5 Computational Properties
6.3 Labeling-based Algorithms
6.3.3 Labeling Algorithm for stage2
In this section we adopt the algorithm presented before forstage2. We first discuss the necessary modifications resulting Algorithm 2 and then present an illustrative example.
FIG. 28. The argumentation frameworkFfrom Example 6.10.
First, asstage2L(F) ⊆ cf2L(F)we can use the propagation rules of Proposition 6.9, also for computingstage2 labelings. Hence, we only make changes in the second part of the algorithm where we branch between the different extensions. Forstage2 semantics we have to consider self-attacking arguments in the AFF|D(Line 10 & 12), as they have an impact on the stage extension of a SCC (while they do not effect the naive extensions). Hence we compute the minimal SCCs including the self-attacking arguments and store them in the set D. Second, instead of computing the naive labelings ofF|C we have to compute the stage labelings ofF|D, a labeling-based procedure for this is presented in [14].
Algorithm 2stage2L(F,L)
Require: AFF = (A, R), labelingL= (Lin,Lout,Lout∆,Lundec);
Ensure: Return allstage2labelings ofF.
1: X ={a∈ Lundec |att(a)⊆ Lout∆};
2: Y ={a∈ Lundec| ∃b∈ Lin,(b, a)∈R, a6⇒A\LF out∆ b};
3: while(X∪Y)6=∅do
4: Lin =Lin∪X,Lout =Lout∪Y,Lundec=Lundec\(X∪Y);
5: updateXandY;
6: end while
7: B={a∈ Lundec| Lin∪ {a} ∈cf(F)};
8: ifB6=∅then
9: C={a∈B |6 ∃b∈B:b⇒A\LF out∆ a, a6⇒A\LF out∆ b};
10: D=C∪ {a∈ Lundec| ∃b∈C, a⇒A\LF out∆ b, b⇒A\LF out∆ a}
11: E=∅;
12: for allL0 ∈stgL(F|D)do
13: Lin =Lin∪ L0in,Lout =Lout∪ L0out,Lundec=Lundec\(L0in∪ L0out);
14: E=E ∪stage2L(F,L);
15: end for
16: return E;
17: else
18: return {(Lin,Lout,Lout∆,Lundec)};
19: end if
To compute allstage2 labelings of an AFF the functionstage2L(F,L)is called with the labelingL= (∅,∅,∅, A).
EXAMPLE6.11
We illustrate the behavior of Algorithm 2 on the AF F pictured in Figure 29. We start
FIG. 29. The argumentation frameworkFfrom Example 6.11.
stage2L(F,L)with the initial labelingL= (∅,∅,∅, A).
In this first call we haveX = ∅,Y = ∅, B = {a, b, d, e, f, g, h, i} andC = {a, b}.
To complete the inner loop we computeD = {a, b, c}which also takes the self-attacking argumentcinto account. Next we call the external procedure to obtain all stage labelings of the restricted AFF|Dwhich gives usL1 = ({a},{b},∅,{c})andL2 = ({b},{c},∅,{a}).
Here we have the first branch where we update the actual labeling to the ones obtained from stgL(F|D).
•ForL1 we callstage2L(F,L)with the updated labelingL = ({a},{b},∅, A\ {a, b}).
This leads us toX =∅,Y =∅andB=C =D={d, e, f, g, h, i}. We callstgL(F|D) which returnsL1,1 = ({e, g, i},{d, f, h},∅,∅)andL1,2 = ({d, f, h},{e, g, i},∅,∅)as the two stage labelings ofF|D. We update the actual labeling with them and branch another time.
–ForL1,1 we callstage2L(F,L)withL = ({a, e, g, i},{b, d, f, h},∅,{c, x}), where we haveX = ∅,Y =∅andB =∅. Thus, Algorithm 2 returns thestage2 labeling ({a, e, g, i},{b, d, f, h},∅,{c, x}).
–ForL1,2we callstage2L(F,L)withL = ({a, d, f, h},{b, e, g, i},∅,{c, x}). Then, X = ∅,Y = {x}and we obtainLout∆ = {x}. AsB = ∅we return({a, d, f, h}, {b, e, g, i},{x},{c}).
