Recently, attention has been given to formally characterize security policies that are enforceable by different kinds of security mechanisms. A very important research problem is the characterization of security policies that are enforceable by execution monitors constrained by memory limitations. This paper contributes to give more precise answers to this research problem. To represent execution monitors constrained by memory limitations, we introduce a new class of automata, bounded history automata. Characterizing memory limitations leads us to define a precise taxonomy of security policies that are enforceable under memory-limitation constraints.