In this thesis, we argue that (order-) lattice-based multi-agent information systems constitute a broad class of networked multi-agent systems in which relational data is passed between nodes. Mathematically modeled as lattice-valued sheaves, we initiate a discrete Hodge theory with a Laplace operator, analogous to the graph Laplacian and the graph connection Laplacian, acting on assignments of data to the nodes of a Tarski sheaf. The Hodge-Tarski theorem (the main theorem) relates the fixed point theory of this operator, called the Tarski Laplacian in deference to the Tarski Fixed Point Theorem, to the global sections (consistent global states) of the sheaf. We present novel applications to signal processing and multi-agent semantics and supply a plethora of examples throughout.
翻译:多智能体系统中的格论
在这篇论文中,我们论证了(有序)基于格的多智能体信息系统构成了一类广泛的网络多智能体系统,在这些系统中,关系型数据在节点之间传递。通过数学建模作为格值扇形,我们引入了离散霍奇理论和拉普拉斯算子,类似于图拉普拉斯算子和图连接拉普拉斯算子,作用在对塔斯基扇形节点的数据赋值上。霍奇-塔斯基定理(主定理)将这个算子的不动点理论,即塔斯基不动点定理中的算子,与扇形的全局剖分(一致的全局状态)联系起来。我们提供了在信号处理和多智能体语义方面的创新应用,并且提供了大量的例子。