## 5 The Labeling Algorithm for Four-Label Enumeration Stable Extension
### Four-Label Algorithm Description
([[http://taas.tweetyproject.org](http://taas.tweetyproject.org)]([http://taas.tweetyproject.org](http://taas.tweetyproject.org)), accessed on 3 August 2024), which participated in ICCMA 2019 and is also based on depth-first search for stable extensions. We then integrated the four-label method into the DREDD solver and tested the configured algorithm (DREDD_4 lab) and the original DREDD algorithm on instances of the main track of ICCMA 2019 and ICCMA 2023. The comparison results are shown in Table 3.

* **DREDD:** The basic DREDD solver using three labels. * **DREDD_4 lab:** DREDD solver with the proposed four-label method applied. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Instance Families**} & \multicolumn{2}{c}{**ArgTools\_4 lab**} & \multicolumn{2}{c}{**ArgTools\_3 lab**} & \multicolumn{2}{c}{**ArgSemSAT**} \\ \cline{2-6}  & **PAR-2 (s)** & **\# Solved** & **PAR-2 (s)** & **\# Solved** & **PAR-2 (s)** & **\# Solved** \\ \hline GroundedGenerator (25) & 0.43 & 25 & 0.5 & 25 & 0.38 & 25 \\ SccGenerator (25) & 759.64 & 19 & 1632.01 & 11 & 4.7 & 25 \\ StableGenerator (25) & 2280.09 & 2 & 2304.98 & 1 & 2123.62 & 22 \\ AdmBuster (15) & 568.49 & 12 & 967.23 & 9 & 9.45 & 15 \\ ABA2 AFs (25) & 1.62 & 25 & 2.7 & 25 & 1.62 & 25 \\ AFBenchGen2 (75) & 718.99 & 52 & 843.2 & 49 & 306.57 & 57 \\ Planning2 AFs (25) & 705.5 & 18 & 1091.34 & 15 & 0.01 & 25 \\ SemBuster (15) & 655.01 & 13 & 854.73 & 10 & 11.64 & 15 \\ Traffic (25) & 73.62 & 25 & 96.79 & 25 & 0.1 & 25 \\ Logic-Based (25) & 586.97 & 19 & 791.07 & 17 & 306.32 & 22 \\ AFGen (17) & 326.97 & 14 & 442.47 & 14 & 5.75 & 17 \\ Crusti\_g2 io (32) & 2400 & 0 & 2400 & 0 & 598.27 & 28 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experiment results on the ICCMA 2023 benchmark of different instance families. Figure 4: (**a**) Cactus plots comparing the number of solved instances by ArgTools_4 lab and ArgTools_3 lab+ on the benchmark of the main track of ICCMA 2023. (**b**) Scatter plots comparing the runtime of ArgTools_4 lab (\(x\)-axis) and ArgTools_3 lab+ (\(y\)-axis) on 17 benchmark families of the main track of ICCMA 2023. If the point is on the dashed line, then the solving effects of both solvers are the same; If the point is above the dashed line, then the solver ArgTools_4 lab performs better; conversely, if it is below the line, the performance is the opposite. In Table 3, we observe that DREDD_4 lab shows a significant improvement in efficiency compared to the original DREDD algorithm. This indicates that, by introducing labels to differentiate between the nature of arguments in the DREDD_4 lab algorithm, the cooperation among labels is enhanced, leading to faster labeling propagation and reduced search space. Therefore, we can conclude that the proposed four-label method is also applicable in other labeling-based solvers. ## 7 Conclusions and Future Work

This paper compares and analyzes two-label and three-label algorithms for solving stable extensions and proposes a four-label algorithm to address their limitations, aiming to enumerate all stable extensions of a given abstract argumentation framework more quickly. Meanwhile, the four-label algorithm has been experimentally validated and analyzed using benchmarks from ICCMA 2019, ICCMA 2021, and ICCMA 2023. The results demonstrate that this additional labeling approach significantly improves algorithm efficiency. Additionally, the four-label method was configured in some state-of-the-art solvers, proving its effectiveness in those contexts as well. In future work, the new labeling strategy and preprocessing strategy proposed in this algorithm could still be applied to solving other semantic extension problems. Although the four-label algorithm significantly outperforms the previous two-label and three-label algorithms, there is still a gap compared to popular reduction-based methods. Therefore, in future work, these strategies can be encoded as SAT problems and combined with popular solvers to solve semantic extension problems more quickly.