A Knowledge-Based Analysis of Global Function Computation

TL;DR

This paper establishes a comprehensive knowledge-based framework for solving global function computation in distributed systems, providing necessary and sufficient conditions, and optimizing message complexity with counterfactual beliefs.

Contribution

It introduces a generalized solvability condition and a knowledge-based program for global function computation, improving message efficiency with counterfactual beliefs.

Findings

A necessary and sufficient condition for global function computation.

A knowledge-based program that solves the problem when possible.

An optimized message protocol using counterfactual beliefs.

Abstract

Consider a distributed system N in which each agent has an input value and each communication link has a weight. Given a global function, that is, a function f whose value depends on the whole network, the goal is for every agent to eventually compute the value f(N). We call this problem global function computation. Various solutions for instances of this problem, such as Boolean function computation, leader election, (minimum) spanning tree construction, and network determination, have been proposed, each under particular assumptions about what processors know about the system and how this knowledge can be acquired. We give a necessary and sufficient condition for the problem to be solvable that generalizes a number of well-known results. We then provide a knowledge-based (kb) program (like those of Fagin, Halpern, Moses, and Vardi) that solves global function computation whenever possible. Finally, we improve the message overhead inherent in our initial kb program by giving a counterfactual belief-based program that also solves the global function computation whenever possible, but where agents send messages only when they believe it is necessary to do so. The latter program is shown to be implemented by a number of well-known algorithms for solving leader election.