SLIDE 1
A Fibrational Framework for Substructural and Modal Logics
Dan Licata Wesleyan University Michael Shulman University of San Diego
1
Mitchell Riley Wesleyan University
2
A Fibrational Framework for Substructural and Modal Logics Dan - - PowerPoint PPT Presentation
A Fibrational Framework for Substructural and Modal Logics Dan - - PowerPoint PPT Presentation
A Fibrational Framework for Substructural and Modal Logics Dan Licata Michael Shulman Wesleyan University University of San Diego Mitchell Riley Wesleyan University 1 Substructural Logic B Weakening ,x:A B ,y:B,x:A B
SLIDE 2
SLIDE 3
Modal Logic
Γ ⊢ ☐A ∅ ⊢ A ♢A ⊢ ♢C A ⊢ ♢C
SLIDE 4
Modal Logic
Γ ⊢ ☐A ∅ ⊢ A ♢A ⊢ ♢C A ⊢ ♢C (♢A) × B vs. ♢(A × B)
SLIDE 5
4
Intuitionistic substructural and modal logics/type systems
Linear/affine: use once (state, sessions) Relevant: strictness annotations Ordered: linguistics Bunched: separation logic Comonads: staging, metavariables, coeffects Monads: effects Interactions between products and modalities
SLIDE 6
5
Homotopy type theory
higher category theory homotopy theory dependent type theory
SLIDE 7
6
Cohesive HoTT
♭ ∫ # ⊣ ⊣
Dependent type theory with modalities ∫A, ♭A, #A [Shulman,Schreiber]
SLIDE 8
6
Cohesive HoTT
♭ ∫ # ⊣ ⊣
Dependent type theory with modalities ∫A, ♭A, #A [Shulman,Schreiber]
♭A ⊢ B A ⊢ # B
SLIDE 9
6
Cohesive HoTT
♭ ∫ # ⊣ ⊣
Dependent type theory with modalities ∫A, ♭A, #A [Shulman,Schreiber] monad ♢
♭A ⊢ B A ⊢ # B
SLIDE 10
6
Cohesive HoTT
♭ ∫ # ⊣ ⊣
Dependent type theory with modalities ∫A, ♭A, #A [Shulman,Schreiber] monad ♢ comonad ☐
♭A ⊢ B A ⊢ # B
SLIDE 11
6
Cohesive HoTT
♭ ∫ # ⊣ ⊣
Dependent type theory with modalities ∫A, ♭A, #A [Shulman,Schreiber] monad ♢ comonad ☐ monad
♭A ⊢ B A ⊢ # B
SLIDE 12
6
Cohesive HoTT
♭ ∫ # ⊣ ⊣
Dependent type theory with modalities ∫A, ♭A, #A [Shulman,Schreiber] monad ♢ comonad ☐ monad (idempotent)
♭♭ A ≅ ♭ A ♭A ⊢ B A ⊢ # B
SLIDE 13
7
𝕋-Cohesion
♮ ♮ ⊣
comonad monad [Finster,L.,Morehouse,Riley]
SLIDE 14
7
𝕋-Cohesion
♮ ♮ ⊣
comonad monad
A ⊢♮A ⊢ A
[Finster,L.,Morehouse,Riley]
SLIDE 15
7
𝕋-Cohesion
♮ ♮ ⊣
comonad monad
A ⊢♮A ⊢ A
[Finster,L.,Morehouse,Riley]
♮(A ⋀ B) ≅ ♮A ×♮B
SLIDE 16
8
Differential Cohesion
[Schreiber; Gross,L.,New,Paykin,Riley,Shulman,Wellen]
♭ ∫ # ⊣ ⊣
SLIDE 17
8
Differential Cohesion
[Schreiber; Gross,L.,New,Paykin,Riley,Shulman,Wellen]
♭ ∫ # ⊣ ⊣ 𝖪 R & ⊣ ⊣
SLIDE 18
8
Differential Cohesion
[Schreiber; Gross,L.,New,Paykin,Riley,Shulman,Wellen]
♭ ∫ # ⊣ ⊣ 𝖪 R & ⊣ ⊣
HoTT/UF Saturday!
