Decomposing minimal models

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 be able to break this task into several subtasks that can be solved separately and in parallel. 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 can be useful for many applications involving computation with minimal models. As an example of such benefits, we introduce new algorithms for minimal model finding and checking that are based on model decomposition. The algorithms' temporal worst-case complexity is exponential in the size s of the largest connected component of the dependency graph, but their actual cost depends on the size of the largest source actually encountered, which can be far smaller than s, and on the class of theories to which sources belong. Indeed, if all sources reduce to an HCF or HEF theory, the algorithms are polynomial in the size of the theory.