Computational Interpretations of Classical Linear Logic
Computational Interpretations of
Classical Linear Logic
Paulo Oliva
Queen Mary, University of London, UK
([email protected])
WoLLIC, Rio de Janeiro
4 July 2007
Computational Interpretations of Classical Linear Logic
Outline
1
Introduction
Functional Interpretations of IL
Linear Logic
2
Functional Interpretation of LL
Motivation: Games
The Interpretation
Relation to Interpretations of IL
3
Conclusions
Characterisation
Summary
Computational Interpretations of Classical Linear Logic
Introduction
Outline
1
Introduction
Functional Interpretations of IL
Linear Logic
2
Functional Interpretation of LL
Motivation: Games
The Interpretation
Relation to Interpretations of IL
3
Conclusions
Characterisation
Summary
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Proof Interpretations
(
Syntax
e.g. “blue car”
⇒
Semantics
real blue cars
)
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Proof Interpretations
(
Syntax
e.g. “blue car”
(
Syntax
e.g. Java
⇒
⇒
Semantics
)
real blue cars
Syntax
Machine code
)
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Proof Interpretations
(
Syntax
e.g. “blue car”
(
Syntax
e.g. Java


 Proofs
e.g. CL


e.g. PA
⇒
⇒
⇒
Semantics
)
real blue cars
Syntax
)
Machine code

Proofs 

IL


PRA
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Functional Interpretations
Formula A
⇒ Set of functionals |A|
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Functional Interpretations
Formula A
Proof π
⇒ Set of functionals |A|
⇒ Functional fπ ∈ |A|
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Functional Interpretations
Formula A
⇒ Set of functionals |A|
Proof π
⇒ Functional fπ ∈ |A|
S 1 `π A
⇒ S2 ` fπ ∈ |A|
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Functional Interpretations
Formula A
∀x∃y(y > x)
⇒ Set of functionals |A|
{f : f x > x}
Proof π
⇒ Functional fπ ∈ |A|
S 1 `π A
⇒ S2 ` fπ ∈ |A|
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Functional Interpretations
Formula A
∀x∃y(y > x)
Proof π
...
S 1 `π A
⇒ Set of functionals |A|
{f : f x > x}
⇒ Functional fπ ∈ |A|
λx.x + 1
⇒ S2 ` fπ ∈ |A|
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Functional Interpretations
Dialectica
Gödel 1958
Relative consistency of arithmetic
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Functional Interpretations
Dialectica
Gödel 1958
Relative consistency of arithmetic
(
Falsity interpreted as empty set (| ⊥ | ≡ ∅)
PA `⊥ ⇒
PRAω ` ∃f (f ∈ ∅)
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Functional Interpretations
Dialectica
Gödel 1958
Relative consistency of arithmetic
(
Falsity interpreted as empty set (| ⊥ | ≡ ∅)
PA `⊥ ⇒
PRAω ` ∃f (f ∈ ∅)
Modified realizability
Kreisel 1959
Independence results for IL
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Functional Interpretations
Dialectica
Gödel 1958
Relative consistency of arithmetic
(
Falsity interpreted as empty set (| ⊥ | ≡ ∅)
PA `⊥ ⇒
PRAω ` ∃f (f ∈ ∅)
Modified realizability
Kreisel 1959
Independence results for IL
(
|P | set of non-computable functionals
IL ` P
⇒
IL ` ∃f (f ∈ |P |)
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Functional Interpretations
Gödel’s Dialectica interpretation
Kreisel’s modified realizability
Diller-Nahm interpretation
Stein’s family of interpretations
Monotone variants of the above (Kohlenbach)
Bounded Dialectica interpretation (Ferreira/O.)
Bounded modified realizability (Ferreira)
...
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Functional Interpretations
Gödel’s Dialectica interpretation
Kreisel’s modified realizability
Diller-Nahm interpretation
Stein’s family of interpretations
Monotone variants of the above (Kohlenbach)
Bounded Dialectica interpretation (Ferreira/O.)
