Simply Typed λ-Calculus

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Simply Typed λ-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.

A. Demaille Simply Typedλ-Calculus 2 / 46

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Simply Typed λ-Calculus

1 Types

2 λ^→: Type Assignments

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Types

1 Types

Untypedλ-calculus Paradoxes

Church vs. Curry

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Types

Alonzo Church (1903–1995) Haskell Curry (1900–1982)

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Types

Types first appeared with

Curry (1934) for Combinatory Logic Church (1940)

Types are syntactic objects assigned to terms:

M :A M has typeA For instance:

I :A→A

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Types

Types first appeared with

Curry (1934) for Combinatory Logic Church (1940)

Types are syntactic objects assigned to terms:

M :A M has typeA For instance:

I :A→A

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Untyped λ-calculus

1 Types

Church vs. Curry

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λ-terms

Λ, set ofλ-terms x ∈ V x ∈Λ

M ∈Λ N∈Λ (MN)∈Λ

M ∈Λ

x∈ V (λx·M)∈Λ

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λβ

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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Properties of λβ

β-reduction is Church-Rosser.

Any term has (at most) a unique NF.

β-reduction is notnormalizing.

Some terms have no NF (Ω).

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Properties of λβ

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Properties of λβ

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Properties of λβ

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Paradoxes

1 Types

Church vs. Curry

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Self application

What is the computational meaning of λx·xx?

Stop considering anything can be applied to anything A function and its argument have different behaviors

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Self application

What is the computational meaning of λx·xx?

Stop considering anything can be applied to anything A function and its argument have different behaviors

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Church vs. Curry

1 Types

Church vs. Curry

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Simple Types

A set of type variables α, β, . . .

A symbol→ for functions

α→α,α →(β→γ),(α→β)→γ, . . . Possibly constants for “primitive” types ιfor integers, etc.

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Simple Types

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Simple Types

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Simple Types

By convention→ is right-associative:

α→β →γ =α→(β →γ)

This matches the right-associativity of λ:

λx·λy·M =λx·(λy·M)

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Simple Types

By convention→ is right-associative:

α→β →γ =α→(β →γ)

This matches the right-associativity of λ:

λx·λy·M =λx·(λy·M)

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Alonzo Style, or Haskell Way?

Church:

Typed λ-calculus x:α λx^α·x :α→α

Curry:

λ-calculus with Types x :α λx·x:α→α

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λ

^→

: Type Assignments

1 Types

2 λ^→: Type Assignments Types

Type Deductions

Subject Reduction Theorem Reducibility

Typability

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Types

1 Types

Type Deductions

Typability

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Simple Types

T V a set of type variables α, β, . . . Simple Types

The set T of typesσ, τ, . . .:

α∈ T V α ∈ T

σ ∈ T τ ∈ T (σ →τ)∈ T

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Type Contexts

Statement

Astatement M :σ is a pair withM ∈Λ, σ∈ T. M is the subject,σ thepredicate.

Type Context, Basis

Atype context Γ is a finite set of statements over distinct variables {x₁ :σ₁, . . .}.

Assignment

The variable x isassigned the typeσ in Γiff x:σ∈Γ.

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Type Contexts

Statement

Type Context, Basis

Assignment

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Type Contexts

Statement

Type Context, Basis

Assignment

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Type Contexts

Type Context Restrictions

Γ−x is theΓwith all assignment x :σ removed.

ΓM is Γ−FV(M).

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Type Deductions

1 Types

Type Deductions

Typability

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A Natural Presentation

Type derivations are trees built from the following nodes.

M :σ→τ N:σ

MN :τ

[

x :σ

]

··

· M :τ

λx·M :σ →τ

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A Natural Presentation

M :σ→τ N:σ MN :τ

[

x :σ

]

··

· M :τ

λx·M :σ →τ

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A Natural Presentation

[

x :σ

]

··

· M :τ

λx·M :σ →τ

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A Natural Presentation

[x :σ]

··

· M :τ λx·M :σ →τ

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Type Statement

A statementM :σ isderivablefrom the type context Γ, Γ`M :σ

if there is a derivation of M :σ whose non-canceled assumptions are inΓ.

