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 6
??
- !A ... - !A
- ω ILLp ILLp
ILLp = Intuit. predicate linear logic (!-free)
On Proof Interpretations and Linear Logic
Overview
- ω IL ... - IL interpretations
IL = Intuitionistic predicate logic 6
??
- !A ... - !A
On Proof Interpretations and Linear Logic
Overview
- ω IL ... - IL interpretations
- ω ILLp ILLp
IL = Intuitionistic predicate logic
ILLp = Intuit. predicate linear logic (!-free) ??
- !A ... - !A
On Proof Interpretations and Linear Logic
Overview
- ω IL ... - IL interpretations 6
- ω ILLp ILLp
IL = Intuitionistic predicate logic
ILLp = Intuit. predicate linear logic (!-free) - !A ... - !A
On Proof Interpretations and Linear Logic
Overview
- ω IL ... - IL interpretations 6
?? - ω ILLp ILLp
IL = Intuitionistic predicate logic
ILLp = Intuit. predicate linear logic (!-free) On Proof Interpretations and Linear Logic
Overview
- ω IL ... - IL interpretations 6
?? - ω ILLp ILLp - !A ... - !A
IL = Intuitionistic predicate logic
ILLp = Intuit. predicate linear logic (!-free) On Proof Interpretations and Linear Logic
Outline
1 Proof Interpretations
2 Linear Logic
3 Unified Interpretation of Linear Logic On Proof Interpretations and Linear Logic Proof Interpretations
Outline
1 Proof Interpretations
2 Linear Logic
3 Unified Interpretation of Linear Logic 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 Proofs
Formulas as set of proofs A 7→ {π : π ` A} 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 Proofs
Formulas as set of proofs A 7→ {π : π ` A} E.g. the set of prime numbers is infinite 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 Proofs
Formulas as set of proofs A 7→ {π : π ` A} E.g. the set of prime numbers is infinite The book “Proofs from THE BOOK” contains six beautiful proofs of this Need to fix the formal system
On Proof Interpretations and Linear Logic Proof Interpretations
Formulas as Sets of Proofs
Formulas as set of proofs A 7→ {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 7→ {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 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 Sets of Programs
Brouwer-Heyting-Kolmogorov Interpretation Formulas as specifications (set of programs) A 7→ {p : p satisfies A} 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 Sets of Programs
Brouwer-Heyting-Kolmogorov Interpretation Formulas as specifications (set of programs) A 7→ {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 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 Sets of Programs
Brouwer-Heyting-Kolmogorov Interpretation Formulas as specifications (set of programs) A 7→ {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 Need to fix formal system and programming language
On Proof Interpretations and Linear Logic Proof Interpretations
Formulas as Sets of Programs
Brouwer-Heyting-Kolmogorov Interpretation Formulas as specifications (set of programs) A 7→ {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 7→ {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 Extension 1: Heyting arith. + G¨odelprimitive rec. Extension 2: Classical logic + Continuation operator ¬A A ¬A · · · · · · · · · ⊥ B B A B
On Proof Interpretations and Linear Logic Proof Interpretations
