Tally NP sets and easy census functions

We study the question of whether every P set has an easy (i.e., polynomial-time computable) census function. We characterize this question in terms of unlikely collapses of language and function classes such as #P ₁ ⊆ FP, where #P₁ is the class of functions that count the witnesses for tally NP sets. We prove that every #P₁ ^PH function can be computed in FP#^p₁ ^#p1. Consequently, every P set has an easy census function if and only if every set in the polynomial hierarchy does. We show that the assumption #P₁ ⊆ FP implies P = BPP and PH ⊆ MOD_kP for each k ≥ 2, which provides further evidence that not all sets in P have an easy census function. We also relate a set's property of having an easy census function to other well-studied properties of sets, such as rankability and scalability (the closure of the rankable sets under P-isomorphisms). Finally, we prove that it is no more likely that the census function of any set in P can be approximated (more precisely, can be n^α -enumerated in time n_β for fixed α and β) than that it can be precisely computed in polynomial time.