Polynomial Hierarchy

The **Polynomial Hierarchy (PH)** is a layered generalization of NP and co-NP that organizes decision problems by how many alternating quantifiers (∃ and ∀) are needed in their logical definitions. Each level, denoted Σₖᴾ or Πₖᴾ, represents problems solvable by a polynomial-time machine with access to an oracle for the previous level. If any two adjacent levels collapse (e.g., Σₖᴾ = Πₖᴾ), the entire hierarchy collapses to that level.