Abelian logic gates

TL;DR

This paper demonstrates that any finite abelian processor can be simulated by simple abelian gates, revealing a fundamental structure and computational capabilities of abelian logic gates.

Contribution

It proves that complex abelian processors can be emulated by simple gates like topplers, simplifying the understanding of abelian computation models.

Findings

Any increasing function from N^k to N^l that is a sum of a linear and a periodic function can be represented using abelian gates.

Finite abelian processors are equivalent to networks of simple abelian gates, such as topplers and adders.

The results provide a structural decomposition of abelian processors into fundamental gate components.

Abstract

An abelian processor is an automaton whose output is independent of the order of its inputs. Bond and Levine have proved that a network of abelian processors performs the same computation regardless of processing order (subject only to a halting condition). We prove that any finite abelian processor can be emulated by a network of certain very simple abelian processors, which we call gates. The most fundamental gate is a "toppler", which absorbs input particles until their number exceeds some given threshold, at which point it topples, emitting one particle and returning to its initial state. With the exception of an adder gate, which simply combines two streams of particles, each of our gates has only one input wire. Our results can be reformulated in terms of the functions computed by processors, and one consequence is that any increasing function from N^k to N^l that is the sum of a linear function and a periodic function can be expressed in terms of (possibly nested) sums of floors of quotients by integers.