🚀 What is β-Reduction?
β-Reduction is the core transformation process in λ-calculus that replaces bound variables with actual arguments to simplify expressions. This tutorial will help you:
- • Identify reducible expressions (β-redex)
- • Apply multi-step β-reduction
- • Recognize normal forms
- • Implement quantum-safe reductions
;; Classic β-reduction
(λx.x) y → y
🛠 β-Reduction Process
1. Redex Identification
- Locate function-application patterns (λx.M) N
- Verify free variable binding conditions
- Check for nested reduction opportunities
2. Step-by-Step Execution
- Replace x with N in M using substitution
- Preserve λ-expression structure integrity
- Repeat until normal form is achieved
🧮 Transformation Examples
Original Expression
(λx.λy.x x) (λz.z)
After β-Reduction
λy.y (λz.z)
;; Normal form achieved
🧩 Try It Yourself
;; Task: Simplify this expression
(λf.λx.f (f x)) (λy.λz.y)
Challenge Steps
- 1. Find the first β-reducible expression
- 2. Apply substitution carefully
- 3. Continue until normal form