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Type inference for monotonicity
Michael Arntzenius
University of Birmingham
S-REPLS 9, 2018
Type inference for monotonicity Michael Arntzenius University of - - PowerPoint PPT Presentation
Type inference for monotonicity Michael Arntzenius University of - - PowerPoint PPT Presentation
Type inference for monotonicity Michael Arntzenius University of Birmingham S-REPLS 9, 2018 x + log x x x log x 4 ::= R | A B types A , B terms M , N ::= x | x . M | M N A Poset R = R A B =
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types
A, B ::= R | A → B
terms M, N ::= x | λx. M | M N
A ∈
Poset
R = R A → B =
monotone maps A → B,
- rdered pointwise
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λx. x + log x : R → R λx. −x :
???? → R
λx. x − log x :
??? → R
λx. 4 :
??? → R
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λx. x + log x : R → R λx. −x :
- p R → R
λx. x − log x : ✷R → R λx. 4 : ♦R → R
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Let ✷A be A, ordered discretely:
x y : ✷A ⇐ ⇒ x = y : A
Then f : ✷A → B is monotone iff
x = y = ⇒ f(x) f(y)
i.e. always!
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λx. x + log x : R → R λx. −x :
- p R → R
λx. x − log x : ✷R → R λx. 4 : ♦R → R
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♦A identifies weakly connected elements in A: x y ∨ y x : A = ⇒ x = y : ♦A
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♦A identifies weakly connected elements in A: x y ∨ y x : A = ⇒ x = y : ♦A
Theorem:
f : ♦A → B ⇐ ⇒ f : A → ✷B
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λx. x + log x : R → R λx. −x :
- p R → R
λx. x − log x : ✷R → R λx. 4 : ♦R → R
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✷ is a necessity modality / monoidal comonad; ♦ is a possibility modality / monoidal monad.
Pfenning & Davies gave typing rules for these in A Judgmental Reconstruction of Modal Logic!
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(λx. x − log x) : ✷R → R
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(λx. x − log x) : ✷R → R (λx. let box u = x in u − box (log u)) : ✷R →
: (
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How to infer monotonicity:
- 1. Infer variable usage modes bottom-up.
- 2. Use modal subtyping for implicit coercion.
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- 1. Bottom-up mode inference
Γ ⊢ M : B
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- 1. Bottom-up mode inference
x : [T] A ⊢ M : B
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- 1. Bottom-up mode inference
x : [T] A ⊢ M : B
modes T ::= id | op | ✷ | ♦
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- 1. Bottom-up mode inference: duplication
x : [T] A ⊢ M : A x : [U] A ⊢ N : B x : [??????] A ⊢ (M, N) : A × B
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- 1. Bottom-up mode inference: duplication
x : [T] A ⊢ M : A x : [U] A ⊢ N : B x : [T ∧ U] A ⊢ (M, N) : A × B ♦ ✷
id
- p
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- 1. Bottom-up mode inference: composition
x : [T] A ⊢ M : B y : [U] B ⊢ N : C x : [UT] A ⊢ let y = M in N
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- 1. Bottom-up mode inference: composition
x : [T] A ⊢ M : B y : [U] B ⊢ N : C x : [UT] A ⊢ let y = M in N
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- 1. Bottom-up mode inference: composition
x : [T] A ⊢ M : B y : [U] B ⊢ N : C x : [UT] A ⊢ let y = M in N
Composition UT is a monoid, with id neutral:
UT T
id
- p
✷ ♦ U
id id
- p
✷ ♦
- p
- p
id
✷ ♦ ✷ ✷ ✷ ✷ ♦ ♦ ♦ ♦ ✷ ♦
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Composition UT and meet T ∧ U form a semiring! Other systems with semiring-valued annotations:
- 1. Linear Haskell (POPL 2018)
- 2. I Got Plenty o’ Nuttin (McBride)
- 3. Monotonicity Types (PMLDC 2017)
- 4. Bounded Linear Types in a Resource Semiring (ESOP 2014)
- 5. The Next 700 Modal Type Assignment Systems (TYPES 2015)
- 6. A Fibrational Framework for Substructural and Modal Logics
(FSCD 2017) massively generalizes this pattern. ... more?
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- 2. Modal subtyping
(λx. let box u = x in u − box (log u)) : ✷R →
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- 2. Modal subtyping: ✷-introduction?
SUBSUMPTION
Γ ⊢ M : A A <: B Γ ⊢ M : B
But A <: ✷A!
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- 2. Modal subtyping!
[U] A <: B
Finds the most general mode U such that UA <: B.
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- 2. Modal subtyping: ✷-introduction!
SUBSUMPTION
x : [T] A ⊢ M : B [U] B <: C x : [UT] A ⊢ M : C
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- 2. Modal subtyping: ✷-introduction!
SUBSUMPTION
x : [T] A ⊢ M : B [U] B <: C x : [UT] A ⊢ M : C
SUBTYPING INTO ✷
[T] A <: B [✷T] A <: ✷B
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- 2. Modal subtyping: ✷-introduction!
SUBSUMPTION
x : [T] A ⊢ M : B [U] B <: C x : [UT] A ⊢ M : C
SUBTYPING INTO ✷
[T] A <: B [✷T] A <: ✷B
✷-INTRODUCTION VIA SUBSUMPTION
x : [T] A ⊢ M : B x : [✷T] A ⊢ M : ✷B
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- 2. Modal subtyping: ✷-elimination
f : ♦A → B ⇐ ⇒ f : A → ✷B
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- 2. Modal subtyping: ✷-elimination
f : ♦A → B ⇐ ⇒ f : A → ✷B
Which suggests this elimination rule for ✷:
x : [T] A ⊢ M : ✷B x : [♦T] A ⊢ unbox M : B
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- 2. Modal subtyping: ✷-elimination
f : ♦A → B ⇐ ⇒ f : A → ✷B
Which suggests this elimination rule for ✷:
x : [T] A ⊢ M : ✷B x : [♦T] A ⊢ unbox M : B
Happily, we can do this with subtyping!
SUBTYPING OUT OF ✷
[T] A <: B [T♦] ✷A <: B
✷-ELIMINATION VIA SUBSUMPTION
x : [T] A ⊢ M : ✷B x : [♦T] A ⊢ M : B
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How to infer monotonicity:
- 1. Infer variable usage modes bottom-up.
- 2. Use modal subtyping for implicit coercion.
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State of work
- 1. Solved: Pattern matching!
- 2. Conjectured: Substitution.
- 3. Unsolved: Internalizing ♦ w/o annotations.
- 4. Open: HM-style type inference?
- 5. Pfenning-Davies elim rule is derivable from