### Abstract

Is there an NP function that, when given a satisfiable formula as input, outputs one satisfying assignment uniquely? That is, can a nondeterministic function cull just one satisfying assignment from a possibly exponentially large collection of assignments? We show that if there is such a nondeterministic function, then the polynomial hierarchy collapses to ZPP^{NP} (and thus, in particular, to NP^{NP}). Because the existence of such a function is known to be equivalent to the statement "every NP function has an NP refinement with unique outputs," our result provides the strongest evidence yet that NP functions cannot be refined. We prove our result via a result of independent interest. We say that a set A is NPSV-selective (NPMV-selective) if there is a 2-ary partial NP function with unique values (a 2-ary partial NP function) that decides which of its inputs (if any) is "more likely" to belong to A; this is a nondeterministic analog of the recursion-theoretic notion of the semirecursive sets and the extant complexity-theoretic notion of P-selectivity. Our hierarchy-collapse result follows by combining the easy observation that every set in NP is NPMV-selective with the following result: If A ∈ NP is NPSV-selective, then A ∈ (NP∩coNP)/poly. Relatedly, we prove that if A ∈ NP is NPSV-selective, then A is Low_{2}. We prove that the polynomial hierarchy collapses even further, namely to NP, if all coNP sets are NPMV-selective. This follows from a more general result we prove: Every self-reducible NPMV-selective set is in NP.

Original language | English (US) |
---|---|

Pages (from-to) | 697-708 |

Number of pages | 12 |

Journal | SIAM Journal on Computing |

Volume | 25 |

Issue number | 4 |

State | Published - Aug 1996 |

Externally published | Yes |

### Fingerprint

### Keywords

- Computational complexity
- Lowness
- Nonuniform complexity
- Semidecision algorithms

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Applied Mathematics
- Theoretical Computer Science

### Cite this

*SIAM Journal on Computing*,

*25*(4), 697-708.

**Computing solutions uniquely collapses the polynomial hierarchy.** / Hemaspaandra, Lane A.; Naik, Ashish V.; Ogihara, Mitsunori; Selman, Alan L.

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 25, no. 4, pp. 697-708.

}

TY - JOUR

T1 - Computing solutions uniquely collapses the polynomial hierarchy

AU - Hemaspaandra, Lane A.

AU - Naik, Ashish V.

AU - Ogihara, Mitsunori

AU - Selman, Alan L.

PY - 1996/8

Y1 - 1996/8

N2 - Is there an NP function that, when given a satisfiable formula as input, outputs one satisfying assignment uniquely? That is, can a nondeterministic function cull just one satisfying assignment from a possibly exponentially large collection of assignments? We show that if there is such a nondeterministic function, then the polynomial hierarchy collapses to ZPPNP (and thus, in particular, to NPNP). Because the existence of such a function is known to be equivalent to the statement "every NP function has an NP refinement with unique outputs," our result provides the strongest evidence yet that NP functions cannot be refined. We prove our result via a result of independent interest. We say that a set A is NPSV-selective (NPMV-selective) if there is a 2-ary partial NP function with unique values (a 2-ary partial NP function) that decides which of its inputs (if any) is "more likely" to belong to A; this is a nondeterministic analog of the recursion-theoretic notion of the semirecursive sets and the extant complexity-theoretic notion of P-selectivity. Our hierarchy-collapse result follows by combining the easy observation that every set in NP is NPMV-selective with the following result: If A ∈ NP is NPSV-selective, then A ∈ (NP∩coNP)/poly. Relatedly, we prove that if A ∈ NP is NPSV-selective, then A is Low2. We prove that the polynomial hierarchy collapses even further, namely to NP, if all coNP sets are NPMV-selective. This follows from a more general result we prove: Every self-reducible NPMV-selective set is in NP.

AB - Is there an NP function that, when given a satisfiable formula as input, outputs one satisfying assignment uniquely? That is, can a nondeterministic function cull just one satisfying assignment from a possibly exponentially large collection of assignments? We show that if there is such a nondeterministic function, then the polynomial hierarchy collapses to ZPPNP (and thus, in particular, to NPNP). Because the existence of such a function is known to be equivalent to the statement "every NP function has an NP refinement with unique outputs," our result provides the strongest evidence yet that NP functions cannot be refined. We prove our result via a result of independent interest. We say that a set A is NPSV-selective (NPMV-selective) if there is a 2-ary partial NP function with unique values (a 2-ary partial NP function) that decides which of its inputs (if any) is "more likely" to belong to A; this is a nondeterministic analog of the recursion-theoretic notion of the semirecursive sets and the extant complexity-theoretic notion of P-selectivity. Our hierarchy-collapse result follows by combining the easy observation that every set in NP is NPMV-selective with the following result: If A ∈ NP is NPSV-selective, then A ∈ (NP∩coNP)/poly. Relatedly, we prove that if A ∈ NP is NPSV-selective, then A is Low2. We prove that the polynomial hierarchy collapses even further, namely to NP, if all coNP sets are NPMV-selective. This follows from a more general result we prove: Every self-reducible NPMV-selective set is in NP.

KW - Computational complexity

KW - Lowness

KW - Nonuniform complexity

KW - Semidecision algorithms

UR - http://www.scopus.com/inward/record.url?scp=0001703687&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001703687&partnerID=8YFLogxK

M3 - Article

VL - 25

SP - 697

EP - 708

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 4

ER -