Algorithm 16 decides all stage extensions. In particular, algorithm 16 decides candidate conflict free subsets ofA(see lines 14-29) in the same way (i.e. using the same set of labels) algorithm 10 does in deciding preferred extensions. Recall that algorithm 10 actually enumerates admissible sets rather than conflict free sets. Thus, algorithm 10 decides that IN arguments make up an admissible set if and only if for eachx∈A x is not UNDEC nor MUST OUT while algorithm 16 reports IN arguments as a conflict free set if and only if for eachx∈A, xis not UNDEC. Then, for every reported conflict free setSalgorithm 16 also determinesS0≡ {x∈A|xis OUT}. After accumulating all candidateS∪S0, algorithm 16 decides that a conflict free setSis a stage extension if and only ifS∪S0is maximal, see lines 7-11.
Heuristics and pruning strategies used in semantics that are based on admissible sets will not be applicable to stage semantics, which are based on conflict free sets. Therefore, as a pruning strategy we skip (see line 27 of algorithm 16) ignoring an argumentyif and only if for eachz∈ {y}+∪ {y}−,zis OUT or MUST OUT or IGNORED. Note that labeling suchyIGNORED is unnecessary as we explain shortly. On selecting the next UNDEC argument to be labeled IN, there are two options. For the first option, denoted byHeu1, we take into account the following rule:
Algorithm 14:Enumerating all stable extensions of an argument system (A,R). 1 N(y)≡ |{z∈A|(y,z)∈R}|;
2 Lab:A→ {IN,OUT,MUST OUT,IGNORED,UNDEC};Lab←φ; 3 foreachx∈AdoLab←Lab∪ {(x,UNDEC)};
4 Estable⊆2A;Estable←φ;
5 callfind-stable-extensions(Lab);
6 reportEstableis the set of all stable extensions; 7 procedurefind-stable-extensions(Lab)
8 while∃y∈A:Lab(y)=UNDECdo
9 selecty∈As.t. Lab(y)=UNDECand∀(z,y)∈R
(Lab(z)∈ {OUT,MUST OUT}), otherwise selecty∈As.t. Lab(y)=UNDEC and∀z∈A:Lab(z)=UNDEC, N(y)≥N(z);
10 Lab0←Lab;
11 Lab0(y)←IN;
12 foreach(y,z)∈RdoLab0(z)←OUT; 13 foreach(z,y)∈Rdo
14 ifLab0(z)∈ {IGNORED,UNDEC}then 15 Lab0(z)←MUST OUT;
16 if@(w,z)∈R:Lab0(w)=UNDECthen 17 if∃(v,y)∈R:Lab(v)=UNDECthen
18 Lab(y)←IGNORED;
19 gotoline 8;
20 else
21 return;
22 callfind-stable-extensions(Lab0); 23 if∃(z,y)∈R:Lab(z)=UNDECthen 24 Lab(y)←IGNORED;
25 else
26 if@(z,y)∈R:Lab(z)=IGNOREDthen
27 Lab←Lab0;
28 else
29 return;
30 if@y∈A:Lab(y)∈ {MUST OUT,IGNORED}then 31 S← {y∈A|Lab(y)=IN};
32 Estable←Estable∪ {S};
Algorithm 15:Enumerating all complete extensions of an argument system (A,R). 1 N(y)≡ |{z∈A|(y,z)∈R}|;
2 Lab:A→ {IN,OUT,MUST OUT,IGNORED,UNDEC};Lab←φ; 3 foreachx∈AdoLab←Lab∪ {(x,UNDEC)};
4 Ecomplete⊆2A;Ecomplete←φ;
5 callfind-complete-extensions(Lab);
6 reportEcompleteis the set of all complete extensions; 7 procedurefind-complete-extensions(Lab)
8 if@y∈A:Lab(y)=MUST OUTthen
9 if@x∈A:Lab(x)∈ {IGNORED,UNDEC}and∀(z,x)∈R Lab(z)=OUTthen 10 S← {w∈A|Lab(w)=IN};
11 Ecomplete←Ecomplete∪ {S}; 12 while∃y∈A:Lab(y)=UNDECdo
13 selecty∈As.t. Lab(y)=UNDEC∧ ∀(z,y)∈R(z∈ {OUT,MUST OUT}), otherwise selecty∈As.t. Lab(y)=UNDECand
∀z∈A:Lab(z)=UNDEC, N(y)≥N(z);
14 Lab0←Lab;
15 Lab0(y)←IN;
16 foreach(y,z)∈RdoLab0(z)←OUT; 17 foreach(z,y)∈Rdo
18 ifLab0(z)∈ {IGNORED,UNDEC}then 19 Lab0(z)←MUST OUT;
20 if@(w,z)∈R:Lab0(w)=UNDECthen
21 Lab(y)←IGNORED;
22 gotoline 12;
23 callfind-complete-extensions(Lab0);
24 if∃(z,y)∈R:Lab(z)∈ {UNDEC,IGNORED}then 25 Lab(y)←IGNORED;
26 else
27 Lab←Lab0;
Table 4.1: The average elapsed time of algorithm 14 versus ASPARTIX, argument systems were generated by using algorithm 11.
