A Four-Label-Based Algorithm for Solving Stable Extension Enumeration in Abstract Argumentation Frameworks
Abstract
:1. Introduction
2. Related Work on Reduction-Based Methods and Direct Solving for AF
3. Background
3.1. Abstract Argumentation Framework and Stable Extensions
3.2. The Relationship between Arguments and Labelings
4. Comparison of Two-Label and Three-Label Enumeration Algorithms for Stable Extensions
4.1. Labeling Model for Stable Extensions
4.2. Comparison of Two-Label and Three-Label Algorithms
- Step 1: Argument is labeled as and propagates the label to argument , with all propagated labels being legal.
- Step 2: Argument is labeled as and propagates the label to arguments and . The arguments , , and , labeled as , are all legal.
- Step 3: Argument is labeled as without propagating any other argument labels, so no labeling legality check is needed. At this point, all arguments have been legally labeled, so the set of arguments labeled as forms a stable extension.
- Step 1: Argument is labeled as and propagates the label to argument . No labeling legality check is needed in this round.
- Step 2: Argument is labeled as and propagates the labels and to arguments and , respectively. A legality check is performed for argument , which is labeled as .
- Step 3: Argument does not propagate any other argument labels, so no legality check is needed.
5. The Labeling Algorithm for Four-Label Enumeration Stable Extension
5.1. The Concept of the Four-Label Enumeration Stable Extension Algorithm
- Step 1: During preprocessing, the initial argument is labeled as , and other ordinary arguments are labeled as . The arguments in the label set are then prioritized for the transition. Consequently, argument is labeled as and propagates the label to argument .
- Step 2: The argument is selected from the ordinary argument set. Then argument is labeled as and propagates labels and to arguments and , respectively. The labeling legality of argument , which is labeled as , is then checked and found to be legal. Next, the algorithm attempts to further propagate labels based on the propagated arguments and . Argument propagates the label to argument , which is then labeled as . At this point, a stable extension is obtained.
Algorithm 1: Four-Label Enumerate All Stable Extensions of . |
5.2. Four-Label Algorithm Description
- Q1:
- Why not directly label initial arguments with the label but rather with the label?
- Q2:
- Why not directly label the self-attacking arguments with the label but rather with the label?
- A1:
- The advantage of using the label is as follows. Since initial arguments are included in all semantic extensions, labeling these arguments with allows the algorithm to focus on arguments that can be further processed, thus avoiding unnecessary argument search issues. Similarly, in the subsequent labeling transition strategy, labels can also be used to label arguments whose label types are determined through inference. Thus, during argument label branching decisions, the algorithm only needs to consider those arguments whose label types are not yet determined and whose label is being attempted to be set as , thereby reducing the number of algorithmic backtracks.
- A2:
- Since self-attacking arguments do not belong to any semantic extension, their final label type is generally . However, in practice, self-attacking arguments may also contain contradictory arguments (e.g., self-attacking arguments that are not attacked by any other arguments) that cannot be legally labeled as . Therefore, self-attacking arguments should be labeled as first to facilitate legality checks. Similarly, contradictory arguments will also arise in the labeling transition strategy, requiring special marking with labels for legality checks, thereby improving the algorithm’s efficiency.
Algorithm 2: Preprocessing Labels |
Algorithm 3: 4lab-in-transitions |
Algorithm 4: 4lab-must_out-transitions |
6. Experiment Analysis
6.1. Experiment Setup
6.2. Experiment Results and Analysis
- ArgTools_3lab: The basic Argtools solver which uses three labels.
- ArgTools_3lab+: Argtools solver with the preprocessing strategy applied.
- ArgTools_3lab++(ArgTools_4lab): Argtools solver with the proposed four-label method applied.
- ArgSemSAT: Stable extension solving algorithm based on the reduction model.
- DREDD: The basic DREDD solver using three labels.
- DREDD_4lab: DREDD solver with the proposed four-label method applied.
7. Conclusions and Future Work
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Instances | Solver | # Solved | PAR-2 (s) |
---|---|---|---|
ICCMA 2019 (326) | ArgTools_2lab | 292 | 260.3 |
ArgTools_3lab | 298 | 217.13 | |
ArgTools_4lab | 326 | 7.36 | |
ICCMA 2021 (587) | ArgTools_2lab | 132 | 1994.39 |
ArgTools_3lab | 136 | 1963.9 | |
ArgTools_4lab | 171 | 1729.37 | |
ICCMA 2023 (329) | ArgTools_2lab | 190 | 1073.15 |
ArgTools_3lab | 201 | 956.82 | |
ArgTools_4lab | 224 | 752.19 |
Instance Families | ArgTools_4lab | ArgTools_3lab | ArgSemSAT | |||
---|---|---|---|---|---|---|
PAR-2 (s) | # Solved | PAR-2 (s) | # Solved | PAR-2 (s) | # Solved | |
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 |
ABA2AFs (25) | 1.62 | 25 | 2.7 | 25 | 1.62 | 25 |
AFBenchGen2 (75) | 718.99 | 52 | 843.2 | 49 | 306.57 | 57 |
Planning2AFs (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_g2io (32) | 2400 | 0 | 2400 | 0 | 598.27 | 28 |
Instances | Solver | # Solved | PAR-2 (s) |
---|---|---|---|
ICCMA 2019 (326) | DREDD | 276 | 375.48 |
DREDD_4lab | 287 | 304.31 | |
ICCMA 2021 (587) | DREDD | 108 | 2118.3 |
DREDD_4lab | 113 | 2079.29 | |
ICCMA 2023 329) | DREDD | 146 | 1407.54 |
DREDD_4lab | 152 | 1376.84 |
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Luo, M.; He, N.; Wu, X.; Xiong, C.; Xu, W. A Four-Label-Based Algorithm for Solving Stable Extension Enumeration in Abstract Argumentation Frameworks. Appl. Sci. 2024, 14, 7656. https://doi.org/10.3390/app14177656
Luo M, He N, Wu X, Xiong C, Xu W. A Four-Label-Based Algorithm for Solving Stable Extension Enumeration in Abstract Argumentation Frameworks. Applied Sciences. 2024; 14(17):7656. https://doi.org/10.3390/app14177656
Chicago/Turabian StyleLuo, Mao, Ningning He, Xinyun Wu, Caiquan Xiong, and Wanghao Xu. 2024. "A Four-Label-Based Algorithm for Solving Stable Extension Enumeration in Abstract Argumentation Frameworks" Applied Sciences 14, no. 17: 7656. https://doi.org/10.3390/app14177656