Post on 25-Sep-2020
A Fibrational Framework for Substructural and Modal Logics
Dan Licata Wesleyan University
Michael ShulmanUniversity of San Diego
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Mitchell RileyWesleyan University
2
Substructural Logic
Γ,x:A ⊢ BΓ ⊢ B
Γ,x:A,y:B ⊢ CΓ,y:B,x:A ⊢ B
Γ,x:A,y:A ⊢ BΓ,x:A ⊢ B
Weakening
Exchange
Contraction
Modal Logic
Γ ⊢ ☐A
∅ ⊢ A
♢A ⊢ ♢C
A ⊢ ♢C
Modal Logic
Γ ⊢ ☐A
∅ ⊢ A
♢A ⊢ ♢C
A ⊢ ♢C
(♢A) × B vs. ♢(A × B)
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Intuitionistic substructural and modal logics/type systems
Linear/affine: use once (state, sessions)Relevant: strictness annotationsOrdered: linguisticsBunched: separation logic Comonads: staging, metavariables, coeffectsMonads: effectsInteractions between products and modalities
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Homotopy type theory
highercategory theory homotopy theory
dependent type theory
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Cohesive HoTT
♭∫ #⊣ ⊣
Dependent type theory with modalities ∫A, ♭A, #A
[Shulman,Schreiber]
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Cohesive HoTT
♭∫ #⊣ ⊣
Dependent type theory with modalities ∫A, ♭A, #A
[Shulman,Schreiber]
♭A ⊢ B A ⊢ # B
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Cohesive HoTT
♭∫ #⊣ ⊣
Dependent type theory with modalities ∫A, ♭A, #A
[Shulman,Schreiber]
monad ♢
♭A ⊢ B A ⊢ # B
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Cohesive HoTT
♭∫ #⊣ ⊣
Dependent type theory with modalities ∫A, ♭A, #A
[Shulman,Schreiber]
monad ♢
comonad ☐
♭A ⊢ B A ⊢ # B
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Cohesive HoTT
♭∫ #⊣ ⊣
Dependent type theory with modalities ∫A, ♭A, #A
[Shulman,Schreiber]
monad ♢
comonad ☐
monad
♭A ⊢ B A ⊢ # B
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Cohesive HoTT
♭∫ #⊣ ⊣
Dependent type theory with modalities ∫A, ♭A, #A
[Shulman,Schreiber]
monad ♢
comonad ☐
monad(idempotent)♭♭ A ≅ ♭ A
♭A ⊢ B A ⊢ # B
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𝕊-Cohesion
♮ ♮⊣
comonad monad
[Finster,L.,Morehouse,Riley]
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𝕊-Cohesion
♮ ♮⊣
comonad monad
A ⊢♮A ⊢ A
[Finster,L.,Morehouse,Riley]
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𝕊-Cohesion
♮ ♮⊣
comonad monad
A ⊢♮A ⊢ A
[Finster,L.,Morehouse,Riley]
♮(A ⋀ B) ≅ ♮A ×♮B
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Differential Cohesion[Schreiber; Gross,L.,New,Paykin,Riley,Shulman,Wellen]
♭∫ #⊣ ⊣
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Differential Cohesion[Schreiber; Gross,L.,New,Paykin,Riley,Shulman,Wellen]
♭∫ #⊣ ⊣
𝖩R &⊣ ⊣
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Differential Cohesion[Schreiber; Gross,L.,New,Paykin,Riley,Shulman,Wellen]
♭∫ #⊣ ⊣
𝖩R &⊣ ⊣
HoTT/UF Saturday!
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What are the common patterns in substructural and modal logics?
