Does the polynomial hierarchy collapse?
An interesting property of the polynomial hierarchy is that if any two classes in the hierarchy are equal, then the hierarchy “collapses.” The following theorem makes this statement precise: Theorem 3 If Σi = Πi, then PH = Σi..
What is a polynomial complexity?
Definition.
An algorithm is said to have polynomial time complexity if its worst-case running time Tworst(n) T worst ( n ) for an input of size n is upper bounded by a polynomial p(n) for large enough n≥n0 n ≥ n 0 ..
What is a polynomial reduction in complexity theory?
A polynomial-time reduction is a way to reduce the task of solving one problem to another.
The way we use reductions in complexity is to argue that if the first problem is hard to solve efficiently, then the second must also be hard..
What is the complexity class in polynomial time?
In computational complexity theory, P, also known as PTIME or DTIME(n), is a fundamental complexity class.
It contains all decision problems that can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time..
What is the complexity of a polynomial function?
Typically, the complexity of a polynomial is measured by its degree, but the complexity could also be the number of monomials.
The survey by Beigel [8] contains many references to papers in which low-complexity circuits are represented via low-complexity polynomials, resulting in lower bounds against those circuits..
What is the complexity of a polynomial?
Typically, the complexity of a polynomial is measured by its degree, but the complexity could also be the number of monomials..
What is the polynomial-time hierarchy theory?
The polynomial-time hierarchy is that subrecursive analog of the Kleene arithmetical hierarchy in which deterministic (nondeterministic) polynomial time plays the role of recursive (recursively enumerable) time.
Known properties of the polynomial-time hierarchy are summarized..
- A polynomial-time reduction is a way to reduce the task of solving one problem to another.
The way we use reductions in complexity is to argue that if the first problem is hard to solve efficiently, then the second must also be hard. - By Wiki, the polynomial hierarchy is the generalization of the classes P, NP, and co-NP to oracle machines.
The addition of oracles (not sure this is a suitable expression) results in the increase of the hierarchy.
On the other hand, another way to define the polynomial hierarchy is to use the quantifiers ∀ and ∃. - polynomial space A way of characterizing the complexity of an algorithm.
If the space complexity (see complexity measure) is polynomially bounded, the algorithm is said to be executable in polynomial space. - Polynomial time complexity refers to a class of problems such that each problem has a polynomial which acts as a bound for that problem's time complexity.
Non- polynomial time complexity refers to any problem whose complexity cannot be bounded by any polynomial.
