Lambda Calculus

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Lambda Calculus

Akim Demailleakim@lrde.epita.fr

EPITA — École Pour l’Informatique et les Techniques Avancées

June 10, 2016

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About these lecture notes

Many of these slides are largely inspired from Andrew D. Ker’s lecture notes [Ker, 2005a, Ker, 2005b]. Some slides are even straightforward copies.

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Lambda Calculus

1 λ-calculus

2 Reduction

3 λ-calculus as a Programming Language

4 Combinatory Logic

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λ-calculus

1 λ-calculus

The Syntax ofλ-calculus Substitution, Conversions

2 Reduction

4 Combinatory Logic

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Why the λ-calculus?

Church, Curry

A theory of functions (1930s).

Turing

A definition of effective computability (1930s).

Brouwer, Heyting, Kolmogorov

A representation of formal proofs (1920–).

McCarthy, Scott, . . .

A basis for functional programming languages (1960s–).

Montague, . . .

Semantics for natural language (1960s–).

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λ-calculus

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λ-calculus

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λ-calculus

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λ-calculus

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What is the λ-calculus?

A mathematical theory of functions A (functional) programming language It allows reasoning on operationalsemantics

Mathematicians are more inclined to denotationalsemantics

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The Syntax of λ-calculus

1 λ-calculus

2 Reduction

4 Combinatory Logic

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The Pure Untyped λ-calculus

The simplestλ-calculus:

Variables x,y,z...

Functions λx·M Application MN

No

Booleans Numbers Types . . .

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The Pure Untyped λ-calculus

The simplestλ-calculus:

Variables x,y,z...

Functions λx·M Application MN

No

Booleans Numbers Types . . .

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The λ-calculus Language

Theλ-terms:

M ::= x | (λx·M) | (MM)

Conventions:

Omit outer parentheses MN = (MN)

Application associates to the left MNL= (MN)L Abstraction associates to the right λx·MN=λx·(MN) Multiple arguments as syntactic sugar λxy ·M =λx·λy·M (Currying EN — Currification FR)

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The λ-calculus Language

Theλ-terms:

M ::= x | (λx·M) | (MM)

Conventions:

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The λ-calculus Language

Theλ-terms:

M ::= x | (λx·M) | (MM)

Conventions:

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The λ-calculus Language

Theλ-terms:

M ::= x | (λx·M) | (MM)

Conventions:

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The λ-calculus Language

Theλ-terms:

M ::= x | (λx·M) | (MM)

Conventions:

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The λ-calculus Language

Theλ-terms:

M ::= x | (λx·M) | (MM)

Conventions:

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Notation

Usual x 7→2x+1 λ-calculus λx·2x+1

Originally ˆx·2x+1 Inpiration xˆ·x =y Transition Λx·2x+1

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Which abstract-syntax trees are correct?

app N app

app x abs

abs var

y y x

app N app

M abs

abs var

x M x

app N M abs

abs var

y y x

app var

v app

var u abs

abs var

y y x

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Which abstract-syntax trees are correct?

app N app

app x abs

abs var

y y x

app N app

M abs

abs var

x M x

app N M abs

abs var

y y x

app var

v app

var u abs

abs var

y y x

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Fully qualified form for λnfx · f (nfx)

3

(λn·(λf ·(λx·(f((nf)x)))))

7

(λx·(λf ·(λn·(f((nf)x)))))

7

(λn·)(λf·)λx·(f((nf)x))

7

(λx·(λf ·(λn·f)))((nf)x))

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Fully qualified form for λnfx · f (nfx)

3 (λn·(λf ·(λx·(f((nf)x))))) 7 (λx·(λf ·(λn·(f((nf)x))))) 7 (λn·)(λf·)λx·(f((nf)x)) 7 (λx·(λf ·(λn·f)))((nf)x))

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The λ-calculus Language: Alternative Presentation

The set Λ ofλ-terms:

x ∈ V x ∈Λ

M ∈Λ N∈Λ (MN)∈Λ

M ∈Λ

x∈ V (λx·M)∈Λ For instance

x∈Λ

(λx·x)∈Λ y∈Λ

((λx·x)y)∈Λ z ∈Λ (((λx·x)y)z)∈Λ

(λz·(((λx·x)y)z))∈Λ x ∈Λ

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The λ-calculus Language: Alternative Presentation

