure, i.e. all obtained labellings are also prescribed by the semantics.
As a sort of counterpart of dependence properties, one may require the result of the combination schema to sometimes depart from the original semantics. Definition 23. A combination schema cs is non symbiotic if 9 m 2 9 P 2 P m such that cs(m)#P * (LSm(P )(AF m))#P .
M IX and 3.2
Another significant requirement for a combination schema is being actually able to identify a non empty set of labellings for the mixing in input. Of course, this cannot be required in general for a reasonable argumentation schema if one of the argumentation semantics of the input mixing S is not universally defined. For instance, the schema returns the (potentially empty) set of labellings prescribed by S when applied w.r.t. a partition of FWH with the unique semantics S involved. As a consequence, we introduce the universal definition requirement under the assumption that all the semantics are universally defined.
Definition 24. A combination schema cs is universally defined if 8 m 2 such that 8 S 2 SEMm S is universally defined, it holds that cs(m) 6= ; . M IX 3.3
Whenever the labellings prescribed by all the involved semantics satisfy a desirable property, it would be nice this property to be satisfied also by the labellings returned by the combination schema. The basic properties we consider for labellings are conflict-freeness, admissibility, and completeness.
Definition 25. A semantics S satisfies the CF (admissibility, completeness) criterion i↵ for any argumentation framework AF , all the labellings in LS(AF ) are conflict-free (admissible, complete) labellings.
Definition 26. A combination schema cs is conflict-freeness (admissibility) (completeness) preserver if 8 m 2 M IX, if 8 S 2 SEMm S satisfies the CF (admissibility, completeness) criterion, then all the labellings in cs(m) are conflict-free (admissible, complete) labellings. 3.4
These properties concern the characterization of the labellings obtained by mixing semantics w.r.t. the skepticism relations between semantics.
We first introduce a criterion of skepticism monotony, basically stating that the level of skepticism of the labellings obtained by a combination schema monotonically depends on that of individual semantics. This can be formalized by requiring that replacing a semantics associated to a subframework with a more commited one should correspondingly result in a more committed set of labellings.
Definition 27. A combination schema cs is monotonic w.r.t. S , with S 2 { SSK , SCR}, i↵ 8 m1, m2 2 M IX such that AF m1 = AF m2 and Pm1 = Pm2 , if 8 Pi 2 P m1 Sm1 (Pi) S Sm2 (Pi) then cs(m1) S cs(m2).
We then consider a criterion of boundedness requiring for any mixing that if the set of involved semantics are lower (upper) bounded by a semantics w.r.t. a skepticism relation, then the resulting set of labellings is also bounded. Definition 28. A combination schema cs is lower-bounded w.r.t. S , with
S 2 { SSK , SCR}, i↵ 8 m 2 M IX, if there is a semantics S0 such that 8 S 2 SEMm S0 S S then LS0 (AF m) S cs(m). A combination schema cs is upperbounded w.r.t. S , with S 2 { SSK , SCR}, i↵ 8 m 2 M IX, if there is a semantics S0 such that 8 S 2 SEMm S S S0 then cs(m) S LS0 (AF m).
It should be noted that, while seemengly related, monotonicity and (lower or upper) boundedness of combination schemas are independent properties. Monotonicity may hold even if the labellings arising from the combination schema are not bounded by a maximal (or minimal) semantics w.r.t. skepticism. 4
An approach to combine di↵erent argumentation semantics generalizes to
multiple semantics the model introduced in [
Definition 29. Given a mixing m = (AF, P, SEM, S), csdb(m) , {SPi2P m LabPi | LabPi 2 FS(Pi)(AF #Pi , Piinp, (Sj=1...n,j6=i LabPj )#Piinp , PiR)}, where FS(Pi) a local function which assigns to the arguments in Pi a set of labellings on the basis of AF #Pi , of the set Piinp = {A 2 / Pi | 9 B 2 Pi : A ! B} (including the external arguments attacking Pi and playing the role of input arguments for the set Pi), of the labels externally assigned to them, i.e. (Sj=1...n,j6=i LabPj )#Piinp , and of the attack relation PiR ⌘ att \ (Piinp ⇥ Pi) from the input arguments of Piinp to Pi.
The question is then how to identify the local function FS of a semantics S to be used in Definition 29. The local function can be univocally identified if the semantics S is complete-compatible.
