In this paper, we study Reiter's propositional default logic when the treewidth of a certain graph representation (semi-primal graph) of the input theory is bounded. We establish a dynamic programming algorithm on tree decompositions that decides whether a theory has a consistent stable extension ( Cons ). Our algorithm can even be used to enumerate all generating defaults ( EnumSD ) that lead to stable extensions. We show that our algorithm decides Cons in linear time in the input theory and triple exponential time in the treewidth Further, our algorithm solves EnumSD with a pre-computation step that is linear in the input theory and triple exponential in the treewidth followed by a linear delay to output solutions. Finally, we take the exponential time hypothesis (ETH) into consideration and establish lower bounds of bounded treewidth algorithms for Cons.

在本文中，我们研究了输入理论的某个图表示（半原始图）的树宽有界时的 Reiter 命题默认逻辑。我们建立了一个关于树分解的动态规划算法，该算法决定一个理论是否具有一致的稳定扩展（Cons）。我们的算法甚至可以用于枚举所有导致稳定扩展的生成默认值 ( EnumSD )。我们证明了我们的算法在输入理论中以线性时间决定Cons并在树宽中决定三倍指数时间 此外，我们的算法解决了EnumSD预计算步骤在输入理论中是线性的，在树宽中是三倍指数，然后是输出解的线性延迟。最后，我们考虑指数时间假设（ETH），并为Cons建立有界树宽算法的下界。