The set Λ ofλ-terms:

x ∈ V x ∈Λ

M ∈Λ N∈Λ (MN)∈Λ

M ∈Λ

x∈ V (λx·M)∈Λ For instance

x∈Λ

(λx·x)∈Λ y∈Λ

((λx·x)y)∈Λ z ∈Λ (((λx·x)y)z)∈Λ

(λz·(((λx·x)y)z))∈Λ x ∈Λ (λz·(((λx·x)y)z))x ∈Λ

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Subterms

The set of subtermsofM,sub(M):

sub(x) :={x}

sub(λx·M) :={λx·M} ∪sub(M) sub(MN) :={MN} ∪sub(M)∪sub(N)

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Variables

The set offree variables ofM,FV(M):

FV(x) :={x}

FV(λx·M) :=FV(M)\ {x}

FV(MN) :=FV(M)∪FV(N)

A variable isfree orbound.

A variable may have bound andfree occurrences: xλx·x.

A term with no free variable isclosed.

Acombinatoris a closed term.

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Variables

FV(x) :={x}

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Variables

FV(x) :={x}

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Variables

FV(x) :={x}

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Variables

FV(x) :={x}

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Substitution, Conversions

1 λ-calculus

2 Reduction

4 Combinatory Logic

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α-Conversion

α-conversion

M andN areα-convertible,M ≡N, iff they differ only by renaming bound variables without introducing captures.

λx·x ≡λy·y xλx·x ≡xλy·y xλx·x 6≡yλy·y λx·λy·xy 6≡λx·λx·xx

From now onα-convertible terms are consideredequal.

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α-Conversion

α-conversion

λx·x ≡λy·y xλx·x ≡xλy·y xλx·x 6≡yλy·y λx·λy·xy 6≡λx·λx·xx From now onα-convertible terms are consideredequal.

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α-Conversion

α-conversion

λx·x ≡λy·y xλx·x ≡xλy·y xλx·x 6≡yλy·y λx·λy·xy 6≡λx·λx·xx From now onα-convertible terms are consideredequal.

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The Variable Convention

To avoid nasty capture issues, we will always silently α-convert terms so that no bound variable of a term is a variable (bound or free) of another.

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Substitution

Thesubstitution ofx by M inN is denoted[M/x]N.

It is anotation, not an operation

Intuitively, all the freeoccurrences of x are replaced by M.

For instance [λz·zz/x]λy·xy =λy·(λz·zz)y. There are many notations for substitution:

[M/x]N N[M/x] N[x:=M] N[x←M]

and even

N[x/M]

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Substitution

and even

N[x/M]

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Substitution

and even

N[x/M]

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Substitution

and even

N[x/M]

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Substitution

[M/x]N N[M/x] N[x:=M] N[x←M] and even

N[x/M]

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Formal Definition of the Substitution

Substitution

[M/x]x:=M

[M/x]y:=y withx 6=y

[M/x](NL) := ([M/x]N)([M/x]L)

[M/x]λy·N:=λy·[M/x]N withx 6=y and y6∈FV(M) The variable convention allows us to “require” that y 6∈FV(M).

Without it:

[M/x]λy·N:=λy·[M/x]N if x 6=y and y6∈FV(M) [M/x]λy·N:=λz·[M/x][z/y]N if x 6=y or y ∈FV(M)

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Formal Definition of the Substitution

Substitution

[M/x]x:=M

[M/x]y:=y withx 6=y

[M/x](NL) := ([M/x]N)([M/x]L)

Without it:

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Formal Definition of the Substitution

Substitution

[M/x]x:=M

[M/x]y:=y withx 6=y

[M/x](NL) := ([M/x]N)([M/x]L)

Without it:

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Substitution

[yy/z](λxy·zy)≡λxu·(yy)u

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β-Conversion

β-conversion

Theβ-convertibility between two terms is the relation β defined as:

(λx·M)N β [N/x]M for any M,N∈Λ.

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The λβ Formal System

It is the “standard” theory of λ-calculus.

The λβ Formal System M =M

M =N N =M

M =N N=L M =L M =M⁰ N =N⁰

MN =M⁰N⁰

M =N λx·M =λx·N

(λx·M)N= [N/x]M

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Reduction

1 λ-calculus

2 Reduction β-Reduction Church-Rosser Reduction Strategies

4 Combinatory Logic

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β-Reduction

1 λ-calculus

4 Combinatory Logic

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Reduction

One step R-Reduction from a relation R The relation→

R is the smallest relation such that:

(M,N)∈R M →

R N

M →

R N ML→

R NL

M →

R N LM →

R LN

M →

R N λx·M →

R λx·N R-Reduction: transitive, reflexive closure

The relation→^∗

M →

R N M →^∗ N N →^∗ L

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Reduction

One step R-Reduction from a relation R The relation→

(M,N)∈R M →

R N

M →

R N ML→

R NL

M →

R N LM →

R LN

M →

R N λx·M →

R λx·N R-Reduction: transitive, reflexive closure

The relation→^∗

M →

R N

M →^∗ N M →^∗

R M

M →^∗

R N N →^∗

R L M →^∗ L

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β-Reduction

β-Redex

Aβ-redexis term under the form(λx·M)N.