Bounded modified realizability (Ferreira)
...
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Functional Interpretations
Gödel’s Dialectica interpretation
Kreisel’s modified realizability
Diller-Nahm interpretation
Stein’s family of interpretations
Monotone variants of the above (Kohlenbach)
Bounded Dialectica interpretation (Ferreira/O.)
Bounded modified realizability (Ferreira)
...
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Functional Interpretations
Gödel’s Dialectica interpretation
Kreisel’s modified realizability
Diller-Nahm interpretation
Stein’s family of interpretations
Monotone variants of the above (Kohlenbach)
Bounded Dialectica interpretation (Ferreira/O.)
Bounded modified realizability (Ferreira)
...
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Functional Interpretations
Gödel’s Dialectica interpretation
Kreisel’s modified realizability
Diller-Nahm interpretation
Stein’s family of interpretations
Monotone variants of the above (Kohlenbach)
Bounded Dialectica interpretation (Ferreira/O.)
Bounded modified realizability (Ferreira)
...
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Goal
Understand different functional interpretations
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Goal
Understand different functional interpretations
Functional interpretation of a refinement of IL
Computational Interpretations of Classical Linear Logic
Introduction
Functional Interpretations of IL
Goal
Understand different functional interpretations
Functional interpretation of a refinement of IL
Linear logic
Computational Interpretations of Classical Linear Logic
Introduction
Linear Logic
Linear Logic (Girard 1987)
Explicit treatment of contraction
Γ, A, A ` B
Γ, A ` B
⇒
Γ, !A, !A ` B
Γ, !A ` B
Computational Interpretations of Classical Linear Logic
Introduction
Linear Logic
Linear Logic (Girard 1987)
Explicit treatment of contraction
Γ, A, A ` B
Γ, A ` B
⇒
Γ, !A, !A ` B
Γ, !A ` B
Refinement of intuitionistic implication
A→B
≡ !A ( B
Computational Interpretations of Classical Linear Logic
Introduction
Linear Logic
Linear Logic (Girard 1987)
Explicit treatment of contraction
Γ, A, A ` B
Γ, A ` B
⇒
Γ, !A, !A ` B
Γ, !A ` B
Refinement of intuitionistic implication
A→B
≡ !A ( B
Refinement of logical connectives
conjunction disjunction
additive
∧
multiplicative
⊗
∨
O
Computational Interpretations of Classical Linear Logic
Introduction
Linear Logic
Linear Logic: Duality
(A ∨ B)⊥ ≡ A⊥ ∧ B ⊥
(∃zA)⊥ ≡ ∀zA⊥
(A ∧ B)⊥ ≡ A⊥ ∨ B ⊥
(∀zA)⊥ ≡ ∃zA⊥
Computational Interpretations of Classical Linear Logic
Introduction
Linear Logic
Linear Logic: Duality
(A ∨ B)⊥ ≡ A⊥ ∧ B ⊥
(∃zA)⊥ ≡ ∀zA⊥
(A ∧ B)⊥ ≡ A⊥ ∨ B ⊥
(∀zA)⊥ ≡ ∃zA⊥
(A O B)⊥ ≡ A⊥ ⊗ B ⊥
(?A)⊥ ≡ !(A⊥ )
(A ⊗ B)⊥ ≡ A⊥ O B ⊥
(!A)⊥
≡ ?(A⊥ )
Computational Interpretations of Classical Linear Logic
Introduction
Linear Logic
Linear Logic: Duality
(A ∨ B)⊥ ≡ A⊥ ∧ B ⊥
(∃zA)⊥ ≡ ∀zA⊥
(A ∧ B)⊥ ≡ A⊥ ∨ B ⊥
(∀zA)⊥ ≡ ∃zA⊥
(A O B)⊥ ≡ A⊥ ⊗ B ⊥
(?A)⊥ ≡ !(A⊥ )
