Lecture 3: Typed Lambda Calculus and...

Lecture 3: Typed Lambda Calculus andCurry-Howard

H. Geuvers

Radboud UniversityNijmegen, NL

21st Estonian Winter School in Computer ScienceWinter 2016

H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 1 / 65

Outline


Typed λ calculus as a basis for logic

λ-term : type

M : A

program : data typeproof : formula

program : (full) specification

Aim:

• Type Theory as an integrated system for proving andprogramming.

• Type Theory as a basis for proof assistants and interactivetheorem proving.


Simple type theory

Simplest system: λ→ or simple type theory, STT. Just arrow types

Typ := TVar | (Typ→ Typ)

• Examples: (α→ β)→ α, (α→ β)→ ((β → γ)→ (α→ γ))

• Brackets associate to the right and outside brackets areomitted:(α→ β)→ (β → γ)→ α→ γ

• Types are denoted by A,B, . . ..


Simple type theory a la Church

Formulation with contexts to declare the free variables:

x1 : A1, x2 : A2, . . . , xn : An

is a context, usually denoted by Γ.Derivation rules of λ→ (a la Church):

x :A ∈ Γ

Γ ` x : A

Γ ` M : A→ B Γ ` N : A

Γ ` M N : B

Γ, x :A ` P : B

Γ ` λx :A.P : A→ B

Γ `λ→ M : A if there is a derivation using these rules withconclusion Γ ` M : A


Examples

` λx : A.λy : B.x : A→ B → A

` λx : A→ B.λy : B → C .λz : A.y (x z) : (A→B)→(B→C )→A→C

` λx : A.λy : (B → A)→ A.y(λz : B.x) : A→ ((B → A)→ A)→ A

Not for every type there is a closed term of that type:

(A→ A)→ A is not inhabited

That is: there is no term M such that

` M : (A→ A)→ A.


Typed Terms versus Type Assignment

• With typed terms also called typing a la Church, we haveterms with type information in the λ-abstraction

λx : A.x : A→ A

• Terms have unique types,• The type is directly computed from the type info in the

variables.

• With typed assignment also called typing a la Curry, weassign types to untyped λ-terms

λx .x : A→ A

• Terms do not have unique types,• A principal type can be computed using unification.


Church vs. Curry typing

• The Curry formulation is especially interesting forprogramming: you want to write as little type information aspossible; let the compiler infer the types for you.

• The Church formulation is especially interesting for proofchecking: terms are created interactively; type structure is sointricate that type inference is undecidable (if you start froman untyped term).[ This lecture]


Formulas-as-Types (Curry, Howard)

Recall: there are two readings of a judgement M : A

1 term as algorithm/program, type as specification:M is a function of type A

2 type as a proposition, term as its proof:M is a proof of the proposition A

• There is a one-to-one correspondence:

typable terms in λ→ ' derivations in minimal propositionlogic

• x1 : B1, x2 : B2, . . . , xn : Bn ` M : A can be read asM is a proof of A from the assumptions B1,B2, . . . ,Bn.


Example

[A→ B → C ]3 [A]1

B → C

[A→ B]2 [A]1

B

C1

A→ C2

(A→ B)→ A→ C3

(A→ B → C )→ (A→ B)→ A→ C

'

λx :A→ B → C .λy :A→ B.λz :A.x z (y z): (A→ B → C )→ (A→ B)→ A→ C


Example

[x : A→ B → C ]3 [z : A]1

x z : B → C

[y : A→ B]2 [z : A]1

y z : B

x z (y z) : C1

λz :A.x z (y z) : A→ C2

λy :A→ B.λz :A.x z (y z) : (A→ B)→ A→ C3

λx :A→ B → C .λy :A→ B.λz :A.x z (y z) : (A→B→C )→(A→B)→A→C

Exercise: Give the derivation that corresponds to

λx :C → E .λy :(C → E )→ E .y(λz .y x) :(C → E )→ ((C → E )→ E )→ E


Typed Combinatory Logic

We have seen Combinatory Logic with the axioms for I, K and S.We now know their typed definition in λ→:

I := λx : A.x : A→ AK := λx : A.λy : B.x : A→ B → AS := λx :A→ B → C .λy :A→ B.λz :A.x z (y z)

: (A→ B → C )→ (A→ B)→ A→ C

• The three axiom schemes A→ A, A→ B → A and(A→ B → C )→ (A→ B)→ A→ C together with thederivation rule Modus Ponens is exactly Hilbert style minimalproposition logic.

• The typed CL terms are exactly the derivations in this logic.

• Modus Ponens corresponds with Application in CL

Exercise: Show that the scheme A→ A is derivable.Cast in CL terminology: I can be defined in terms of S and K. Tobe precise: I = SKK.


