Class Description

βP: Limited-Nondeterminism NP

β_kP is the class of decision problems solvable by a polynomial-time Turing machine that makes O(log^kn) nondeterministic transitions, with the same acceptance mechanism as NP. Equivalently, the machine receives a purported proof of size O(log^kn) that the answer is 'yes.'

Then βP is the union of β_kP over all constant k.

Defined in [KF84]. See also the survey [GLM96].

There exist oracles relative to which basically any consistent inclusion structure among the β_kP's can be realized [BG98].

β₂P contains LOGNP and LOGSNP.

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LOGNP: Logarithmically-Restricted NP

The class of decision problems expressible in logical form as

The set of I for which there exists a subset S={s₁,...,s_{log n}} of {1,...,n} of size log n, such that for all x there exists y such that for all j in S, the predicate φ(I,s_j,x,y,j) holds. Here x and y are logarithmic-length strings, or equivalently polynomially bounded numbers, and φ is computable in P.

LOGNP₀ is the subclass in which φ is a first-order predicate without quantifiers and x and y are bounded lists of indices of input bits. LOGNP is also the closure of LOGNP₀ under polynomial-time many-one reductions.

The motivation is that the analogue of LOGNP₀ without the logarithmic bound on |S| is SO-E, which by Fagin's theorem equals NP [Fag74].

Defined in [PY96], where it was also shown that the following problem is complete for LOGNP under many-one reductions:

Vapnik-Chervonenkis (V-C) Dimension. Given a family F of subsets of a set U, find a subset of S of U of maximum cardinality, such that every subset of S can be written as the intersection of S with some set in F.

Contains LOGSNP, and is contained in βP (indeed β₂P).

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Σ₂P: NP With NP Oracle

Contains languages whose complements are in Π₂P.

Along with Π₂P, comprises the second level of PH, the polynomial hierarchy.

Note that this is equal to NP with a coNP oracle.

[Uma98] has shown that the following problems are complete for Σ₂P:

• Minimum equivalent DNF. Given a DNF formula F and integer k, is there a DNF formula equivalent to F with k or fewer occurences of literals?
• Shortest implicant. Given a formula F and integer k, is there a conjunction of k or fewer literals that implies F? (Note that this problem cannot be Σ₂P-complete for DNF formulas unless Σ₂P equals βP^NP.)

The problem of deciding if a perfect graph is 2-clique-colorable (defined in [FMF16]) has been shown to be complete for Σ₂P.

For any fixed k, there is a problem in Σ₂P ∩ Π₂P that cannot be solved by circuits of size n^k [Kan82].

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