Computational Models

Understanding what is computable and how efficiently is fundamental to AGI. This page explores the theoretical models that define the boundaries and capabilities of computation.

Turing Machines

Definition and Variants

• Standard Turing Machine: Tape, head, state machine
• Universal Turing Machine: Interpreters and self-reference
• Non-deterministic Turing Machines: Exploring multiple computation paths
• Oracle Machines: Computation beyond current capabilities

Church-Turing Thesis

The limits of what can be computed algorithmically. Understanding uncomputability and its implications for AGI.

Lambda Calculus

Core Concepts

• Functions as First-Class Citizens: Foundation of functional programming
• Beta Reduction: Computation as term rewriting
• Fixed Points and Recursion: Y-combinator and self-reference
• Typed Lambda Calculus: Type systems and proof theory

Relevance to AGI

• Functional approaches to AI (LISP, Scheme, Haskell)
• Higher-order reasoning
• Program synthesis
• MeTTa language foundations

Computational Complexity

Complexity Classes

• P vs NP: Practical vs verifiable problems
• NP-Complete Problems: Hardest problems in NP (SAT, graph coloring)
• PSPACE: Planning and game-playing complexity
• Approximation Algorithms: Trading optimality for efficiency

Resource Bounds

• Time Complexity: Big-O notation, asymptotic analysis
• Space Complexity: Memory requirements
• Communication Complexity: Distributed systems
• Circuit Complexity: Hardware considerations

Applications to AGI

• Understanding tractability of reasoning problems
• When to use heuristics vs exact methods
• Resource allocation in cognitive architectures
• Complexity of learning

Automata Theory

Finite State Machines

• Deterministic and non-deterministic automata
• Regular languages and expressions
• State-based agent modeling

Pushdown Automata

• Context-free languages
• Parsing and language understanding
• Hierarchical planning

Computational Logic

Propositional Logic

• SAT solvers
• Boolean satisfiability in planning
• Logic minimization

First-Order Logic

• Quantifiers and predicates
• Resolution and unification
• Automated theorem proving
• Prolog and logic programming

Higher-Order Logic

• Types and polymorphism
• Dependent types
• Proof assistants (Coq, Lean)

Kolmogorov Complexity

Algorithmic Information Theory

• Minimal Description Length: Simplicity as a principle
• Incompressibility: Random strings and information content
• Solomonoff Induction: Universal prediction

Applications

• Occam's razor in model selection
• Compression and generalization
• Minimum description length learning

Quantum Computing Basics

Quantum vs Classical

• Superposition and entanglement
• Quantum gates and circuits
• Grover's and Shor's algorithms

Potential for AGI

• Quantum machine learning
• Speed advantages for specific problems
• Theoretical limits

Recommended Resources

• Introduction to the Theory of Computation - Michael Sipser
• Computational Complexity - Christos Papadimitriou
• The Lambda Calculus - Barendregt
• Algorithmic Information Theory - Gregory Chaitin

---

Next: Knowledge Representation