Selection and Uniformization Problems in the Monadic Theory of Ordinals: A Survey

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, <) 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.

• Type: Conference
• Year: 2008
• Where: BIRTHDAY
• Authors: Alexander Rabinovich, Amit Shomrat
• Keywords: Applied Computing, BIRTHDAY 2008, Fundamental Algorithmic Questions, Monadic Logic, Unique