SLIDE 1
A type system for monotonicity
Michael Arntzenius
University of Birmingham
ICFP 2018
It’s just the simply-typed λ-calculus!
A ∈ Poset A → B = monotone maps A → B,
- rdered pointwise
A type system for monotonicity Michael Arntzenius University of - - PowerPoint PPT Presentation
A type system for monotonicity Michael Arntzenius University of - - PowerPoint PPT Presentation
A type system for monotonicity Michael Arntzenius University of Birmingham ICFP 2018 Its just the simply-typed -calculus! A Poset A B = monotone maps A B , ordered pointwise op A A A b b
SLIDE 2
SLIDE 3
a b c
A
a b c
- p A
a b c
✷A
f : ✷A → B is monotone iff x = y = ⇒ f(x) f(y)
i.e. always!
SLIDE 4
setMap : ✷(✷A → B) → Set A → Set B setMap f xs = let box g = f in do x ← xs let box y = x return (box (g (box y)))
SLIDE 5
setMap : ✷(✷A → B) → Set A → Set B setMap f xs = do x ← xs return
(f x)
SLIDE 6
A <: B
id :
A → B
SLIDE 7
[T]A <: B
id : TA → B
T ∈ {id, op, ✷, ...}
SLIDE 8
f : TA → B g : UB → C g ◦ f : (UT)A → C
SLIDE 9
Monotonicity tames dragons!
- 1. Eventual consistency in distributed systems
http://bloom-lang.net/calm/
- 2. Determinism in parallel programs
LVars: Lattice-based Data Structures for Deterministic Parallelism, Lindsey Kuper & Ryan Newton
- 3. Recursive queries in Datalog & Datafun
http://www.rntz.net/datafun
- 4. Paradoxes of self-reference
SLIDE 10
fin
SLIDE 11
subtractEach : List (Z × op Z) → List Z subtractEach xs = [x − y | (x, y) ← xs]
SLIDE 12
a b : id A ⇐ ⇒ a b : A a b : op A ⇐ ⇒ a b : A a b : ✷A ⇐ ⇒ a b ∧ a b : A a b : ♦A ⇐ = a b ∨ b a : A ♦
✷
id
- p
UT T
id
- p
✷ ♦
U
id id
- p
✷ ♦
- p op
id
✷ ♦ ✷ ✷ ✷ ✷ ♦
♦ ♦ ♦
✷ ♦
SLIDE 13