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The access matrix model as formalized by Harrison, Ruzzo, and Ullman (HRU) has broad expressive power. Unfortunately, HRU has weak safety properties (ie, the determination of whether or not a given subject can ever acquire access to a given object). Most security policies of practical interest fall into the undecidable cases of HRU. This is true even for monotonic policies (ie, where access rights can be deleted only if the deletion is itself reversible). In this paper we de ne the typed access matrix (TAM) model by introducing strong typing into HRU (ie, each subject or object is created to be of a particular type which thereafter does not change). We prove that monotonic TAM (MTAM) has strong safety properties similar to Sandhu's Schematic Protection Model. Safety in MTAM's decidable case is, however, NP-hard. We develop a model called ternary MTAM which has polynomial safety for its decidable case, and which nevertheless retains the full expressive power of MTAM. There is compelling evidence that the decidable safety cases of ternary MTAM are quite adequate for modeling practical monotonic security policies.