Fifty Years of the Spectrum Problem: Survey and New Results - Mathematics > LogicReport as inadecuate

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Abstract: In 1952, Heinrich Scholz published a question in the Journal of SymbolicLogic asking for a characterization of spectra, i.e., sets of natural numbersthat are the cardinalities of finite models of first order sentences. G\-unterAsser asked whether the complement of a spectrum is always a spectrum. Theseinnocent questions turned out to be seminal for the development of finite modeltheory and descriptive complexity. In this paper we survey developments overthe last 50-odd years pertaining to the spectrum problem. Our presentationfollows conceptual developments rather than the chronological order. Originallya number theoretic problem, it has been approached in terms of recursiontheory, resource bounded complexity theory, classification by complexity of thedefining sentences, and finally in terms of structural graph theory. AlthoughScholz- question was answered in various ways, Asser-s question remains open.One appendix paraphrases the contents of several early and not easily accesiblepapers by G. Asser, A. Mostowski, J. Bennett and S. Mo. Another appendixcontains a compendium of questions and conjectures which remain open.

Author: Arnaud Durand, Neil Jones, Johann Makowsky, Malika More


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