Selection and uniformization problems in the monadic theory of ordinals: a survey

Authors:
Alexander Rabinovich;Amit Shomrat
Affiliations:
Sackler Faculty of Exact Sciences, Tel Aviv University, Israel;Sackler Faculty of Exact Sciences, Tel Aviv University, Israel
Venue:
Pillars of computer science
Year:
2008

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Abstract

A formula ψ(Y) is a selector for a formula φ(Y) in a structure M if there exists a unique Y that satisfies ψ in M and this Y also satisfies φ. A formula ψ(X, Y) uniformizes a formula φ(X, Y) in a structure M if for every X there exists a unique Y such that ψ(X, Y) holds in M and for this Y, φ(X, Y) also holds in 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 M = (α, X, Y). We present what is known regarding this generalization of Church's problem.