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Type Statements

Prove

`λfx·f(fx) : (σ→σ)→σ→σ

[f :σ →σ]⁽²⁾

[f :σ→σ]⁽²⁾ [x :σ]⁽¹⁾ fx:σ

f(fx) :σ

(1) λx·f(fx) :σ →σ

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λfx·f(fx) : (σ →σ)→σ→σ

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Type Statements

Prove

[f :σ →σ]⁽²⁾

[f :σ→σ]⁽²⁾ [x:σ]⁽¹⁾ fx:σ

f(fx) :σ

(1) λx·f(fx) :σ →σ

(2)

λfx·f(fx) : (σ →σ)→σ→σ

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Type Statements

Prove

[f :σ →σ]⁽²⁾

[f :σ→σ]⁽²⁾ [x:σ]⁽¹⁾ fx:σ

f(fx) :σ

(1) λx·f(fx) :σ →σ

(2) λfx·f(fx) : (σ →σ)→σ→σ

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Type Statements

Prove

`λxy·x :σ→τ →σ

[x:σ]⁽¹⁾ λy·x :τ →σ

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λxy ·x :σ→τ →σ

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Type Statements

Prove

[x :σ]⁽¹⁾ λy·x :τ →σ

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λxy ·x :σ→τ →σ

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Type Statements

Prove

[x :σ]⁽¹⁾ λy·x :τ →σ

(1) λxy ·x :σ→τ →σ

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Alternative Presentation of Type Derivations

Type Derivations

{x :σ} 7→x :σ Γ7→M :σ→τ ∆7→N:σ

Γ∪∆7→MN :τ

Γ,∆consistent

Γ7→M :τ

Γ\ {x :σ} 7→λx·M :σ →τ Γ,{x :σ}consistent Γ7→M :τ

Γ,∆`M :τ

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Alternative Presentation of Type Derivations

Type Derivations

{x :σ} 7→x :σ Γ7→M :σ→τ ∆7→N:σ

Γ∪∆7→MN :τ

Γ,∆consistent

Γ7→M :τ

Γ\ {x :σ} 7→λx·M :σ →τ Γ,{x :σ}consistent Γ7→M :τ

Γ,∆`M :τ

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Type Statements

Prove

{x:σ} 7→x:σ {x :σ} 7→λy·x:τ →σ

7→λxy ·x :σ→τ →σ

[x:σ]⁽¹⁾ λy·x :τ →σ

(1) λxy·x:σ →τ →σ

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Type Statements

Prove

{x :σ} 7→x:σ {x:σ} 7→λy·x:τ →σ

[x:σ]⁽¹⁾ λy·x :τ →σ

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Type Statements

Prove

{x :σ} 7→x:σ {x:σ} 7→λy·x:τ →σ

[x:σ]⁽¹⁾ λy·x :τ →σ

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Type Statements

Prove

{x :σ} 7→x:σ {x:σ} 7→λy·x:τ →σ

[x:σ]⁽¹⁾ λy·x :τ →σ

(1) λxy·x :σ →τ →σ

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Type Statements

Type ω=λx·xx.

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Type Statements

Type ω=λx·xx.

··

· 7→λx·xx :σ

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Type Statements

Type ω=λx·xx.

··

·

{x :σ₁} 7→xx :σ₂

σ =σ1 →σ2

7→λx·xx :σ

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Type Statements

Type ω=λx·xx.

··

·

{x :σ1} 7→x:τ →σ2

··

·

{x :σ1} 7→x :τ {x :σ1} 7→xx :σ2

σ =σ₁→σ₂ 7→λx·xx :σ

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Typability

A termM is typable if there exists a typeσ such that `M :σ.

ω,Ωare not typable.

S,K,I are typable.

Y is not typable!

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Typability

S,K,I are typable.

