A Note on Polynomial-size Monotone Proofs of the Pigeon Hole Principle
نویسنده
چکیده
We see that the version of the pigeonhole principle in which every hole is forced to receive a pigeon (called onto) and the version in which every pigeon is mapped into exactly one hole (called functional) have polynomial-size proofs in the tree-like monotone sequent calculus. The proofs are surprisingly simple reductions to the non-monotone case.
منابع مشابه
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...
متن کاملIAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Advanced Course on Computational Complexity Lecture 13: Polynomial-Size Frege Proofs of the Pigeonhole Principle
The pigeonhole principle states that there is no one-to-one function from a set of size n to a set of size n − 1. In other words, if n pigeons are put into n − 1 holes, then at least one hole will be occupied by more than one pigeon. This simple fact has an astonishing variety of applications in mathematics. It also corresponds to a tautology that has been used extensively in the study of the c...
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