Proof Mining in Subsystems of Analysis

This dissertation studies the use of methods of proof theory in extracting new information from proofs in subsystems of classical analysis. We focus mainly on ineffective proofs, i.e. proofs which make use of ineffective principles ranging from weak König’s lemma to full comprehension. The main contributions of the dissertation can be divided into four parts: (1) A discussion of how monotone functional interpretation provides the right notion of “numerical implication” (cf. [21]) in analysis. We show among other things that monotone functional interpretation naturally creates well-studied moduli when applied to various classes of statements (e.g. uniqueness, convexity, contractivity, continuity and monotonicity) and that the interpretation of implications between those statements corresponds to translations between the different moduli. (2) A case study in L1-approximation, in which we analyze Cheney’s proof of Jackson’s theorem, concerning uniqueness of the best approximation, w.r.t. L1-norm, of continuous functions f ∈ C[0, 1] by polynomials of bounded degree. The result of our analysis provides the first effective modulus of uniqueness for L1-approximation. Moreover, using this modulus we give the first complexity analysis of the sequence of the best L1-approximations for polynomial-time computable functions f ∈ C[0, 1]. (3) A comparison between three different forms of bar recursion, in which we show among other things that the type structure of strongly majorizable functionals is a model of modified bar recursion, that modified bar recursion is not S1-S9 computable over the type structure of total continuous functions and that modified bar recursion defines (primitive recursively, in the sense of Kleene) Spector’s bar recursion. (4) An adaptation of functional interpretation to handle ineffective proofs in feasible analysis, which provides the first modular procedure for extracting polynomial-time realizers from ineffective proofs (i.e. proofs involving weak König’s lemma) of Π2-theorems in feasible analysis.