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AUTOMATIC AND POLYNOMIAL-TIME ALGEBRAIC STRUCTURES

Published online by Cambridge University Press:  29 April 2019

NIKOLAY BAZHENOV ,
MATTHEW HARRISON-TRAINOR ,
ISKANDER KALIMULLIN ,
ALEXANDER MELNIKOV  and
KENG MENG NG
NIKOLAY BAZHENOV
Affiliation:
SOBOLEV INSTITUTE OF MATHEMATICS NOVOSIBIRSK, RUSSIA and NOVOSIBIRSK STATE UNIVERSITY NOVOSIBIRSK, RUSSIA E-mail: nickbazh@yandex.ruURL: http://bazhenov.droppages.com
MATTHEW HARRISON-TRAINOR
Affiliation:
DEPARTMENT OF PURE MATHEMATICS UNIVERSITY OF WATERLOO ON N2L 3G1, CANADA E-mail: maharris@uwaterloo.caURL: http://www.math.uwaterloo.ca/∼maharris/
ISKANDER KALIMULLIN
Affiliation:
DEPARTMENT OF MATHEMATICS KAZAN FEDERAL UNIVERSITY KAZAN, RUSSIAE-mail: ikalimul@gmail.com
ALEXANDER MELNIKOV
Affiliation:
SCHOOL OF NATURAL AND COMPUTATIONAL SCIENCES MASSEY UNIVERSITY AUCKLAND, NEW ZEALAND E-mail: alexander.g.melnikov@gmail.com
KENG MENG NG
Affiliation:
DIVISION OF MATHEMATICAL SCIENCES SCHOOL OF PHYSICAL AND MATHEMATICAL SCIENCES NANYANG TECHNOLOGICAL UNIVERSITY 21 NANYANG LINK, SINGAPORE637371E-mail: kmng@ntu.edu.sg
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Abstract

A structure is automatic if its domain, functions, and relations are all regular languages. Using the fact that every automatic structure is decidable, in the literature many decision problems have been solved by giving an automatic presentation of a particular structure. Khoussainov and Nerode asked whether there is some way to tell whether a structure has, or does not have, an automatic presentation. We answer this question by showing that the set of Turing machines that represent automata-presentable structures is ${\rm{\Sigma }}_1^1 $-complete. We also use similar methods to show that there is no reasonable characterisation of the structures with a polynomial-time presentation in the sense of Nerode and Remmel.

Keywords

Primary 03C5703D05Secondary 03D2003D4503D8068Q45automatic structuresclassificationindex sets
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 4 , December 2019 , pp. 1630 - 1669
Copyright
Copyright © The Association for Symbolic Logic 2019 

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