Tally NP sets and easy census functions

Judy Goldsmith, Mitsunori Ogihara, Jörg Rothe

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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 #P1 ⊆ FP, where #P1 is the class of functions that count the witnesses for tally NP sets. We prove that every #P1PH 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. 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)
Pages (from-to)29-52
Number of pages24
JournalInformation and Computation
Issue number1
StatePublished - 2000
Externally publishedYes

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Information Systems
  • Computer Science Applications
  • Computational Theory and Mathematics


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