Source: J. Symbolic Logic
Volume 73, Issue 3
A monadic formula ψ(Y) is a selector for a formula φ(Y)
in a structure ℳ if there exists a unique subset P of ℳ
which satisfies ψ and this P also satisfies φ.
We show that for every ordinal α≥ ωω there are formulas having no selector
in the structure (α, <).
For α ≤ ω1, 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.
J. R. Büchi and L. H. Landweber, Solving sequential conditions by finite-state strategies, Transactions of American Mathematical Society, vol. 138 (1969), pp. 295--311.
Mathematical Reviews (MathSciNet): MR280205
J. R. Büchi and D. Siefkes, The monadic second-order theory of all countable ordinals, Decidable Theories, Vol. 2 (J. R. Büchi and D. Siefkes, editors), Lecture Notes in Mathematics, vol. 328, Springer, 1973, pp. 1--126.
Mathematical Reviews (MathSciNet): MR345804
A. Church, Logic, arithmetic and automata, Proceedings of the International Congress of Mathematicians, Almquist and Wilksells, Uppsala, 1963, pp. 21--35.
Mathematical Reviews (MathSciNet): MR175789
S. Feferman and R. L. Vaught, The first-order properties of products of algebraic systems, Fundamenta Mathematicae, vol. 47 (1959), pp. 57--103.
Mathematical Reviews (MathSciNet): MR108455
Y. Gurevich, Monadic second-order theories, Model-Theoretic Logics (J. Barwise and S. Feferman, editors), Springer-Verlag, 1985, pp. 479--506.
Mathematical Reviews (MathSciNet): MR819544
Y. Gurevich and A. Rabinovich, Definability in the rationals with the real order in the background, Journal of Logic and Computation, vol. 12 (2002), no. 1, pp. 1--11.
J. Hintikka, Distributive normal forms in the calculus of predicates, Acta Philosophica Fennica, vol. 6 (1953), pp. 3--71.
Mathematical Reviews (MathSciNet): MR69778
K. Hrbacek and T. Jech, Introduction to Set Theory, 3rd, revised and expanded ed., Marcel Dekker, New York, 1999.
P. B. Larson and S. Shelah, The stationary set splitting game, Mathematical Logic Quarterly, vol. 54 (2008), no. 2, pp. 187--193.
H. Läuchli and J. Leonard, On the elementary theory of linear order, Fundamenta Mathematicae, vol. 59 (1966), pp. 109--116.
Mathematical Reviews (MathSciNet): MR199108
S. Lifsches and S. Shelah, Uniformization and skolem functions in the class of trees, Journal of Symbolic Logic, vol. 63 (1998), no. 1, pp. 103--127.
R. McNaughton, Testing and generating infinite sequences by a finite automaton, Information and Control, vol. 9 (1966), pp. 521--530.
Mathematical Reviews (MathSciNet): MR213241
I. Neeman, Finite state automata and monadic definability of ordinals, preprint. Available at: http://www.math.ucla.edu/$\sim$ineeman.
A. Rabinovich, The Church synthesis problem over countable ordinals, submitted.
A. Rabinovich and A. Shomrat, Selection over classes of countable ordinals expanded by monadic predicates, submitted.
J. G. Rosenstein, Linear Orderings, Academic Press, New York, 1982.
Mathematical Reviews (MathSciNet): MR662564
S. Shelah, The monadic theory of order, Annals of Mathematics, Ser. 2, vol. 102 (1975), pp. 379--419.
Mathematical Reviews (MathSciNet): MR491120
W. Thomas, 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 (J. Mycielski, G. Rozenberg, and A. Salomaa, editors), Lecture Notes in Computer Science, vol. 1261, Springer, 1997, pp. 118--143.