The Untyped Lambda Calculus

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Delia KESNER IRIF, CNRS et Universit ´e Paris kesner@irif.fr www.irif.fr/˜kesner

The Untyped Lambda Calculus

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Alonzo Church (1903 - 1995)

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λ -calculus: syntax

Grammar for terms:

t, u, v ::= x (variable) | t u (application) | λx.t (abstraction) Notation :

Application is left-associative so that t

1

t

2

. . . t

n

means (. . . (t

1

t

2

) . . . t

n

) .

λx

1

. . . x

n

.t means λx

1

. . . . λx

n

.t .

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Inductive definition of λ -terms

x is a variable (Var) x is a λ -term

t is a λ -term u is a λ -term (App) t u is a λ -term

t is a λ -term x is a variable

(Lamb)

λx.t is a λ -term

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Size of λ -terms

size(x) := 1

size(uv) := 1 + size(u) + size(v)

size(λx.u) := 1 + size(u)

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Free and bound variables

λx.(z x y (λz.z y)) The set of free and bound variables are defined as follows

fv(x) : = { x } bv(x) : = ∅

fv(t u) := fv(t) ∪ fv(u) bv(t u) := bv(t) ∪ bv(u) fv(λx.t) := fv(t) \ {x} bv(λx.t) := bv(t) ∪ {x}

But for t = x (λx.x)

fv(x (λx.x)) = {x} = bv(x (λx.x))

A term is closed iff fv(t) = ∅ .

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Alpha-conversion

The relation =

α

, usually called renaming or alpha-conversion, is inductively defined as:

y is a fresh variable λx.t =

α

λy.t[x\\y]

t=

α

t

⁰

λx.t=

α

λx.t

⁰

t=

α

t

⁰

u=

α

u

⁰

t u=

α

t

⁰

u

⁰

where the operation t[x\\y] denotes the replacement of all the free occurrences of x in t by a fresh variable y . Formally,

x[x\\y] : = y

z[x\\y] := z x , z

(t u)[x\\y] := t[x\\y] u[ x\\y]

(λx.t)[x \\ y] : = λx.t

(λz.t)[x\\y] := λz.t[x\\y] x , z

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Examples

λx.(λx.x z) =

α

λy.(λx.x z) =

α

λy.(λy.y z)

x (λx.x) =

α

x (λz.z)

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Barendregt variable convention

From now on we assume the following variable convention:

1

No variable is both free and bound.

2

Bound variables have all different names.

Example

x (λz.z) is OK, x (λx.x) is not OK, λx.λy.x z is OK but λx.λx.x z is not OK.

Theorem

For every λ -term t there is λ -term u verifying the Barendregt convention such that t =

α

u .

Indeed, x (λx.x) =

α

x (λz.z) and λx.λx.x z =

α

λy.λx.x z .

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Operational semantics of λ -calculus

A one-step β -reduction is given inductively by

(λx.t) u →

β

t{x\u}

t →

β

t

⁰

λx.t →

β

λx.t

⁰

t →

β

t

⁰

t u →

_β

t

⁰

u

u →

β

u

⁰

t u →

_β

t u

⁰

What is exactly { \ } ?

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Towards a notion of substitution: warning!

(λx.(λy.x)) y →

β

(λy.x){x\y} = λy.y Incorrect

(λx.(λy.x)) y =

α

(λx.(λz.x)) y →

β

(λz.x){ x\y} = λz.y

Correct

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A simple notion of higher-order substitution

t{x/u} means replace all the free occurrences of x in t by u . This operation is defined modulo α -conversion as follows:

x{x/u} := u

y { x/u } : = y

(λy.v){x/u} := λy.v{x/u} if x , y and y < fv(u) (no capture holds)

(t v){x/u} := (t{x/u} v{x/u})

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A terminating β -reduction sequences

(λx.λ f.x f y) (λz.z) (λw.w w) →

_β

(λ f.x f y) { x \ λz.z } (λw.w w) = (λ f.(λz.z) f y) (λw.w w) →

_β

(λ f. f y) (λw.w w) →

_β

( f y){ f \λw.w w} =

(λw.w w) y →

_β

(w w) { w \ y } =

(y y)

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A non terminating β -reduction sequences

Let ∆

f

= λx. f (x x) .

∆

f

∆

f

→

β

( f (x x)){x\ ∆

f

} =

f (∆

f

∆

f

) →

_β

f ( f (x x)){ x\ ∆

f

} =

f ( f ( ∆

f

∆

f

)) →

_β

.. .

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Expressivity

Arithmetic.

Booleans.

Recursion.

The λ -calculus is Turing complete.

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Properties of λ -calculus

[Free variables decrease] If t →

β

t

⁰

, then f v(t) ⊇ f v(t

⁰

) .

Example

In t

0

= (λx.x)(λz.zz) →

_β

λz.zz = t

1

we have fv(t

0

) = fv(t

1

) .

In t

0

= (λx.y)(zz) →

_β

y = t

1

we have fv(t

0

) ⊇ fv(t

1

) .

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Properties of λ -calculus

[Confluence] The reduction relation →

β