Tally NP sets and easy census functions

Judy Goldsmith, Mitsunori Ogihara, Jörg Rothe

Research output: Chapter in Book/Report/Conference proceedingConference contribution


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.

Original languageEnglish (US)
Title of host publicationMathematical Foundations of Computer Science 1998 - 23rd International Symposium, MFCS 1998, Proceedings
EditorsLubos Brim, Jozef Gruska, Jiri Zlatuska
PublisherSpringer Verlag
Number of pages10
ISBN (Print)3540648275, 9783540648277
StatePublished - 1998
Externally publishedYes
Event23rd International Symposium on the Mathematical Foundations of Computer Science, MFCS 1998 - Brno, Czech Republic
Duration: Aug 24 1998Aug 28 1998

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume1450 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Other23rd International Symposium on the Mathematical Foundations of Computer Science, MFCS 1998
Country/TerritoryCzech Republic

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)


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