SLIDE 19
9
What are the common patterns in substructural and modal logics?
SLIDE 20
S4 ☐
Γ ; Δ ⊢☐A Γ; Δ,☐A ⊢ C Γ,A ; Δ,☐A ⊢ C Γ ; ⋅ ⊢ A Γ,A; Δ ⊢ C Γ,A; Δ,A ⊢ C Γ; Δ ⊢ C
SLIDE 21
S4 ☐
Γ ; Δ ⊢☐A Γ; Δ,☐A ⊢ C Γ,A ; Δ,☐A ⊢ C Γ ; ⋅ ⊢ A Γ,A; Δ ⊢ C Γ,A; Δ,A ⊢ C
morally ☐Γ× Δ → C
Γ; Δ ⊢ C
SLIDE 22
S4 ☐
Γ ; Δ ⊢☐A Γ; Δ,☐A ⊢ C Γ,A ; Δ,☐A ⊢ C Γ ; ⋅ ⊢ A Γ,A; Δ ⊢ C Γ,A; Δ,A ⊢ C
context is all boxed formulae (up to weakening) morally ☐Γ× Δ → C
Γ; Δ ⊢ C
SLIDE 23
S4 ☐
Γ ; Δ ⊢☐A Γ; Δ,☐A ⊢ C Γ,A ; Δ,☐A ⊢ C Γ ; ⋅ ⊢ A Γ,A; Δ ⊢ C Γ,A; Δ,A ⊢ C
context is all boxed formulae (up to weakening) morally ☐Γ× Δ → C
Γ; Δ ⊢ C
because ☐A → A
SLIDE 24
Γ ; ⋅ ⊢ ! A Γ; Δ,!A ⊢ C Γ,A ; Δ ⊢ C Γ ; ⋅ ⊢ A Γ,A; Δ ⊢ C Γ,A; Δ,A ⊢ C
Linear Logic !
context is all !’ed formulae (no weakening) cartesian/ structural linear
SLIDE 25
Linear Logic ⊗
Γ,Δ ⊢ A ⊗ B Γ ⊢ A Δ ⊢ B Γ,A⊗B ⊢ C Γ,A,B ⊢ C
SLIDE 26
Linear Logic ⊗
Γ,Δ ⊢ A ⊗ B Γ ⊢ A Δ ⊢ B Γ,A⊗B ⊢ C Γ,A,B ⊢ C
context “is” a ⊗
SLIDE 27
Linear Logic ⊗
Γ,Δ ⊢ A ⊗ B Γ ⊢ A Δ ⊢ B Γ,A⊗B ⊢ C Γ,A,B ⊢ C
context “is” a ⊗
Γ0 ⊢ A ⊗ B Γ ⊢ A Δ ⊢ B Γ0 ≡ Γ, Δ
context is a ⊗, up to some structural rules
SLIDE 28
Types inherit properties
A,B ⊢ B ⊗ A A ⊢ A B ⊢ B A,B ≡ B,A A⊗B ⊢ B ⊗ A
SLIDE 29
14
Types inherit properties
⋅ ; ! A ⊗ ! B ⊢ ! (A ⊗ B) ⋅; ! A, ! B ⊢ ! (A ⊗ B) A ; ! B ⊢ ! (A ⊗ B) A,B ; ⋅ ⊢ ! (A ⊗ B) A,B; ⋅ ⊢ A ⊗ B
SLIDE 30
15
Pattern for ☐ ! ⊗
Operation on contexts, with explicit or admissible structural properties Type constructor that “internalizes” the context
- peration, inherits the structural properties
SLIDE 31
16
This paper
Products and left-adjoints (⊗,F) all one connective Negatives and right adjoints (⊸,U) another Cut elimination for all instances at once Equational theory: differ by structural rules Categorical semantics A framework that abstracts the common aspects of many intuitionistic substructural and modal logics
SLIDE 32
17
Examples
Non-assoc, ordered, linear, affine, relevant, cartesian, bunched products and implications N-linear variables [Reed,Abel,McBride] Monoidal, lax, non- left adjoints Non-strong, strong, ☐-strong monads Cohesion
SLIDE 33
17
Examples
Non-assoc, ordered, linear, affine, relevant, cartesian, bunched products and implications N-linear variables [Reed,Abel,McBride] Monoidal, lax, non- left adjoints Non-strong, strong, ☐-strong monads Cohesion Logical adequacy: sequent is provable iff its encoding is
SLIDE 34
18
Closely Related Work
SLIDE 35
18
Closely Related Work
Display logic, Lambek calculus, resource semantics
SLIDE 36
18
Closely Related Work
Display logic, Lambek calculus, resource semantics Adjoint linear logic [Benton&Wadler,95]
