Mathematical Foundations
Formal systems and mathematical principles that power the εεΡΑ.λς computation engine.
λ Lambda Calculus Foundations
The core of the system is built on Church's λ-calculus with β-reduction and α-conversion. Implemented as a pure functional system with:
- • Unrestricted β-reduction with capture-avoiding substitution
- • Hindley-Milner type inference
- • Currying and higher-order functions
- • Church encoding of basic data types
{
λx.λy.x y → β y
λf.λx.f x → I (Identity combinator)
λf.λg.λx.f(g x) → Ζ (Z-combinator)
}
Combinator Logic
Combinatory logic implementation includes primitive combinators and derived operators:
- • S: λxyz.x z(y z)
- • K: λxy.x
- • I: λx.x (Derived from SK)
- • B: λxyz.x(y z)
- • C: λxyz.x z y
Combinator reduction visualizer available in advanced mode
Type Theory Implementation
Type system features include:
- • Hindley-Milner style polymorphism
- • Type inference with unification algorithm
- • Algebraic data types
- • Recursive types
- • Subtyping with covariance/contravariance
// Example type inference
λx:Num. x + 3 : Num → Num