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Uniformization and skolem functions in the class of trees

  • Shmuel Lifsches (a1) and Saharon Shelah (a2)

Abstract

The monadic second-order theory of trees allows quantification over elements and over arbitrary subsets. We classify the class of trees with respect to the question: does a tree T have definable Skolem functions (by a monadic formula with parameters)? This continues [6] where the question was asked only with respect to choice functions. A natural subclass is defined and proved to be the class of trees with definable Skolem functions. Along the way we investigate the spectrum of definable well orderings of well ordered chains.

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[1]Büchi, J. R. and Landweber, L. H., Solving sequential conditions by finite-state strategies, Transactions of the American Mathematical Society, vol. 138 (1969), pp. 295311.
[2]Feferman, S. and Vaught, R. L., The first order properties of products of algebraic systems, Fundamenta Mathematicae, vol. 47 (1959), pp. 57103.
[3]Gurevich, Y., Monadic second-order theories, Model theoretic logics (Barwise, J. and Feferman, S., editors), Springer-Verlag, Berlin, 1985, pp. 479506.
[4]Gurevich, Y. and Shelah, S., Rabin's uniformization problem, this Journal, vol. 48 (1983), pp. 11051119.
[5]Hausdorff, F., Grundzüge einer Theorie der geordnetn Mengen, Mathematische Annalen, vol. 65 (1908), pp. 435505.
[6]Lifsches, S. and Shelah, S., Uniformization, choice functions and well orders in the class of trees, this Journal, vol. 61 (1996), pp. 12061227.
[7]Rabin, M. O., Decidability of second-order theories and automata on infinite trees, Transactions of the American Mathematical Society, vol. 141 (1969), pp. 135.
[8]Shelah, S., The monadic theory of order, Annals of Mathematics. Second Series, vol. 102 (1975), pp. 379419.

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Uniformization and skolem functions in the class of trees

  • Shmuel Lifsches (a1) and Saharon Shelah (a2)

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