Definition 30. A semantics S is complete-compatible if it satisfies the following constraints: 1. For any argumentation framework AF = (Ar , att ), every labelling Lab 2 LS(AF ) satisfies the following conditions: – if A 2 Ar is initial, then Lab(A) = in – if B 2 Ar and there is an initial argument A such that A ! B, then
Lab(B) = out – if C 2 Ar is self-attacking, and there are no attackers of C besides C itself, then Lab(C) = undec 2. for any set of arguments I and any labelling LabI 2 LI , the argumentation framework AF 0 = (I0, att 0), where I0 = I [ { A0 | A 2 out(LabI )} and att 0 = {(A0, A) | A 2 out(LabI )} [ { (A, A) | A 2 undec(LabI )} admits a labelling4, i.e. |LS(AF 0)| > 0.
In this case, the local function of S can be univocally determined by
considering each input tuple (AFL, I, LabI , att I ), and constructing a corresponding
standard argumentation framework [
It is easy to verify that CO, GR, PR and ID are complete-compatible. The corresponding local functions can be explicitly expressed as in the following proposition.
Proposition 1. The canonical local functions of complete, grounded, preferred and ideal semantics are as follows: 4 Due to the first point, this labelling is necessary unique, i.e. |LS(AF 0)| = 1.
D
Stable semantics is not complete-compatible, since self-attacking arguments otherwise unattacked prevent any stable labelling to exist (see the second item of Definition 30). It turns out that several local functions guarantee the combination schema csdb to be adherent to a unique semantics. In particular, for all input labellings with at least one undec-labelled argument, the output of the local function can be choosen arbitrarily, as long as it includes no labellings with undec-labelled arguments. To maximize the set of labellings, the following local function can be adopted for ST:
FST(AF, I, LabI , attI ) , {Lab 2 FCO(AF, I, LabI , attI ) | undec(Lab) = ;}
As a simple example of application of csdb, consider the partitioned argumentation as in Figure 2.
Considering a corresponding mixing m such that S(P1) = PR and S(P2) = GR, the application of FPR returns for AF #P1 the labellings {(A, in), (B, out)} and {(A, out), (B, in)}. Taking into account that A and B are the input arguments of P2, it is easy to see that FGR returns for AF #P2 the labelling {(C, out), (D, in)} both with the input labelling {(A, in), (B, out)} and with the input labelling {(A, out), (B, in)}. Summing up, it turns out that csdb(m) = P1: GR
P2: ID {{(A, in), (B, out), (C, out), (D, in)}, {(A, out), (B, in), (C, out), (D, in)}}. This example shows in particular that csdb is non symbiotic.
Considering instead a mixing m such that S(P1) = GR and S(P2) = PR, the application of FGR returns for AF #P1 the labelling {(A, undec), (B, undec)}, and the application of FPR with such an input returns for AF #P2 a singleton including the labelling {(C, undec), (D, undec)}. Summing up, csdb(m) = {{(A, undec), (B, undec), (C, undec), (D, undec}}.
In general, csdb is not universally defined, since in particular examples where the partition elements are related by a cyclic relation induced by attacks the resulting set of labellings is empty even if the involved semantics are universally defined. As an example, consider the mixing depicted in Figure 3. First, note that P2 has B as its only input argument, and FID returns for AF #P2 the labelling {(C, out), (D, in)} with the input labelling (B, undec), the labelling {(C, undec), (D, undec)} with the input labelling (B, out), and the labelling {(C, out), (D, in)} with the input labelling (B, in). As to P1, FGR returns for AF #P1 the labelling {(A, undec), (B, undec)} with the input labelling (D, undec), the labelling {(A, undec), (B, undec)} with the input labelling (D, out), and the labelling {(A, in), (B, out)} with the input labelling (D, in). It can be seen that there is no labelling satisfying Defnition 29. In particular, if B is undeclabelled then D is in-labelled according to ID, but in this case GR labels B as out. If instead B is out-labelled then D is undec-labelled according to ID, but in this case GR labels B as undec. Finally, if B is in-labelled then D is in-labelled according to ID, but in this case GR labels B as out.
The outcome of the previous example is due to the peculiar behavior of FID which is not monotonic, i.e. the outcome labellings does not respect the v relations between input labellings. In particular, considering the input labellings LabI 1 = (B, undec) and LabI 2 = (B, out) for which LabI 1 v LabI 2, it turns out that FID(AF #P2 , {B}, LabI 1, {(B, C)}) = {(C, out), (D, in)}, and FID(AF #P2 , {B}, LabI 2, {(B, C)}) = {(C, undec), (D, undec)}. Thus it does not hold that FID(AF #P2 , {B}, LabI 1, {(B, C)}) v FID(AF #P2 , {B}, LabI 2, {(B, C)}).
On the other hand, csdb is universally defined under semantics with monotonic canonical local functions.
