Graph-based construction of 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 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.