(A ⊗ B)⊥ ≡ A⊥ O B ⊥
(!A)⊥
A ( B ≡ A⊥ O B
AOB
≡ A⊥ ( B
≡ ?(A⊥ )
Computational Interpretations of Classical Linear Logic
Introduction
Linear Logic
Linear Logic: Structural
Γ ` A ∆, A ` B
(cut)
Γ, ∆ ` B
Γ, A ` B
(⊥)
Γ, B ⊥ ` A⊥
A`A
(id)
Γ`A
(per)
π{Γ} ` A
Computational Interpretations of Classical Linear Logic
Introduction
Linear Logic
Linear Logic: Structural
Γ ` A ∆, A ` B
(cut)
Γ, ∆ ` B
Γ, A ` B
(⊥)
Γ, B ⊥ ` A⊥
A`A
(id)
Γ`A
(per)
π{Γ} ` A
Computational Interpretations of Classical Linear Logic
Introduction
Linear Logic
Linear Logic: Connectives and Quantifiers
Γ`A Γ`B
Γ`A∧B
Γ`A
Γ`A∨B
Γ`A ∆`B
Γ, ∆ ` A ⊗ B
Γ, A ` B
Γ`A(B
Γ`A
Γ ` ∀zA
Γ ` A[t/z]
Γ ` ∃zA
Computational Interpretations of Classical Linear Logic
Introduction
Linear Logic
Linear Logic: Connectives and Quantifiers
Γ`A Γ`B
Γ`A∧B
Γ`A
Γ`A∨B
Γ`A ∆`B
Γ, ∆ ` A ⊗ B
Γ, A ` B
Γ`A(B
Γ`A
Γ ` ∀zA
Γ ` A[t/z]
Γ ` ∃zA
Computational Interpretations of Classical Linear Logic
Introduction
Linear Logic
Linear Logic: Modalities
Γ`B
Γ, !A, !A ` B
(con)
(wkn)
Γ, !A ` B
Γ, !A ` B
!Γ ` A
(!)
!Γ ` !A
Γ`A
(?)
Γ ` ?A
Computational Interpretations of Classical Linear Logic
Functional Interpretation of LL
Outline
1
Introduction
Functional Interpretations of IL
Linear Logic
2
Functional Interpretation of LL
Motivation: Games
The Interpretation
Relation to Interpretations of IL
3
Conclusions
Characterisation
Summary
Mathematicians are happy with proof or counter-example
Mathematicians are happy with proof or counter-example
Mathematics is like a game,
mathematicians are always winners
because they play both roles
Mathematicians are happy with proof or counter-example
Mathematics is like a game,
mathematicians are always winners
because they play both roles
View a mathematical statement as the description of a game
∀n ≥ 2 ∃x, y, z(
4
1 1 1
= + + )
n
x y z
Paul Erdös
∀n ≥ 2 ∃x, y, z(
4
1 1 1
= + + )
n
x y z
Paul Erdös
f0 , f1 , f2 : N → N∗
n ∈ {2, . . .}
∀n ≥ 2 ∃x, y, z(
4
1 1 1
= + + )
n
x y z
Paul Erdös
f0 , f1 , f2 : N → N∗
n ∈ {2, . . .}
4
1
1
1
=
+
+
n
f0 (n) f1 (n) f2 (n)
Games: Formal Description
Game G ≡ (D1 , D2 , R ⊆ D1 × D2 )
Games: Formal Description
Game G ≡ (D1 , D2 , R ⊆ D1 × D2 )
Two players
Eloise and Abelard
Two domains of moves
x ∈ D1 and y ∈ D2
Games: Formal Description
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
(usually |G|xy )
Computational Interpretations of Classical Linear Logic
Functional Interpretation of LL
Motivation: Games
Games: Examples
Domain 1
Domain 2
Adjudication
x ∈ {0, 1, 2}
y ∈ {0, 1, 2}
x + 1 = y mod 3
Computational Interpretations of Classical Linear Logic
Functional Interpretation of LL
Motivation: Games
Games: 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
Computational Interpretations of Classical Linear Logic
Functional Interpretation of LL
Motivation: Games
Games: 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∈N
y∈N
x + y is even
x≥y
Computational Interpretations of Classical Linear Logic
Functional Interpretation of LL