Computation = Cut-elimination

• β-reduction: (λx :A.M)P →β M[x := P]

Cut-elimination in minimal logic = β-reduction in λ→.

[A]1

D1

B1

A→ B

D2

A

B

−→

D2

AD1

B

[x : A]1

D1

M : B1

λx :A.M : A→ B

D2

P : A

(λx :A.M)P : B

−→β

D2

P : AD1

M[x := P] : B


Example

Proof of A→ A→ B, (A→ B)→ A ` B with a cut.

[A]1

[A]1 A→ A→ B

A→ B

B

A→ B

(A→ B)→ A

[A]1

[A]1 A→ A→ B

A→ B

B

A→ B

A

B

It contains a cut: a →-i directly followed by an →-e.


Example proof with term information

[y : A]1

[y : A]1 p : A→ A→ B

p y : A→ B

p y y : B

λy :A.p y y : A→ B

q : (A→ B)→ A

[x : A]1

[x : A]1 p : A→ A→ B

p x : A→ B

p x x : B

λx :A.p x x : A→ B

q(λx :A.p x x) : A

(λy :A.p y y)(q(λx :A.p x x)) : B

Term contains a β-redex: (λx :A.p x x) (q(λx :A.p x x))


Extension with other connectives

Adding product types × to λ→. (Proposition logic withconjunction ∧.)

Γ ` M : A× B

Γ ` π1M : A

Γ ` M : A× B

Γ ` π2M : B

Γ ` P : A Γ ` Q : B

Γ ` 〈P,Q〉 : A× B

With reduction rules

π1〈P,Q〉 → P

π2〈P,Q〉 → Q

Similar rules can be given for sum-types A + B, corresponding todisjunction A ∨ B.


Extension to predicate logic

• First order language: domain D, with variables x , y , z : D andpossibly functions over D, e.g. f : D → D, g : D → D → D.

• Rules for ∀x :D.φ and ∃x :D.φ.

• NB There are two “kinds” of variables: the first ordervariables (ranging over the domain D) and the “proofvariables” (used as [local] assumptions of formulas).

• Formulas and domain are both types. What is the type of apredicate or relation?

• A predicate P is a map from D to the collection of types, ∗• P : D → ∗ for P a predicate and R : D → D → ∗ for R a

binary relation on D.

• We will have to make this more precise . . .


Idea of extending to ∀

Term rules for the ∀-quantifier in predicate logic.

Γ ` M : ∀x :D.Aif t : D

Γ ` M t : A[x := t]

Γ ` M : Ax not free in Γ

Γ ` λx :D.M : ∀x :D.A

With the usual β-reduction rule

(λx :D.M)t → M[x := t]

.This conforms with cut-elimination (or “detour elimination”) onlogical derivations.


Example

Deriving irreflexivity from anti-symmetry

AntiSymR := ∀x , y :D.(Rxy)→ (Ryx)→ ⊥IrreflR := ∀x :D.(Rxx)→ ⊥

Derivation in predicate logic:

∀x , y :D.R x y → R y x → ⊥

∀y :D.R x y → R y x → ⊥

R x x → R x x → ⊥ [R x x ]1

R x x → ⊥ [R x x ]1

⊥1

R x x → ⊥

∀x :D.R x x → ⊥


Example derivation in type theory, with terms

H : ∀x , y :D.R x y → R y x → ⊥

H x : ∀y :D.R x y → R y x → ⊥

H x x : R x x → R x x → ⊥ [H ′ : R x x ]1

H x x H ′ : R x x → ⊥ [H ′ : R x x ]1

H x x H ′H ′ : ⊥1

λH ′:(R x x).H x x H ′H ′ : R x x → ⊥

λx :A.λH ′:(R x x).H x x H ′H ′ : ∀x :D.R x x → ⊥


Dependent Type Theory

• We have seen informally “dependent types at work” in thepredicate logic example.

• Now: the rules

With dependent types:

• everything depends on everything

• we can’t first define the types, then the terms

• two universes: ∗ and �

• ∗ is the universe of types

• We can’t have ∗ : ∗, so we have another universe: ∗ : �.

NB The Coq system uses “Set” and “Prop” for what I call ∗ and“Type” for what I call �.


First order Dependent Type theory, λP

Derive judgements of the form

Γ ` M : B

• Γ is a context

x1 : B1, x2 : B2, . . . , xn : Bn

• M and B are termstaken from the set of pseudoterms

T ::= Var | ∗ |� | (T T) | (λx :T.T) |Πx :T.T

Auxiliary judgementΓ `

denoting that Γ is a correct context.H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 22 / 65

Derivation rules of λP

s ranges over {∗,�}.

