Abstract
A formula ψ(Y) is a selector for a formula ϕ(Y) in a structure \(\mathcal{M}\) if there exists a unique Y that satisfies ψ in \(\mathcal{M}\) and this Y also satisfies ϕ. A formula ψ(X,Y) uniformizes a formula ϕ(X,Y) in a structure \(\mathcal{M}\) if for every X there exists a unique Y such that ψ(X,Y) holds in \(\mathcal{M}\) and for this Y, ϕ(X,Y) also holds in \(\mathcal{M}\). In this paper we survey some fundamental algorithmic questions and recent results regarding selection and uniformization, when the formulas ψ and ϕ are formulas of the monadic logic of order and the structure \(\mathcal{M}=(\alpha,<)\) is an ordinal α equipped with its natural order. A natural generalization of the Church problem to ordinals is obtained when some additional requirements are imposed on the uniformizing formula ψ(X,Y). We present what is known regarding this generalization of Church’s problem.
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References
Büchi, J.R.: Weak second-order arithmetic and finite automata. Zeit. Math. Logik und Grundl. Math. 6, 66–92 (1960)
Büchi, J.R.: On a decision method in restricted second order arithmetic. In: Proc. Int. Congress on Logic, Method, and Philosophy of Science. 1960, pp. 1–12. Stanford University Press (1962)
Büchi, J.R., Landweber, L.H.: Solving sequential conditions by finite-state strategies. Trans. Amer. Math. Soc. 138, 295–311 (1969)
Büchi, J.R., Siefkes, D.: The Monadic Second-Order Theory of all Countable Ordinals. In: Bloomfield, R.E., Jones, R.B., Marshall, L.S. (eds.) VDM 1988. LNCS, vol. 328, pp. 1–126. Springer, Heidelberg (1988)
Church, A.: Logic, arithmetic and automata. In: Proc. Inter. Cong. Math. 1963, Almquist and Wilksells, Uppsala (1963)
Elgot, C.: Decision problems of finite-automata design and related arithmetics. Trans. Amer. Math. Soc. 98, 21–51 (1961)
Feferman, S., Vaught, R.L.: The first-order properties of products of algebraic systems. Fundamenta Mathematicae 47, 57–103 (1959)
Gurevich, Y., Shelah, S.: Rabin’s uniformization problem. Jou. of Symbolic Logic 48, 1105–1119 (1983)
Gurevich, Y.: Monadic second-order theories. In: Barwise, J., Feferman, S. (eds.) Model-Theoretic Logics, pp. 479–506. Springer, Heidelberg (1985)
Larson, P.B., Shelah, S.: The stationary set splitting game. Mathematical Logic Quarterly (to appear)
Lifsches, S., Shelah, S.: Uniformization and Skolem functions in the class of trees. Jou. of Symbolic Logic 63(1), 103–127 (1998)
McNaughton, R.: Testing and generating infinite sequences by a finite automaton. Information and Control 9, 521–530 (1966)
Rabin, M.O.: Decidability of second-order theories and automata on infinite trees. Trans. Amer. Math. Soc. 141, 1–35 (1969)
Rabin, M.O.: Automata on infinite objects and Church’s problem. Amer. Math. Soc., Providence, RI (1972)
Rabinovich, A.: The Church synthesis problem over countable ordinals (submitted)
Rabinovich, A., Shomrat, A.: Selection in the monadic theory of a countable ordinal (submitted)
Rabinovich, A., Shomrat, A.: Selection over classes of ordinals expanded by monadic predicates (submitted)
Shelah, S.: The monadic theory of order. Annals of Math., Ser. 2 102, 379–419 (1975)
Shomrat, A.: Uniformization Problems in the Monadic Theory of Countable Ordinals. M.Sc. Thesis,Tel Aviv University (2007)
Trakhtenbrot, B.A.: The synthesis of logical nets whose operators are described in terms of one-place predicate calculus. Doklady Akad. Nauk SSSR 118(4), 646–649 (1958)
Trakhtenbrot, B.A.: Certain constructions in the logic of one-place predicates. Doklady Akad. Nauk SSSR 138, 320–321 (1961)
Trakhtenbrot, B.A.: Finite automata and monadic second order logic. Siberian Math. J 3, 101–131 (1962) Russian; English translation in: AMS Transl. 59, 23–55 (1966)
Trakhtenbrot, B.A., Barzdin, Y.M.: Finite Automata. North Holland, Amsterdam (1973)
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Rabinovich, A., Shomrat, A. (2008). Selection and Uniformization Problems in the Monadic Theory of Ordinals: A Survey. In: Avron, A., Dershowitz, N., Rabinovich, A. (eds) Pillars of Computer Science. Lecture Notes in Computer Science, vol 4800. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78127-1_31
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