Artificial Intelligence with PHP
  • Getting Started
    • Introduction
    • Audience
    • How to Read This Book
    • Glossary
    • Contributors
    • Resources
  • Artificial Intelligence
    • Introduction
    • Overview of AI
      • History of AI
      • Structure of AI
      • Types of AI
        • Reactive AI
        • Limited Memory AI
        • Theory of Mind AI
        • Self-Aware AI
      • How Does AI Work?
      • Will AI Take Over the World?
    • AI Capabilities in PHP
      • Overview of AI Libraries in PHP
      • Introduction to LLM Agents PHP SDK
    • Theoretical Foundations of AI
      • Introduction to Theoretical Foundations of AI
      • Problem Solving in AI
        • Introduction
        • Types of Search Algorithms
          • Uninformed (Blind) Search
            • Global Search
              • Breadth-First Search (BFS)
              • Depth-First Search (DFS)
              • Uniform Cost Search (UCS)
              • Iterative Deepening Depth-First Search (IDDFS)
              • Bidirectional Search (BDS)
            • Local Search
              • Depth-Limited Search (DLS)
              • Random Walk Search (RWS)
          • Informed (Heuristic) Search
            • Global Search
              • Greedy Search
              • A* Search
                • A* Tree Search
                • A* Graph Search
                • A* Graph vs A* Tree Search
              • Iterative Deepening A* Search
              • Beam Search
            • Local Search
              • Hill Climbing Search
                • Introduction
                • Implementation in PHP
                • Practical Applications
                • Best Practices and Optimization
              • Simulated Annealing Search
              • Local Beam Search
              • Genetic Algorithms Search
              • Tabu Search
          • Comparison of Search Algorithms
        • Adversarial Search
        • Means-Ends Analysis
      • Knowledge & Uncertainty in AI
        • Knowledge-Based Agents
        • Knowledge Representation
          • Introduction
          • Types of Knowledge in AI
          • The KR Cycle in AI
          • KR Techniques
          • Approaches to KR in AI
          • Propositional Logic
          • First-Order Logic
        • Reasoning in AI
        • Uncertain Knowledge Representation
        • The Wumpus World
        • Applications and Challenges
      • Cybernetics and AI
      • Philosophical and Ethical Foundations of AI
    • Mathematics for AI
      • Computational Theory in AI
      • Logic and Reasoning
      • Set Theory and Discrete Mathematics
      • Decision Making & Game Theory in AI
    • AI Agents
      • Introduction to AI Agents
      • Structure of AI Agent
      • Components of AI Agents
      • Types of AI Agents
      • AI Agent Environment
      • AI Agent Architecture
      • Application of Agents in AI
      • LLM AI Agents
        • Introduction to LLM AI Agents
        • Implementation in PHP
      • Challenges in AI Agent Development
      • Turing Test in AI
      • Future of AI Agents
    • Key Application of AI
      • AI in Astronomy
      • AI in Healthcare
      • AI in Gaming
      • AI in Finance
      • AI in Data Security
      • AI in Social Media
      • AI in Travel and Transport
      • AI in Automotive Industry
      • AI in Robotics
      • AI in Entertainment
      • AI in Agriculture
      • AI in E-commerce
      • AI in Education
      • AI in Software Development
      • AI in Dating
      • AI in Adult Entertainment
      • AI in Criminal Justice
      • AI in Criminal World
      • AI in Terrorist Activities
      • AI in Military Domain
      • AI in Transforming our World
    • Practice
  • Machine Learning
    • Introduction
    • Overview of ML
      • History of ML
        • Origins and Early Concepts
        • 19th Century
        • 20th Century
        • 21st Century
        • Coming Years
      • Key Principles
      • Problems and Challenges
      • Machine Learning Life Cycle
    • ML Capabilities in PHP
      • Overview of ML Libraries in PHP
      • Configuring an Environment for PHP
        • Direct Installation
        • Using Docker
        • Additional Notes
      • Introduction to PHP-ML
      • Introduction to Rubix ML
    • Mathematics for ML
      • Linear Algebra
        • Scalars
          • Definition and Operations
          • Scalars with PHP
        • Vectors
          • Definition and Operations
          • Vectors in Machine Learning
          • Vectors with PHP
        • Matrices
          • Definition and Types
          • Matrix Operations
          • Determinant of a Matrix
          • Cofactor Matrices
          • Adjugate Matrices
          • Inverse Matrices
          • Matrices in Machine Learning
          • Matrices with PHP
        • Tensors
          • Definition of Tensors
          • Tensor Properties
            • Tensor Dimension
            • Tensor Rank
            • Tensor Shape
            • Tensor Types
          • Tensor Operations
          • Practical Applications
          • Tensors in Machine Learning
          • Tensors with PHP
        • Linear Transformations
          • Introduction
          • LT Role in Neural Networks
          • LT with PHP
        • Eigenvalues and Eigenvectors
        • Norms and Distances
        • Linear Algebra in Optimization
      • Calculus
      • Probability and Statistics
      • Information Theory
      • Optimization Techniques
      • Graph Theory and Networks
      • Discrete Mathematics and Combinatorics
      • Advanced Topics
    • Data Fundamentals
      • General Data Processing
        • Introduction
        • Data Types and Formats
          • Data Types
          • Structured Data Formats
          • Unstructured Data Formats
        • Storage and Management
          • Database Systems for AI
          • Efficient Data Storage Techniques
          • Big Data Considerations
            • Introduction
            • Big Data Techniques in PHP
          • Data Security and Privacy
          • Optimizing Data Retrieval for AI Algorithms
          • Data Serialization and Deserialization in PHP
          • Data Versioning and Management
      • ML Data Processing
        • Introduction
        • Types of Data in ML
        • Stages of Data Processing
          • Data Acquisition
            • Data Collection
            • Ethical Considerations in Data Preparation
          • Data Cleaning
            • Data Cleaning Types
            • Data Cleaning Examples
            • Implementation with PHP
          • Data Transformation
            • Data Transformation Types
            • Data Transformation Examples
            • Implementation with PHP ?..
          • Data Integration
          • Data Reduction
          • Data Validation and Testing
            • Data Splitting and Sampling
          • Data Representation
            • Data Structures in PHP
            • Data Visualization Techniques
    • ML Algorithms
      • Classification of ML Algorithms
        • By Learning Types
        • By Model Depth
        • By Tasks Resolved
        • By Methods Used
        • By Feature Types
      • Supervised Learning
        • Regression
          • Linear Regression
            • Types of Linear Regression ?..
            • Finding Best Fit Line
            • Gradient Descent
            • Assumptions of Linear Regression
            • Evaluation Metrics for Linear Regression
            • How It Works by Math
            • Implementation in PHP
              • Simple Linear Regression
              • Multiple Linear Regression
              • Polynomial Regression
        • Classification
        • Recommendation Systems
          • User-Based Collaborative Filtering
          • Matrix Factorization
      • Unsupervised Learning
        • Clustering
        • Dimension Reduction
        • Search and Optimization
        • Recommendation Systems
          • Item-Based Collaborative Filtering
          • Popularity-Based Recommendations
      • Semi-Supervised Learning
        • Regression
        • Classification
        • Clustering
      • Reinforcement Learning
      • Distributed Learning
    • Integrating ML into Web
      • Open-Source Projects
      • Introduction to EasyAI-PHP
    • Key Applications of ML
    • Practice
  • Neural Networks
    • Introduction
    • Overview of NN
      • History of NN
      • Basic Components of NN
        • Neurons
        • Layers
        • Connections and Weights ?..
      • Problems and Challenges
      • How NN Works
    • NN Capabilities in PHP
    • Mathematics for NN
    • NN Types
      • Classification of NN Types
      • Basic NN
        • Simple Perceptron
        • Implementation in PHP
          • Simple Perceptron with Pure PHP
          • Simple Perceptron with Libraries
      • NN with Hidden Layers
      • Deep Learning
      • Convolutional Neural Networks (CNN)
      • Recurrent Neural Networks (RNN)
      • Bayesian Neural Networks
    • Integrating NN into Web
    • Key Applications of NN
    • Practice
  • Natural Language Processing
    • Introduction
    • Overview of NLP
      • History of NLP
        • Ancient Times
        • Medieval Period
        • 15th-16th Century
        • 17th-18th Century
        • 19th Century
        • 20th Century
        • 21st Century
        • Coming Years
      • Key Concepts in NLP
      • Machine Learning Role in NLP
      • Common Challenges in NLP
    • NLP Capabilities in PHP
      • Overview of NLP Libraries in PHP
      • Challenges in NLP with PHP
    • Mathematics for NLP
    • NLP Techniques
      • Techniques and Algorithms in NLP
        • Basic NLP Techniques
        • Advanced NLP Techniques
      • NLP Workflow
      • Popular Tools and Frameworks for NLP
      • Basic Text Processing with PHP
      • Advanced NLP Topics
    • Integrating NLP into Web
    • Key Applications of NLP
    • Practice
  • Computer Vision
    • Introduction
  • Robotics
    • Introduction
  • Cognitive Computing
    • Introduction
  • Expert Systems
    • Introduction
  • AI Ethics and Safety
    • Introduction
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  • What is Theoretical AI?
  • Historical Evolution
  • Key Theoretical Concepts
  1. Artificial Intelligence
  2. Theoretical Foundations of AI

