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A Complexity Analysis of Functional Interpretations
Mircea-Dan Hernest
February 2003 |
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.
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