On the Computation of Distributed Knowledge as the Greatest Lower Bound of Knowledge

TL;DR

This paper develops efficient algorithms for computing the greatest lower bound of join endomorphisms in finite lattices, with applications to distributed knowledge in multi-agent systems, improving computational complexity and characterizing knowledge structures.

Contribution

It introduces faster algorithms for distributive lattices, new methods for arbitrary lattices, and characterizes distributed knowledge as a greatest lower bound in the lattice of endomorphisms.

Findings

Distributive case solved in O(n) time, improving from O(n^2).

New algorithms for arbitrary lattices outperform existing methods.

Deciding distributed knowledge in S5 systems can be done in near-linear time.

Abstract

Let $L$ be a finite lattice and $\mathcal{E}(L)$ be the set of join endomorphisms of $L$. We consider the problem of given $L$ and $f,g \in \mathcal{E}(L)$, finding the greatest lower bound $f \sqcap_{{\scriptsize \mathcal{E}(L)}} g$ in the lattice $\mathcal{E}(L)$. (1) We show that if $L$ is distributive, the problem can be solved in time $O(n)$ where $n=| L |$. The previous upper bound was $O(n^2)$. (2) We provide new algorithms for arbitrary lattices and give experimental evidence that they are significantly faster than the existing algorithm. (3) We characterize the standard notion of distributed knowledge of a group as the greatest lower bound of the join-endomorphisms representing the knowledge of each member of the group. (4) We show that deciding whether an agent has the distributed knowledge of two other agents can be computed in time $O(n^2)$ where $n$ is the size of the underlying set of states. (5) For the special case of $S5$ knowledge, we show that it can be decided in time $O(n\alpha_{n})$ where $\alpha_{n}$ is the inverse of the Ackermann function.