Class Description

LOGSNP: Logarithmically-Restricted SNP

The class of decision problems expressible in logical form as

LOGSNP0 is the subclass in which φ is a first-order predicate without quantifiers and x is a bounded lists of indices of input bits. LOGSNP is also the closure of LOGSNP0 under polynomial-time many-one reductions. See LOGNP and SNP for the motivation.

Defined in [PY96].

Contained in LOGNP, and therefore QPLIN.

If P = LOGSNP, then for every constructible f(n) > n, NTIME(f(n)) is contained in DTIME(g(n)sqrt(g(n))), where g(n) = O(f(n) logf(n)) [FK97].

Linked From

βP: Limited-Nondeterminism NP

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

Then βP is the union of βkP 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 βkP'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

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:

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

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