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 #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.
| Original language | English (US) |
|---|---|
| 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 |
|---|---|
| Country | Czech Republic |
| City | Brno |
| Period | 8/24/98 → 8/28/98 |
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ASJC Scopus subject areas
- Computer Science(all)
- Theoretical Computer Science
Cite this
Tally NP sets and easy census functions. / Goldsmith, Judy; Ogihara, Mitsunori; Rothe, Jörg.
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 1450 LNCS 1998. p. 483-492 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1450 LNCS).Research output: Chapter in Book/Report/Conference proceeding › Conference contribution
}
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.
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M3 - Conference contribution
AN - SCOPUS:84896789776
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 -