A model of PCF in Guarded Type Theory Marco Paviotti 1 Rasmus - - PowerPoint PPT Presentation

A model of PCF in Guarded Type Theory

Marco Paviotti1 Rasmus Møgelberg1 Lars Birkedal2

1IT University of Copenhagen 2Aarhus University

June 23th, 2015 MFPS 2015 Nijmegen, Netherlands

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Guarded Type Theory

Birkedal and Møgelberg ’12

In Type Theory unrestricted fix-point fix: (A → A) → A is inconsistent

e.g. fix(id) : A leads to every type to be inhabited

In Guarded Type Theory restricted fix-points are allowed by using the ⊲ operator

• next : A → ⊲A
• ⊛: ⊲(A → B) → ⊲A → ⊲B
• fix: (⊲A → A) → A

s.t. f (next(fix(f ))) = fix(f )

• X ∼

= A × ⊲X

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Guarded Type Theory

Birkedal and Møgelberg ’12

In Type Theory unrestricted fix-point fix: (A → A) → A is inconsistent

e.g. fix(id) : A leads to every type to be inhabited

In Guarded Type Theory restricted fix-points are allowed by using the ⊲ operator

• next : A → ⊲A
• ⊛: ⊲(A → B) → ⊲A → ⊲B
• fix: (⊲A → A) → A

s.t. f (next(fix(f ))) = fix(f )

• X ∼

= A × ⊲X

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Streams StrA ∼ = A × StrA

Streams in Coq

• ones = 1 : : ones

✔

• bad = tail bad

✘

• nats = 0 : : map (1 +) nats

✘

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Streams StrA ∼ = A × StrA

Streams in Coq

• ones = 1 : : ones

✔

• bad = tail bad

✘

• nats = 0 : : map (1 +) nats

✘ Strg

A ∼

= A × ⊲ Strg

A

Guarded Streams : : : A → ⊲ Strg

A → Strg A

head : Strg

A → A

tail : Strg

A → ⊲ Strg A

• ones = 1 : : ones : Strg

A

✔

• bad = tail bad : Strg

A

✘

• nats = 0 : : next(map (1 +)) ⊛ nats : Strg

A

✔

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Model of Guarded Type Theory

Birkedal and Møgelberg ’12

The category of presheaves over ω X X(1) ✛ r1 X(2) .. ✛ rn−1 X(n) ✛ rn .. ⊲X 1 ✛ ! X(1) .. ✛ rn−2 X(n − 1) ✛ rn .. Strg

A ∼

= A × ⊲ Strg

A

Guarded Streams Strg

A

A × 1 ✛ r1 A × (A × 1) ✛ r2 A × (A × A × 1) ⊲ Strg

A

1 ✛ ! A × 1 ✛ r2 A × A × 1 A × ⊲ Strg

A

A × 1 ✛ r1 A × A × 1 ✛ r2 A × A × A × 1

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Can we do denotational semantics in Guarded Type Theory ?

in particular, is it possible to model recursion with guarded recursion ?

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Can we do denotational semantics in Guarded Type Theory ?

in particular, is it possible to model recursion with guarded recursion ?

• Motivations Mechanising denotational semantics in a

proof-assistant

• Contributions

+ Model of PCF in GTT + Adequacy Theorem proved in GTT

Similar to Escardo’s metric model 1, but here the whole development is entirely carried out within guarded type theory

1M.H. Escardo, “A metric model of PCF”. Presented at the Workshop on

Realizability Semantics and Applications, 1999

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Outline

• Operational Semantics of PCF
• Denotational Semantics
• Computational Adequacy
• Discussion

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PCF

σ, τ := nat | σ → τ L, M, N := n | x | λx.M | pred M | succ M | Y M | ifz L M N Γ, x : σ, ∆ ⊢ x : σ Γ ⊢ n : nat Γ, x : σ ⊢ M : τ Γ ⊢ (λx : σ.M) : σ → τ Γ ⊢ M : σ → τ Γ ⊢ N : σ Γ ⊢ MN : τ Γ ⊢ M : nat Γ ⊢ succ M : nat Γ ⊢ M : nat Γ ⊢ pred M : nat Γ ⊢ M : σ → σ Γ ⊢ Yσ M : σ Γ ⊢ L : nat Γ ⊢ M : σ Γ ⊢ N : σ Γ ⊢ ifz L M N : σ

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Big-step semantics

The big-step relation is defined by induction on terms and indexes:

M ⇓k Q

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Big-step semantics

The big-step relation is defined by induction on terms and indexes:

M ⇓k Q

explicit step counting

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Big-step semantics

The big-step relation is defined by induction on terms and indexes:

M ⇓k Q

explicit step counting Predicates on values can define M ⇓k v as M ⇓k λv′.v = v′

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Big-step semantics

The big-step relation is defined by induction on terms and indexes:

M ⇓k Q

explicit step counting Predicates on values can define M ⇓k v as M ⇓k λv′.v = v′

v ⇓0 Q def = = Q(v)

