Computational Complexity Theory

Computational Complexity Theory is a central field of computer science that classifies computational problems according to their inherent difficulty and the resources required to solve them. Rather than analyzing the performance of a specific algorithm, this theory seeks to understand the minimum amount of resources—primarily time (computation steps) and space (memory)—that any algorithm would need to solve a particular problem, measured as a function of the input size. It formally defines and studies complexity classes, such as P (problems solvable in polynomial time, considered "tractable") and NP (problems whose solutions can be verified in polynomial time), in order to categorize problems and understand the fundamental limits of computation, with the P versus NP problem being its most famous unsolved question.

1. Introduction to Computational Complexity Theory
   1. Core Questions and Motivation
      1. Defining Computational Problems
      2. Computational Hardness
      3. Resource Requirements Classification
      4. Problems versus Algorithms
      5. Efficiency and Feasibility Measures
   2. Historical Development
      1. Early Computability Theory
         1. Turing's Computable Numbers
         2. Church's Lambda Calculus
         3. Gödel's Incompleteness Theorems
      2. Emergence of Complexity Theory
         1. Decision Problems Focus
         2. Computability to Efficiency Transition
         3. Foundational Milestones
   3. Connections to Related Fields
      1. Computability Theory
      2. Algorithm Design and Analysis
      3. Cryptography
      4. Artificial Intelligence
      5. Operations Research
      6. Mathematical Logic