Type Theory

Type theory is a formal system for specifying and studying type systems, providing foundations for programming languages, constructive mathematics, and formal verification.

• Types:
  • Basic types
  • Type constructors
  • Type judgments
  • Type rules

• Untyped:

  • Lambda abstraction
  • Beta reduction
  • Church encodings
  • Fixed points

• Typed:

  • Simply typed lambda calculus
  • Type inference
  • Normalization
  • Curry-Howard correspondence

• Features:
  • Function types
  • Product types
  • Sum types
  • Unit and void types

• Varieties:
  • Parametric polymorphism
  • Ad-hoc polymorphism
  • Subtype polymorphism
  • Row polymorphism

• Concepts:
  • Dependent functions (Π-types)
  • Dependent pairs (Σ-types)
  • Identity types
  • Universes

• Features:
  • Type operators
  • Kind system
  • Higher-kinded types
  • Type-level programming

• Properties:
  • Resource tracking
  • Uniqueness typing
  • Session types
  • Quantum computing applications

• Components:
  • Effect annotations
  • Effect inference
  • Handler systems
  • Algebraic effects

• Features:
  • Tactics language
  • Extraction
  • Program verification
  • Mathematical proofs

• Characteristics:
  • Dependent types
  • Pattern matching
  • Inductive families
  • Universe polymorphism

• Capabilities:
  • Mathematical library
  • Metaprogramming
  • Automation
  • Community packages

• Implementation:
  • Type checkers
  • Type inference
  • Subtyping
  • Generic programming

• Methods:
  • Program correctness
  • Protocol verification
  • Security properties
  • Hardware verification

• Areas:
  • Constructive mathematics
  • Homotopy type theory
  • Category theory
  • Foundations

• GHC: Haskell compiler
• OCaml: ML family language
• Rust: Systems programming
• TypeScript: Typed JavaScript

• LSP: Language Server Protocol
• Tree-sitter: Parsing toolkit
• Z3: SMT solver
• Metamath: Formal math

• "Types and Programming Languages" (Benjamin Pierce)
• "Practical Foundations for Programming Languages" (Robert Harper)
• "Homotopy Type Theory" (HoTT Book)
• "Software Foundations" (Benjamin Pierce et al.)

• Oregon Programming Languages Summer School
• Type Theory Foundations
• Certified Programming with Dependent Types
• Programming Language Foundations in Agda

• Try Agda
• Learn TT
• Coq Interactive Tutorial
• Lean Game

• Homotopy Type Theory
• Cubical Type Theory
• Linear Type Systems
• Gradual Typing
• Effect Systems

• Smart contracts
• Quantum computing
• Machine learning
• Security verification
• Distributed systems

1. Safety properties
2. Expressiveness
3. Inference capabilities
4. Implementation complexity
5. User experience

• Bidirectional typing
• Constraint solving
• Modular design
• Performance optimization
• Error reporting

• Verified compilation
• Automated reasoning
• Program synthesis
• Type-based optimization
• Formal mathematics

• Higher-dimensional type theory
• Computational effects
• Refinement types
• Dependent linear types
• Modal type theory

• Types Working Group
• LFCS
• Research groups
• Conferences

• Types Forum
• Coq Users
• Agda Mailing List
• Lean Zulip

• Journal of Functional Programming
• Logical Methods in Computer Science
• TYPES Conference
• POPL Conference
• ICFP Conference