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Big-step semantics

The big-step relation is defined by induction on terms and indexes:

M ⇓k Q

explicit step counting Predicates on values can define M ⇓k v as M ⇓k λv′.v = v′

v ⇓0 Q def = = Q(v) MN ⇓k+m Q def = = M ⇓k Q′ where Q′(λx.L) = L[N/x] ⇓m Q

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Big-step semantics

The big-step relation is defined by induction on terms and indexes:

M ⇓k Q

explicit step counting Predicates on values can define M ⇓k v as M ⇓k λv′.v = v′

v ⇓0 Q def = = Q(v) MN ⇓k+m Q def = = M ⇓k Q′ where Q′(λx.L) = L[N/x] ⇓m Q Yσ M ⇓k+1 Q def = = ⊲(M(Yσ M) ⇓k Q)

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Big-step semantics

The big-step relation is defined by induction on terms and indexes:

M ⇓k Q

explicit step counting Predicates on values can define M ⇓k v as M ⇓k λv′.v = v′ Synchronising with the type theory

v ⇓0 Q def = = Q(v) MN ⇓k+m Q def = = M ⇓k Q′ where Q′(λx.L) = L[N/x] ⇓m Q Yσ M ⇓k+1 Q def = = ⊲(M(Yσ M) ⇓k Q)

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Small-Step Operational Semantics

(λx : σ.M)(N) →0 M[N/x] Yσ M →1 M(Yσ M) M →k M′ M(N) →k M′(N) Let →0

∗ be the reflexive, transitive closure of →0.

M ⇒0 Q def = = ΣN : TermPCF.M →0

∗ N and Q(N)

M ⇒k+1 Q def = = ΣM′, M′′ : TermPCF.M →0

∗ M′

and M′ →1 M′′ and ⊲(M′′ ⇒k Q) Define M ⇒k v as M ⇒k λv′.v = v′

Lemma

M ⇓k v ⇔ M ⇒k v

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Outline

• Operational Semantics of PCF
• Denotational Semantics
• Computational Adequacy
• Discussion

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Lifting Monad

LA ∼ = A + ⊲LA

Lifting monad

• η : A → LA

θ : ⊲LA → LA

• Time step operation : δ = θ ◦ next : LA → LA
• Bottom element ⊥ = fix(θ)
• LA is a free ⊲–algebra on A
• L is the guarded recursive version of Capretta’s partiality

monad1

1Venanzio Capretta, “General Recursion via Co-Inductive Types”, In Logical

Methods in Computer Science, 2005

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Lifting monad

LA ∼ = A + ⊲LA

Lifting monad LN ∼ = N + ⊲LN LN N + 1 ✛r1 N + N + 1 ✛ r2 N + N + N + 1 ⊲LN 1 ✛ ! N + 1 ✛ r1 N + N + 1 N + ⊲LN N + 1 ✛r1 N + N + 1 ✛ r2 N + N + N + 1

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Interpreting PCF

• Interpreting Types

nat def = = LN τ → σ def = = τ → σ

• All types are ⊲–algebras with θσ : ⊲σ → σ
• Interpreting terms t : Γ → σ

Γ ⊢ Yσ M(γ) = (fixσ)(λx : ⊲σ.θσ(next(M(γ))) ⊛ x))

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Interpreting PCF

• Interpreting Types

nat def = = LN τ → σ def = = τ → σ

• All types are ⊲–algebras with θσ : ⊲σ → σ
• Interpreting terms t : Γ → σ

Γ ⊢ Yσ M(γ) = (fixσ)(λx : ⊲σ.θσ(next(M(γ))) ⊛ x))

can be thought of

θ ◦ ⊲M

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Interpreting PCF

• Interpreting Types

nat def = = LN τ → σ def = = τ → σ

• All types are ⊲–algebras with θσ : ⊲σ → σ
• Interpreting terms t : Γ → σ

Γ ⊢ Yσ M(γ) = (fixσ)(λx : ⊲σ.θσ(next(M(γ))) ⊛ x))

Lemma

Let Γ ⊢ M : σ → σ then Yσ M = δσ ◦ M(Yσ M)

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Soundness

Theorem (Soundness)

Let M be a closed term of type τ, if M ⇓k v then M(∗) = δkv(∗)

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Outline

• Operational Semantics of PCF
• Denotational Semantics
• Computational Adequacy

if M(∗) = δk v(∗) then M ⇓k v

• Discussion

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Logical Relation

Adequacy proved by (proof-relevant) logical relation

d Rτ M

Define Rτ by induction on τ η(v) Rnat M def = = M ⇓0 v θnat(r) Rnat M def = = ΣM′, M′′ : TermPCF.M →0

∗ M′

and M′ →1 M′′ and r ⊲Rnat next(M′′)

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Logical Relation

Adequacy proved by (proof-relevant) logical relation

d Rτ M

Define Rτ by induction on τ η(v) Rnat M def = = M ⇓0 v θnat(r) Rnat M def = = ΣM′, M′′ : TermPCF.M →0

