Theory of Computation

Theory of Computation is a fundamental branch of computer science that explores the capabilities and limitations of algorithms and computation itself using abstract mathematical models. It addresses three central questions: what are the formal models of computation (automata theory and formal languages), what problems can be solved by a computer in principle (computability theory), and of those solvable problems, how much time and memory do they require (complexity theory). This field provides the theoretical underpinnings for understanding the limits of what computers can do and the inherent difficulty of computational problems.

1. Introduction to Theory of Computation
   1. Historical Context and Motivation
      1. Origins of Computational Theory
      2. Hilbert's Decision Problem
      3. Church-Turing Thesis Historical Development
   2. Core Questions of Computation
      1. What Can Be Computed
         1. Computability Concepts
         2. Decidable vs Undecidable Problems
         3. Effective Procedures
      2. How Efficiently Can Problems Be Solved
         1. Computational Resources
         2. Time Complexity
         3. Space Complexity
         4. Trade-offs Between Resources
      3. Models of Computation
         1. Purpose of Formal Models
         2. Equivalence of Models
         3. Computational Power Hierarchy
   3. Mathematical Foundations
      1. Set Theory
         1. Basic Set Operations
            1. Union
            2. Intersection
            3. Difference
            4. Complement
         2. Set Relations
            1. Subset Relations
            2. Power Sets
            3. Cartesian Products
         3. Infinite Sets
            1. Countable Sets
            2. Uncountable Sets
            3. Diagonalization
      2. Relations and Functions
         1. Binary Relations
            1. Reflexivity
            2. Symmetry
            3. Transitivity
            4. Antisymmetry
         2. Equivalence Relations
            1. Definition and Properties
            2. Equivalence Classes
            3. Partitions
         3. Functions
            1. Injective Functions
            2. Surjective Functions
            3. Bijective Functions
            4. Inverse Functions
            5. Composition of Functions
      3. Graph Theory Basics
         1. Graph Definitions
            1. Vertices and Edges
            2. Directed vs Undirected Graphs
            3. Graph Representation
         2. Paths and Connectivity
            1. Simple Paths
            2. Cycles
            3. Connected Components
         3. Trees
            1. Tree Properties
            2. Rooted Trees
            3. Binary Trees
            4. Tree Traversal
      4. Formal Language Theory Foundations
         1. Alphabets and Strings
            1. Alphabet Definition
            2. String Construction
            3. Empty String
            4. String Length
         2. String Operations
            1. Concatenation
            2. Reversal
            3. Substring Relations
            4. Prefixes and Suffixes
         3. Languages
            1. Language as Set of Strings
            2. Language Operations
            3. Closure Properties
      5. Proof Techniques
         1. Direct Proof
         2. Proof by Contradiction
         3. Mathematical Induction
            1. Base Case
            2. Inductive Step
            3. Strong Induction
         4. Structural Induction
         5. Proof by Construction
         6. Diagonalization Arguments