One step β-Reduction

(λx·M)N→

β [N/x]M · · ·

β-Reduction The relation→^∗

β is the transitive, reflexive closure of →

β.

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β-Reduction

β-Redex

(λx·M)N→

β [N/x]M · · ·

β. β-Conversion

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β-Reduction

β-Redex

(λx·M)N→

β [N/x]M · · ·

β.

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β-Reduction

β-Redex

(λx·M)N→

β [N/x]M · · ·

β. β-Conversion

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β-Reductions

(λx·x)y →

y (λx·xx)y → yy

(λx·xx)(λx·xx) → (λx·xx)(λx·xx)

(λx·x(xx))(λx·x(xx)) → (λx·x(xx)) (λx·x(xx))(λx·x(xx))

Omega Combinators

ω ≡ λx·xx

Ω ≡ ωω

Ωe ≡ λx·x(xx)

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β-Reductions

(λx·x)y → y (λx·xx)y →

yy

Omega Combinators

ω ≡ λx·xx

Ω ≡ ωω

Ωe ≡ λx·x(xx)

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β-Reductions

(λx·x)y → y (λx·xx)y → yy (λx·xx)(λx·xx) →

(λx·xx)(λx·xx)

Omega Combinators

ω ≡ λx·xx

Ω ≡ ωω

Ωe ≡ λx·x(xx)

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β-Reductions

(λx·x)y → y (λx·xx)y → yy

(λx·xx)(λx·xx) → (λx·xx)(λx·xx) (λx·x(xx))(λx·x(xx)) →

(λx·x(xx)) (λx·x(xx))(λx·x(xx))

Omega Combinators

ω ≡ λx·xx

Ω ≡ ωω

Ωe ≡ λx·x(xx)

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β-Reductions

Omega Combinators

ω ≡ λx·xx

Ω ≡ ωω

Ωe ≡ λx·x(xx)

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β-Reductions

Omega Combinators

ω ≡ λx·xx

Ω ≡ ωω

Ωe ≡ λx·x(xx)

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More β-Reductions

(λx·xyx)λz·z →

(λz·z)y(λz·z) (λx·x)((λy·y)x) → (λx·x)(x) (λx·x)((λy·y)x) → ((λy·y)x) (λx·x)((λy·y)x) →^∗ x

(λx·xx)((λx·xx)y) →^∗ yy(yy) (λx·xx)((λx·x)y) →^∗ yy (λx·x)((λx·xx)y) →^∗ yy

Therefore

λβ`(λx·xx)((λx·x)y) = (λx·x)((λx·xx)y)

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More β-Reductions

(λx·xyx)λz·z → (λz·z)y(λz·z) (λx·x)((λy·y)x) →

(λx·x)(x) (λx·x)((λy·y)x) → ((λy·y)x) (λx·x)((λy·y)x) →^∗ x

Therefore

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More β-Reductions

(λx·xyx)λz·z → (λz·z)y(λz·z) (λx·x)((λy·y)x) → (λx·x)(x) (λx·x)((λy·y)x) →

((λy·y)x) (λx·x)((λy·y)x) →^∗ x

Therefore

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More β-Reductions

(λx·xyx)λz·z → (λz·z)y(λz·z) (λx·x)((λy·y)x) → (λx·x)(x) (λx·x)((λy·y)x) → ((λy·y)x) (λx·x)((λy·y)x) →^∗

x (λx·xx)((λx·xx)y) →^∗ yy(yy)

(λx·xx)((λx·x)y) →^∗ yy (λx·x)((λx·xx)y) →^∗ yy

Therefore

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More β-Reductions

(λx·xyx)λz·z → (λz·z)y(λz·z) (λx·x)((λy·y)x) → (λx·x)(x) (λx·x)((λy·y)x) → ((λy·y)x) (λx·x)((λy·y)x) →^∗ x

