Automata Theory Questions Long
The concept of the polynomial hierarchy (PH) is a fundamental idea in computational complexity theory that helps classify and understand the complexity of decision problems. It provides a hierarchy of complexity classes that extends beyond the well-known classes such as P, NP, and co-NP.
To understand the polynomial hierarchy, we first need to define the concept of an oracle machine. An oracle machine is a theoretical model of computation that has access to an oracle, which is an additional black box that can solve a specific decision problem in a single step. The oracle machine can query the oracle with a specific input and receive an answer in constant time.
Now, let's define the polynomial hierarchy. The polynomial hierarchy is a hierarchy of complexity classes that is built using the concept of oracle machines. The classes in the polynomial hierarchy are denoted as ΣiP and ΠiP, where i is a non-negative integer.
The class ΣiP represents the set of decision problems that can be solved by a non-deterministic Turing machine with an oracle for a problem in Σi-1P. In other words, a problem is in ΣiP if there exists a non-deterministic Turing machine that can solve it by making a polynomial number of queries to an oracle for a problem in Σi-1P.
Similarly, the class ΠiP represents the set of decision problems that can be solved by a non-deterministic Turing machine with an oracle for a problem in Πi-1P. A problem is in ΠiP if there exists a non-deterministic Turing machine that can solve it by making a polynomial number of queries to an oracle for a problem in Πi-1P.
The polynomial hierarchy is defined as the union of all the classes in the hierarchy, i.e., PH = ΣiP ∪ ΠiP for all non-negative integers i.
The polynomial hierarchy provides a way to classify decision problems based on their complexity. As i increases, the problems in ΣiP and ΠiP become more complex. The class Σ0P corresponds to the class P, which contains decision problems that can be solved in polynomial time by a deterministic Turing machine. The class Π0P corresponds to the class co-NP, which contains decision problems for which the complement can be solved in polynomial time.
The polynomial hierarchy extends beyond P and NP by introducing additional levels of complexity. For example, Σ1P contains decision problems that can be solved in polynomial time by a non-deterministic Turing machine with access to an NP oracle. Similarly, Π1P contains decision problems that can be solved in polynomial time by a non-deterministic Turing machine with access to a co-NP oracle.
The polynomial hierarchy continues to higher levels, with each level representing an increase in complexity. It is still an open question whether the polynomial hierarchy collapses to a finite level or extends infinitely.
In summary, the polynomial hierarchy is a hierarchy of complexity classes that extends beyond P and NP. It provides a way to classify decision problems based on their complexity and their relationship to other complexity classes. The polynomial hierarchy helps us understand the inherent difficulty of problems and the resources required to solve them.