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Rabin's uniformization problem1

  • Yuri Gurevich (a1) and Saharon Shelah (a2)

Abstract

The set of all words in the alphabet {l, r} forms the full binary tree T. If xT then xl and xr are the left and the right successors of x respectively. We consider the monadic second-order language of the full binary tree with the two successor relations. This language allows quantification over elements of rand over arbitrary subsets of T. We prove that there is no monadic second-order formula ϕ*(X, y) such that for every nonempty subset X of T there is a unique yX that satisfies ϕ*(X, y) in T.

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2Department of Computer and Communication Sciences, University of Michigan, Ann Arbor, MI 48109, USA

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1

The work was done in principal during the 1980–81 academic year when both authors were fellows in the Institute for Advanced Studies of the Hebrew University in Jerusalem.

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References

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Büchi, J.R. [1962], On a decision method in second order arithmetic, Logic, methodology and philosophy of science (Proceedings of the 1960 international congress), Stanford University Press, Stanford, California, pp. 111.
Büchi, J.R. and Landweber, L.H. [1969], Solving sequential conditions by finite-state strategies, Transactions of the American Mathematical Society, vol. 138, pp. 295311.
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Gurevich, Y. and Harrington, L.A. [1982], Automata, trees, and games, Proceedings of the Fourteenth ACM Symposium on the Theory of Computing (San Francisco, 05, 1982) (to appear).
Läuchli, H. [1968], A decision procedure for the weak second order theory of linear order, Contributions to mathematical logic(Proceedings of the logic colloquium, Hannover, 1966), North-Holland, Amsterdam, pp. 189197.
Rabin, M.O. [1969], Decidability of second-order theories and automata on infinite trees, Transactions of the American Mathematical Society, vol. 141, pp. 135.
Rabin, M.O. [1972], Automata on infinite objects and Church's problem, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, no. 13, American Mathematical Society, Providence, R.I.
Shelah, S. [1975], The monadic theory of order, Annals of Mathematics, ser. 2, vol. 102, pp. 379419.
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Rabin's uniformization problem1

  • Yuri Gurevich (a1) and Saharon Shelah (a2)

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