Motivation: Games
Games: 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
Computational Interpretations of Classical Linear Logic
Functional Interpretation of LL
Motivation: Games
Games: 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
fi ∈ N → N∗
n≥2
4
n
=
1
f0 n
+
1
f1 n
+
1
f2 n
Computational Interpretations of Classical Linear Logic
Functional Interpretation of LL
Motivation: Games
Goal
A is true (is provable)
iff
Eloise has winning move in game |A|xy
Computational Interpretations of Classical Linear Logic
Functional Interpretation of LL
Motivation: Games
Symmetry
Game A⊥ should be game A with roles reversed
|A⊥ |xy ≡ ¬|A|yx
|(A⊥ )⊥ |xy ≡ |A|xy
Computational Interpretations of Classical Linear Logic
Functional Interpretation of LL
The Interpretation
Interpretation
|A ⊗ B|x,v
f,g
:≡ |A|xfv and |B|vgx
Computational Interpretations of Classical Linear Logic
Functional Interpretation of LL
The Interpretation
Interpretation
|A ⊗ B|x,v
f,g
:≡ |A|xfv and |B|vgx
x
gx
|A ( B|f,g
x,w :≡ |A|f w implies |B|w
Computational Interpretations of Classical Linear Logic
Functional Interpretation of LL
The Interpretation
Interpretation
|A ⊗ B|x,v
f,g
:≡ |A|xfv and |B|vgx
x
gx
|A ( B|f,g
x,w :≡ |A|f w implies |B|w
|∀zA(z)|fy,z
:≡ |A(z)|fy z
Computational Interpretations of Classical Linear Logic
Functional Interpretation of LL
The Interpretation
Interpretation
|A ⊗ B|x,v
f,g
:≡ |A|xfv and |B|vgx
x
gx
|A ( B|f,g
x,w :≡ |A|f w implies |B|w
|∀zA(z)|fy,z
:≡ |A(z)|fy z
|∃zA(z)|x,z
f
:≡ |A(z)|xfz
Exponential Games
One of the players plays first (break symmetry)
Exponential Games
One of the players plays first (break symmetry)
The other player responds with (multiple copies of game)
Exponential Games
One of the players plays first (break symmetry)
The other player responds with (multiple copies of game)
(1) unlimited number of copies
(
|!A|x :≡ ∀y|A|xy
|?A|y :≡ ∃x|A|xy
Exponential Games
One of the players plays first (break symmetry)
The other player responds with (multiple copies of game)
(1) unlimited number of copies
(
|!A|x :≡ ∀y|A|xy
|?A|y :≡ ∃x|A|xy
(2) finitely many copies
(
|!A|x :≡ ∀y ∈ f x|A|xy
|?A|y :≡ ∃x ∈ f y|A|xy
Exponential Games
One of the players plays first (break symmetry)
The other player responds with (multiple copies of game)
(1) unlimited number of copies
(
|!A|x :≡ ∀y|A|xy
|?A|y :≡ ∃x|A|xy
(2) finitely many copies
(
|!A|x :≡ ∀y ∈ f x|A|xy
|?A|y :≡ ∃x ∈ f y|A|xy
(3) no copying allowed (make single move)
(
|!A|xf :≡ |A|xfx
|?A|fy :≡ |A|fy y
Computational Interpretations of Classical Linear Logic
Functional Interpretation of LL
The Interpretation
Soundness
Theorem
If
Γ ` A is provable in linear logic
then
Eloise wins game Γ ( A
Computational Interpretations of Classical Linear Logic
Functional Interpretation of LL
The Interpretation
Soundness
Theorem
If
Γ ` A is provable in linear logic
then
Eloise wins game Γ ( A , i.e.
she has moves t, r such that for all v, y
|Γ|vr(y) ` |A|t(v)
y
IL
Q
Q
Q
Q
?
