We develop approximate counting of sets definable by Boolean circuits in bounded arithmetic using the dual weak pigeonhole principle (dWPHP(PV)), as a generalization of results from . We discuss applications to formalization of randomized complexity classes (such as BPP, APP, MA, AM) in PV1 + dWPHP(PV).
"Approximate counting in bounded arithmetic." J. Symbolic Logic 72 (3) 959 - 993, September 2007. https://doi.org/10.2178/jsl/1191333850