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On Proof Interpretations and Linear Logic

On Proof Interpretations and Linear Logic

Paulo Oliva

Queen Mary, University of London, UK
([email protected])

Heyting Day
Utrecht, 27 February 2015

To Anne Troelstra and Dick de Jongh

On Proof Interpretations and Linear Logic

Overview

-- ILω

. . .

interpretations

IL
IL = Intuitionistic predicate logic

On Proof Interpretations and Linear Logic

Overview

-- ILω

. . .

interpretations

IL
ILLωp

-

ILLp
IL = Intuitionistic predicate logic
ILLp = Intuit. predicate linear logic (!-free)

On Proof Interpretations and Linear Logic

Overview

-

. . .

interpretations

IL

- ILω
6

ILLωp

-

ILLp
IL = Intuitionistic predicate logic
ILLp = Intuit. predicate linear logic (!-free)

On Proof Interpretations and Linear Logic

Overview

-

. . .

interpretations

IL

- ILω
6

??

ILLωp

-

ILLp
IL = Intuitionistic predicate logic
ILLp = Intuit. predicate linear logic (!-free)

On Proof Interpretations and Linear Logic

Overview

-

. . .

interpretations

IL

- ILω
6

??

ILLωp

-

ILLp

-- !A

. . .
!A

IL = Intuitionistic predicate logic
ILLp = Intuit. predicate linear logic (!-free)

On Proof Interpretations and Linear Logic

Outline

1

Proof Interpretations

23

Linear Logic Unified Interpretation of Linear Logic

On Proof Interpretations and Linear Logic Proof Interpretations

Outline

1

Proof Interpretations

23

Linear Logic Unified Interpretation of Linear Logic

On Proof Interpretations and Linear Logic Proof Interpretations

Formulas as Sets of Proofs

Formulas as set of proofs

A

  • →
  • {π : π ` A}

On Proof Interpretations and Linear Logic Proof Interpretations

Formulas as Sets of Proofs

Formulas as set of proofs

A

  • →
  • {π : π ` A}

E.g. the set of prime numbers is infinite

On Proof Interpretations and Linear Logic Proof Interpretations

Formulas as Sets of Proofs

Formulas as set of proofs

A

  • →
  • {π : π ` A}

E.g. the set of prime numbers is infinite The book “Proofs from THE BOOK” contains six beautiful proofs of this

On Proof Interpretations and Linear Logic Proof Interpretations

Formulas as Sets of Proofs

Formulas as set of proofs

A

  • →
  • {π : π ` A}

E.g. the set of prime numbers is infinite The book “Proofs from THE BOOK” contains six beautiful proofs of this

Non-theorems map to the empty set E.g. the set of even numbers is finite

On Proof Interpretations and Linear Logic Proof Interpretations

Formulas as Sets of Proofs

Formulas as set of proofs

A

  • →
  • {π : π ` A}

E.g. the set of prime numbers is infinite The book “Proofs from THE BOOK” contains six beautiful proofs of this

Non-theorems map to the empty set E.g. the set of even numbers is finite

Need to fix the formal system

On Proof Interpretations and Linear Logic Proof Interpretations

Formulas as Sets of Programs

Brouwer-Heyting-Kolmogorov Interpretation

Formulas as specifications (set of programs)

A

→

{p : p satisfies A}

On Proof Interpretations and Linear Logic Proof Interpretations

Formulas as Sets of Programs

Brouwer-Heyting-Kolmogorov Interpretation

Formulas as specifications (set of programs)

A

→

{p : p satisfies A}
E.g. the set of prime numbers is infinite Specifies a program that on input n: N outputs a prime number p ≥ n and a certificate that p is prime

On Proof Interpretations and Linear Logic Proof Interpretations

Formulas as Sets of Programs

Brouwer-Heyting-Kolmogorov Interpretation

Formulas as specifications (set of programs)

A

→

{p : p satisfies A}
E.g. the set of prime numbers is infinite Specifies a program that on input n: N outputs a prime number p ≥ n and a certificate that p is prime

If A is provable then the set should be non-empty

On Proof Interpretations and Linear Logic Proof Interpretations

Formulas as Sets of Programs

Brouwer-Heyting-Kolmogorov Interpretation

Formulas as specifications (set of programs)