(λx·xx)((λx·xx)y) →^∗

yy(yy) (λx·xx)((λx·x)y) →^∗ yy (λx·x)((λx·xx)y) →^∗ yy

Therefore

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More β-Reductions

(λx·xx)((λx·xx)y) →^∗ yy(yy) (λx·xx)((λx·x)y) →^∗

yy (λx·x)((λx·xx)y) →^∗ yy

Therefore

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More β-Reductions

(λx·xx)((λx·xx)y) →^∗ yy(yy) (λx·xx)((λx·x)y) →^∗ yy (λx·x)((λx·xx)y) →^∗

yy

Therefore

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More β-Reductions

Therefore

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More β-Reductions

(λx·xx)((λx·xx)y) →^∗ yy(yy) (λx·xx)((λx·x)y) →^∗ yy (λx·x)((λx·xx)y) →^∗ yy Therefore

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Other rules

η-reduction

λx·Mx →

η M η-expansion

M →

ηexp

λx·Mx

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Church-Rosser

1 λ-calculus

4 Combinatory Logic

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Normal Forms

GivenR, a relation on terms.

R-Normal Form (R-NF)

A termM is in R-Normal Form if there is noN such that M →

R N.

R-Normalizable Term

A termM is R-Normalizable(or has anR-Normal Form) if there exists a termN inR-NF such thatM →^∗

R N.

R-Strongly Normalization Term

A termM is R-Strongly Normalizable there is no infinite one-step

reduction sequence starting from M. I.e., any one-step reduction sequence

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Normal Forms

R N.

R-Normalizable Term

R N.

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Normal Forms

R N.

R-Normalizable Term

R N.

reduction sequence starting from M. I.e., any one-step reduction sequence

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β-Normal Terms

I =λx·x is in β-NF II has aβ-NF β-reduces to I

II is β-strongly normalizing Ωis not (weakly) normalizable

Ω = (λx·xx)(λx·xx)→(λx·xx)(λx·xx) = Ω KIΩis weakly normalizable (K =λx·(λy·x)) KIΩ→I

KIΩis not strongly normalizable KIΩ→KIΩ

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β-Normal Terms

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β-Normal Terms

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β-Normal Terms

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β-Normal Terms

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β-Normal Terms

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Normalizing Relation

R isweakly normalizingif every term isR-normalizable.

R isstrongly normalizing if every term isR-strongly normalizable.

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β-Reduction

Ωis not weakly normalizable

β-reduction is not weakly normalizing!

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Reduction Strategy

With a weakly normalizing relation that is not strongly normalizing:

some terms are not weakly normalizable but not strongly i.e., some termscan be reducedifyou reduce them “properly”

Reduction Strategy

Areduction strategy is a function specifying what is the next one-step reduction to perform.

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Reduction Strategy

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Reduction Strategy

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Confluence

Diamond property

→

R satisfies the diamond propertyifM →

R N1,M →

R N2 implies the existence of Lsuch that N1→

R L,N2 →

R L.

Church-Rosser

→

R is Church-Rosser if→^∗

R satisfies the diamond property.

→

R is Church-Rosser if M →^∗

R N1,M →^∗

R N2 implies the existence ofLsuch that N1

→∗ R L,N2

→∗ R L.

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Confluence

Diamond property

→

R N1,M →

R L,N2 →

R L.

Church-Rosser

→

R N1,M →^∗

R N2 implies the existence ofLsuch

∗ ∗

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Confluence

Diamond property

→

R N1,M →

R L,N2 →

R L.

Church-Rosser

→

R N1,M →^∗

R N2 implies the existence ofLsuch that N1

→∗ R L,N2

→∗ R L.

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Confluence

Unique Normal Form Property

→

R has the unique normal form propertyifM →^∗

R N₁,M →^∗

R N₂ withN₁,N₂ in normal form, implies N1≡N2.

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Properties

The diamond property implies Church-Rosser.

If R is Church-Rosser thenM =

R N iff there exists Lsuch that M →^∗

R LandN →^∗

R L.

If R is Church-Rosser then it has the unique normal form property.

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Properties

R LandN →^∗

R L.

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Properties

R LandN →^∗

R L.

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λ-calculus has the Church-Rosser Property

β-reduction is Church-Rosser.

Any term has (at most) a unique NF.

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λ-calculus has the Church-Rosser Property

β-reduction is Church-Rosser.

Any term has (at most) a unique NF.

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Reduction Strategies

1 λ-calculus

4 Combinatory Logic

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Reduction Strategy

Areduction strategy is a (partial)functionfrom term to term.