Definition 31. Given a semantics S, its canonical local function FS is monotonic if for every AF , I, att I and for every pair of labellings LabI 1 and LabI 2
C
P2 such that LabI 1 v LabI 2, it holds that for every Lab1 2 FS(AF, I, LabI 1, att I ) there is a labelling Lab2 2 FS(AF, I, LabI 2, att I ) such that Lab1 v Lab2. Proposition 2. The combination schema csdb is universally defined under any semantics S such that its canonical local function FS is monotonic.
In particular, since both FGR and FPR are monotonic, csdb is universally defined under {GR, PR}.
Taking into account the relationship between csdb and the decomposability
schema proposed in [
Proposition 3. csdb is conflict-freeness, admissibility and completeness preserver under any set of semantics SSEM [ { ST} where all the semantics in SSEM are complete-compatible.
As to semantics dependence, it is immediate to see that csdb is reasonable.
On the other hand, csdb is not in general adherent to a unique semantics, while
the following results can be obtained on the basis of [
As a simple example showing that csdb is not adherent to a unique semantics under {GR, PR}, consider the partitioned argumentation framework depicted in Figure 4. In a corresponding mixing m1 such that Sm1 (P1) = Sm1 (P2) = PR as well as in a mixing m2 such that Sm2 (P1) = Sm2 (P2) = GR, csdb returns {(A, in), (B, out), (C, in), (D, out)}, {(A, out), (B, in), (C, out), (D, in)}, and {(A, undec), (B, undec), (C, undec), (D, undec)} as output labellings. On
P2 the other hand, LGR(AF ) = {{(A, undec), (B, undec), (C, undec), (D, undec)}}, and LPR(AF ) includes the labellings {(A, in), (B, out), (C, in), (D, out)} and {(A, out), (B, in), (C, out), (D, in)}.
Skepticism-related properties turn out to be poorly satisfied, in particular according to the skeptical perspective. As to monotonicity, consider the partitioned argumentation framework of Figure 5, and two corresponding mixings m1 aPnRd.mC2lesaurclyh, tShmat1 (SPmi)1 (PSSK2)S=m2S(Pmi2)(Pfo2r) =i 2 P{R1,,S2}m.1 (HPo1w)e=verG, Rit aisndnoStmt2h(eP1c)as=e that csdb(m1) SSK csdb(m2), thus csdb is not monotonic w.r.t. SSK . In fact, it turns out that csdb(m1) = {{(A, undec), (B, undec), (C, out), (D, in)}}, while csdb(m2) = {{(A, in), (B, out), (C, in), (D, out)}, {(A, in), (B, out), (C, out), (D, in)}, {(A, out), (B, in), (C, out), (D, in)}}. csdb is not upper-bounded, since 8 S 2 SEMm1 S SSK PR, and it is not the case that csdb(m1) SSK LPR(AF ). The example of Figure 4 shows that csdb is not lower-bounded either, since for m1 it does not hold that LPR(AF ) SSK csdb(m1).
As to the credulous perspective, the following result has been obtained. Proposition 4. csdb is lower-bounded w.r.t. SCR under {GR, PR}. 5
We believe the considerations and results presented in this paper open the way to several interesting investigations, both at a conceptual and a technical level.
From a technical level, some other semantics can be considered, including e.g. semi-stable and CF2 semantics.
At a more general level, one might wonder whether the set of identified
principles should be enlarged or, conversely, whether some principles should be
weakened or given up. On the one hand, a significant set of properties has been
identified for argumentation semantics [
S
SK concerns the semantics rather than csdb, in particular the beaviour of PR that prescribes for AF #P2 with an undecided input a more committed outcome than the one that would be obtained with a more decided input. To put it in other terms, while PR leaves all arguments undecided in the argumentation framework of Figure 5, it yields argument D as justified if the two-lenght cycle in P1 is replaced with a self attacking argument which attacks C. According to these considerations, skepticism-related properties can be considered not appropriate, or at least they should be refined to take into account the behavior of the semantics involved.
A relevant question concerns the analysis of the relationship holding between principles. For instance, under a set of semantics prescribing labellings for which v is coreflexive, if a combination schema satisfies both lower-boundedness and upper-boundedness then it is adherent to a unique semantics. This again confirms that skepticism-related properties in the current form are dicult to satisfy.
More generally, an interesting question is whether the proposed principles can actually be satisfied altogether. This in turn relates to the identification of the space of combination schemas that are worth considering. For instance, a combination schema which takes into account a greater set of input arguments w.r.t. csdb may be able to be adherent to a unique semantics.
Another interesting issue to consider is the case when the same argument is interpreted according to di↵erent semantics with respect to di↵erent attackers, which may require an extension of the notion of mixing.