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Fifty years of the spectrum problem: survey and new results

Published online by Cambridge University Press:  05 September 2014

Arnaud Durand ,
Neil D. Jones ,
Johann A. Makowsky  and
Malika More
Arnaud Durand
Affiliation:
IMJ – Projet Logique, Université Paris Diderot, Paris 7, France E-mail: durand@logique.jussieu.fr
Neil D. Jones
Affiliation:
DIKU, University of Copenhagen, Denmark E-mail: neil@diku.dk
Johann A. Makowsky
Affiliation:
Faculty of Computer Science, Technion, Haifa, Israel E-mail: janos@cs.technion.ac.il
Malika More
Affiliation:
Univ Clermont 1, Laic, France E-mail: malika.more@u-clermont1.fr
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Abstract

In 1952, Heinrich Scholz published a question in The Journal of Symbolic Logic asking for a characterization of spectra, i.e., sets of natural numbers that are the cardinalities of finite models of first order sentences. Günter Asser in turn asked whether the complement of a spectrum is always a spectrum. These innocent questions turned out to be seminal for the development of finite model theory and descriptive complexity. In this paper we survey developments over the last 50-odd years pertaining to the spectrum problem. Our presentation follows conceptual developments rather than the chronological order. Originally a number theoretic problem, it has been approached by means of recursion theory, resource bounded complexity theory, classification by complexity of the defining sentences, and finally by means of structural graph theory. Although Scholz' question was answered in various ways, Asser's question remains open.

Type
Articles
Information
Bulletin of Symbolic Logic , Volume 18 , Issue 4 , December 2012 , pp. 505 - 553
Copyright
Copyright © Association for Symbolic Logic 2012

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