SLIDE 37
18
Closely Related Work
Display logic, Lambek calculus, resource semantics Adjoint linear logic [Benton&Wadler,95]
cartesian linear
F U ⊣
SLIDE 38
18
Closely Related Work
Display logic, Lambek calculus, resource semantics Adjoint linear logic [Benton&Wadler,95]
cartesian linear
F U ⊣ A ::= F C | A ⊗ B | … C ::= U A | C × D | …
SLIDE 39
18
Closely Related Work
Display logic, Lambek calculus, resource semantics Adjoint linear logic [Benton&Wadler,95]
cartesian linear
F U ⊣ !A := FU A A ::= F C | A ⊗ B | … C ::= U A | C × D | …
SLIDE 40
18
Closely Related Work
Display logic, Lambek calculus, resource semantics Adjoint linear logic [Benton&Wadler,95] λ-calculus for Resource Separation [Atkey,2004] Adjoint logic [Reed,2009]
cartesian linear
F U ⊣ !A := FU A A ::= F C | A ⊗ B | … C ::= U A | C × D | …
SLIDE 41
19
Technique
Γ ⊢α A ψ ⊢ α : p Sequent Context descriptor
A substructural/modal typing judgement is an ordinary structural judgement, annotated with a term that describes the tree structure of the context
SLIDE 42
20
Sequent Calculus
SLIDE 43
21
Mode Theory
x1:p1, … , xn:pn ⊢ α : q α ⇒ β p,q,… Modes Context Descriptors Structural Properties
SLIDE 44
21
Mode Theory
Types p,q are “modes” of types/contexts x1:p1, … , xn:pn ⊢ α : q α ⇒ β p,q,… Modes Context Descriptors Structural Properties
SLIDE 45
21
Mode Theory
Types p,q are “modes” of types/contexts Terms α are descriptions of the context x1:p1, … , xn:pn ⊢ α : q α ⇒ β p,q,… Modes Context Descriptors Structural Properties
SLIDE 46
21
Mode Theory
Types p,q are “modes” of types/contexts Terms α are descriptions of the context “Transformations” α ⇒ β are structural properties x1:p1, … , xn:pn ⊢ α : q α ⇒ β p,q,… Modes Context Descriptors Structural Properties
SLIDE 47
21
Mode Theory
Types p,q are “modes” of types/contexts Terms α are descriptions of the context “Transformations” α ⇒ β are structural properties x1:p1, … , xn:pn ⊢ α : q α ⇒ β p,q,… Modes Context Descriptors Structural Properties
cartesian/ structural
SLIDE 48
22
a:A,b:B,c:C,d:D ⊢(a ⊗ b) ⊗ (c ⊗ d) X
SLIDE 49
22
a:A,b:B,c:C,d:D ⊢(a ⊗ b) ⊗ (c ⊗ d) X
Non-associative logic: no equations/transformations
SLIDE 50
22
a:A,b:B,c:C,d:D ⊢(a ⊗ b) ⊗ (c ⊗ d) X
Non-associative logic: no equations/transformations Ordered logic: (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c)
SLIDE 51
22
a:A,b:B,c:C,d:D ⊢(a ⊗ b) ⊗ (c ⊗ d) X
Non-associative logic: no equations/transformations Ordered logic: (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c) Linear logic: a ⊗ b = b ⊗ a
SLIDE 52
22
a:A,b:B,c:C,d:D ⊢(a ⊗ b) ⊗ (c ⊗ d) X