S4 ☐
Γ ; Δ ⊢☐A Γ; Δ,☐A ⊢ C
Γ,A ; Δ,☐A ⊢ CΓ ; ⋅ ⊢ A
Γ,A; Δ ⊢ CΓ,A; Δ,A ⊢ C
Γ; Δ ⊢ C
S4 ☐
Γ ; Δ ⊢☐A Γ; Δ,☐A ⊢ C
Γ,A ; Δ,☐A ⊢ CΓ ; ⋅ ⊢ A
Γ,A; Δ ⊢ CΓ,A; Δ,A ⊢ C
morally ☐Γ× Δ → CΓ; Δ ⊢ C
S4 ☐
Γ ; Δ ⊢☐A Γ; Δ,☐A ⊢ C
Γ,A ; Δ,☐A ⊢ CΓ ; ⋅ ⊢ A
Γ,A; Δ ⊢ CΓ,A; Δ,A ⊢ C
context is all boxed formulae (up to weakening)
morally ☐Γ× Δ → CΓ; Δ ⊢ C
S4 ☐
Γ ; Δ ⊢☐A Γ; Δ,☐A ⊢ C
Γ,A ; Δ,☐A ⊢ CΓ ; ⋅ ⊢ A
Γ,A; Δ ⊢ CΓ,A; Δ,A ⊢ C
context is all boxed formulae (up to weakening)
morally ☐Γ× Δ → CΓ; Δ ⊢ C
because ☐A → A
Γ ; ⋅ ⊢ ! A Γ; Δ,!A ⊢ C
Γ,A ; Δ ⊢ CΓ ; ⋅ ⊢ A
Γ,A; Δ ⊢ CΓ,A; Δ,A ⊢ C
Linear Logic !
context is all !’ed formulae (no weakening)
cartesian/structural
linear
Linear Logic ⊗
Γ,Δ ⊢ A ⊗ BΓ ⊢ A Δ ⊢ B
Γ,A⊗B ⊢ CΓ,A,B ⊢ C
Linear Logic ⊗
Γ,Δ ⊢ A ⊗ BΓ ⊢ A Δ ⊢ B
Γ,A⊗B ⊢ CΓ,A,B ⊢ C
context “is” a ⊗
Linear Logic ⊗
Γ,Δ ⊢ A ⊗ BΓ ⊢ A Δ ⊢ B
Γ,A⊗B ⊢ CΓ,A,B ⊢ C
context “is” a ⊗
Γ0 ⊢ A ⊗ BΓ ⊢ A Δ ⊢ BΓ0 ≡ Γ, Δ
context is a ⊗, up to some structural rules
Types inherit properties
A,B ⊢ B ⊗ AA ⊢ AB ⊢ BA,B ≡ B,A
A⊗B ⊢ B ⊗ A
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Types inherit properties
⋅ ; ! A ⊗ ! B ⊢ ! (A ⊗ B)⋅; ! A, ! B ⊢ ! (A ⊗ B)
A ; ! B ⊢ ! (A ⊗ B)A,B ; ⋅ ⊢ ! (A ⊗ B)A,B; ⋅ ⊢ A ⊗ B
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Pattern for ☐ ! ⊗
Operation on contexts, with explicit or admissible structural properties
Type constructor that “internalizes” the context operation, inherits the structural properties
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This paper
Products and left-adjoints (⊗,F) all one connectiveNegatives and right adjoints (⊸,U) anotherCut elimination for all instances at onceEquational theory: differ by structural rulesCategorical semantics
A framework that abstracts the common aspects of many intuitionistic
substructural and modal logics
17
ExamplesNon-assoc, ordered, linear, affine, relevant, cartesian, bunched products and implicationsN-linear variables [Reed,Abel,McBride]Monoidal, lax, non- left adjointsNon-strong, strong, ☐-strong monadsCohesion
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ExamplesNon-assoc, ordered, linear, affine, relevant, cartesian, bunched products and implicationsN-linear variables [Reed,Abel,McBride]Monoidal, lax, non- left adjointsNon-strong, strong, ☐-strong monadsCohesion
Logical adequacy: sequent is provable iff its encoding is
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Closely Related Work
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Closely Related WorkDisplay logic, Lambek calculus, resource semantics
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Closely Related WorkDisplay logic, Lambek calculus, resource semanticsAdjoint linear logic [Benton&Wadler,95]
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Closely Related WorkDisplay logic, Lambek calculus, resource semanticsAdjoint linear logic [Benton&Wadler,95]
cartesian
linear
F U⊣
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Closely Related WorkDisplay logic, Lambek calculus, resource semanticsAdjoint linear logic [Benton&Wadler,95]
cartesian
linear
F U⊣A ::= F C | A ⊗ B | …C ::= U A | C × D | …
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Closely Related WorkDisplay logic, Lambek calculus, resource semanticsAdjoint linear logic [Benton&Wadler,95]
cartesian
linear
F U⊣
!A := FU A
A ::= F C | A ⊗ B | …C ::= U A | C × D | …
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Closely Related WorkDisplay logic, Lambek calculus, resource semanticsAdjoint 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 | …
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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
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Sequent Calculus
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Mode Theory
x1:p1, … , xn:pn ⊢ α : q
α ⇒ β
p,q,… Modes
Context Descriptors
Structural Properties
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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
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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
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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