If → is a reduction strategy, then any term has a unique maximal reduction sequence.

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Reduction Strategy

Areduction strategy is a (partial)functionfrom term to term.

If → is a reduction strategy, then any term has a unique maximal reduction sequence.

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Head Reduction

The head reduction →^h on terms is defined by:

λ~x·(λy·M)N~L→^h λ~x·[N/y]M~L

λx1. . .xn·(λy·M)NL1. . .Lm

→h λx1. . .xn·[N/y]ML1. . .Lm n,m≥0 Note that any term has one of the following forms:

λ~x·(λy·M)~L λ~x·y~L

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Head Reduction

The head reduction →^h on terms is defined by:

λ~x·(λy·M)N~L→^h λ~x·[N/y]M~L

λx1. . .xn·(λy·M)NL1. . .Lm

→h λx1. . .xn·[N/y]ML1. . .Lm n,m≥0 Note that any term has one of the following forms:

λ~x·(λy·M)~L λ~x·y~L

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Head Reduction

KIΩ→^h I KΩI →^h ΩI

→h II

→h I xIx 6→^h xx Normal terms have the form:

λ~x·y~L

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Leftmost Reduction

Theleftmost reduction →^l performs a single step ofβ-conversion on the leftmost λx·M.

Any head reduction is a leftmost reduction (but not conversly).

Leftmost reduction is normalizing.

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Leftmost Reduction

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Leftmost Reduction

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λ-calculus as a Programming Language

1 λ-calculus

2 Reduction

3 λ-calculus as a Programming Language Booleans

Natural Numbers Pairs

Recursion

4 Combinatory Logic

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Booleans

1 λ-calculus

2 Reduction

Recursion

4 Combinatory Logic

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Booleans

How would you code Booleans inλ-calculus?

How would you translateif M then N else L?

ifMNL

Do we need if?

What if Booleanswerethe if?

MNL

What is true?

What is false?

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Booleans

ifMNL

Do we need if?

MNL

What is true?

What is false?

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Booleans

ifMNL

Do we need if?

MNL

What is true?

What is false?

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Booleans

ifMNL

Do we need if?

MNL

What is true?

What is false?

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Booleans

ifMNL

Do we need if?

MNL

What is true?

What is false?

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Booleans

ifMNL

Do we need if?

MNL

What is true?

What is false?

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Booleans

ifMNL

Do we need if?

MNL

What is true?

What is false?

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Booleans

ifMNL

Do we need if?

MNL

What is true?

What is false?

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Boolean Combinators

Boolean Combinators (Church Booleans) T :=λxy ·x F :=λxy ·y

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Natural Numbers

1 λ-calculus

2 Reduction

Recursion

4 Combinatory Logic

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Church’s Integers

Integers

n:=λf ·λx·fⁿx =λf ·λx·(f · · ·(f

| {z }

ntimes

x )· · ·)

| {z }

ntimes

2=λf ·λx·f(fx) 3=λf ·λx·f(f(fx))

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Church’s Integers

Operations

succ

succ:=λn·λf ·λx·f(nfx) plus

plus:=λm·λn·λf ·λx·mf(nfx) plus:=λm·λn·n succ m

plus:=λn·n succ

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Pairs

1 λ-calculus

2 Reduction

Recursion

4 Combinatory Logic

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Church’s pairs

Pairs

pair :=λxy ·λf ·fxy first :=λp·pT second :=λp·pF

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Recursion

1 λ-calculus

2 Reduction

Recursion

4 Combinatory Logic

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Fixed point Combinators

Curry’s Y Combinator

Y :=λf ·(λx·f(xx))(λx·f(xx)) Turing’s Θ Combinator

Θ := (λxy·y(xxy))(λxy ·y(xxy))

There are infinitely many fixed-point combinators.

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Fixed point Combinators

Y :=λf ·(λx·f(xx))(λx·f(xx)) Turing’s Θ Combinator

Θ := (λxy·y(xxy))(λxy ·y(xxy)) There are infinitely many fixed-point combinators.

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Fixed point Combinators

Y :=λf ·(λx·f(xx))(λx·f(xx))

Y g = (λf ·(λx·f(xx))(λx·f(xx)))g

→_β (λx·g(xx))(λx·g(xx))

→_β g((λx·g(xx))(λx·g(xx))) g(Y g)→_β g(λf ·((λx·f(xx))(λx·f(xx)))g)

→_β g(λf ·((λx·f(xx))(λx·f(xx)))g)

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Reduction strategies in Programming Languages

Full beta reductions Reduce any redex.