Formulas as Types
Curry-Howard Correspondence Minimal Logic + Simply-typed λ-calculus
π `ML A ⇔ tπ : A Extension 2: Classical logic + Continuation operator ¬A A ¬A · · · · · · · · · ⊥ B B A B
On Proof Interpretations and Linear Logic Proof Interpretations
Formulas as Types
Curry-Howard Correspondence Minimal Logic + Simply-typed λ-calculus
π `ML A ⇔ tπ : A Extension 1: Heyting arith. + G¨odelprimitive rec. On Proof Interpretations and Linear Logic Proof Interpretations
Formulas as Types
Curry-Howard Correspondence Minimal Logic + Simply-typed λ-calculus
π `ML A ⇔ tπ : A Extension 1: Heyting arith. + G¨odelprimitive rec. Extension 2: Classical logic + Continuation operator ¬A A ¬A · · · · · · · · · ⊥ B B A B ∀x∃y(y > x) {f : fx > x}
Proof π ⇒ Functional fπ ∈ |A|
... λx.x + 1
S1 `π A ⇒ S2 ` fπ ∈ |A|
On Proof Interpretations and Linear Logic Proof Interpretations
Functional Interpretations
Formula A ⇒ Set of functionals |A| ∀x∃y(y > x) {f : fx > x}
... λx.x + 1
S1 `π A ⇒ S2 ` fπ ∈ |A|
On Proof Interpretations and Linear Logic Proof Interpretations
Functional Interpretations
Formula A ⇒ Set of functionals |A|
Proof π ⇒ Functional fπ ∈ |A| ∀x∃y(y > x) {f : fx > x}
... λx.x + 1
On Proof Interpretations and Linear Logic Proof Interpretations
Functional Interpretations
Formula A ⇒ Set of functionals |A|
Proof π ⇒ Functional fπ ∈ |A|
S1 `π A ⇒ S2 ` fπ ∈ |A| ... λx.x + 1
On Proof Interpretations and Linear Logic Proof Interpretations
Functional Interpretations
Formula A ⇒ Set of functionals |A|
∀x∃y(y > x) {f : fx > x}
Proof π ⇒ Functional fπ ∈ |A|
S1 `π A ⇒ S2 ` fπ ∈ |A| On Proof Interpretations and Linear Logic Proof Interpretations
Functional Interpretations
Formula A ⇒ Set of functionals |A|
∀x∃y(y > x) {f : fx > x}
Proof π ⇒ Functional fπ ∈ |A|
... λx.x + 1
S1 `π A ⇒ S2 ` fπ ∈ |A| ( |P | set of non-computable functionals HA ` P ⇒ HAω ` ∃f(f ∈ |P |)
( 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 (G¨odel1958) Relative consistency of Peano arithmetic PA ( |P | set of non-computable functionals HA ` P ⇒ HAω ` ∃f(f ∈ |P |)
Modified realizability (Kreisel 1959) Independence results for Heyting arithmetic HA
On Proof Interpretations and Linear Logic Proof Interpretations
Functional Interpretations
Dialectica (G¨odel1958) Relative consistency of Peano arithmetic PA ( Falsity interpreted as empty set (| ⊥ | ≡ ∅) PA `⊥ ⇒ T ` ∃f(f ∈ ∅) ( |P | set of non-computable functionals HA ` P ⇒ HAω ` ∃f(f ∈ |P |)
On Proof Interpretations and Linear Logic Proof Interpretations
Functional Interpretations
Dialectica (G¨odel1958) 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 (G¨odel1958) 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 |) Conservation results S + P ` A ⇒ S ` A Proof mining
S + A ` B ⇒ S + Aw ` Bs Build models M |= S
On Proof Interpretations and Linear Logic Proof Interpretations
Functional Interpretations
Relative consistence S `⊥ ⇒ T `⊥ Independence results S 6` P Proof mining
S + A ` B ⇒ S + Aw ` Bs Build models M |= S
On Proof Interpretations and Linear Logic Proof Interpretations
Functional Interpretations
Relative consistence S `⊥ ⇒ T `⊥ Independence results S 6` P Conservation results S + P ` A ⇒ S ` A Build models M |= S