Q
Q
(·)
Q
Q
Q
?
LL
Q
Q
s
Q
-
G
Modified realizability
IL Q
(Kreisel’1959)
Q
Q
Q
?
Q
Q
(·)
Q
Q
Q
Q
Q
s
Q
-
?
LL
(1) ∀y|A|xy
G
Diller-Nahm interpretation (Diller-Nahm’1974)
Modified realizability
IL Q
(Kreisel’1959)
Q
Q
Q
?
Q
Q
(·)
Q
Q
Q
Q
Q
s
Q
-
?
LL
(1) ∀y|A|xy
(2) ∀y ∈ f x |A|xy
G
Dialectica interpretation
(Gödel’1958)
Diller-Nahm interpretation (Diller-Nahm’1974)
Modified realizability
IL Q
(Kreisel’1959)
Q
Q
Q
?
Q
Q
(·)
Q
Q
Q
Q
Q
s
Q
-
?
LL
(1) ∀y|A|xy
(2) ∀y ∈ f x |A|xy
(3) |A|xfx
G
Computational Interpretations of Classical Linear Logic
Functional Interpretation of LL
Relation to Interpretations of IL
Parametrised Interpretation
∗
∗
Bounded quantifiers ∀xρ @ aρ A and ∃xρ @ aρ A
(∀x @ a A)⊥ ≡ ∃x @ a A⊥
(∃x @ a A)⊥ ≡ ∀x @ a A⊥
Parametrised interpretation:
|!A|xf :≡ ∀y @ f x |A|xy
|?A|fy :≡ ∃x @ f y |A|xy
Computational Interpretations of Classical Linear Logic
Functional Interpretation of LL
Relation to Interpretations of IL
Parametrised Interpretation
Γ`B
Γ, !A ` B
Γ, !A, !A ` B
Γ, !A ` B
cA :
ρ∗
εA : ρ∗ × ρ∗ ,→
ρ∗
Γ`A
Γ `?A
ηA :
ρ∗
!Γ ` A
!Γ `!A
µA : ρ → τ ∗ ,→ ρ∗ → τ ∗
ρ
,→
Computational Interpretations of Classical Linear Logic
Conclusions
Outline
1
Introduction
Functional Interpretations of IL
Linear Logic
2
Functional Interpretation of LL
Motivation: Games
The Interpretation
Relation to Interpretations of IL
3
Conclusions
Characterisation
Summary
Computational Interpretations of Classical Linear Logic
Conclusions
Characterisation
Characterisation
We have seen that if A is provable in LL then
Eloise has a winning move
Computational Interpretations of Classical Linear Logic
Conclusions
Characterisation
Characterisation
We have seen that if A is provable in LL then
Eloise has a winning move
What about the other way around?
Computational Interpretations of Classical Linear Logic
Conclusions
Characterisation
Characterisation
We have seen that if A is provable in LL then
Eloise has a winning move
What about the other way around?
For which extension of LL do we have the
converse?
