### 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 | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Pages | 483-492 |

Number of pages | 10 |

Volume | 1450 LNCS |

State | Published - 1998 |

Externally published | Yes |

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) |
---|---|

Volume | 1450 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### 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 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 1450 LNCS, pp. 483-492). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1450 LNCS).

**Tally NP sets and easy census functions.** / Goldsmith, Judy; Ogihara, Mitsunori; Rothe, Jörg.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 1450 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1450 LNCS, pp. 483-492, 23rd International Symposium on the Mathematical Foundations of Computer Science, MFCS 1998, Brno, Czech Republic, 8/24/98.

}

TY - GEN

T1 - Tally NP sets and easy census functions

AU - Goldsmith, Judy

AU - Ogihara, Mitsunori

AU - Rothe, Jörg

PY - 1998

Y1 - 1998

N2 - 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#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 #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.

AB - 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#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 #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.

UR - http://www.scopus.com/inward/record.url?scp=84896789776&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84896789776&partnerID=8YFLogxK

M3 - Conference contribution

SN - 3540648275

SN - 9783540648277

VL - 1450 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 483

EP - 492

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

ER -