We construct quasipolynomial-size proofs of the propositional pigeonhole principle in the deep inference system KS, addressing an open problem raised in previous works and matching the best known upper bound for the more general class of monotone proofs. We make significant use of monotone formulae computing boolean threshold functions, an idea previously considered in works of Atserias et al. The main construction, monotone proofs witnessing the symmetry of such functions, involves an implementation of merge-sort in the design of proofs in order to tame the structural behaviour of atoms, and so the complexity of normalization. Proof transformations from previous work on atomic flows are then employed to yield appropriate KS proofs. As further results we show that our constructions can be applied to provide quasipolynomial-size KS proofs of the parity principle and the generalized pigeonhole principle. These bounds are inherited for the class of monotone proofs, and we are further able to construct n-size monotone proofs of the weak pigeonhole principle with (1 + ε)n pigeons and n holes for ε = 1= logk n, thereby also improving the best known bounds for monotone proofs.
|Title of host publication||Proceedings of the Joint Meeting of the 23rd EACSL Annual Conference on Computer Science Logic, CSL 2014 and the 29th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2014|
|Publisher||Association for Computing Machinery|
|Publication status||Published - 2014|
|Event||Joint Meeting of the 23rd Annual EACSL Conference on Computer Science Logic, CSL 2014 and the 29th Annual ACM/ IEEE Symposium on Logic in Computer Science, LICS 2014 - Vienna , Austria|
Duration: 14 Jul 2014 → 18 Jul 2014
|Conference||Joint Meeting of the 23rd Annual EACSL Conference on Computer Science Logic, CSL 2014 and the 29th Annual ACM/ IEEE Symposium on Logic in Computer Science, LICS 2014|
|Period||14/07/14 → 18/07/14|