###### Abstract  
In abstract argumentation frameworks, the computation of stable extensions is an important semantic task for evaluating the acceptability of arguments. The current approaches for the computation of stable extensions are typically conducted through methodologies that are either label-based or extension-based. Label-based algorithms operate by assigning labels to each argument, thus reducing the attack relations between arguments to constraint relations among the labels. This paper analyzes the existing two-label and three-label enumeration algorithms for stable extensions through case studies. It is found that both the two-label and three-label algorithms are not precise enough in defining types of arguments. To address these issues, this paper proposes a four-label enumeration algorithm for stable extensions. This method introduces \(amust\_in\) label to pre-mark certain \(in\)-type arguments, thereby achieving a finer classification of \(in\)-type arguments. This enhances the labelings' propagation ability and reduces the algorithm's search space. Our proposed four-label algorithm was tested on authoritative benchmark sets of abstract argumentation framework problems: ICCMA 2019, ICCMA 2021, and ICCMA 2023. Experimental results show that the four-label algorithm significantly improves solving efficiency compared to existing two-label and three-label algorithms. Additionally, ablation experiments confirm that both the four-label transition strategy and preprocessing strategy enhance the algorithm's performance.  
abstract argumentation framework; labelings; stable extensions; depth-first search +
Footnote †: journal: Applied  
to the two-label algorithm to differentiate argument types and reduce the search space. Although the three-label algorithm partially optimizes the issues present in the two-label algorithm, it still has problems with redundant searches. To our knowledge, there is currently no comprehensive, efficient, and direct DFS within backtracking algorithms for enumerating stable extensions. Therefore, this paper introduces a four-label enumeration algorithm for stable extensions by adding the _must_in_ label.  
The main contributions of this paper can be summarized in two aspects. First, this paper conducts a comprehensive analysis of the existing two-label and three-label enumeration stable extension methods, illustrating their limitations through sample analyses. Second, by introducing the _must_in_ label, a four-label algorithm is proposed. In the four-label algorithm, we first propose a preprocessing strategy that can preemptively label and address arguments concerning confirmed labeling types, thereby achieving the target state (i.e., stable extensions) earlier. Additionally, a four-label transition strategy is introduced to enhance collaboration between labels, thereby accelerating labeling propagation and reducing the search space of the algorithm.  
The paper is organized as follows. Section 2 gives an introduction to the semantic extension model of the abstract argumentation framework and the related algorithms for solving stable extensions. Section 3 presents the definitions of the argumentation framework, labeling model, and stable extensions, as well as their interrelationships. Section 4 gives a comparative analysis of the existing two-label and three-label algorithms for solving stable extensions and highlights their limitations. Section 5 elaborates on the concepts and details of the four-label algorithm. Section 6 demonstrates the experimental results of the four-label algorithm and other label algorithms on the ICCMA benchmark sets and analyzes the impact of the preprocessing strategy and labeling transition strategy on the algorithm's efficiency through ablation experiments. Section 7 contains the remarks and future work.