Non-associative logic: no equations/transformations Ordered logic: (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c) Linear logic: a ⊗ b = b ⊗ a Relevant logic: a ⟹ a ⊗ a
SLIDE 53
22
a:A,b:B,c:C,d:D ⊢(a ⊗ b) ⊗ (c ⊗ d) X
Non-associative logic: no equations/transformations Ordered logic: (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c) Linear logic: a ⊗ b = b ⊗ a Relevant logic: a ⟹ a ⊗ a Affine logic: a ⟹ 1
SLIDE 54
22
a:A,b:B,c:C,d:D ⊢(a ⊗ b) ⊗ (c ⊗ d) X
Non-associative logic: no equations/transformations Ordered logic: (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c) Linear logic: a ⊗ b = b ⊗ a Relevant logic: a ⟹ a ⊗ a Affine logic: a ⟹ 1 BI: two function symbols ✻ and ⋀: (a ✻ b) ⋀ (c ✻ d)
SLIDE 55
22
a:A,b:B,c:C,d:D ⊢(a ⊗ b) ⊗ (c ⊗ d) X
Non-associative logic: no equations/transformations Ordered logic: (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c) Linear logic: a ⊗ b = b ⊗ a Relevant logic: a ⟹ a ⊗ a Affine logic: a ⟹ 1 BI: two function symbols ✻ and ⋀: (a ✻ b) ⋀ (c ✻ d) Modalities: unary function symbols: r(a) ⊗ r(b) ⊗ c ⊗ d
SLIDE 56
22
a:A,b:B,c:C,d:D ⊢(a ⊗ b) ⊗ (c ⊗ d) X
cartesian
Non-associative logic: no equations/transformations Ordered logic: (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c) Linear logic: a ⊗ b = b ⊗ a Relevant logic: a ⟹ a ⊗ a Affine logic: a ⟹ 1 BI: two function symbols ✻ and ⋀: (a ✻ b) ⋀ (c ✻ d) Modalities: unary function symbols: r(a) ⊗ r(b) ⊗ c ⊗ d
SLIDE 57
23
Weakening over weakening
SLIDE 58
23
Weakening over weakening
Γ,x:A ⊢α B Γ ⊢α B
SLIDE 59
23
Weakening over weakening
Γ,x:A ⊢α B Γ ⊢α B
a:A,b:B,c:C,d:D ⊢(a ⊗ b) ⊗ (c ⊗ d) X a:A,b:B,c:C,d:D,e:E ⊢(a ⊗ b) ⊗ (c ⊗ d) X
SLIDE 60
24
Structural Rules (Admiss)
Γ,x:A ⊢α B Γ ⊢α B
SLIDE 61
24
Structural Rules (Admiss)
Γ,x:A ⊢α B Γ ⊢α B Γ,x:A,y:B ⊢α C Γ,y:B,x:A ⊢α B
SLIDE 62
24
Structural Rules (Admiss)
Γ ⊢α A Γ,x:A ⊢β B Γ ⊢β[α/x] B Γ,x:A ⊢α B Γ ⊢α B Γ,x:A,y:B ⊢α C Γ,y:B,x:A ⊢α B
SLIDE 63
24
Structural Rules (Admiss)
Γ ⊢α A Γ,x:A ⊢β B Γ ⊢β[α/x] B Γ,x:A ⊢α B Γ ⊢α B Γ,x:A ⊢x A Γ,x:A,y:B ⊢α C Γ,y:B,x:A ⊢α B
SLIDE 64
24
Structural Rules (Admiss)
Γ ⊢α A Γ,x:A ⊢β B Γ ⊢β[α/x] B Γ,x:A ⊢α B Γ ⊢α B Γ,x:A ⊢x A Γ,x:A,y:B ⊢α C Γ,y:B,x:A ⊢α B Γ
SLIDE 65
25
Hypothesis
Γ ⊢β P β ⇒ x x : P ∊ Γ
SLIDE 66
25
Hypothesis
Γ ⊢β P β ⇒ x x : P ∊ Γ
up to whatever structural properties you’ve asserted, the context is just x (typically x⊗y ⇒ x)
SLIDE 67
26
F types
Fα (x1:A1,…,xn:An)
SLIDE 68
26
F types
the “product” of A1 … An structured according to α
Fα (x1:A1,…,xn:An)
SLIDE 69
26
F types
the “product” of A1 … An structured according to α
Fα (x1:A1,…,xn:An) A ⊗ B := F(x ⊗ y) (x:A, y:B) e.g.