(base) ∅ ` (ctxt)Γ ` A : s

Γ, x :A `if x not in Γ (ax)

Γ `

Γ ` ∗ : �

(proj)Γ `

Γ ` x : Aif x :A ∈ Γ (Π)

Γ ` A : ∗ Γ, x :A ` B : s

Γ ` Πx :A.B : s

(λ)Γ, x :A ` M : B Γ ` Πx :A.B : s

Γ ` λx :A.M : Πx :A.B(app)

Γ ` M : Πx :A.B Γ ` N : A

Γ ` MN : B[x := N]

(conv)Γ ` M : B Γ ` A : s

Γ ` M : AA =βη B

Notation: write A→ B for Πx :A.B if x /∈ FV(B).


The use of the Π-type

• The Π rule allows to form two forms of function types.

(Π)Γ, x :A ` B : s Γ ` A : ∗

Γ ` Πx :A.B : s

Πx :A.B ' {f | ∀a : A(f a : B[x := a])}

Write A→ B if x /∈ FV(B)

• With s = ∗, we can form D→ D and Πx :D.x = x , etc.• With s = �, we can form D→ D→ ∗ and D→ ∗.


Representation of PRED (minimal predicate logic) into λP

Represent both the domains of the logic and the formulas as types.

A : ∗,P : A→ ∗,R : A→ A→ ∗,

Now implication is represented as → and ∀ is represented as Π:

∀x :A.P x 7→ Πx :A.P x

Intro and elim rules are just λ-abstraction and application


Example

A : ∗,R : A→ A→ ∗ ` λz :A.λh:(Πx , y :A.R x y).h z z

: Πz :A.(Πx , y :A.R x y)→ R z z

This term is a proof of ∀z :A.(∀x , y :A.R(x , y))→ R(z , z)Exercise: Find terms of the following types (NB → bindsstrongest)

(Πx :A.P x → Q x)→ (Πx :A.P x)→ Πx :A.Q x

and

(Πx :A.P x → Πz .R z z)→ (Πx :A.P x)→ Πz :A.R z z).

Also write down the contexts in which these terms are typed.


Direct embedding of logic in type theory

For λ→ and λP we have seen

Direct representations of logic in type theory.

• Connectives each have a counterpart in the type theory:implication ∼ →-typeuniversal quantification ∼ ∀-type

• Logical rules have their direct counterpart in type theoryλ-abstraction ∼ →-introductionapplication ∼ →- elimination λ-abstraction ∼ ∀-introductionapplication ∼ ∀-elimination

• Context declares signature, local varibales and assumptions.


LF embedding of logic in type theory

Second way of interpreting logic in type theory De Bruijn:

Logical framework encoding of logic in type theory.

• Type theory used as a meta system for encoding ones ownlogic.

• Choose an appropriate context ΓL, in which the logic L(including its proof rules) is declared.

• Context used as a signature for the logic.

• Use the type system as the ‘meta’ calculus for dealing withsubstitution and binding.


Direct and LF embedding

proof formula

direct embedding λx :A.x A→ ALF embedding imp intrAAλx :T A.x T (A⇒ A)

• Direct representation: One type system : One logic, Logicalrules ∼ type theoretic rules

• LF encoding One type system : Many logics, Logical rules ∼context declarations


Examples of the Deep embedding

The encoding of logics in a logical framework is shown by threeexamples:

1 Minimal proposition logic

2 Minimal predicate logic (just {⇒, ∀})3 Untyped λ-calculus


Minimal propositional logic

Fix the signature (context) of minimal propositional logic.

prop : ∗imp : prop→ prop→ prop

Notation:A⇒ B for impAB

The type prop is the type of ‘names’ of propositions.NB : A term of type propcan not be inhabited (proved), as it isnot a type.We ‘lift’ a name p : prop to the type of its proofs by introducingthe following map:

T : prop→ ∗.

Intended meaning of Tp is ‘the type of proofs of p’.We interpret ‘p is valid’ by ‘Tp is inhabited’.


Encoding of derivations

To derive Tp we also encode the logical derivation rules

imp intr : Πp, q : prop.(Tp → Tq)→ T(p ⇒ q),

imp el : Πp, q : prop.T(p ⇒ q)→ Tp → Tq.

New phenomenon: Π-type:

Πx :A.B(x) ' the type of functions f such that

f a : B(a) for all a:A

imp intr takes two (names of) propositions p and q and a termf : T p → T q and returns a term of type T(p ⇒ q)Indeed A⇒ A, becomes valid:

imp intrAA(λx :TA.x) : T(A⇒ A)

Exercise: Construct a term of type T(A⇒ (B ⇒ A))


Signature of PROP in LF

To encode proposition logic in LF we need a context (signature)ΣPROP:

prop : ∗⇒ : prop→ prop→ prop

T : prop→ ∗imp intr : (A,B : prop)(TA→ TB)→ T(A⇒ B)

imp el : (A,B : prop)T(A⇒ B)→ TA→ TB.

