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Posttranslational protein modifications play a key role in regulating cellular processes. We present a general model of reversible protein modification networks and demonstrate that a single protein modified by several enzymes is capable of integrating multiple signals into robust digital decisions by switching between multiple forms that can activate distinct cellular processes. First we consider two competing protein modification cycles and show that in the saturated regime, the protein is concentrated into a single form determined by the enzyme activities. We generalize this to protein modification networks with tree structure controlled by multiple enzymes that can be characterized by their phase diagram, which is a partition of the space of enzyme activities into regions corresponding to different dominant forms. We show that the phase diagram can be obtained analytically from the wiring diagram of the modification network by recursively solving a set of balance equations for the steady-state distributions and then applying a positivity condition to determine the regions corresponding to different responses. We also implement this method in a computer algebra system that automatically generates the phase diagram as a set of inequalities. Based on this theoretical framework, we determine some general properties of protein modification systems.