SLIDE 70
26
F types
the “product” of A1 … An structured according to α
Fα (x1:A1,…,xn:An) A ⊗ B := F(x ⊗ y) (x:A, y:B) e.g. ♭A := F(r x) (x:A)
SLIDE 71
27
F Left
Γ,A⊗B,Δ ⊢ C Γ,A,B,Δ ⊢ C Γ[A∗B] ⊢ C Γ[A,B] ⊢ C Γ; Δ,!A ⊢ C Γ,A ; Δ ⊢ C Γ; Δ,☐A ⊢ C Γ,A ; Δ,☐A ⊢ C
SLIDE 72
28
F Left
Γ, x:Fα(Δ) ⊢β B Γ,Δ ⊢β[α/x] B
SLIDE 73
28
F Left
Γ, x:Fα(Δ) ⊢β B Γ,Δ ⊢β[α/x] B
remember where in the tree Δ variables occur
SLIDE 74
29
F Right
Γ0 ⊢ A ⊗ B Γ ⊢ A Δ ⊢ B Γ0 ≡ Γ, Δ Γ0 ⊢ !A Γ ; ⋅ ⊢ A Γ0 ≡ (Γ; ⋅)
SLIDE 75
30
F Right
Γ ⊢β Fα Δ β ⇒ α[γ] Γ ⊢γ Δ
SLIDE 76
31
Exchange
x:A⊗B ⊢x B⊗A
SLIDE 77
31
Exchange
x : F(y⊗z)(y:A,z:B) ⊢x F(z’⊗y’)(z’:B,y’:A) x:A⊗B ⊢x B⊗A
SLIDE 78
31
Exchange
x : F(y⊗z)(y:A,z:B) ⊢x F(z’⊗y’)(z’:B,y’:A) y:A,z:B ⊢y⊗z F(z’⊗y’)(z’:B,y’:A) x:A⊗B ⊢x B⊗A
SLIDE 79
31
Exchange
x : F(y⊗z)(y:A,z:B) ⊢x F(z’⊗y’)(z’:B,y’:A) y:A,z:B ⊢y⊗z F(z’⊗y’)(z’:B,y’:A) x:A⊗B ⊢x B⊗A
SLIDE 80
31
Exchange
x : F(y⊗z)(y:A,z:B) ⊢x F(z’⊗y’)(z’:B,y’:A) y:A,z:B ⊢y⊗z F(z’⊗y’)(z’:B,y’:A) y⊗z ⇒ (z’⊗y’)[?] x:A⊗B ⊢x B⊗A
SLIDE 81
31
Exchange
x : F(y⊗z)(y:A,z:B) ⊢x F(z’⊗y’)(z’:B,y’:A) y:A,z:B ⊢y⊗z F(z’⊗y’)(z’:B,y’:A) x:A⊗B ⊢x B⊗A
SLIDE 82
31
Exchange
x : F(y⊗z)(y:A,z:B) ⊢x F(z’⊗y’)(z’:B,y’:A) y:A,z:B ⊢y⊗z F(z’⊗y’)(z’:B,y’:A) y⊗z ⇒ (z’⊗y’)[z/z’,y/y’] x:A⊗B ⊢x B⊗A
SLIDE 83
31
Exchange
x : F(y⊗z)(y:A,z:B) ⊢x F(z’⊗y’)(z’:B,y’:A) y:A,z:B ⊢y⊗z F(z’⊗y’)(z’:B,y’:A) x:A⊗B ⊢x B⊗A
SLIDE 84
31
Exchange
x : F(y⊗z)(y:A,z:B) ⊢x F(z’⊗y’)(z’:B,y’:A) y:A,z:B ⊢y⊗z F(z’⊗y’)(z’:B,y’:A) y⊗z ⇒ z⊗y x:A⊗B ⊢x B⊗A
SLIDE 85
31
Exchange
x : F(y⊗z)(y:A,z:B) ⊢x F(z’⊗y’)(z’:B,y’:A) y:A,z:B ⊢y⊗z F(z’⊗y’)(z’:B,y’:A) y:A ⊢y A y⊗z ⇒ z⊗y x:A⊗B ⊢x B⊗A
SLIDE 86
31
Exchange
x : F(y⊗z)(y:A,z:B) ⊢x F(z’⊗y’)(z’:B,y’:A) y:A,z:B ⊢y⊗z F(z’⊗y’)(z’:B,y’:A) y:A ⊢y A z:B ⊢z B y⊗z ⇒ z⊗y x:A⊗B ⊢x B⊗A
SLIDE 87
32
Right Adjoints
Γ ⊢ A ⊸ B Γ,A ⊢ B Γ ⊢c U A Γ;⋅⊢l A Γ[A⊸B,Δ] ⊢ C Δ ⊢ A Γ[B] ⊢ C Γ,U A;Δ ⊢l B Γ;Δ,A ⊢l B
