Distributed Automata and Logic

TL;DR

This thesis characterizes the logical expressiveness of various classes of distributed automata operating on graphs, extending previous work to include global acceptance, nondeterminism, and asynchrony, with implications for decidability and message loss.

Contribution

It provides new logical characterizations for extended classes of distributed automata, including those with global acceptance, nondeterminism, and asynchronous operation, linking automata behavior to monadic second-order and modal -calculus logics.

Findings

Extended automata with global acceptance are equivalent to monadic second-order logic.

Nondeterministic or deterministic variants have decidable emptiness problems.

Asynchronous automata's expressive power is unaffected by message loss.

Abstract

Distributed automata are finite-state machines that operate on finite directed graphs. Acting as synchronous distributed algorithms, they use their input graph as a network in which identical processors communicate for a possibly infinite number of synchronous rounds. For the local variant of those automata, where the number of rounds is bounded by a constant, Hella et al. (2012, 2015) have established a logical characterization in terms of basic modal logic. In this thesis, we provide similar logical characterizations for two more expressive classes of distributed automata. The first class extends local automata with a global acceptance condition and the ability to alternate between nondeterministic and parallel computations. We show that it is equivalent to monadic second-order logic on graphs. By restricting transitions to be nondeterministic or deterministic, we also obtain two strictly weaker variants for which the emptiness problem is decidable. Our second class transfers the standard notion of asynchronous algorithm to the setting of nonlocal distributed automata. The resulting machines are shown to be equivalent to a small fragment of least fixpoint logic, and more specifically, to a restricted variant of the modal {\mu}-calculus that allows least fixpoints but forbids greatest fixpoints. Exploiting the connection with logic, we additionally prove that the expressive power of those asynchronous automata is independent of whether or not messages can be lost.