Complex systems, ranging from developing embryos to systems of locally communicating agents, display an apparent capability of “programmable” pattern formation: They reproducibly form target patterns, but those targets can be readily changed. A distinguishing feature of such 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 local signals to update their discrete state following logical rules. We study systems with different update rules, topologies, and control schemes, assessing their capability of programmable pattern formation and their susceptibility to errors. Only a fraction permits local organizers to dictate any target pattern, by transcribing temporal patterns into spatial patterns, reminiscent of the principle underlying vertebrate somitogenesis. An alternative scheme employing variable rules cannot reach all patterns but is insensitive to the timing of organizer inputs. Our results establish a basis for designing synthetic systems and models of programmable pattern formation closer to real systems.

复杂的系统，从发育中的胚胎到局部通信代理系统，显示出明显的“可编程”模式形成能力：它们可重复地形成目标模式，但这些目标可以很容易地改变。这种系统的一个显着特征是它们的子单元能够进行信息处理。在这里，我们探索了在理论框架内形成可编程模式的方案，其中子单元处理本地信号以遵循逻辑规则更新其离散状态。我们研究具有不同更新规则、拓扑和控制方案的系统，评估它们形成可编程模式的能力及其对错误的敏感性。只有一小部分允许本地组织者通过将时间模式转录为空间模式来规定任何目标模式，让人想起脊椎动物体节发生的基本原理。采用可变规则的替代方案不能达到所有模式，但对组织者输入的时间不敏感。我们的结果为设计更接近真实系统的可编程模式形成的合成系统和模型奠定了基础。