On Proof Interpretations and Linear Logic Proof Interpretations
Functional Interpretations
Relative consistence S `⊥ ⇒ T `⊥ Independence results S 6` P Conservation results S + P ` A ⇒ S ` A Proof mining
S + A ` B ⇒ S + Aw ` Bs On Proof Interpretations and Linear Logic Proof Interpretations
Functional Interpretations
Relative consistence S `⊥ ⇒ T `⊥ Independence results S 6` P Conservation results S + P ` A ⇒ S ` A Proof mining
S + A ` B ⇒ S + Aw ` Bs Build models M |= S q-realizability (Kleene 1969 / Troelstra 1971) t qr (A → B) ≡ ∀x((x qr A) ∧ A → tx qr B) Related to Kleene’s slash translation (1962) Realizability with truth (Smorynski / Troelstra 1998) t rt (A → B) ≡ ∀x(x rt A → tx rt B) ∧ (A → B) Related to Aczel’s slash translation (1968)
On Proof Interpretations and Linear Logic Proof Interpretations
Other Interpretations
Diller-Nahm (1974) Decidability of prime formulas not required Realizability with truth (Smorynski / Troelstra 1998) t rt (A → B) ≡ ∀x(x rt A → tx rt B) ∧ (A → B) Related to Aczel’s slash translation (1968)
On Proof Interpretations and Linear Logic Proof Interpretations
Other Interpretations
Diller-Nahm (1974) Decidability of prime formulas not required q-realizability (Kleene 1969 / Troelstra 1971) t qr (A → B) ≡ ∀x((x qr A) ∧ A → tx qr B) Related to Kleene’s slash translation (1962) On Proof Interpretations and Linear Logic Proof Interpretations
Other Interpretations
Diller-Nahm (1974) Decidability of prime formulas not required q-realizability (Kleene 1969 / Troelstra 1971) t qr (A → B) ≡ ∀x((x qr A) ∧ A → tx qr B) Related to Kleene’s slash translation (1962) Realizability with truth (Smorynski / Troelstra 1998) t rt (A → B) ≡ ∀x(x rt A → tx rt B) ∧ (A → B) Related to Aczel’s slash translation (1968) On Proof Interpretations and Linear Logic Proof Interpretations
Troelstra on Proof Interpretations
(1971) A. S. Troelstra. Notions of realizability for intuitionistic analysis 2nd Scandinavian Logic Symposium, pp. 369 – 405 (1973) A. S. Troelstra. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis Lecture Notes in Mathematics 344 (1988) A. S. Troelstra and D. van Dalen. Constructivism in Mathematics North-Holland, Amsterdam. Two volumes (1998) A. S. Troelstra. Realizability Handbook of Proof Theory, pages 408–473 On Proof Interpretations and Linear Logic Linear Logic
Outline
1 Proof Interpretations
2 Linear Logic
3 Unified Interpretation of Linear Logic Refinement of intuitionistic implication A → B ≡ !A ( B Refinement of logical connectives conjunction disjunction
additive u t multiplicative ? +
On Proof Interpretations and Linear Logic Linear Logic
Linear Logic (Girard 1987) Explicit treatment of contraction Γ, A, A ` B Γ, !A, !A ` B ⇒ Γ,A ` B Γ, !A ` B Refinement of logical connectives conjunction disjunction
additive u t multiplicative ? +
On Proof Interpretations and Linear Logic Linear Logic
Linear Logic (Girard 1987) Explicit treatment of contraction Γ, A, A ` B Γ, !A, !A ` B ⇒ Γ,A ` B Γ, !A ` B
Refinement of intuitionistic implication A → B ≡ !A ( B On Proof Interpretations and Linear Logic Linear Logic
Linear Logic (Girard 1987) Explicit treatment of contraction Γ, A, A ` B Γ, !A, !A ` B ⇒ Γ,A ` B Γ, !A ` B
