Bulletin of Symbolic Logic

Fifty years of the spectrum problem: survey and new results

Arnaud Durand,Neil D. Jones,Johann A. Makowsky, and Malika More

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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.

Article information

Source
Bull. Symbolic Logic Volume 18, Issue 4 (2012), 505-553.

Dates
First available: 13 November 2012

Permanent link to this document
http://projecteuclid.org/euclid.bsl/1352802980

Digital Object Identifier
doi:10.2178/bsl.1804020

Zentralblatt MATH identifier
06126491

Mathematical Reviews number (MathSciNet)
MR3053068

Citation

Durand, Arnaud; Jones, Neil D.; Makowsky, Johann A.; More, Malika. Fifty years of the spectrum problem: survey and new results. Bulletin of Symbolic Logic 18 (2012), no. 4, 505--553. doi:10.2178/bsl.1804020. http://projecteuclid.org/euclid.bsl/1352802980.


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Supplemental materials

  • Supplementary material: Appendix to "Fifty years of the spectrum problem: Survey and new results". The appendix contains a detailed presentation of the material of Section~4.