Enumerative counting is hard

J.-Y. Cai, L.A. Hemachandra
1988 [1988] Proceedings. Structure in Complexity Theory Third Annual Conference  
An n-variable Boolean formula may have anywhere from 0 to 2" satisfying assignments. Can a polynomial-time machine, given such a formula, reduce this exponential number of possibilities to a small number of possibilities? We call such a machine an enumerator and prove that if there is a good polynomial-time enumerator for # P (i.e., one where for every Boolean formula A the small set has at most o(lf['-") numbers), then P = NP = P*' and probabilistic polynomial time equals polynomial time.
more » ... ermore, we show that #P polynomial-time Turing reduces to enumerating # P. fJ>
doi:10.1109/sct.1988.5279 dblp:conf/coco/CaiH88 fatcat:mzratsooqjg5riutgc6x2inpua