Refinement of intuitionistic implication A → B ≡ !A ( B Refinement of logical connectives conjunction disjunction
additive u t multiplicative ? + On Proof Interpretations and Linear Logic Linear Logic
Intuitionistic Linear Logic
Γ ` A Γ ` B Γ ` A Γ ` A u B Γ ` A t B
Γ ` A ∆ ` B Γ,A ` B Γ, ∆ ` A?B Γ ` A ( B Γ ` A Γ ` A[t/z] Γ ` ∀zA Γ ` ∃zA On Proof Interpretations and Linear Logic Linear Logic
Intuitionistic Linear Logic
Γ ` A Γ ` B Γ ` A Γ ` A u B Γ ` A t B
Γ ` A ∆ ` B Γ,A ` B Γ, ∆ ` A?B Γ ` A ( B Γ ` A Γ ` A[t/z] Γ ` ∀zA Γ ` ∃zA On Proof Interpretations and Linear Logic Linear Logic
Structural Rules
A ` A (id)
Γ ` A ∆,A ` B (cut) Γ, ∆ ` B
Γ ` A (per) π{Γ} ` A On Proof Interpretations and Linear Logic Linear Logic
Structural Rules
A ` A (id)
Γ ` A ∆,A ` B (cut) Γ, ∆ ` B
Γ ` A (per) π{Γ} ` A On Proof Interpretations and Linear Logic Linear Logic
Exponential !A
Γ, !A, !A ` B Γ ` B (con) (wkn) Γ, !A ` B Γ, !A ` B
!Γ ` A Γ ` !A (!I) (!E) !Γ ` !A Γ ` A Translation (·)◦ : IL → ILL (A ∧ B)◦ ≡ A◦ ?B◦ (∀xA)◦ ≡ !∀xA◦ (A ∨ B)◦ ≡ A◦ t B◦ (∃xA)◦ ≡ ∃xA◦ ◦ ◦ ◦ (A → B) ≡ !(A ( B ) Thm. IL ` A iff ILL ` A∗ iff ILL ` A◦
On Proof Interpretations and Linear Logic Linear Logic
Girard Translations
Translation (·)∗ : IL → ILL (A ∧ B)∗ ≡ A∗ u B∗ (∀xA)∗ ≡ ∀xA∗ (A ∨ B)∗ ≡ !A∗ t !B∗ (∃xA)∗ ≡ ∃x!A∗ ∗ ∗ ∗ (A → B) ≡ !A ( B Thm. IL ` A iff ILL ` A∗ iff ILL ` A◦
On Proof Interpretations and Linear Logic Linear Logic
Girard Translations
Translation (·)∗ : IL → ILL (A ∧ B)∗ ≡ A∗ u B∗ (∀xA)∗ ≡ ∀xA∗ (A ∨ B)∗ ≡ !A∗ t !B∗ (∃xA)∗ ≡ ∃x!A∗ ∗ ∗ ∗ (A → B) ≡ !A ( B Translation (·)◦ : IL → ILL (A ∧ B)◦ ≡ A◦ ?B◦ (∀xA)◦ ≡ !∀xA◦ (A ∨ B)◦ ≡ A◦ t B◦ (∃xA)◦ ≡ ∃xA◦ ◦ ◦ ◦ (A → B) ≡ !(A ( B ) On Proof Interpretations and Linear Logic Linear Logic
Girard Translations
Translation (·)∗ : IL → ILL (A ∧ B)∗ ≡ A∗ u B∗ (∀xA)∗ ≡ ∀xA∗ (A ∨ B)∗ ≡ !A∗ t !B∗ (∃xA)∗ ≡ ∃x!A∗ ∗ ∗ ∗ (A → B) ≡ !A ( B Translation (·)◦ : IL → ILL (A ∧ B)◦ ≡ A◦ ?B◦ (∀xA)◦ ≡ !∀xA◦ (A ∨ B)◦ ≡ A◦ t B◦ (∃xA)◦ ≡ ∃xA◦ ◦ ◦ ◦ (A → B) ≡ !(A ( B ) Thm. IL ` A iff ILL ` A∗ iff ILL ` A◦ On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Outline
1 Proof Interpretations
2 Linear Logic
3 Unified Interpretation of Linear Logic On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
One-Move Two-Player Games
Game G ≡ (D1,D2,R ⊆ D1 × D2) Two players Eloise and Abelard Two domains of moves
x ∈ D1 and y ∈ D2 Adjudication of Winner Relation R(x, y) between players’ moves x (usually |G|y ) x ∈ {0,..., 5} y ∈ {0,..., 5} x + y is even x ∈ N y ∈ N x ≥ y f ∈ N → N y ∈ N f(y) ≥ y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Examples
Domain 1 Domain 2 Adjudication
x ∈ {0, 1, 2} y ∈ {0, 1, 2} x + 1 = y mod 3 x ∈ N y ∈ N x ≥ y f ∈ N → N y ∈ N f(y) ≥ y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Examples
Domain 1 Domain 2 Adjudication
x ∈ {0, 1, 2} y ∈ {0, 1, 2} x + 1 = y mod 3 x ∈ {0,..., 5} y ∈ {0,..., 5} x + y is even f ∈ N → N y ∈ N f(y) ≥ y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Examples
Domain 1 Domain 2 Adjudication
x ∈ {0, 1, 2} y ∈ {0, 1, 2} x + 1 = y mod 3 x ∈ {0,..., 5} y ∈ {0,..., 5} x + y is even x ∈ N y ∈ N x ≥ y On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Examples
Domain 1 Domain 2 Adjudication
