DNA-grafted nanoparticles have been called "programmable atom-equivalents": Like atoms, they form three-dimensional crystals, but unlike atoms, the particles themselves carry information (the sequences of the grafted strands) that can be used to "program" the equilibrium crystal structures. We show that the programmability of these colloids can be generalized to the full temperature-dependent phase diagram, not just the crystal structures themselves. We add information to the buffer in the form of soluble DNA strands designed to compete with the grafted strands through strand displacement. Using only two displacement reactions, we program phase behavior not found in atomic systems or other DNA-grafted colloids, including arbitrarily wide gas-solid coexistence, reentrant melting, and even reversible transitions between distinct crystal phases.