∗ M′

and M′ →1 M′′ and r ⊲Rnat next(M′′)

LN ∼ = N + ⊲LN an element in this type is ei- ther of the form η(v) or θnat(r)

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Logical Relation

Adequacy proved by (proof-relevant) logical relation

d Rτ M

Define Rτ by induction on τ η(v) Rnat M def = = M ⇓0 v θnat(r) Rnat M def = = ΣM′, M′′ : TermPCF.M →0

∗ M′

and M′ →1 M′′ and r ⊲Rnat next(M′′)

Delayed Relation ⊲R t ⊲Rnat u delayed version of R

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Logical Relation

Adequacy proved by (proof-relevant) logical relation

d Rτ M

Define Rτ by induction on τ η(v) Rnat M def = = M ⇓0 v θnat(r) Rnat M def = = ΣM′, M′′ : TermPCF.M →0

∗ M′

and M′ →1 M′′ and r ⊲Rnat next(M′′) f Rτ→σ M def = = Πα: τ, N : TermPCF.α Rτ N = ⇒ f (α) Rσ (MN)

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Adequacy

Lemma (Fundamental Lemma)

Let Γ ⊢ t : τ, suppose Γ ≡ x1 : τ1, · · · , xn : τn and ti : τi, αi : τi and αi Rτi ti for i ∈ {1, . . . , n}, then t( α) Rτ t[ t/ x]

Theorem (Computational Adequacy)

If M is a closed term of type nat then M ⇓k v iff M(∗) = δkv.

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Outline

• Operational Semantics of PCF
• Denotational Semantics
• Computational Adequacy
• Discussion

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Type theory vs. Topos logic

Setωop also models the topos logic. The following is derivable in the non-proof-relevant topos logic:

∃k.∃v. Ynat (λx.x) ⇓k v

Proof (Sketch)

• The argument is by Guarded Recursion: assume

⊲(∃k.∃v. Ynat (λx.x) ⇓k v)

• by property of Setωop ∃k.∃v.⊲(Ynat (λx.x) ⇓k v)
• which implies ∃k.∃v. Ynat (λx.x) ⇓k+1 v

In Type Theory

Σk.Σv. Ynat λx.x ⇓k v

is not globally inhabited

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Type theory vs. Topos logic

Setωop also models the topos logic. The following is derivable in the non-proof-relevant topos logic:

∃k.∃v. Ynat (λx.x) ⇓k v

Proof (Sketch)

• The argument is by Guarded Recursion: assume

⊲(∃k.∃v. Ynat (λx.x) ⇓k v)

• by property of Setωop ∃k.∃v.⊲(Ynat (λx.x) ⇓k v)
• which implies ∃k.∃v. Ynat (λx.x) ⇓k+1 v

In Type Theory

Σk.Σv. Ynat λx.x ⇓k v

is not globally inhabited

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Type theory vs. Topos logic

Setωop also models the topos logic. The following is derivable in the non-proof-relevant topos logic:

∃k.∃v. Ynat (λx.x) ⇓k v

Proof (Sketch)

• The argument is by Guarded Recursion: assume

⊲(∃k.∃v. Ynat (λx.x) ⇓k v)

• by property of Setωop ∃k.∃v.⊲(Ynat (λx.x) ⇓k v)
• which implies ∃k.∃v. Ynat (λx.x) ⇓k+1 v

In Type Theory

Σk.Σv. Ynat λx.x ⇓k v

is not globally inhabited

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Type theory vs. Topos logic

Setωop also models the topos logic. The following is derivable in the non-proof-relevant topos logic:

∃k.∃v. Ynat (λx.x) ⇓k v

Proof (Sketch)

• The argument is by Guarded Recursion: assume

⊲(∃k.∃v. Ynat (λx.x) ⇓k v)

• by property of Setωop ∃k.∃v.⊲(Ynat (λx.x) ⇓k v)
• which implies ∃k.∃v. Ynat (λx.x) ⇓k+1 v

In Type Theory

Σk.Σv. Ynat λx.x ⇓k v

is not globally inhabited

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Conclusions

We presented a model for PCF that is adequate w.r.t. the operational semantics. The work has been carried out entirely in guarded type theory:

• Operational Semantics with explicit step-indexing is synchronised

with the time steps in the type theory

• Denotational semantics with proof of adequacy

Main message Guarded type theory as a meta theory for deno- tational semantics of programming languages.

Future work

• Using the model to reason about contextual equivalence
• FPC in guarded type theory

Thanks!

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Conclusions

We presented a model for PCF that is adequate w.r.t. the operational semantics. The work has been carried out entirely in guarded type theory:

• Operational Semantics with explicit step-indexing is synchronised

with the time steps in the type theory

• Denotational semantics with proof of adequacy

Main message Guarded type theory as a meta theory for deno- tational semantics of programming languages.

Future work

• Using the model to reason about contextual equivalence
• FPC in guarded type theory

Thanks!

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