Formal Foundations of Computation

Explore the mathematical underpinnings of computation, from lambda calculus to category theory, in interactive visualizations.

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The Lambda Calculus

β-Reduction

The fundamental operation of computation in lambda calculus, defined by the rule:
(λx.M) N → M[x := N]

Reduction in action

Currying

The transformation of functions with multiple arguments into nested single-argument functions:
f(x, y) → f(x)(y)

Functional composition

Fixed Points

The Y combinator enables recursion through the magical property:
Y f = f (Y f)

Recursive patterns

Real-World Applications

Programming Languages

  • Haskell's type system rooted in category theory
  • JavaScript's prototype inheritance modeled as catamorphisms

Machine Learning

  • Neural networks as differentiable lambda expressions
  • Type theory foundations for safe AI systems

Interactive Models

Type Theory Playground

Build and verify type systems using Coq and Agda frameworks

Interactive type checker

Category Visualizer

Explore category theory concepts through real-time visual transformations

Interactive graph editor

λ Interpreter

Write and execute lambda expressions in your browser

Live λ calculus evaluator