@inproceedings{8c837aa1aaa441a4ae3760b4bcdf3237,

title = "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 ⊆ 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 ⊆ MODkP 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 nα -enumerated in time nβ for fixed α and β) than that it can be precisely computed in polynomial time.",

author = "Judy Goldsmith and Mitsunori Ogihara and J{\"o}rg Rothe",

note = "Funding Information: 1Supported in part by NSF Grant CCR-9610348. 2Supported in part by NSF CAREER Award CCR-9701911. 3Supported in part by Grants NSF-INT-9513368 DAAD-315-PRO-fo-ab, NSF-CCR-9322513, and NSF-INT-9815095 DAAD-315-PPP-gu-ab, and by a NATO Postdoctoral Science Fellowship from the Deutscher Akademischer Austauschdienst ({\textquoteleft}{\textquoteleft}Gemeinsames Hochschulsonderprogramm III von Bund und Landern{\textquoteright}{\textquoteright}). Work done in part while visiting the University of Rochester and the University of Kentucky. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.; 23rd International Symposium on the Mathematical Foundations of Computer Science, MFCS 1998 ; Conference date: 24-08-1998 Through 28-08-1998",

year = "1998",

doi = "10.1007/bfb0055798",

language = "English (US)",

isbn = "3540648275",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

publisher = "Springer Verlag",

pages = "483--492",

editor = "Lubos Brim and Jozef Gruska and Jiri Zlatuska",

booktitle = "Mathematical Foundations of Computer Science 1998 - 23rd International Symposium, MFCS 1998, Proceedings",

}