Skip to main content Accessibility help
×
Home

Selection in the monadic theory of a countable ordinal

  • Alexander Rabinovich (a1) and Amit Shomrat (a2)

Abstract

A monadic formula ψ(Y) is a selector for a formula φ(Y) in a structure if there exists a unique subset P of which satisfies ψ and this P also satisfies φ. We show that for every ordinal αωω there are formulas having no selector in the structure (α, <). For αω1, we decide which formulas have a selector in (α, <) , and construct selectors for them. We deduce the impossibility of a full generalization of the Büchi-Landweber solvability theorem from (ω, <) to (ωω, <). We state a partial extension of that theorem to all countable ordinals. To each formula we assign a selection degree which measures “how difficult it is to select”. We show that in a countable ordinal all non-selectable formulas share the same degree.

Copyright

References

Hide All
[1]Büchi, J. R. and Landweber, L. H., Solving sequential conditions by finite-state strategies, Transactions of American Mathematical Society, vol. 138 (1969), pp. 295311.
[2]Büchi, J. R. and Siefkes, D., The monadic second-order theory of all countable ordinals, Decidable Theories, Vol. 2 (Büchi, J. R. and Siefkes, D., editors), Lecture Notes in Mathematics, vol. 328, Springer, 1973, pp. 1126.
[3]Church, A., Logic, arithmetic and automata, Proceedings of the International Congress of Mathematicians, Almquist and Wilksells, Uppsala, 1963, pp. 2135.
[4]Feferman, S. and Vaught, R. L., The first-order properties of products of algebraic systems, Fundamenta Mathematical vol. 47 (1959), pp. 57103.
[5]Gurevich, Y., Monadic second-order theories, Model-Theoretic Logics (Barwise, J. and Feferman, S., editors), Springer-Verlag, 1985, pp. 479506.
[6]Gurevich, Y. and Rabinovich, A., Definability in the rationals with the real order in the background, Journal of Logic and Computation, vol. 12 (2002), no. 1, pp. 111.
[7]Hintikka, J., Distributive normal forms in the calculus of predicates, Acta Philosophica Fennica, vol. 6 (1953), pp. 371.
[8]Hrbacek, K. and Jech, T., Introduction to Set Theory, 3rd, revised and expanded ed., Marcel Dekker, New York, 1999.
[9]Larson, P. B. and Shelah, S., The stationary set splitting game, Mathematical Logic Quarterly, vol. 54 (2008), no. 2, pp. 187193.
[10]Läuchli, H. and Leonard, J., On the elementary theory of linear order, Fundamenta Mathematicae, vol. 59 (1966), pp. 109116.
[11]Lifsches, S. and Shelah, S., Uniformization and skolem functions in the class of trees, this Journal, vol. 63 (1998), no. 1, pp. 103127.
[12]McNaughton, R., Testing and generating infinite sequences by a finite automaton, Information and Control, vol. 9 (1966), pp. 521530.
[13]Neeman, I., Finite state automata and monadic definability of ordinals, preprint. Available at: http://www.math.ucla.edu/~ineeman.
[14]Rabinovich, A., The Church synthesis problem over countable ordinals, submitted.
[15]Rabinovich, A. and Shomrat, A., Selection over classes of countable ordinals expanded by monadic predicates, submitted.
[16]Rosenstein, J. G., Linear Orderings, Academic Press, New York, 1982.
[17]Shelah, S., The monadic theory of order, Annals of Mathematics, Ser. 2, vol. 102 (1975), pp. 379419.
[18]Thomas, W., Ehrenfeucht games, the composition method, and the monadic theory of ordinal words, Structures in Logic and Computer Science, A Selection of Essays in Honor of A. Ehrenfeucht (Mycielski, J., Rozenberg, G., and Salomaa, A., editors), Lecture Notes in Computer Science, vol. 1261, Springer, 1997, pp. 118143.

Related content

Powered by UNSILO

Selection in the monadic theory of a countable ordinal

  • Alexander Rabinovich (a1) and Amit Shomrat (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.