Polynomial Hierarchy Article Index for
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Polynomial Hierarchy
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DEFINITIONS

There are multiple equivalent definitions of the classes of the polynomial hierarchy.



  1. For the oracle definition of the polynomial hierarchy, define

    :\Delta_0^{ m P} := \Sigma_0^{ m P} := \Pi_0^{ m P} := \mbox{P},

    where P is the set of Decision Problem s solvable in Polynomial Time . Then for i ≥ 0 define

    :\Delta_{i+1}^{ m P} := \mbox{P}^{\Sigma_i^{ m P}}
    :\Sigma_{i+1}^{ m P} := \mbox{NP}^{\Sigma_i^{ m P}}
    :\Pi_{i+1}^{ m P} := \mbox{coNP}^{\Sigma_i^{ m P}}

    where AB is the set of Decision Problem s solvable by a Turing Machine in class A augmented by an Oracle for some problem in class B. For example, \Sigma_1^{ m P} = { m NP}, \Pi_1^{ m P} = { m coNP} , and \Delta_2^{ m P} = { m P^{NP}} is the class of problems solvable in polynomial time with an oracle for some problem in NP.




  : <math> Orall^p L : \left\{ x \in \{0,1\}^ \ \left \ \left( orall w \in \{0,1\}^{\leq p(x)} ight) \langle x,w angle \in L ight ight\} </math>
  :<math>\exists^{ M P} \mathcal{C} : \left\{\exists^p L \ \ p \mbox{ is a polynomial and } L \in \mathcal{C} ight\}</math>
  :<math> Orall^{ M P} \mathcal{C} : \left\{ orall^p L \ \ p \mbox{ is a polynomial and } L \in \mathcal{C} ight\}</math>
  Note That DeMorgan's Laws Hold: <math> { M Co} \exists^{ M P} \mathcal{C} orall^{ m P} { m co} \mathcal{C} </math> and <math> { m co} orall^{ m P} \mathcal{C} = \exists^{ m P} { m co} \mathcal{C} </math>, where <math>{ m co}\mathcal{C} = \left\{ L^c L \in \mathcal{C} ight\}</math>




PROBLEMS IN POLYNOMIAL HIERARCHY