We show how to formalize approximate counting via hash functions insubsystems of bounded arithmetic, using variants of the weakpigeonhole principle. We discuss several applications, including aproof of the tournament principle, and an improvement on the knownrelationship of the collapse of the bounded arithmetic hierarchy tothe collapse of the polynomial-time hierarchy.
"Approximate counting by hashing in bounded arithmetic." J. Symbolic Logic 74 (3) 829 - 860, September 2009. https://doi.org/10.2178/jsl/1245158087