Desired properties of the encoding:

• Adequacy (soundness) of the encoding:

`PROP A =⇒ ΣPROP, a1:prop, . . . , an:prop ` p : TA for some p.

{a, . . . , an} is the set of proposition variables in A.• Faithfulness (or completeness) is the converse. It also holds,

but more involved to prove.


Minimal predicate logic over one domain A

Signature:

prop : ∗,A : ∗,T : prop→ ∗f : A→ A,

R : A→ A→ prop,

⇒ : prop→ prop→ prop,



Now encode ∀: ∀ takes a P : A→ prop and returns a proposition,so we add:

∀ : (A→ prop)→ prop


Minimal predicate logic over one domain A

Signature: ΣPRED

prop : ∗,A : ∗,

...



Now encode ∀: ∀ takes a P : A→ prop and returns a proposition,so:

∀ : (A→ prop)→ prop

Universal quantification is translated as follows.

∀x :A.(Px) 7→ ∀(λx :A.(Px))


Intro and Elim rules for ∀

∀ : (A→ prop)→ prop,

∀ intr : ΠP:A→ prop.(Πx :A.T(Px))→ T(∀P),

∀ elim : ΠP:A→ prop.T(∀P)→ Πx :A.T(Px).

The proof of∀z :A(∀x , y :A.Rxy)⇒ Rzz

is now mirrored by the proof-term

∀ intr[ ]( λz :A.imp intr[ ][ ](λh:T(∀x , y :A.Rxy).∀ elim[ ](∀ elim[ ]hz)z) )

We have replaced the instantiations of the Π-type by [ ].This term is of type

T(∀(λz :A.imp(∀(λx :A.(∀(λy :A.Rxy))))(Rzz)))


Intro and Elim rules for ∀

∀ : (A→ prop)→ prop,

∀ intr : ΠP:A→ prop.(Πx :A.T(Px))→ T(∀P),

∀ elim : ΠP:A→ prop.T(∀P)→ Πx :A.T(Px).

The proof of∀z :A(∀x , y :A.Rxy)⇒ Rzz

is now mirrored by the proof-term

∀ intr[ ]( λz :A.imp intr[ ][ ](λh:T(∀x , y :A.Rxy).∀ elim[ ](∀ elim[ ]hz)z) )

Exercise: Construct a proof-term that mirrors the (obvious) proofof

∀x(P x ⇒ Q x)⇒ ∀x .P x ⇒ ∀x .Q x


Untyped λ-calculus

Signature Σlambda : D : ∗;app : D→ (D→ D);abs : (D→ D)→ D.

• A variable x in λ-calculus becomes x : D in the type system.

• The translation [−] : Λ→ Term(D) is defined as follows.

[x ] = x ;

[PQ] = app [P] [Q];

[λx .P] = abs (λx :D.[P]).

Examples: [λx .xx ] := abs(λx :D.app x x)[(λx .xx)(λy .y)] := app(abs(λx :D.app x x))(abs(λy :D.y)).


Introducing β-equality

eq:D→ D→ ∗.

Notation P = Q for eq P Q.Rules for proving equalities.

refl : Πx :D.x = x ,

sym : Πx , y :D.x = y → y = x ,

trans : Πx , y , z :D.x = y → y = z → x = z ,

mon : Πx , x ′, z , z ′:D.x = x ′ → z = z ′ → (app z x) = (app z ′ x ′),

xi : Πf , g :D→ D.(Πx :D.(fx) = (gx))→ (abs f ) = (abs g),

beta : Πf :D→ D.Πx :D.(app(abs f )x) = (fx).


Properties of λP

• Uniqueness of typesIf Γ ` M : σ and Γ ` M : τ , then σ=βητ .

• Subject ReductionIf Γ ` M : σ and M →βη N, then Γ ` N : σ.

• Strong NormalizationIf Γ ` M : σ, then all βη-reductions from M terminate.

Proof of SN is by defining a reduction preserving map from λP toλ→.


Decidability Questions

Γ ` M : σ? TCPΓ ` M : ? TSPΓ `? : σ TIP

For λP:

• TIP is undecidable

• TCP/TSP: simultaneously with Context checking


Curry-Howard-de Bruijn

logic ∼ type theory

formula ∼ typeproof ∼ term

detour elimination ∼ β-reduction

proposition logic ∼ simply typed λ-calculuspredicate logic ∼ dependently typed λ-calculus λP

intuitionistic logic ∼ . . . + inductive typeshigher order logic ∼ . . . + higher types and polymorphism

classical logic ∼ . . . + exceptions


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