We introduce a variant of Spector's bar recursion in finite types (which we call "modified bar recursion") to give a realizability interpretation of the classical axiom of dependent choice allowing for the extraction of witnesses from proofs of ∀∃-formulas in classical analysis. As another application, we show that the fan functional can be defined by modified bar recursion together with a version of bar recursion due to Kohlenbach. We also show that the type structure M of strongly majorizable functionals is a model for modified bar recursion. §1. Introduction. In [22], Spector extended Gödel's Dialectica Interpretation of Peano Arithmetic [10] to classical analysis using bar recursion in finite types. Although considered questionable from an intuitionistic point of view ([1], 6.6), there has been considerable interest in bar recursion, and several variants of this definition scheme and their interrelations have been studied by, e.g., Schwichtenberg [19], Bezem [8] and Kohlenbach [14]. In this paper we add another variant of bar recursion and use it to give a realizability interpretation of the negatively translated axiom of dependent choice that can be used to extract witnesses from proofs of ∀∃-formulas in full classical analysis. Our interpretation is inspired by a paper by Berardi, Bezem and Coquand [2] who use a similar kind of recursion in order to interpret dependent choice. The main difference to our paper is that in [2] a rather ad-hoc infinitary term calculus and a non-standard notion of realizability are used whereas we work with a straightforward combination of negative translation, A-translation, modified realizability, and Plotkin's adequacy result for the partial continuous functional semantics of PCF [18]. As a second application of bar recursion, we show that the definition of the fan functional within PCF given in [3] and [17] can be derived from Kohlenbach's and our variant of bar recursion. Furthermore, we prove that our version of bar recursion exists in the model of majorizable functions. The relation between modified bar recursion and Spector's original definition is established in [5].

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