A

→

{p : p satisfies A}
E.g. the set of prime numbers is infinite Specifies a program that on input n: N outputs a prime number p ≥ n and a certificate that p is prime

If A is provable then the set should be non-empty Non-theorems map to un-implementable specifications

On Proof Interpretations and Linear Logic Proof Interpretations

Formulas as Sets of Programs

Brouwer-Heyting-Kolmogorov Interpretation

Formulas as specifications (set of programs)

A

→

{p : p satisfies A}
E.g. the set of prime numbers is infinite Specifies a program that on input n: N outputs a prime number p ≥ n and a certificate that p is prime

If A is provable then the set should be non-empty Non-theorems map to un-implementable specifications

Need to fix formal system and programming language

On Proof Interpretations and Linear Logic Proof Interpretations

Formulas as Types

Curry-Howard Correspondence Minimal Logic + Simply-typed λ-calculus

  • π `ML tπ : A
  • A

⇔

On Proof Interpretations and Linear Logic Proof Interpretations

Formulas as Types

Curry-Howard Correspondence Minimal Logic + Simply-typed λ-calculus

  • π `ML tπ : A
  • A

⇔

Extension 1: Heyting arith. + G¨odel primitive rec.

On Proof Interpretations and Linear Logic Proof Interpretations

Formulas as Types

Curry-Howard Correspondence Minimal Logic + Simply-typed λ-calculus

  • π `ML tπ : A
  • A

⇔

Extension 1: Heyting arith. + G¨odel primitive rec. Extension 2: Classical logic + Continuation operator

  • ¬A
  • A

·

B
¬A

···
···
··

⊥

B

  • A
  • B

On Proof Interpretations and Linear Logic Proof Interpretations

Functional Interpretations

  • Formula A
  • ⇒ Set of functionals |A|

On Proof Interpretations and Linear Logic Proof Interpretations

Functional Interpretations

  • Formula A
  • ⇒ Set of functionals |A|

  • Proof π
  • ⇒ Functional fπ ∈ |A|

On Proof Interpretations and Linear Logic Proof Interpretations

Functional Interpretations

Formula A Proof π
⇒ Set of functionals |A| ⇒ Functional fπ ∈ |A|

S1 `π A

⇒ S2 ` fπ ∈ |A|

On Proof Interpretations and Linear Logic Proof Interpretations

Functional Interpretations

Formula A

∀x∃y(y > x)

Proof π
⇒ Set of functionals |A|

{f : fx > x}

⇒ Functional fπ ∈ |A|

S1 `π A

⇒ S2 ` fπ ∈ |A|

On Proof Interpretations and Linear Logic Proof Interpretations

Functional Interpretations

Formula A

∀x∃y(y > x)

Proof π

. . .

⇒ Set of functionals |A|

{f : fx > x}

⇒ Functional fπ ∈ |A|

λx.x + 1

S1 `π A

⇒ S2 ` fπ ∈ |A|

On Proof Interpretations and Linear Logic Proof Interpretations

Functional Interpretations

Dialectica (Go¨del 1958)
Relative consistency of Peano arithmetic PA

On Proof Interpretations and Linear Logic Proof Interpretations

Functional Interpretations

Dialectica (Go¨del 1958)
Relative consistency of Peano arithmetic PA

(

Falsity interpreted as empty set (| ⊥ | ≡ ∅)

PA `⊥ ⇒ T ` ∃f(f ∈ ∅)

On Proof Interpretations and Linear Logic Proof Interpretations

Functional Interpretations

Dialectica (Go¨del 1958)
Relative consistency of Peano arithmetic PA

(

Falsity interpreted as empty set (| ⊥ | ≡ ∅)

PA `⊥ ⇒ T ` ∃f(f ∈ ∅)

Modified realizability (Kreisel 1959)

Independence results for Heyting arithmetic HA

On Proof Interpretations and Linear Logic Proof Interpretations

Functional Interpretations

Dialectica (Go¨del 1958)
Relative consistency of Peano arithmetic PA

(

Falsity interpreted as empty set (| ⊥ | ≡ ∅)

PA `⊥ ⇒ T ` ∃f(f ∈ ∅)

Modified realizability (Kreisel 1959)

Independence results for Heyting arithmetic HA

(

|P| set of non-computable functionals

HA ` P

  • ⇒
  • HAω ` ∃f(f ∈ |P|)

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