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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
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a:A,b:B,c:C,d:D ⊢(a ⊗ b) ⊗ (c ⊗ d) X
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a:A,b:B,c:C,d:D ⊢(a ⊗ b) ⊗ (c ⊗ d) X
Non-associative logic: no equations/transformations
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a:A,b:B,c:C,d:D ⊢(a ⊗ b) ⊗ (c ⊗ d) X
Non-associative logic: no equations/transformationsOrdered logic: (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c)
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a:A,b:B,c:C,d:D ⊢(a ⊗ b) ⊗ (c ⊗ d) X
Non-associative logic: no equations/transformationsOrdered logic: (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c)Linear logic: a ⊗ b = b ⊗ a
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a:A,b:B,c:C,d:D ⊢(a ⊗ b) ⊗ (c ⊗ d) X
Non-associative logic: no equations/transformationsOrdered logic: (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c)Linear logic: a ⊗ b = b ⊗ aRelevant logic: a ⟹ a ⊗ a
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a:A,b:B,c:C,d:D ⊢(a ⊗ b) ⊗ (c ⊗ d) X
Non-associative logic: no equations/transformationsOrdered logic: (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c)Linear logic: a ⊗ b = b ⊗ aRelevant logic: a ⟹ a ⊗ aAffine logic: a ⟹ 1
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a:A,b:B,c:C,d:D ⊢(a ⊗ b) ⊗ (c ⊗ d) X
Non-associative logic: no equations/transformationsOrdered logic: (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c)Linear logic: a ⊗ b = b ⊗ aRelevant logic: a ⟹ a ⊗ aAffine logic: a ⟹ 1BI: two function symbols ✻ and ⋀: (a ✻ b) ⋀ (c ✻ d)
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a:A,b:B,c:C,d:D ⊢(a ⊗ b) ⊗ (c ⊗ d) X
Non-associative logic: no equations/transformationsOrdered logic: (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c)Linear logic: a ⊗ b = b ⊗ aRelevant logic: a ⟹ a ⊗ aAffine logic: a ⟹ 1BI: two function symbols ✻ and ⋀: (a ✻ b) ⋀ (c ✻ d)Modalities: unary function symbols: r(a) ⊗ r(b) ⊗ c ⊗ d
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a:A,b:B,c:C,d:D ⊢(a ⊗ b) ⊗ (c ⊗ d) X
cartesian
Non-associative logic: no equations/transformationsOrdered logic: (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c)Linear logic: a ⊗ b = b ⊗ aRelevant logic: a ⟹ a ⊗ aAffine logic: a ⟹ 1BI: two function symbols ✻ and ⋀: (a ✻ b) ⋀ (c ✻ d)Modalities: unary function symbols: r(a) ⊗ r(b) ⊗ c ⊗ d
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Weakening over weakening
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Weakening over weakening
Γ,x:A ⊢α BΓ ⊢α B
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Weakening over weakening
Γ,x:A ⊢α BΓ ⊢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
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Structural Rules (Admiss)
Γ,x:A ⊢α BΓ ⊢α B
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Structural Rules (Admiss)
Γ,x:A ⊢α BΓ ⊢α B
Γ,x:A,y:B ⊢α CΓ,y:B,x:A ⊢α B
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Structural Rules (Admiss)
Γ ⊢α A Γ,x:A ⊢β BΓ ⊢β[α/x] B
Γ,x:A ⊢α BΓ ⊢α B
Γ,x:A,y:B ⊢α CΓ,y:B,x:A ⊢α B
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Structural Rules (Admiss)
Γ ⊢α A Γ,x:A ⊢β BΓ ⊢β[α/x] B
Γ,x:A ⊢α BΓ ⊢α B
Γ,x:A ⊢x A
Γ,x:A,y:B ⊢α CΓ,y:B,x:A ⊢α B
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Structural Rules (Admiss)
Γ ⊢α A Γ,x:A ⊢β BΓ ⊢β[α/x] B
Γ,x:A ⊢α BΓ ⊢α B
Γ,x:A ⊢x A
Γ,x:A,y:B ⊢α CΓ,y:B,x:A ⊢α B
Γ ⊢β A
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Hypothesis
Γ ⊢β Pβ ⇒ x x : P ∊ Γ
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Hypothesis
Γ ⊢β Pβ ⇒ x x : P ∊ Γ
up to whatever structural properties you’ve asserted, the context is just x (typically x⊗y ⇒ x)
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F types
Fα (x1:A1,…,xn:An)
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F types the “product” of A1 … Anstructured according to α
Fα (x1:A1,…,xn:An)
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F types the “product” of A1 … Anstructured according to α
Fα (x1:A1,…,xn:An)
A ⊗ B := F(x ⊗ y) (x:A, y:B)e.g.
