Computational Complexity Theory

  1. Polynomial Hierarchy
    1. Oracle Computation
      1. Oracle Turing Machines
        1. Definition and Operation
        2. Oracle Access Models
          1. Query Complexity
          2. Polynomial Hierarchy Definition
            1. Alternating Quantifier Characterization
              1. Oracle-Based Definition
                1. Level Structure
                  1. Σₖᴾ Classes
                    1. Πₖᴾ Classes
                      1. Δₖᴾ Classes
                      2. Relationships Between Levels
                      3. Low-Level Hierarchy
                        1. Level 0: P
                          1. Level 1: NP and co-NP
                            1. Level 2 and Beyond
                            2. Hierarchy Collapse
                              1. Collapse Conditions
                                1. P = NP Implications
                                  1. Partial Collapse Scenarios
                                  2. Complete Problems
                                    1. Σₖᴾ-Complete Problems
                                      1. Πₖᴾ-Complete Problems
                                        1. Construction Techniques

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                                      8. Hierarchy Theorems

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                                      10. Randomized Complexity