## 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^{#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 #P_{1} ⊆ FP implies P = BPP and PH ⊆ MOD_{k}P 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 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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Pages (from-to) | 29-52 |

Number of pages | 24 |

Journal | Information and Computation |

Volume | 158 |

Issue number | 1 |

DOIs | |

State | Published - 2000 |

### 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
- Information Systems
- Computer Science Applications
- Computational Theory and Mathematics