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F types the “product” of A1 … Anstructured 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)
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F Left
Γ,A⊗B,Δ ⊢ CΓ,A,B,Δ ⊢ C
Γ[A∗B] ⊢ CΓ[A,B] ⊢ C
Γ; Δ,!A ⊢ C
Γ,A ; Δ ⊢ C
Γ; Δ,☐A ⊢ CΓ,A ; Δ,☐A ⊢ C
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F Left
Γ, x:Fα(Δ) ⊢β B Γ,Δ ⊢β[α/x] B
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F Left
Γ, x:Fα(Δ) ⊢β B Γ,Δ ⊢β[α/x] B
remember where in the tree Δ variables occur
29
F Right
Γ0 ⊢ A ⊗ BΓ ⊢ A Δ ⊢ BΓ0 ≡ Γ, Δ
Γ0 ⊢ !A
Γ ; ⋅ ⊢ AΓ0 ≡ (Γ; ⋅)
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F Right
Γ ⊢β Fα Δ β ⇒ α[γ] Γ ⊢γ Δ
31
Exchange
x:A⊗B ⊢x B⊗A
31
Exchange
x : F(y⊗z)(y:A,z:B) ⊢x F(z’⊗y’)(z’:B,y’:A)x:A⊗B ⊢x B⊗A
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
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
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
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
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
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
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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
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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
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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 Az:B ⊢z By⊗z ⇒ z⊗y
x:A⊗B ⊢x B⊗A
32
Right Adjoints
Γ ⊢ A ⊸ BΓ,A ⊢ B
Γ ⊢c U AΓ;⋅⊢l A
Γ[A⊸B,Δ] ⊢ CΔ ⊢ A Γ[B] ⊢ C
Γ,U A;Δ ⊢l B
Γ;Δ,A ⊢l B
33
Right Adjoints
Uc.α(Δ|A) typeq
A typepΔ ctxφφ,c:q ⊢ α : p
A ⊸ B := Uc.(c ⊗ y)(y:A|B)
33
Right Adjoints
Uc.α(Δ|A) typeq
A typepΔ ctxφφ,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)
34
Right Adjoints
Uc.α(Δ|A) typeq
A typepΔ ctxφφ,c:q ⊢ α : p
Γ ⊢β Uc.α(Δ|A)Γ,Δ ⊢α[β/c] A
35
Right Adjoints
x:X ⊢x Uc ⊗ a(a:A|Y)A ⊸ Y
35
Right Adjoints
x:X ⊢x Uc ⊗ a(a:A|Y)
x:X,a:A ⊢x⊗a Y
A ⊸ Y
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
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
36
ExamplesNon-assoc, ordered, linear, affine, relevant, cartesian, bunched products and implicationsN-linear variables [Reed,Abel,McBride]Monoidal, lax, non- left adjointsNon-strong, strong, ☐-strong monadsSpatial type theory
Logical adequacy: sequent is provable iff its encoding is
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 𝜋
38
Equational Theoryβη for F and Uequations governing action of 2-cells: when do two terms differ by placement of structural properties?
A,B ⊢ B × AA ⊢ AB ⊢ BA,B ≡ B,A
A,B ⊢ B × AA,B ⊢ AA,B ⊢ BA,B ≡ A,B,A,B
39
This paper
Products and left-adjoints (⊗,F) all one connectiveNegatives and right adjoints (⊸,U) anotherCut elimination for all instances at onceEquational theory: differ by structural rulesCategorical semantics
A framework that abstracts the common aspects of many intuitionistic
substructural and modal logics