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Abelian Logic Gates

Part of: Operations research and management science Theory of computing

Published online by Cambridge University Press:  13 March 2019

ALEXANDER E. HOLROYD ,
LIONEL LEVINE  and
PETER WINKLER
ALEXANDER E. HOLROYD
Affiliation:
University of Washington, Seattle, WA 98195, USAhttp://aeholroyd.org
LIONEL LEVINE
Affiliation:
Cornell University, Ithaca, NY 14853, USAhttp://www.math.cornell.edu/~levine
PETER WINKLER
Affiliation:
Dartmouth College, Hanover, NH 03755, USAhttp://math.dartmouth.edu/~pw
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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, which sends letters (‘particles’) from a unary alphabet. Our results can be reformulated in terms of the functions computed by processors, and one consequence is that any increasing function from ℕk to ℕℓ 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.

MSC classification

Primary: 68Q10: Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.)
Secondary: 68Q45: Formal languages and automata 68Q85: Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.) 90B10: Network models, deterministic
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 3 , May 2019 , pp. 388 - 422
Copyright
Copyright © Cambridge University Press 2019 

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Footnotes

†

The second author is supported by NSF grant DMS-1455272 and a Sloan Fellowship.

‡

The third author is supported by NSF grant DMS-1162172.

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