Mechanizing Proof Theory: Resource-Aware Logics and Proof Transformations toExtract Implicit Information

摘要

Few systems for mechanical proof-checking have been used so far to transformformal proofs rather than to formalize informal arguments and to verify correctness. The unwinding of proofs, namely, the process of applying lemmata and extracting explicit values for the parameters within a proof, is an obvious candidate for mechanization. It corresponds to the procedures of Cut-elimination and functional interpretation in proof-theory and allows the extraction of the constructive content of a proof, sometimes yielding information useful in mathematics and in computing. Resource-aware logics restrict the number of times an assumption may be used in a proof and are of interest for proof-checking not only in relation to their decidability or computational complexity, but also because they efficiently solve the practical problem of representing the structure of relevance in a derivation. In particular, in Direct Logic only one subformula-occurrence of the input is allowed, and the connections established during a successful proof-verification can be represented on the input without altering it. In addition, the values for the parameters obtained from unwinding are read off directly. In Linear Logic, where classical logic is regarded as the limit of a resource-aware logic, long-standing issues in proof-theory have been successfully attacked. We are particularly interested in the system of proof-nets as a multiple-conclusion Natural Deduction system for Linear Logic. (Author) (kr)