Proof Compression with the CIRes Method [Abstract]
Résumé
It is well-known that eliminating cuts from sequent calculus proofs frequently increases the size and length of proofs. In the worst case, cut-elimination can produce non-elementarily larger and longer proofs [3, 2]. Given this fact, it is natural to attempt to devise methods that could introduce cuts and compress cut-free proofs. However, this has been a notoriously difficult task. Indeed, the problem of answering, given a proof φ and a number l such that l ≤ length(φ), whether there is a proof ψ (possibly with cuts) of the same theorem and such that length(ψ) < l is known to be undecidable [1]. Besides compression, cut-introduction can also be used for structuring and extracting interesting concepts from proofs. In the formalization of mathematical proofs, lemmas correspond to cuts, and hence the automatic introduction of cuts is, in a formal level, the automatic discovery of lemmas that are potentially useful for structuring mathematical knowledge. This talk presents a new method for the introduction of atomic cuts, CIRes, and shows that it is capable of providing exponential compression in the length of proofs. This compression is illustrated by an example sequence of proofs encoding computations of Fibonacci numbers.