Introduction to Theoretical Foundations of AI

PreviousTheoretical Foundations of AINextProblem Solving in AI

Last updated 26 days ago

Artificial Intelligence is built on a foundation of theoretical principles that define how machines learn, reason, and solve problems. Theoretical AI focuses on the mathematical and logical frameworks that guide the development of intelligent systems. By understanding these foundations, we can create AI that is more efficient, adaptable, and capable of handling complex tasks.

What is Theoretical AI?

Theoretical AI explores the fundamental questions of intelligence—both human and artificial. It involves developing algorithms, models, and formal methods that allow machines to process information, make decisions, and learn from data. Unlike applied AI, which focuses on real-world applications, theoretical AI provides the blueprint for building smarter and more reliable systems.

Historical Evolution

The study of AI has evolved over decades, beginning with early symbolic reasoning in the 1950s, where logic-based approaches dominated. In the 1980s, expert systems and rule-based AI gained popularity, followed by breakthroughs in machine learning in the 2000s. Today, deep learning and neural networks have revolutionized AI, but theoretical research continues to refine these models, improving their efficiency and understanding their limitations.

Key Theoretical Concepts

Theoretical AI is grounded in several core concepts. Search algorithms help AI navigate problem spaces efficiently, while knowledge representation enables machines to store and process information. Uncertainty and probabilistic reasoning allow AI to make informed decisions even with incomplete data. Optimization techniques improve performance, ensuring AI systems function effectively in dynamic environments.

By studying these principles, we gain deeper insights into AI’s capabilities and constraints. This chapter provides the foundation for exploring advanced AI topics, equipping you with the knowledge needed to understand and develop intelligent systems.

Introduction to Theoretical Foundations of AI