Y is not typable!

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Typability

S,K,I are typable.

Y is not typable!

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Typability

S,K,I are typable.

Y is not typable!

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Subject Construction Lemma I

Consider the derivationπ forM :σ.

If M =x, then Γ ={x :σ} and

π ={x :σ} 7→x :σ If M =NL, then

π =ΓN7→N:τ →σ ΓL7→L:τ Γ7→NL:σ

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Subject Construction Lemma I

If M =x, then Γ ={x :σ} and

π ={x :σ} 7→x :σ If M =NL, then

π =ΓN7→N:τ →σ ΓL7→L:τ Γ7→NL:σ

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Subject Construction Lemma I

If M =λx·N, thenσ=σ1 →σ2 and Ifx ∈FV(N)

π=

Γ∪ {x:σ1} 7→N:σ2

Γ7→λx·N:σ1→σ2

Ifx 6∈FV(N)

π=

Γ7→N:σ2

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Subject Construction Lemma I

π=

Γ∪ {x:σ1} 7→N:σ2

Ifx 6∈FV(N)

π=

Γ7→N:σ2

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Subject Construction Lemma I

π=

Γ∪ {x:σ1} 7→N:σ2

Ifx 6∈FV(N)

π=

Γ7→N:σ2

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Derivations are not Unique

They are for β-normal forms.

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Subject Reduction Theorem

1 Types

Type Deductions

Typability

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Conversions and Types

α-Invariance

If Γ7→M :σ andM ≡_αN thenΓ7→N :σ.

Substitution If

Γ∪ {x :τ}`M :σ,

∆`N:τ, Γ,∆consistent, x 6∈FV(N) then

Γ∪∆`[N/x]M :σ

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Conversions and Types

α-Invariance

Substitution If

Γ∪ {x :τ}`M :σ,

Γ∪∆`[N/x]M :σ

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Conversions and Types

α-Invariance

Substitution If

Γ∪ {x :τ}`M :σ,

Γ∪∆`[N/x]M :σ

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Subject Reduction Theorem

If Γ`M :σ andM →_β N thenΓ`N:σ.

What about the converse?

Subject Expansion

If Γ`N :σ andM →_β N thenΓ`M :σ.

Ω is not typable, butKIΩ→I.

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Subject Reduction Theorem

Subject Expansion

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Subject Reduction Theorem

Subject Expansion

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Subject Reduction Theorem

Subject Expansion

Ωis not typable, but KIΩ→I.

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Reducibility

1 Types

Type Deductions

Typability

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Types and Normalization

λ^→ is Strongly Normalizing

All typable terms are β-strongly normalizing.

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Typability

1 Types

Type Deductions

Typability

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` M : σ suffices

Note that with Γ ={x₁:σ₁, . . . ,x_n:σ_n} Γ`M :τ is equivalent to

`λx₁. . .x_n·M :σ₁ →. . .→σ_n→τ

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Decidability of Type Assignment

Type checking GivenM, σ, does `M :σ?

`M :σ?

Typability GivenM, does there exist σ such that `M :σ?

`M :?

Inhabitation Givenσ, does there exist M such that `M :σ?

`? :σ

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Decidability of Type Assignment

Type Checking is Decidable

It is decidable whether a statement of λ^→ is provable.

Typability is Decidable

It is decidable whether a term of λ^→ has a type.

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Decidability of Type Assignment

Type Checking is Decidable

It is decidable whether a statement of λ^→ is provable.

Typability is Decidable

It is decidable whether a term of λ^→ has a type.

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Types are not Unique...

Typable terms do not have unique types.

{x:σ} 7→x :σ {x :σ} 7→λy·x :τ →σ

7→K :σ →τ →σ

is valid for anyσ, τ, including σ=σ⁰→σ⁰⁰, τ = (τ⁰ →τ⁰⁰)→τ⁰ etc.

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Types are not Unique. . .

but some are more Unique than others ...

All the types of K are instances ofσ →τ →σ.

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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.

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