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|)