Abelian Networks I. Foundations and Examples

Abstract

Abstract. In Deepak Dhar's model of abelian distributed processors, automata occupy the vertices of a graph and communicate via the edges. We show that two simple axioms ensure that the final output does not depend on the order in which the automata process their inputs. A collection of automata obeying these axioms is called an abelian network. We prove a least action principle for abelian networks. As an application, we show how abelian networks can solve certain linear and nonlinear integer programs asynchronously. In most previously studied abelian networks, the input alphabet of each automaton consists of a single letter; in contrast, we propose two non-unary examples of abelian networks: oil and water and abelian mobile agents.

Abelian networks II: halting on all inputs

“…We now recall the definition of an abelian network, refering the reader to [BL13] for details. In an abelian network on a directed graph G = (V, E), each vertex v ∈ V has a processor P v which is an automaton with input alphabet A v and state space Q v .…”

“…The commutativity conditions in the previous paragraph hold vacuously for a toppling network because it is unary: each alphabet A v has just one letter. See [BL13] for many other examples of abelian networks, including non-unary examples.…”

“…It follows readily from the axioms of an abelian network (see [BL13,Lemma 4.2]) that π a •π b = π b •π a for all a, b ∈ A and hence π w depends only on |w|. For y ∈ N A , write π y to mean π w for any word w such that |w| = y.…”

“…In this section we continue the development of the foundations of abelian networks begun in [BL13]. We associate two algebraic objects to an abelian network, the total kernel K and production matrix P .…”

“…However, there is a sizable class of automata networks, proposed by Dhar in [Dha99] and termed abelian networks in [BL13], for which the output is uniquely determined by the input.…”