Lattice Theory in Multi-Agent Systems

TL;DR

This thesis develops a lattice-theoretic framework for multi-agent systems using sheaf theory, introducing a Tarski Laplacian and a Hodge-Tarski theorem to analyze global states and consistency in networked data.

Contribution

It introduces a novel lattice-based sheaf model for multi-agent systems, along with a discrete Hodge theory and Tarski Laplacian, connecting fixed point theory to global state consistency.

Findings

Established a Tarski Hodge theorem relating fixed points to global sections.

Developed a lattice-theoretic Hodge Laplacian analogous to graph Laplacians.

Applied the theory to signal processing and multi-agent semantics.

Abstract

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.