## 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 |
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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 |

Pages | 483-492 |

Number of pages | 10 |

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 |

### Conference

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

City | Brno |

Period | 8/24/98 → 8/28/98 |

### Bibliographical 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 (‘‘Gemeinsames Hochschulsonderprogramm III von Bund und Landern’’). Work done in part while visiting the University of Rochester and the University of Kentucky.

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science