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...

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

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

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

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

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

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...

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

1145/2421119.2421135. S2CID 13367514. Aaronson, Scott (2010). "BQP and the polynomial hierarchy". In Schulman, Leonard J. (ed.). Proceedings of the 42nd ACM Symposium...

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...

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

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...

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...

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...

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...

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...

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

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]...

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

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...

vertex cover. These problems are only expressible in Datalog if the polynomial hierarchy collapses. The DLV (DataLog with Disjunction, where the logical disjunction...

{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...

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

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...

complexity hierarchies: Polynomial hierarchy Exponential hierarchy Chomsky hierarchy Ineffective complexity hierarchies: Arithmetical hierarchy Hyperarithmetical...

that a polynomial-time machine with a #P oracle (P#P) can solve all problems in PH, the entire polynomial hierarchy. In fact, the polynomial-time machine...