Reasoning with minimal models is at the heart of many knowledge representation systems. Yet, it turns out that this task is formidable even when very simple theories are considered. It is, therefore, crucial to devise methods that attain good performances in most cases. To this end, a path to follow is to find ways to break the task at hand into several sub-tasks that can be solved separately and in parallel. And, in fact, we show that minimal models of positive propositional theories can be decomposed based on the structure of the dependency graph of the theories: this observation turns out to be useful for many applications involving computation with minimal models. In particular, we introduce a new algorithm for minimal model finding based on model decomposition. The algorithm temporal worst-case complexity is exponential in the size s of the largest connected component of the dependency graph, but the actual cost depends on the size of the largest component actually encountered at run time that can be far smaller than s, and on the class of theories to which components belong. For example, if all components reduce to either an Head Cycle Free or an Head Elementary-set Free theory, the algorithm is polynomial in the size of the theory.

使用最小模型进行推理是许多知识表示系统的核心。然而，事实证明，即使考虑非常简单的理论，这项任务也是艰巨的。因此，设计在大多数情况下都能获得良好性能的方法至关重要。为此，一条可以遵循的路径是想办法将手头的任务分解为几个可以分别并行解决的子任务。而且，事实上，我们证明了正命题理论的最小模型可以根据理论的依赖图的结构进行分解：这一观察结果对许多涉及使用最小模型进行计算的应用程序很有用。特别是，我们介绍了一种基于模型分解的最小模型查找新算法。算法时间最坏情况复杂度在大小上是指数级的s依赖图的最大连通组件，但实际成本取决于运行时实际遇到的最大组件的大小，可能远小于s，以及组件所属的理论类别。例如，如果所有分量都简化为 Head Cycle Free 或 Head Elementary-Set Free 理论，则该算法在理论大小上是多项式的。