Mathematical Logic

1. Computability Theory
   1. Recursive Functions
      1. Primitive Recursive Functions
         1. Definition and Properties
            1. Closure Properties
            2. Base Functions
               1. Zero Function
               2. Successor Function
               3. Projection Functions
            3. Constructing New Functions
               1. Composition
               2. Primitive Recursion Schema
            4. Examples
               1. Addition
               2. Multiplication
               3. Factorial Function
         2. Limitations
            1. Non-expressibility of Certain Functions
            2. Comparison with General Recursive Functions
      2. μ-Recursive Functions
         1. Definition and Characteristics
            1. Inclusion of Minimization
            2. μ-Operator and its Role
         2. Comparison with Primitive Recursive Functions
            1. Extension by Minimization
            2. Expressiveness and Power
         3. Examples
            1. Computable Functions beyond Primitive
            2. Solving Equations through Search
         4. Equivalence to Turing Computability
            1. Church-Turing Thesis
            2. Constructive Proofs
   2. Turing Machines
      1. Definition and Components
         1. Infinite Tape Concept
         2. State Machine Model
         3. Alphabet and Symbols
         4. Transition Functions
      2. Types of Turing Machines
         1. Deterministic Turing Machines
         2. Non-deterministic Turing Machines
            1. Equivalence in Power to Deterministic
         3. Universal Turing Machines
            1. Concept of Universality
            2. Simulation of Other Turing Machines
      3. Decidability
         1. Definition of Decidable Problems
         2. Decision Problem Examples
            1. Halting Problem
            2. Membership Problem
         3. Methods for Proving Decidability
            1. Reductions
            2. Recursive Algorithms
      4. Undecidability
         1. Definition and Recognition of Problems
            1. Non-computable Functions
            2. Rice's Theorem
            3. Post Correspondence Problem
         2. Implications of Undecidability
            1. Limits of Algorithmic Solutions
            2. Impact on Formal Systems
   3. Complexity Theory
      1. Theoretical Foundations
         1. Definition of Complexity Classes
         2. Measuring Computational Complexity
            1. Time Complexity
            2. Space Complexity
      2. P vs NP Problem
         1. P Class
            1. Polynomial-Time Solvable Problems
            2. Examples and Relevance
         2. NP Class
            1. Non-deterministic Polynomial-Time
            2. Definition by Verification
         3. NP-Complete Problems
            1. Cook-Levin Theorem
            2. Common Problems in NP-Complete
         4. Implications and Significance
            1. Computational Intractability
            2. Open Questions in the Field
      3. Complexity Class Hierarchies
         1. P, NP, and co-NP
         2. Polynomial Hierarchy
         3. Exploration of PSPACE and EXPTIME
      4. Reduction Techniques
         1. Polynomial-time Reductions
         2. Use in Proving Complexity Class Membership
         3. Concepts of Completeness and Hardness