## 2 Related Work on Reduction-Based Methods and Direct Solving for AF  
In early abstract argumentation frameworks, the concepts of admissible set, complete extension, grounded extension, preferred extension, and stable extensions were generally represented through extension models [15]. Furthermore, these concepts can also be expressed through labeling models [16; 17]. In the current research on abstract argumentation frameworks, the design of algorithms for solving stable extensions primarily revolves around these two models. These algorithms can generally be divided into reduction-based algorithms (or methods) and direct solving algorithms [12].  
Reduction-based methods transform the problem of solving abstract argumentation frameworks (such as stable extensions) into another classical NP problem, by using existing mature algorithms to solve this NP problem. These reduction-based methods can be further subdivided into reduction methods based on extension models and those based on labeling models. The reduction methods based on extension models leverage the properties of extension models to transform the abstract argumentation framework problem into a classical NP problem. For example, ASPARTIX-V [18; 19] encodes the problem of finding stable extensions in an argumentation framework into an answer set problem (ASP). Since there is a direct correspondence between answer sets and stable models, this method can use a standard ASP solver to directly model and solve the semantics of the abstract argumentation framework. The solutions to the ASP problem can then be decoded to obtain solutions to the original problem. Similarly, \(\mu\)-toksia [20] encodes the problem of solving semantic extensions into a SAT problem, then uses the SAT solver CryptoMinSat [21] as the underlying solver. The reduction methods based on labelings combine the definition of the semantics with their corresponding labelings to reduce the problem to a classic NP problem. For instance, ArgSemSAT uses the Glucose [22] SAT solver, while Fudge uses the CaDiCaL [23; 24] SAT solver. Essentially, mature and general solvers like ASP solvers and SAT solvers employ complex and efficient algorithmic strategies, such as conflict-driven clause learning [25; 26].  
Direct methods for solving semantic problems in abstract argumentation frameworks can be divided into incomplete search algorithms [27] and complete search algorithms. A representative example of incomplete algorithms is the local search algorithm. This kind of algorithm first initializes a set of labels for the arguments, then continually selects and flips the labels of some conflicting arguments using heuristic strategies until a legal solution is found [28]. Complete algorithms, on the other hand, are based on a depth-first search framework and traverse the entire solution space to find the solution. To enhance the efficiency of complete algorithms, effective branching strategies, labeling propagation methods, and pruning strategies are commonly employed. Notable solvers of this type include ArgTools [29] and DREDD [30]. The four-label algorithm presented in this paper is also a direct solving algorithm. By introducing a new label, our proposed algorithm optimizes the labeling transition strategy, thereby accelerating labeling propagation and reducing the search space.