Diverse complex systems, ranging from developing embryos to systems of locally communicating agents, display an apparent capability of "programmable" pattern formation: They reproducibly form a target pattern, but this target can be readily changed. A distinguishing feature of such systems, as compared to simpler physical pattern forming systems, is that their subunits are capable of information processing. Here, we explore schemes for programmable pattern formation within a theoretical framework, in which subunits process discrete local signals to update their internal state according to logical rules. We study systems with different update rules, different topologies, and different control schemes, to assess their ability to perform programmable pattern formation and their susceptibility to errors. Only a small subset of systems permits local organizer cells to dictate any target pattern. These systems follow a common principle, whereby a temporal pattern is transcribed into a spatial pattern, reminiscent of the clock-and-wavefront mechanism underlying vertebrate somitogenesis. An alternative scheme employing several different rules can only form a fraction of patterns but is robust with respect to the timing of organizer cell inputs. Our results establish a basis for the design of synthetic systems, and for more detailed models of programmable pattern formation closer to real systems.
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