# Computing solutions uniquely collapses the polynomial hierarchy

Lane A. Hemaspaandra, Ashish V. Naik, Mitsunori Ogihara, Alan L. Selman

Research output: Contribution to journalArticle

39 Citations (Scopus)

### 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 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.

Original language English (US) 697-708 12 SIAM Journal on Computing 25 4 Published - Aug 1996 Yes

### Fingerprint

Polynomial Hierarchy
Polynomials
Computing
Assignment
Partial
Output
Selectivity
Recursion
Refinement
Likely
Analogue

### Keywords

• Computational complexity
• Lowness
• Nonuniform complexity
• Semidecision algorithms

### ASJC Scopus subject areas

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

### Cite this

Hemaspaandra, L. A., Naik, A. V., Ogihara, M., & Selman, A. L. (1996). Computing solutions uniquely collapses the polynomial hierarchy. 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.

In: SIAM Journal on Computing, Vol. 25, No. 4, 08.1996, p. 697-708.

Research output: Contribution to journalArticle

Hemaspaandra, LA, Naik, AV, Ogihara, M & Selman, AL 1996, 'Computing solutions uniquely collapses the polynomial hierarchy', SIAM Journal on Computing, vol. 25, no. 4, pp. 697-708.
Hemaspaandra, Lane A. ; Naik, Ashish V. ; Ogihara, Mitsunori ; Selman, Alan L. / Computing solutions uniquely collapses the polynomial hierarchy. In: SIAM Journal on Computing. 1996 ; Vol. 25, No. 4. pp. 697-708.
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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 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.",
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