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.

Extracting Herbrand Disjunctions by Functional Interpretation

“…Later presentations of Statman's theorem are due to Orevkov and Pudlak [15,16,17]. The short proofs given by Pudlak are of size polynomial in n, yielding FI-extracted terms of size exponential in n (by [10]). The formulas occurring in the proof can be shown to have ¬-depth at most n, but by careful analysis of the extracted FI terms one can bound their degree by n − 1.…”

Gödel's Functional Interpretation and its Use in Current Mathematics*

“… The extraction of the terms t in both theorems is carried out by recursion over the given proof. The complexity of this extraction procedure is rather low: the size of the extracted terms is linear in the size of the given proof, the extraction algorithm has a cubic worst‐time complexity and the depth of the verifying proof is linear in the depth of the given proof and the maximal size of formulas occurring in that proof (see Hernest and Kohlenbach 2005 for this and much more detailed information). The extraction algorithm has been further optimized in the ‘Light Functional Interpretation’ of Hernest 2005 and is implemented in Hernest n.d. In general, (A′) D causes unnecessarily high types.…”

Gödel's Functional Interpretation and Its Use in Current Mathematics

“…The complexity of this extraction procedure is rather low: the size of the extracted terms is linear in the size of the given proof, the extraction algorithm has a cubic worst-time complexity and the depth of the verifying proof is linear in the depth of the given proof and the maximal size of formulas occurring in that proof (see [47] for this and much more detailed information). The extraction algorithm has been further optimized in the 'Light Functional Interpretation' of [45] and is implemented in [46].…”