SLIDE 88
33
Right Adjoints
Uc.α(Δ|A) typeq A typep Δ ctxφ φ,c:q ⊢ α : p A ⊸ B := Uc.(c ⊗ y)(y:A|B)
SLIDE 89
33
Right Adjoints
Uc.α(Δ|A) typeq A typep Δ ctxφ φ,c:q ⊢ α : p A ⊸ B := Uc.(c ⊗ y)(y:A|B) A \ B := Uc.(y ⊗ c)(y:A|B) A / B := Uc.(c ⊗ y)(y:A|B)
SLIDE 90
34
Right Adjoints
Uc.α(Δ|A) typeq A typep Δ ctxφ φ,c:q ⊢ α : p Γ ⊢β Uc.α(Δ|A) Γ,Δ ⊢α[β/c] A
SLIDE 91
35
Right Adjoints
x:X ⊢x Uc ⊗ a(a:A|Y) A ⊸ Y
SLIDE 92
35
Right Adjoints
x:X ⊢x Uc ⊗ a(a:A|Y) x:X,a:A ⊢x⊗a Y A ⊸ Y
SLIDE 93
35
Right Adjoints
x:X ⊢x Uc ⊗ a(a:A|Y) x:X,a:A ⊢x⊗a Y z:Fx⊗a(x:X,a:A) ⊢z Y A ⊸ Y
SLIDE 94
35
Right Adjoints
x:X ⊢x Uc ⊗ a(a:A|Y) x:X,a:A ⊢x⊗a Y z:Fx⊗a(x:X,a:A) ⊢z Y A ⊸ Y X ⊗ A
SLIDE 95
36
Examples
Non-assoc, ordered, linear, affine, relevant, cartesian, bunched products and implications N-linear variables [Reed,Abel,McBride] Monoidal, lax, non- left adjoints Non-strong, strong, ☐-strong monads Spatial type theory Logical adequacy: sequent is provable iff its encoding is
SLIDE 96
37
Bifibrations
locally discrete fibration (action of structural rules) Fα Δ makes this into an opfibration Uα(Δ|A) makes this into a fibration Sound/Complete: Syntax forms a bifibration and can be interpreted in any
D M
derivations modes cartesian 2- multicategories
𝜌
SLIDE 97
38
Equational Theory
βη for F and U equations governing action of 2-cells: when do two terms differ by placement of structural properties?
A,B ⊢ B × A A ⊢ A B ⊢ B A,B ≡ B,A A,B ⊢ B × A A,B ⊢ A A,B ⊢ B A,B ≡ A,B,A,B
SLIDE 98
39
This paper
Products and left-adjoints (⊗,F) all one connective Negatives and right adjoints (⊸,U) another Cut elimination for all instances at once Equational theory: differ by structural rules Categorical semantics A framework that abstracts the common aspects of many intuitionistic substructural and modal logics