x ∈ {0, 1, 2} y ∈ {0, 1, 2} x + 1 = y mod 3 x ∈ {0,..., 5} y ∈ {0,..., 5} x + y is even x ∈ N y ∈ N x ≥ y f ∈ N → N y ∈ N f(y) ≥ y x A iff ∃x∀y|A|y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Goal
A is true (is provable) iff x Eloise has winning move in game |A|y On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Goal
A is true (is provable) iff x Eloise has winning move in game |A|y
x A iff ∃x∀y|A|y f,g x gx |A ( B|x,w :≡ |A|fxw ( |B|w
f fa |∀zA(z)|y,a :≡ |A(a)|y
x,a x |∃zA(z)|y :≡ |A(a)|y
x v Inductively: Assume |A|y and |B|w defined, then
x,v x v |A?B|y,w :≡ |A|y ? |B|w
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretation
x Formula A mapped to game |A|y f,g x gx |A ( B|x,w :≡ |A|fxw ( |B|w
f fa |∀zA(z)|y,a :≡ |A(a)|y
x,a x |∃zA(z)|y :≡ |A(a)|y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretation
x Formula A mapped to game |A|y x v Inductively: Assume |A|y and |B|w defined, then
x,v x v |A?B|y,w :≡ |A|y ? |B|w f fa |∀zA(z)|y,a :≡ |A(a)|y
x,a x |∃zA(z)|y :≡ |A(a)|y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretation
x Formula A mapped to game |A|y x v Inductively: Assume |A|y and |B|w defined, then
x,v x v |A?B|y,w :≡ |A|y ? |B|w
f,g x gx |A ( B|x,w :≡ |A|fxw ( |B|w x,a x |∃zA(z)|y :≡ |A(a)|y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretation
x Formula A mapped to game |A|y x v Inductively: Assume |A|y and |B|w defined, then
x,v x v |A?B|y,w :≡ |A|y ? |B|w
f,g x gx |A ( B|x,w :≡ |A|fxw ( |B|w
f fa |∀zA(z)|y,a :≡ |A(a)|y On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretation
x Formula A mapped to game |A|y x v Inductively: Assume |A|y and |B|w defined, then
x,v x v |A?B|y,w :≡ |A|y ? |B|w
f,g x gx |A ( B|x,w :≡ |A|fxw ( |B|w
f fa |∀zA(z)|y,a :≡ |A(a)|y
x,a x |∃zA(z)|y :≡ |A(a)|y On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Soundness
Theorem If π ILL ` A there is t (extracted from π) such that ω t ILL ` ∀y|A|y (1) Singleton x x |!A|{y} :≡ |A|y (2) Finite x x |!A|a :≡ ∀y ∈a |A|y (3) Arbitrary x x |!A| :≡ ∀y|A|y
What kind of sets?
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Exponential Game !A
Opponent can choose a “set” a of moves
x x |!A|a :≡ ∀y ∈a |A|y (1) Singleton x x |!A|{y} :≡ |A|y (2) Finite x x |!A|a :≡ ∀y ∈a |A|y (3) Arbitrary x x |!A| :≡ ∀y|A|y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Exponential Game !A
Opponent can choose a “set” a of moves
x x |!A|a :≡ ∀y ∈a |A|y What kind of sets? (2) Finite x x |!A|a :≡ ∀y ∈a |A|y (3) Arbitrary x x |!A| :≡ ∀y|A|y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Exponential Game !A
Opponent can choose a “set” a of moves
x x |!A|a :≡ ∀y ∈a |A|y What kind of sets? (1) Singleton x x |!A|{y} :≡ |A|y (3) Arbitrary x x |!A| :≡ ∀y|A|y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Exponential Game !A
Opponent can choose a “set” a of moves
x x |!A|a :≡ ∀y ∈a |A|y What kind of sets? (1) Singleton x x |!A|{y} :≡ |A|y (2) Finite x x |!A|a :≡ ∀y ∈a |A|y On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Exponential Game !A
Opponent can choose a “set” a of moves
x x |!A|a :≡ ∀y ∈a |A|y What kind of sets? (1) Singleton x x |!A|{y} :≡ |A|y (2) Finite x x |!A|a :≡ ∀y ∈a |A|y (3) Arbitrary x x |!A| :≡ ∀y|A|y (Diller-Nahm) (realizability) (q- or truth-real.)
