Rigorous and Empirical Analysis of Random Cellular Automata

9626152格里菲思大卫·格里菲思目前和计划的研究集中在复杂、随机和确定性晶格动力学的数学理论和计算机可视化之间的相互作用。格里菲斯在他对细胞自动机动力学和其他非线性空间相互作用的理论和经验研究中利用了这种“共生”。在理论方面，这项工作正在产生一种日益丰富和具有启发性的严格的阈值增长理论，这是一种用于离散时间并行更新晶格构型的新范式。在经验方面，Griffeath正在开发用于研究复杂空间系统的有效、交互的模拟工具，并行设计了两个平台：(A)基于麻省理工学院新型CAM8细胞自动机的高端模拟器，他拥有世界上为数不多的几个原型之一；以及(B)低端32位Windows 95模拟器和图形前端WinCA。在接下来的三年里，Griffeath打算使用这种混合方法研究各种类型的动力系统，包括各种元胞自动机、相互作用的粒子系统和基于个体的模型，这些模型作为复杂现象的原型，如成核、在可激发介质中形成螺旋、晶体生长的形状理论、丛生、宿主-寄生虫相互作用、树枝生长和表面张力相分离。自20世纪50年代中期S以来，增长和竞争的数学模型在空间复杂系统理论的发展中发挥了关键作用。特别是在过去的20年里，对随机系统和确定性系统都发展了广泛的理论和应用。对聚集、形状出现、界面竞争和液滴相互作用等基本现象的建模为应用科学的许多领域带来了新的曙光。Griffeath在下一个资助期的工作将特别受到空间数学生态社区引发的问题的推动，正如他在1996年1月在圣达菲研究所组织的一次跨学科会议上所确认的那样。他预计，他正在开发的各种模拟工具将能够为客户真正感兴趣的问题提供新的见解，并将产生大量有趣的新纯数学。

9626152 Griffeath The current and planned research of David Griffeath focuses on an interplay between mathematical theory and computer visualization for complex, random and deterministic lattice dynamics. Griffeath exploits this "symbiosis" in his theoretical and empirical studies of cellular automaton dynamics and other nonlinear spatial interactions. On the theoretical side, this work is giving rise to an increasingly rich and illuminating rigorous theory of Threshold Growth, a new paradigm for discrete-time parallel updating of lattice configurations. On the empirical side, Griffeath is developing effective, interactive simulation tools for the study of complex spatial systems, designing two platforms in parallel: (a) a high-end simulator based on the new CAM8 cellular automaton machine from MIT, of which he has one of the first few prototypes in the world; and (b) a low-end 32-bit Windows 95 simulator and graphical front end called WinCA. Over the next three years, among the types of dynamical system Griffeath intends to study using this hybrid methodology are various cellular automata, interacting particle systems, and individual-based models that serve as prototypes for such complex phenomena as nucleation, spiral formation in excitable media, the shape theory of crystal growth, flocking, host-parasite interactions, dendritic growth, and phase separation by surface tension. Mathematical models for growth and competition have played a key role in the development of spatial complex systems theory since the mid 1950's. An extensive body of theory and applications, for both random and deterministic systems, has been developed, especially over the past twenty years. Modeling of such fundamental phenomena as aggregation, emergence of shape, interfacial competition, and droplet interaction sheds new light on many areas of applied science. Griffeath's work during the next funding period will be motivated especially by problems elicited from the spatial mathematical ecology community, as identi fied during an interdisciplinary conference that he organized at the Santa Fe Institute during January, 1996. He expects that the various simulation tools he is developing will be able to provide new insights into questions of genuine interest to that clientele, and that a substantial body of interesting new pure mathematics will result as well.