•ForL2we callstage2L(F,L)withL= ({b},{c},∅, A\ {b, c}). ThenX=∅,Y ={g}
andLout∆ ={g}. Next,X ={h},Y =∅andLin ={b, h}. In the next iteration we haveX =∅,Y ={i}andLout ={g, i}and thenX ={d},Y =∅andLin ={b, d, h}.
We continue withX =∅,Y ={e, x}andLout ={e, g, i, x}andX ={f},Y =∅and Lin ={b, d, f, h}. FinallyX =∅,Y =∅andB=∅and the algorithm returns the last stage2 labeling ofF, namely({b, d, f, h},{c},{e, g, i, x},{a}).
3
7 Discussion
In this paper we studied abstract argumentation semantics which treat odd- and even-length cycles in a similar fashion, i.e. semantics which are able to select arguments from odd-length cycles. We first highlighted shortcomings of the existing semanticscf2 and stage, and then,
stg cf2 stage2 I-max. Yes Yes Yes
Reinst. No No No
Weak reinst. No Yes Yes CF-reinst. Yes Yes Yes
Direct. No Yes Yes
Succinct. No Yes Yes
TABLE6. Properties of Naive-based Semantics.
to overcome these shortcomings, we proposed to use the SCC-recursive schema ofcf2 and instantiate the base case with stage semantics, instead of only naive semantics. Thus, we obtained a new sibling semantics ofcf2 which we calledstage2. We showed that this novel semantics solves the problematic behavior ofcf2 on longer cycles, in particular on cycles of length≥ 6. Moreover, on coherent AFs (and in particular on odd-cycle free AFs)stage2 semantics coincide with the standard admissibility based semantics. This guarantees that we do not get an unintended behavior on this class of AFs, and in general on parts of AFs that are coherent. Furthermore,stage2 satisfies the directionality property as well as the weak reinstatement property which was not the case for stage semantics. A comparison of the properties of stage,cf2andstage2 semantics is provided in Table 6.
The analysis of equivalence showed thatstage2 is the second semantics considered so far where strong equivalence coincides with syntactic equivalence, i.e. there are no redundant attacks at all. Thus,stage2 semantics also satisfies the succinctness property, which allows to relate the semantics according to how much meaning every attack has for the computation of the extensions.
We provided a comprehensive complexity analysis forcf2 andstage2 semantics. A sum-mary of the obtained results for the standard reasoning problems for argumentation semantics and for the investigation of tractable fragments is pictured in Table 7. It turned out that both semantics are computationally hard andstage2semantics is even located on the second level of the polynomial hierarchy. Thusstage2it is among the hardest but also most expressiveness argumentation semantics. Faced with this in general intractable complexity, we were able to identify tractable fragments for both semantics, namely acyclic, bipartite and symmetric self-attack free frameworks. However, we showed that recent techniques [32] to augment such fragments are not applicable here.
In the implementation part we gave the ASP encodings ofstage2 semantics, where the alternative characterization facilitated this step. Asstage2is located at the second level of the polynomial hierarchy, we needed more involved programming techniques like the saturation encodings. To simplify those encodings we applied the novelmetaspoptimization front-end from the ASP systemgringo/claspD. All these encodings are incorporated in the system ASPARTIX and available on the web10. We also provided labeling based algorithms forcf2 andstage2 to directly compute the respective extensions.
10Seehttp://rull.dbai.tuwien.ac.at:8080/ASPARTIX/for a web front-end.
cf2 stage2 stg
Verσ inP coNP-c coNP-c
Credσ NP-c ΣP2-c ΣP2-c
Skeptσ coNP-c ΠP2-c ΠP2-c
Exists¬∅σ inP inP inP
Credσacycl
inP inP P-c
Skeptσacycl inP inP P-c
Credσeven−free
NP-c coNP-h ΣP2-c Skeptσeven−free coNP-c coNP-h ΠP2-c Credσbipart
inP inP P-c
Skeptσbipart inP inP P-c
Credσsym
inP inP/ΣP2-c∗ inP/ΣP2-c∗ Skeptσsym inP inP/ΠP2-c∗ inP/ΠP2-c∗
TABLE7: Summary of complexity results (∗ with self-attacking arguments). An entryC-c denotes completeness for the classC.