A Complexity Analysis of Functional Interpretations

Abstract

We give a quantitative analysis of Gödel's functional interpretation and its monotone variant. The two have been used for the extraction of programs and numerical bounds as well as for conservation results. They apply both to (semi-)intuitionistic as well as (combined with negative translation) classical proofs. The proofs may be formalized in systems ranging from weak base systems to arithmetic and analysis (and numerous fragments of these). We give upper bounds in basic proof data on the depth, size, maximal type degree and maximal type arity of the extracted terms as well as on the depth of the verifying proof. In all cases terms of size linear in the size of the proof at input can be extracted and the corresponding extraction algorithms have cubic worst-time complexity. The verifying proofs have depth linear in the depth of the proof at input and the maximal size of a formula of this proof.