Applicative order

The leftmost, innermost redex is always reduced first. Intuitively reduce function “arguments” before the function itself. Applicative order always attempts to apply functions to normal forms, even when this is not possible.

Normal order

The leftmost, outermost redex is reduced first.

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Reduction strategies in Programming Languages

Call by name

As normal order, but no reductions are performed inside abstractions.

λx·(λx·x)x is in NF.

Call by value

Only the outermost redexes are reduced: a redex is reduced only when its right hand side has reduced to a value (variable or lambda

abstraction).

Call by need

As normal order, but function applications that would duplicate terms instead name the argument, which is then reduced only “when it is needed”. Called in practical contexts “lazy evaluation”.

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λ-calculus as a Programming Language

Lisp (xkcd 224)

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Combinatory Logic

1 λ-calculus

2 Reduction

4 Combinatory Logic

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Moses Ilyich Schönfinkel (1889–1942)

Russian logician and mathematician.

Member of David Hilbert’s group at the University of Göttingen.

Mentally ill and in a sanatorium in 1927.

His papers were burned by his neighbors for heating.

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Combinatory Logic

λ-reduction is complex

its implementation is full of subtle pitfalls invented in 1936 by Alonzo Church

Combinatory Logic a simpler alternative

invented by Moses Schönfinkel in 1920’s developed by Haskell Curry in 1925

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Combinatory Logic

λ-reduction is complex

its implementation is full of subtle pitfalls invented in 1936 by Alonzo Church Combinatory Logic

a simpler alternative

invented by Moses Schönfinkel in 1920’s developed by Haskell Curry in 1925

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Combinators

Classic Combinators

S := (λx·(λy·(λz ·((xz)(yz))))) K := (λx·(λy·x))

I := (λx·x) We no longer need λ!

SXYZ →XZ(YZ) KXY →X

IX →X

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Combinators

The Combinator I

I := (λx·x)

IX →X

SKKX →KX(KX)→X

I =SKK

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Combinators

The Combinator I

I := (λx·x)

IX →X

SKKX →KX(KX)→X

I =SKK

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Combinators

The Combinator I

I := (λx·x)

IX →X

SKKX →KX(KX)→X

I =SKK

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Combinators

The Combinator I

I := (λx·x)

IX →X

SKKX →KX(KX)→X

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Combinatory Logic

S SXYZ →XZ(YZ)

(λx·(λy·(λz·((xz)(yz))))) K KXY →X

(λx·(λy·x)) I IX →X

(λx·x)

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Combinatory Logic

Combination is left-associative:

SKKX = (((SK)K)X)→KX(KX)→X I.e., I =SKK: two symbols and two rules suffice.

Same expressive power as λ-calculus.

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Boolean Combinators

T =K F =KI

TXY →X FXY →Y

KIXY = (((KI)X)Y)→IY →Y

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The Y Combinator in SKI

Y =S(K(SII))(S(S(KS)K)(K(SII))) The simplest fixed point combinator in SK

Y =SSK(S(K(SS(S(SSK))))K

by Jan Willem Klop:

Yk = (LLLLLLLLLLLLLLLLLLLLLLLLLLLL)

where:

L=λabcdefghijklmnopqstuvwxyzr(r(thisisafixedpointcombinator))

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The Y Combinator in SKI

Y =SSK(S(K(SS(S(SSK))))K by Jan Willem Klop:

Yk = (LLLLLLLLLLLLLLLLLLLLLLLLLLLL)

where:

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The Y Combinator in SKI

Y =SSK(S(K(SS(S(SSK))))K by Jan Willem Klop:

Yk = (LLLLLLLLLLLLLLLLLLLLLLLLLLLL) where:

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Bibliography Notes

[Ker, 2005a] Complete and readable lecture notes on λ-calculus. Uses conventions different from ours.

[Ker, 2005b] Additional information, including slides.

[Barendregt and Barendsen, 2000] A classical introduction to λ-calculus.

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Bibliography I

Barendregt, H. and Barendsen, E. (2000).

Introduction to lambda calculus.

http:

//www.cs.ru.nl/~erikb/onderwijs/T3/materiaal/lambda.pdf.

Ker, A. D. (2005a).

Lambda calculus and types.

http://web.comlab.ox.ac.uk/oucl/work/andrew.ker/

lambda-calculus-notes-full-v3.pdf.

Ker, A. D. (2005b).

Lambda calculus notes.