computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize...

computer science In computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems. NP is...

as Hard as the Polynomial-Time Hierarchy" and was given the 1998 Gödel Prize. The theorem states that the entire polynomial hierarchy PH is contained...

or any language in the polynomial hierarchy. ∃ R {\displaystyle \exists \mathbb {R} } is the set of problems having a polynomial-time many-one reduction...

Sampling is a problem that exists within BQP, but not within the polynomial hierarchy. Recent conjectures have provided evidence that a similar problem...

them. For example, PH, the union of all complexity classes in the polynomial hierarchy, is precisely the class of languages expressible by statements of...

the arithmetical hierarchy. Analytical hierarchy Lévy hierarchy Hierarchy (mathematics) Interpretability logic Polynomial hierarchy P. G. Hinman, Recursion-Theoretic...

NP-complete, the polynomial time hierarchy collapses to its second level. Since it is widely believed that the polynomial hierarchy does not collapse...

bounded-error probabilistic polynomial time (BPP) is the class of decision problems solvable by a probabilistic Turing machine in polynomial time with an error...

complexity class P/poly, then this assumption implies the collapse of the polynomial hierarchy at its second level. Such a collapse is believed unlikely, so the...

existence of a short (polynomial length) solution. The correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force...

theory, the exponential hierarchy is a hierarchy of complexity classes that is an exponential time analogue of the polynomial hierarchy. As elsewhere in complexity...

machine in polynomial time, with an error probability of less than 1/2 for all instances. The abbreviation PP refers to probabilistic polynomial time. The...

for FPNP (see the function problem article) or for FΔ2P (see the polynomial hierarchy article). An example of an NP-easy problem is the problem of sorting...

practical problems, including all of BPP. If it contains NP, then the polynomial hierarchy collapses to the second level. On the other hand, it also contains...

NP-complete, the polynomial time hierarchy collapses to its second level. Since it is widely believed that the polynomial hierarchy does not collapse...

over NP predicates. A collapse of the boolean hierarchy would imply a collapse of the polynomial hierarchy. BH is defined as follows: BH1 is NP. BH2k is...

level of the polynomial hierarchy When combined these two facts along with Toda's theorem result in the collapse of the polynomial hierarchy, which as mentioned...

being equal to NP and on a more far-reaching conjecture that the polynomial time hierarchy of complexity classes is infinite. The compression theorem is...

BQP still remain, such as the connection between BQP and the polynomial-time hierarchy, whether or not BQP contains NP-complete problems, and the exact...

coefficients in certain polynomials, those polynomials should have the polynomial SOS property. When fixing the maximum degree of the polynomials involved, sum-of-squares...

hierarchy is a hierarchy contained in NP, the A hierarchy more closely mimics the polynomial-time hierarchy from classical complexity. It is known that A[1]...

{NP}}} . P/poly is also helpful in investigating properties of the polynomial hierarchy. For example, if NP ⊆ P/poly, then PH collapses to Σ 2 P {\displaystyle...

Arthur can validate deterministically in polynomial time. Both MA and AM are contained in the polynomial hierarchy. In particular, MA is contained in the...

It is however known that if the problem is NP-complete then the polynomial hierarchy collapses to a finite level. In November 2015, László Babai, a mathematician...

second levels of the polynomial hierarchy. A language L is in S 2 P {\displaystyle {\mathsf {S}}_{2}^{P}} if there exists a polynomial-time predicate P such...

believed they are different, and this leads to the definition of the polynomial hierarchy. Oracle machines are useful for investigating the relationship between...

_{n}^{1}} , sets in the analytical hierarchy Σ i P {\displaystyle \Sigma _{i}^{P}} , a set in the polynomial hierarchy Harish-Chandra's σ function Weierstrass...

Δ i P {\displaystyle \Delta _{i}^{P}} , a classification in the polynomial hierarchy δ13C, a measure of the ratio of stable isotopes 13C:12C δ15N, a measure...

1 edition, ISBN 978-0-521-42426-4 "S2P: Second Level of the Symmetric Hierarchy". Stanford University Complexity Zoo. Archived from the original on 2012-10-14...