### 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 #P_{1} is the class of functions that count the witnesses for tally NP sets. We prove that every #P_{1} ^{PH} function can be computed in FP#^{p}_{1} ^{#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 #P_{1} ⊆ FP implies P = BPP and PH ⊆ 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.

Original language | English (US) |
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Title of host publication | Mathematical Foundations of Computer Science 1998 - 23rd International Symposium, MFCS 1998, Proceedings |

Editors | Lubos Brim, Jozef Gruska, Jiri Zlatuska |

Publisher | Springer Verlag |

Pages | 483-492 |

Number of pages | 10 |

ISBN (Print) | 3540648275, 9783540648277 |

DOIs | |

State | Published - 1998 |

Event | 23rd International Symposium on the Mathematical Foundations of Computer Science, MFCS 1998 - Brno, Czech Republic Duration: Aug 24 1998 → Aug 28 1998 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 1450 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 23rd International Symposium on the Mathematical Foundations of Computer Science, MFCS 1998 |
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Country | Czech Republic |

City | Brno |

Period | 8/24/98 → 8/28/98 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Mathematical Foundations of Computer Science 1998 - 23rd International Symposium, MFCS 1998, Proceedings*(pp. 483-492). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1450 LNCS). Springer Verlag. https://doi.org/10.1007/bfb0055798