Abstract
The spectrum of a first-order sentence is the set of cardinalities of its finite models. This paper is concerned with spectra of sentences over languages that contain only unary function symbols. In particular, it is shown that a set S of natural numbers is the spectrum of a sentence over the language of one unary function symbol precisely if S is an eventually periodic set.
This research was done while the author was a Research Fellow at the IBM Haifa Research Laboratory.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
G. Asser, Das Reprä sentantenproblem im Prädikatenkalkül der ersten Stufe mit Identität, Z. Math. Logik Grundlag. Math. 1, 1955, pp. 252–263.
A. Durand, C. Lautemann and T. Schwentick, Subclasses of binary NP, Journal of Logic and Computation, to appear.
A. Durand and S. Ranaivoson, First-order spectra with one binary predicate, Theoretical Computer Science 160, 1–2, 1996, pp. 305–320.
S. Eilenberg, Automata, Languages, and Machines, Vol. A, Academic Press, New York and London, 1974.
H. Enderton, A Mathematical Introduction to Logic, Academic Press, New York and London, 1972.
R. Fagin, Contributions to the model theory of finite structures, Ph.D. Thesis, University of California at Berkeley, 1973.
R. Fagin, Generalized first-order spectra and polynomial-time recognizable sets, in: R. M. Karp, ed., Complexity of Computation, SIAM-AMS Proc. 7, 1974, pp. 43–73.
R. Fagin, A spectrum hierarchy, Z. Math. Logik Grundlag. Math. 21, 1975, pp. 123–134.
R. Fagin, Finite-model theory—a. personal perspective, Theoretical Computer Science 116, 1993, pp. 3–31.
R. Fagin, L. Stockmeyer and M. Y. Vardi, On monadic NP vs. monadic NP, Information and Computation 120, 1, 1995, pp. 78–92.
E. Grandjean, Universal quantifiers and time complexity of random access machines, Math. Systems Theory 18, 1985, pp. 171–187.
E. Grandjean, First-order spectra with one variable, J. Comput. Systems Sci. 40, 2, 1990, pp. 136–153.
Y. Gurevich and S. Shelah, The monadic second-order theory of one unary function, in preparation.
K. Harrow, Sub-elementary classes of functions and relations, Doctoral Dissertation, New York University, Department of Mathematics, 1973.
N. G. Jones and A. L. Selman, Turing machines and the spectra of first-order formulas, J. Symbolic Logic 39, 1974, pp. 139–150.
B. Loescher, One unary function says less than two in existential second order logic, Information Processing Letters 61, 1997, pp. 69–75.
B. Loescher and A. Sharell, The expressive power of quantification over functions in existential second order logic, in preparation.
J. Lynch, Complexity classes and theories of finite models, Math. Systems Theory 15, 1982, pp. 127–144.
F. Olive, Caractérisation logique des problèmes NP: robustesse et normalisation, Ph.D. Thesis, University de Caen, 1996.
H. Scholz, Problem #1: Ein ungelöstes Problem in der symbolischen Logik, J. Symbolic Logic 17, 1952, p. 160.
J. R. Shoenfield, Mathematical Logic, Addison-Wesley, Reading, MA, 1967.
A. Woods, Some problems in logic and number theory and their connections, Ph.D. Thesis, University of Manchester, 1981.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Durand, A., Fagin, R., Loescher, B. (1998). Spectra with only unary function symbols. In: Nielsen, M., Thomas, W. (eds) Computer Science Logic. CSL 1997. Lecture Notes in Computer Science, vol 1414. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0028015
Download citation
DOI: https://doi.org/10.1007/BFb0028015
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64570-2
Online ISBN: 978-3-540-69353-6
eBook Packages: Springer Book Archive