Abstract
We compute that the index set of PAC-learnable concept classes is m-complete \({\Sigma^{0}_{3}}\) within the set of indices for all concept classes of a reasonable form. All concept classes considered are computable enumerations of computable \({\Pi^{0}_{1}}\) classes, in a sense made precise here. This family of concept classes is sufficient to cover all standard examples, and also has the property that PAC learnability is equivalent to finite VC dimension.
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Calvert, W. PAC learning, VC dimension, and the arithmetic hierarchy. Arch. Math. Logic 54, 871–883 (2015). https://doi.org/10.1007/s00153-015-0445-8
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DOI: https://doi.org/10.1007/s00153-015-0445-8