Proof theory and computational analysis

In this survey paper we start with a discussion how functionals of finite type can be used for the proof-theoretic extraction of numerical data (e.g. effective uniform bounds and rates of convergence) from non-constructive proofs in numerical analysis. We focus on the case where the extractability of polynomial bounds is guaranteed. This leads to the concept of hereditarily polynomial bounded analysis PBA. We indicate the mathematical range of PBA which turns out to be surprisingly large. Finally we discuss the relationship between PBA and so-called feasible analysis FA. It turns out that both frameworks are incomparable. We argue in favor of the thesis that PBA offers the more useful approach for the purpose of extracting mathematically interesting bounds from proofs. In a sequel of appendices to this paper we indicate the expressive power of PBA. ∗Basic Research in Computer Science, Centre of the Danish National Research Foundation.