- Untyped λ Calculus
- Simply Typed λ Calculus
Term with non-terminating sequence of reductions:
(λ x . x x) (λ y . y y)
↝ [x↦(λ y . y y)](x x)
= (λ y . y y) (λ y . y y)
↝ [y↦(λ y . y y)](y y)
= (λ y . y y) (λ y . y y)
↝ [y↦(λ y . y y)](y y)
= (λ y . y y) (λ y . y y)
↝ [y↦(λ y . y y)](y y)
= (λ y . y y) (λ y . y y)
↝ [y↦(λ y . y y)](y y)
= (λ y . y y) (λ y . y y)
↝ [y↦(λ y . y y)](y y)
= (λ y . y y) (λ y . y y)
...
Term with non-terminating sequence of reductions but has a normal form!
(λ x . a) ((λ y . y y) (λ y . y y))
↝ a
(λ x . a) ((λ y . y y) (λ y . y y))
↝ (λ x . a) ((λ y . y y) (λ y . y y))
↝ (λ x . a) ((λ y . y y) (λ y . y y))
↝ (λ x . a) ((λ y . y y) (λ y . y y))
↝ (λ x . a) ((λ y . y y) (λ y . y y))
↝ (λ x . a) ((λ y . y y) (λ y . y y))
↝ (λ x . a) ((λ y . y y) (λ y . y y))
↝ (λ x . a) ((λ y . y y) (λ y . y y))
↝ a
Definition: a term is in normal form if no reductions apply.
Definition: a λ calculus is confluent if:
Given some term e, and two reduction sequences
e ↝...↝ e1
e ↝...↝ e2
there exists a term e3 and reduction sequences
e1 ↝...↝ e3
e2 ↝...↝ e3
This is the diamond property.
If a λ calculus is confluent then it is normalizing.
Theorem: The Simply Typed λ Calculus is strongly normalizing. I.e., no infinite chains of reductions.
The Curry-Howard Isomorphism says:
- Type inference is logic inference (plus stuff).
Γ ⊢ e1 : A → B Γ ⊢ e2 : A
----------------------------------
Γ ⊢ e1 e2 : B
If we scribble out e1 and e2 (the terms) leaving just the types, we have the Modus Ponens rule.
Γ ⊢ A → B Γ ⊢ A
------------------------
Γ ⊢ B
-
Types are Predicate Calculus Theorems
Take
e = (λ f . λ g . λ x . f ( g x ))applied to the three argumentsisZero round π.Which has the type
e : (int → bool) → (ℝ → int) -> ℝ -> bool. Considered as a logic statement instead of a type, this is a tautology. -
Types of basis objects are axioms.
(b ↦ T) ∈ Γ
---------
Γ ⊢ b : T
Existence of term of type T is a proof of the logical statement T.