In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. Each class in the hierarchy is contained within PSPACE..
What is the polynomial hierarchy of complexity theory?
In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. Each class in the hierarchy is contained within PSPACE..
To really appreciate complexity theory, one should be familiar with the following topics: Regular languages, context-free grammars, and context-free languages. These topics provide the vocabulary for describing problems that complexity theory deals with.
There are often general hierarchies of complexity classes; for example, it is known that a number of fundamental time and space complexity classes relate to Basic complexity classesRelationships between Other models of computation
Complexity theory hierarchy
Hierarchy of complexity classes for formulas defining sets
In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical. The arithmetical hierarchy was invented independently by Kleene (1943) and Mostowski (1946).
In computational complexity theory, the exponential hierarchy is a hierarchy of complexity classes that is an exponential time analogue of the polynomial hierarchy. As elsewhere in complexity theory, “exponential” is used in two different meanings, leading to two versions of the exponential hierarchy. This hierarchy is sometimes also referred to as the weak exponential hierarchy, to differentiate it from the strong exponential hierarchy.
Ordinal-indexed family of rapidly increasing functions: ℕ→ℕ
In computability theory, computational complexity theory and proof theory, a fast-growing hierarchy is an ordinal-indexed family of rapidly increasing functions fα: N → N. A primary example is the Wainer hierarchy, or Löb–Wainer hierarchy, which is an extension to all α < ε0. Such hierarchies provide a natural way to classify computable functions according to rate-of-growth and computational complexity.
Both deterministic and nondeterministic machines can solve more problems given more space
In computational complexity theory, the space hierarchy theorems are separation results that show that both deterministic and nondeterministic machines can solve more problems in (asymptotically) more space, subject to certain conditions. For example, a deterministic Turing machine can solve more decision problems in space n log n than in space n. The somewhat weaker analogous theorems for time are the time hierarchy theorems.
Given more time, a Turing machine can solve more problems
In computational complexity theory, the time hierarchy theorems are important statements about time-bounded computation on Turing machines. Informally, these theorems say that given more time, a Turing machine can solve more problems. For example, there are problems that can be solved with n2 time but not n time.
