Tally NP sets and easy census functions

Abstract

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 $\# P_1 \subseteq FP$, where #P₁ is the class of functions that count the witnesses for tally NP sets. We prove that every #P ^PH₁ function can be computed in $FP^{\# P_1 ^{\# P_1 } }$. 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 $\# P_1 \subseteq FP$ implies P = BPP and $PH \subseteq MOD_k P$ 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.

Supported in part by NSF grant CCR-9315354.

Supported in part by the National Science Foundation under grants CCR-9701911 and INT-9726724.

Supported in part by grants NSF-INT-9513368/DAAD-315-PRO-fo-ab and NSF-CCR-9322513 and by a NATO Postdoctoral Science Fellowship from the Deutscher Akademischer Austauschdienst (“Gemeinsames Hochschulsonderprogramm III von Bund und Ländern”).

Work done in part while visiting the University of Kentucky and the University of Rochester.