|A| Range of|R| ASPARTIX algorithm 14 61 61-1952 30.80 13.30 62 62-1842 31.30 12.90 63 94-2101 34.40 14.80 64 64-2208 34.10 14.30 65 65-2458 38.20 15.40 66 66-2334 39.30 15.00 67 67-2477 37.30 18.70 68 68-2526 41.10 17.50 69 100-2628 45.60 16.40 70 70-2795 50.80 19.80
Table 4.2: The average elapsed time of algorithm 14 versus ASPARTIX, argument systems were generated by setting attacks with a specific probability.
|A| probability= 2×loge|A| |A| probability= 3×loge|A| |A| probability= 4×loge|A| |A| ASPARTIX Alg. 14 ASPARTIX Alg. 14 ASPARTIX Alg. 14
61 20.30 36.20 86.9 43.2 66.90 12.50 62 20.50 42.00 89.8 45.2 69.50 13.10 63 20.40 47.40 90.4 51.6 74.70 13.60 64 22.00 52.10 95.4 55.6 76.00 16.30 65 24.50 63.60 101.8 63.1 82.50 19.40 66 23.30 68.20 101.9 68.6 87.60 21.40 67 25.00 81.20 110 75.6 91.20 23.10 68 24.90 92.00 121.6 82.8 92.90 25.60 69 28.10 112.20 126.9 93.8 102.70 30.30 70 28.20 122.20 132.3 103.9 111.40 34.30
1. select an UNDEC argumentys.t. for eachz∈ {y}+∪ {y}−,zis OUT or MUST OUT or IGNORED.
2. otherwise select an UNDEC argumentysuch that|{y}+|is maximal. For the second possibility, denoted byHeu2, we consider the following rule:
1. select an UNDEC argumentys.t. for eachz∈ {y}+∪ {y}−,zis OUT or MUST OUT or IGNORED.
2. otherwise select an UNDEC argumentysuch that|{y}+|+|{y}−|is maximal. The aim of the first part ofHeu1&Heu2, which is identical in both selection rules,
Table 4.3: The average elapsed time of algorithm 15 versus ASPARTIX, argument systems were generated by using algorithm 11.
|A| Range of|R| ASPARTIX Algorithm 15 56 56-1849 43.90 15.10 57 57-1786 47.90 16.50 58 91-1880 52.70 20.80 59 59-1893 57.80 12.90 60 60-2047 61.00 20.30 61 61-1989 61.10 19.10 62 62-2037 68.60 24.10 63 63-2275 75.10 21.30 64 92-2160 77.60 33.20 65 65-2154 86.00 37.70
Table 4.4: The average elapsed time of algorithm 15 versus ASPARTIX, argument systems were generated by setting attacks with a specific probability.