(Dialectica)
x x |!A|a :≡ !∀y ∈a |A|y x x |!A| :≡ !∀y|A|y x x |!A| :≡ !∀y|A|y ? !A
, possible interpretations of !A
x x |!A|y :≡ !|A|y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretation
x Given A 7→ |A|y (Diller-Nahm) (realizability) (q- or truth-real.)
(Dialectica)
x x |!A|a :≡ !∀y ∈a |A|y x x |!A| :≡ !∀y|A|y x x |!A| :≡ !∀y|A|y ? !A
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretation
x Given A 7→ |A|y , possible interpretations of !A
x x |!A|y :≡ !|A|y (realizability) (q- or truth-real.)
(Dialectica) (Diller-Nahm)
x x |!A| :≡ !∀y|A|y x x |!A| :≡ !∀y|A|y ? !A
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretation
x Given A 7→ |A|y , possible interpretations of !A
x x |!A|y :≡ !|A|y x x |!A|a :≡ !∀y ∈a |A|y (q- or truth-real.)
(Dialectica) (Diller-Nahm) (realizability)
x x |!A| :≡ !∀y|A|y ? !A
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretation
x Given A 7→ |A|y , possible interpretations of !A
x x |!A|y :≡ !|A|y x x |!A|a :≡ !∀y ∈a |A|y x x |!A| :≡ !∀y|A|y (Dialectica) (Diller-Nahm) (realizability) (q- or truth-real.)
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretation
x Given A 7→ |A|y , possible interpretations of !A
x x |!A|y :≡ !|A|y x x |!A|a :≡ !∀y ∈a |A|y x x |!A| :≡ !∀y|A|y x x |!A| :≡ !∀y|A|y ? !A (Diller-Nahm) (realizability) (q- or truth-real.)
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretation
x Given A 7→ |A|y , possible interpretations of !A
x x |!A|y :≡ !|A|y (Dialectica) x x |!A|a :≡ !∀y ∈a |A|y x x |!A| :≡ !∀y|A|y x x |!A| :≡ !∀y|A|y ? !A (realizability) (q- or truth-real.)
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretation
x Given A 7→ |A|y , possible interpretations of !A
x x |!A|y :≡ !|A|y (Dialectica) x x |!A|a :≡ !∀y ∈a |A|y (Diller-Nahm) x x |!A| :≡ !∀y|A|y x x |!A| :≡ !∀y|A|y ? !A (q- or truth-real.)
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretation
x Given A 7→ |A|y , possible interpretations of !A
x x |!A|y :≡ !|A|y (Dialectica) x x |!A|a :≡ !∀y ∈a |A|y (Diller-Nahm) x x |!A| :≡ !∀y|A|y (realizability) x x |!A| :≡ !∀y|A|y ? !A On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretation
x Given A 7→ |A|y , possible interpretations of !A
x x |!A|y :≡ !|A|y (Dialectica) x x |!A|a :≡ !∀y ∈a |A|y (Diller-Nahm) x x |!A| :≡ !∀y|A|y (realizability) x x |!A| :≡ !∀y|A|y ? !A (q- or truth-real.) 6 (·)? forget
? ILL - ILLω x x |!A|y ≡ !|A|y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretations of IL
Dialectica IL - ILω 6 forget
- ILLω x x |!A|y ≡ !|A|y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretations of IL
Dialectica IL - ILω
(·)?
? ILL 6 forget
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretations of IL
Dialectica IL - ILω
(·)?