Computational Interpretations of Classical Linear Logic
Conclusions
Characterisation
Characterisation (modified realizability)
|A ⊗ B|x,v
f,g
:≡ |A|xfv ⊗ |B|vgx
x
gx
|A ( B|f,g
x,w :≡ |A|f w ( |B|w
|∀zA(z)|fy,z
:≡ |A(z)|fy z
|∃zA(z)|x,z
f
:≡ |A(z)|xfz
|!A|x
:≡ !∀y|A|xy
|?A|y
:≡ ?∃x|A|xy
Computational Interpretations of Classical Linear Logic
Conclusions
Characterisation
Characterisation (modified realizability)
t[v]
|Γ|vr[y] ` |A|y
Computational Interpretations of Classical Linear Logic
Conclusions
Characterisation
Characterisation (modified realizability)
t[v]
|Γ|vr[y] ` |A|y
∀w∃v|Γ|vw `
∃x∀y|A|xy
Computational Interpretations of Classical Linear Logic
Conclusions
Characterisation
Characterisation (modified realizability)
t[v]
|Γ|vr[y] ` |A|y
axiom v
∀w∃v|Γ|w ` ∃x∀y|A|xy
Computational Interpretations of Classical Linear Logic
Conclusions
Characterisation
Characterisation (modified realizability)
t[v]
|Γ|vr[y] ` |A|y
axiom v
∀w∃v|Γ|w ` ∃x∀y|A|xy
J
J
J
J
J
^
J
∃v∀w|Γ|vw ` ∀y∃x|A|xy
Computational Interpretations of Classical Linear Logic
Conclusions
Characterisation
Characterisation (modified realizability)
t[v]
|Γ|vr[y] ` |A|y
axiom v
∀w∃v|Γ|w ` ∃x∀y|A|xy
J
J
J cut rule
J
J
^
J
∃v∀w|Γ|vw ` ∀y∃x|A|xy
Computational Interpretations of Classical Linear Logic
Conclusions
Characterisation
Characterisation (modified realizability)
t[v]
|Γ|vr[y] ` |A|y
axiom v
∀w∃v|Γ|w ` ∃x∀y|A|xy
J
J
J cut rule
J
J
^
J
∃v∀w|Γ|vw ` ∀y∃x|A|xy
?
`
x
x
y |A|y
Æ
v
v
w |Γ|w
Æ
Computational Interpretations of Classical Linear Logic
Conclusions
Characterisation
Simultaneous Quantifier
Æ
A0 (a0 , y0 ), . . . , An (an , yn )
x0
xn
y0 A0 (x0 , y0 ), . . . , yn An (xn , yn )
( )
Æ
Æ
yi may only appear free in the terms aj , for j 6= i
Computational Interpretations of Classical Linear Logic
Conclusions
Characterisation
Simultaneous Quantifier
Æ
A0 (a0 , y0 ), . . . , An (an , yn )
x0
xn
y0 A0 (x0 , y0 ), . . . , yn An (xn , yn )
( )
Æ
Æ
yi may only appear free in the terms aj , for j 6= i
(x = y), (x 6= y)
( )
= y), xy (x 6= y)
Æ
Æ
x
y (x
Æ
New Principles
f
y,z A(f z, y, z)
Æ
Sequential choice
∀z yx A(x, y, z) (
Æ
New Principles
f
y,z A(f z, y, z)
Æ
Æ
Parallel choice
x
v
y A(x) O w B(v) (
f,g
y,w (A(f w)
Æ
Sequential choice
∀z yx A(x, y, z) (
O B(gy))
Æ
Æ
New Principles
f
y,z A(f z, y, z)
Æ
Æ
Parallel choice
x
v
y A(x) O w B(v) (
Æ
Æ
Trump advantage
! yx A ( ∃x!∀yA
f,g
y,w (A(f w)
Æ
Sequential choice
∀z yx A(x, y, z) (
O B(gy))
Æ
New Principles
f
y,z A(f z, y, z)
Æ
Æ
Parallel choice
x
v
y A(x) O w B(v) (
f,g
y,w (A(f w)
Æ
Sequential choice
∀z yx A(x, y, z) (
O B(gy))
Æ
Æ
Trump advantage
! yx A ( ∃x!∀yA
Æ
Theorem
These are sufficient for deriving the equivalence
between A and its interpretation yx |A|xy .
Æ
Computational Interpretations of Classical Linear Logic
Conclusions
Summary
Summary
Functional interpretations of linear logic
Usual interpretations of IL derivable
Interesting use of (simple) branching quantifier
Characterisation using branching quantifier
Sound extensions of linear logic