Proof Complexity of the Cut-free Calculus of Structures

We investigate the proof complexity of analytic subsystems of the deep inference proof system SKSg (the calculus of structures). Exploiting the fact that the cut rule (i↑) of SKSg corresponds to the ¬-left rule in the sequent calculus, we establish that the ‘analytic'system KSg+c↑ has essentially the same complexity as the monotone Gentzen calculus MLK. In particular, KSg+c↑ quasipolynomially simulates SKSg, and admits polynomial-size proofs of some variants of the pigeonhole principle.