ελββΩΩΑ Documentation
Comprehensive guide to the epsilon-beta-omega formal system and its implementations.
1. Core Concepts
Epsilon Verification
Epsilon normalization guarantees completeness in lambda-calculus expressions through type-theoretic validation. Ensures β-reductions maintain Ω-consistency.
Omega Normalization
Final form of expressions in Ω-closed systems. Enables distributed computation through canonical form preservation.
2. Syntax Reference
Basic Reductions
λx.ε(β(x))
→ ω-reduction:
→ Ω-normal form
Type Normalization
Ω: (λx:ε → β(x))
≡ ε: (λx:β)
→ Ω-normalized
3. Implementation
Step-by-Step Setup
- Clone the repository:
git clone https://github.com/ελββΩΩΑ - Install dependencies:
npm install epsilon-core omega-runtime - Run verifier:
ε-verify --input=λ-expression --output=Ω-form
4. API Reference
ε-Verify
POST /api/verify
{
"expression": "λx.β(x)"
}
Returns Ω-normal form
β-Resolver
GET /api/resolve?term=λx.y
Returns reduction steps
5. Validation Methods
Completeness Check
Ensures all reductions terminate in finite steps using ordinal recursion limits.
Consistency Proof
Proves β-reduction paths remain within Ω-normal bounds.
Concurrency
Distributed verification across multiple ω-nodes.
6. Interactive Demo
λx.λy.ε(β(x) ⨁ β(y))
→ ω-reduction:
→ λx.λy.Ω(x) * Ω(y)
Apply epsilon verification to a beta-OR function
Run Interactive Example →