In this paper, we propose a mathematical framework for automated bug localization. This framework can be briefly summarized as follows. A program execution can be represented as a rooted acyclic directed graph. We define an execution snapshot by a cut-set on the graph. A program state can be regarded as a conjunction of labels on edges in a cut-set. Then we argue that a debugging task is a pruning process of the execution graph by using cut-sets. A pruning algorithm, i.e., a debugging task, is also presented.