|A| probability= 2×loge|A|
|A| probability=
4×loge|A| |A| ASPARTIX Algorithm 15 ASPARTIX Algorithm 15
56 30.20 34.70 137.70 9.90 57 33.10 41.60 167.10 10.50 58 38.60 47.60 173.70 10.20 59 40.30 57.00 189.20 10.10 60 44.60 72.80 198.90 10.70 61 42.90 76.50 221.00 14.20 62 44.70 90.80 245.20 17.70 63 53.50 107.90 283.00 19.50 64 54.90 127.10 303.70 22.70 65 62.70 150.30 337.40 25.30
argument in the the first part of the rules, based on the following property: if a conflict free setSwill be captured while such yis ignored thenS∪ {y}is conflict free as well, and so, there is no need to ignore y in the first place. Consequently, the earlier we label suchyIN, the bigger part of the search tree that will be bypassed. Turning to the second part ofHeu1 &Heu2. Recall that the aim of heuristics in our algorithms is to
accelerate achieving a goal state. In algorithm 16 a goal state is a conflict free set with a maximal range such that there is nox∈A: xis UNDEC. So, we noteHeu2is potentially
more powerful, because, by maximizing the number of OUT/MUST OUT arguments the number of UNDEC arguments is minimized more thanHeu1 which maximizes
only the number of OUT arguments. See tabels 4.5 and 4.6 that reflect the efficiency of algorithm 16 by usingHeu1andHeu2versus DLV solving ASPARTIX encodings.
Algorithm 16:Enumerating stage extensions of an argument system (A,R). 1 N(y)≡ |{z∈A|(y,z)∈R}|;
2 O(y)≡ |{z∈A|(z,y)∈R}|;
3 Lab:A→ {IN,OUT,MUST OUT,IGNORED,UNDEC};Lab←φ; 4 foreachx∈AdoLab←Lab∪ {(x,UNDEC)};
5 Estage: (A→ {IN,OUT,MUST OUT,IGNORED,UNDEC})×Z; Estage←φ;
6 callfind-conflict-free-sets(Lab);
/* The next loop is to collect conflict free sets, those which have
a maximal range */
7 foreach(Lab1,i)∈Estagedo
8 foreach(Lab2,j)∈Estage:j,ido
9 if(|{x:Lab1(x)∈ {IN,OUT}}|,|{z:Lab2(z)∈ {IN,OUT}}| ∨Lab1=Lab2)and ∀y∈A:Lab1(y)∈ {IN,OUT}(Lab2(y)∈ {IN,OUT})then
10 Estage←Estage\ {(Lab1,i)};
11 continue to next iteration from line 7; 12 foreach(Lab1,i)∈Estagedo
13 report{x:Lab1(x)=IN}as a stage extension; 14 procedurefind-conflict-free-sets(Lab)
15 while∃y∈A:Lab(y)=UNDECdo 16 selecty∈As.t. Lab(y)=UNDECand
∀z∈ {y}+∪ {y}−(Lab(z)∈ {OUT,MUST OUT,IGNORED}), otherwise select y:Lab(y)=UNDEC satis f ying∀z:Lab(z)=UNDEC, (N(y)+O(y))≥
(N(z)+O(z));
17 Lab0←Lab;
18 Lab0(y)←IN;
19 foreach(y,z)∈RdoLab0(z)←OUT; 20 foreach(z,y)∈Rdo
21 ifLab0(z)∈ {IGNORED,UNDEC}then 22 Lab0(z)←MUST OUT;
23 callfind-conflict-free-sets(Lab0);
24 if∃z∈ {y}+∪ {y}−and Lab(z)=UNDECthen 25 Lab(y)←IGNORED;
26 else
27 Lab←Lab0;
28 Estage←Estage∪ {(Lab,|Estage|+1)};
Table 4.5: The average elapsed time of algorithm 16 versus ASPARTIX, argument systems were generated by using algorithm 11.
|A| Range of|R| ASPARTIX Algorithm 16 usingHeu1 Algorithm 16 usingHeu2
21 21-245 302.90 1.80 1.10 22 22-293 392.10 3.60 1.50 23 23-332 541.40 6.10 2.40 24 24-322 724.50 8.50 4.60 25 25-379 813.40 19.40 6.70 26 26-471 1,328.90 26.90 11.70 27 27-396 1,619.00 39.30 13.10 28 28-463 2,143.80 80.00 17.10 29 29-552 4,027.90 93.30 47.20 30 30-483 3,717.70 140.00 45.40