Section: New Results

On the Pigeonhole and Related Principles in Deep Inference and Monotone Systems

Participant : Anupam Das.

The size of proofs of the propositional pigeonhole principle over various systems is a topic of much interest in the proof complexity literature. In particular, it has received notable attention in recent years from the deep inference community, where its classification over the system KS appears as an open problem in numerous publications. In [21] we construct quasipolynomial-size proofs of the propositional pigeonhole principle in the deep inference system KS, addressing this question by 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 behavior 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 $nO(loglogn)$-size monotone proofs of the weak pigeonhole principle, thereby also improving the best known bounds for monotone proofs.