Monotone Proofs of the Pigeon Hole Principle

We study the complexity of proving the Pigeon Hole Principle (PHP) in a monotone variant of the Gentzen Calculus, also known as Geometric Logic. We show that the standard encoding of the PHP as a monotone sequent admits quasipolynomial-size proofs in this system. This result is a consequence of deriving the basic properties of certain quasipolynomial-size monotone formulas computing the boolean threshold functions. Since it is known that the shortest proofs of the PHP in systems such as Resolution or Bounded Depth Frege are exponentially long, it follows from our result that these systems are exponentially separated from the monotone Gentzen Calculus. We also consider the monotone sequent (CLIQUE) expressing the clique-coclique principle deened by Bonet, Pitassi and Raz (1997). We show that monotone proofs for this sequent can be easily reduced to monotone proofs of the one-to-one and onto PHP, and so CLIQUE also has quasipolynomial-size monotone proofs. As a consequence, Cutting Planes with polynomially bounded coeecients is also exponentially separated from the monotone Gentzen Calculus. Finally, a simple simulation argument implies that these results extend to the Intuitionistic Gentzen Calculus. Our results partially answer some questions left open by P. Pudll ak.