? ILL - ILLω x x |!A|y ≡ !|A|y On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretations of IL
Dialectica IL - ILω 6 (·)? forget
? ILL - ILLω x x |!A|y ≡ !|A|y 6 (·)∗ forget
? ILL - ILLω x x |!A|a ≡ !∀y ∈a |A|y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretations of IL
Diller-Nahm IL - ILω 6 forget
- ILLω x x |!A|a ≡ !∀y ∈a |A|y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretations of IL
Diller-Nahm IL - ILω
(·)∗
? ILL 6 forget
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretations of IL
Diller-Nahm IL - ILω
(·)∗
? ILL - ILLω x x |!A|a ≡ !∀y ∈a |A|y On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretations of IL
Diller-Nahm IL - ILω 6 (·)∗ forget
? ILL - ILLω x x |!A|a ≡ !∀y ∈a |A|y 6 (·)◦ forget
? ILL - ILLω x x |!A| ≡ !∀y|A|y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretations of IL
Realizability IL - ILω 6 forget
- ILLω x x |!A| ≡ !∀y|A|y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretations of IL
Realizability IL - ILω
(·)◦
? ILL 6 forget
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretations of IL
Realizability IL - ILω
(·)◦
? ILL - ILLω x x |!A| ≡ !∀y|A|y On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretations of IL
Realizability IL - ILω 6 (·)◦ forget
? ILL - ILLω x x |!A| ≡ !∀y|A|y On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretations of IL
Real. with truth IL - ILω 6 (·)◦ forget
? ILL - ILLω x x |!A| ≡ !∀y|A|y ? !A On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Interpretations of IL
q-realizability IL - ILω 6 (·)∗ forget
? ILL - ILLω x x |!A| ≡ !∀y|A|y ? !A New: Connection between A◦ ∼ !A∗ and
Γ|aA iff Γ|kA and Γ ` A
New: Diller-Nahm with truth (and q-variant)
x x |!A|a :≡ ∀y ∈ a |A|y ? !A
New: Hybrid functional interpretation (exploring the fact that !A not canonical in LL)
x x x x |!DA|y :≡ |A|y |!rA|y :≡ ∀y|A|y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Conclusions New: ‘Explanation’ of different interpretations New: Diller-Nahm with truth (and q-variant)
x x |!A|a :≡ ∀y ∈ a |A|y ? !A
New: Hybrid functional interpretation (exploring the fact that !A not canonical in LL)
x x x x |!DA|y :≡ |A|y |!rA|y :≡ ∀y|A|y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Conclusions New: ‘Explanation’ of different interpretations
New: Connection between A◦ ∼ !A∗ and
Γ|aA iff Γ|kA and Γ ` A New: Hybrid functional interpretation (exploring the fact that !A not canonical in LL)
x x x x |!DA|y :≡ |A|y |!rA|y :≡ ∀y|A|y
On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Conclusions New: ‘Explanation’ of different interpretations
New: Connection between A◦ ∼ !A∗ and
Γ|aA iff Γ|kA and Γ ` A
New: Diller-Nahm with truth (and q-variant)
x x |!A|a :≡ ∀y ∈ a |A|y ? !A On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
Conclusions New: ‘Explanation’ of different interpretations
New: Connection between A◦ ∼ !A∗ and
Γ|aA iff Γ|kA and Γ ` A
New: Diller-Nahm with truth (and q-variant)
x x |!A|a :≡ ∀y ∈ a |A|y ? !A
New: Hybrid functional interpretation (exploring the fact that !A not canonical in LL)
x x x x |!DA|y :≡ |A|y |!rA|y :≡ ∀y|A|y On Proof Interpretations and Linear Logic Unified Interpretation of Linear Logic
References
K. G¨odel. Uber¨ eine bisher noch nicht ben¨utzteErweiterung des finiten Standpunktes. Dialectica, 12:280–287, 1958 V. C. V. de Paiva. A Dialectica-like model of linear logic Category Theory and Computer Science, 341–356 (LNCS 389), 1989 A. Blass. A cat. arising in LL, comp. theory, and set theory Advances in Linear Logic, 222, 61–81. LMS Lecture Notes, 1995 J. M. E. Hyland. Proof theory in the abstract Annals of Pure and Applied Logic, 114:43–78, 2002 M. Shirahata. The Dialectica interpretation of first-order CLL Theory and Applications of Categories, 17(4):49–79, 2006 P. Oliva. Unifying functional interpretations Notre Dame Journal of Formal Logic, 47(2):263–290, 2006