Semantic Characterisations of Second-Order Computability over the Real Numbers

We propose semantic characterisations of second-order computability over the reals based on Σ-definability theory. Notions of computability for operators and real-valued functionals defined on the class of continuous functions are introduced via domain theory. We consider the reals with and without equality and prove theorems which connect computable operators and real-valued functionals with validity of finite Σ-formulas.

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