Emptiness Problems for Distributed Automata

TL;DR

This paper studies the decidability of the emptiness problem for distributed automata operating on graphs, identifying decidable cases for forgetful automata and proving undecidability in general and for certain restricted classes.

Contribution

It introduces a decidable class of forgetful automata on graphs, relates their expressiveness to classical automata, and establishes undecidability results for other classes.

Findings

Decidability in LogSpace for forgetful automata on certain graph classes.

Forgetful automata are more expressive than finite tree automata but equivalent to finite word automata in specific cases.

Undecidability of the emptiness problem for general and some restricted classes of distributed automata.

Abstract

We investigate the decidability of the emptiness problem for three classes of distributed automata. These devices operate on finite directed graphs, acting as networks of identical finite-state machines that communicate in an infinite sequence of synchronous rounds. The problem is shown to be decidable in LogSpace for a class of forgetful automata, where the nodes see the messages received from their neighbors but cannot remember their own state. When restricted to the appropriate families of graphs, these forgetful automata are equivalent to classical finite word automata, but strictly more expressive than finite tree automata. On the other hand, we also show that the emptiness problem is undecidable in general. This already holds for two heavily restricted classes of distributed automata: those that reject immediately if they receive more than one message per round, and those whose state diagram must be acyclic except for self-loops.