## Notre Dame Journal of Formal Logic

### Some Remarks on Real Numbers Induced by First-Order Spectra

#### 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 In the present note, we answer these open problems as well as other open problems from an earlier, unpublished version of the survey.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
Accepted: 9 October 2013
First available in Project Euclid: 24 March 2016

https://projecteuclid.org/euclid.ndjfl/1458829375

Digital Object Identifier
doi:10.1215/00294527-3489987

Mathematical Reviews number (MathSciNet)
MR3521485

Zentralblatt MATH identifier
06621294

#### 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

#### References

• [1] Adamczewski, B., and J. Cassaigne, “Diophantine properties of real numbers generated by finite automata,” Compositio Mathematica, vol. 142 (2006), pp. 1351–72.
• [2] Allouche, J.-P., and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003.
• [3] Béal, M.-P., F. Mignosi, A. Restivo, and M. Sciortino, “Forbidden words in symbolic dynamics,” Advances in Applied Mathematics, vol. 25 (2000), pp. 163–93.
• [4] Brouwer, L., “Besitzt jede reelle Zahl eine Dezimalbruchentwicklung?” Mathematische Annalen, vol. 83 (1921), pp. 201–10.
• [5] Champernowne, D. G., “The construction of decimals normal in the scale of ten,” Journal of the London Mathematical Society (2), vol. S1-8 (1933), pp. 254–60.
• [6] Chomsky, N., and G. A. Miller, “Finite state languages,” Information and Control, vol. 1 (1958), pp. 91–112.
• [7] Durand, A., N. D. Jones, J. Makowsky, and M. More, “Fifty years of the spectrum problem: Survey and new results,” Bulletin of Symbolic Logic, vol. 18 (2012), pp. 505–53.
• [8] Durand, A., N. D. Jones, J. Makowsky, and M. More, “Fifty years of the spectrum problem: Survey and new results,” preprint, arXiv:0907.5495v1 [math.LO].
• [9] Ebbinghaus, H.-D., J. Flum, and W. Thomas, Mathematical Logic, 2nd ed., Undergraduate Texts in Mathematics, Springer, New York, 1994.
• [10] Enderton, H. B., A Mathematical Introduction to Logic, 2nd ed., Harcourt/Academic Press, Burlington, Mass., 2001.
• [11] Hardy, G. H, and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, Oxford, 1938.
• [12] Hartmanis, J., and R. E. Stearns, “On the computational complexity of algorithms,” Transactions of the American Mathematical Society, vol. 117 (1965), pp. 285–306.
• [13] Hertling, P., and C. Spandl, “Shifts with decidable language and non-computable entropy,” Discrete Mathematics and Theoretical Computer Science, vol. 10 (2008), pp. 75–93.
• [14] Jones, N. D., Computability and Complexity from a Programming Perspective, Foundations of Computing, MIT Press, 1997.
• [15] Jones, N. D., and A. L. Selman, “Turing machines and the spectra of first-order formulas,” Journal of Symbolic Logic, vol. 39 (1974), pp. 139–50.
• [16] Jürgensen, H., H.-J. Shyr, and G. Thierrin, “Disjunctive $\omega$-languages,” Elektronische Informationsverarbeitung und Kybernetik, vol. 19 (1983), pp. 267–78.
• [17] Jürgensen, H., and G. Thierrin, “On $\omega$-languages whose syntactic monoid is trivial,” International Journal of Computer and Information Sciences, vol. 12 (1983), pp. 359–65.
• [18] Ko, K.-I., Complexity Theory of Real Functions, Birkhäuser, Boston, 1991.
• [19] Lind, D., and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.