Notre Dame Journal of Formal Logic

Some Remarks on Real Numbers Induced by First-Order Spectra

Sune Kristian Jakobsen and Jakob Grue Simonsen

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Abstract

The spectrum of a first-order sentence is the set of natural numbers occurring as the cardinalities of finite models of the sentence. In a recent survey, Durand et al. introduce a new class of real numbers, the spectral reals, induced by spectra and pose two open problems associated to this class. In the present note, we answer these open problems as well as other open problems from an earlier, unpublished version of the survey.

Specifically, we prove that (i) every algebraic real is spectral, (ii) every automatic real is spectral, (iii) the subword density of a spectral real is either 0 or 1, and both may occur, and (iv) every right-computable real number between 0 and 1 occurs as the subword entropy of a spectral real.

In addition, Durand et al. note that the set of spectral reals is not closed under addition or multiplication. We extend this result by showing that the class of spectral reals is not closed under any computable operation satisfying some mild conditions.

Article information

Source
Notre Dame J. Formal Logic Volume 57, Number 3 (2016), 355-368.

Dates
Received: 19 May 2011
Accepted: 9 October 2013
First available in Project Euclid: 24 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1458829375

Digital Object Identifier
doi:10.1215/00294527-3489987

Mathematical Reviews number (MathSciNet)
MR3521485

Zentralblatt MATH identifier
06621294

Subjects
Primary: 03C13: Finite structures [See also 68Q15, 68Q19] 68Q15: Complexity classes (hierarchies, relations among complexity classes, etc.) [See also 03D15, 68Q17, 68Q19] 68Q19: Descriptive complexity and finite models [See also 03C13]
Secondary: 11U05: Decidability [See also 03B25] 11U09: Model theory [See also 03Cxx]

Keywords
first-order logic spectral reals computability theory computational complexity

Citation

Jakobsen, Sune Kristian; Simonsen, Jakob Grue. Some Remarks on Real Numbers Induced by First-Order Spectra. Notre Dame J. Formal Logic 57 (2016), no. 3, 355--368. doi:10.1215/00294527-3489987. https://projecteuclid.org/euclid.ndjfl/1458829375.


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