1
Space-Efficient Gradual Typing
David HermanNortheastern University
Aaron Tomb, Cormac FlanaganUniversity of California, Santa Cruz
2
The point
Naïve type conversions in functional programming languages are not safe for space.
But they can and should be.
3
Gradual Typing:
Software evolution via hybrid type checking
4
Dynamic vs. static typing
DynamicTyping
StaticTyping
5
Gradual typing
DynamicTyping
StaticTyping
6
Type checking
let x = f() in … let y : Int = x - 3 in …
7
Type checking
let x : ? = f() in … let y : Int = x - 3 in …
8
Type checking
let x : ? = f() in … let y : Int = x - 3 in …
- : Int × Int → Int
9
Type checking
let x : ? = f() in … let y : Int = <Int>x - 3 in …
10
Type checking
let x : ? = f() in … let y : Int = <Int>x - 3 in …
Int
11
Evaluation
let x : ? = f() in … let y : Int = <Int>x - 3 in …
12
Evaluation
let x : ? = 45 in … let y : Int = <Int>x - 3 in …
13
Evaluation
let y : Int = <Int>45 - 3 in …
14
Evaluation
let y : Int = 45 - 3 in …
15
Evaluation
let y : Int = 42 in …
16
Evaluation (take 2)
let x : ? = f() in … let y : Int = <Int>x - 3 in …
17
Evaluation (take 2)
let x : ? = true in … let y : Int = <Int>x - 3 in …
18
Evaluation (take 2)
let y : Int = <Int>true - 3 in …
19
Evaluation (take 2)
error: “true is not an Int”
20
Space Leaks
21
Space leaks
fun even(n) = if (n = 0) then true else odd(n - 1)
fun odd(n) = if (n = 0) then false else even(n - 1)
22
Space leaks
fun even(n : Int) = if (n = 0) then true else odd(n - 1)
fun odd(n : Int) : Bool = if (n = 0) then false else even(n - 1)
23
Space leaks
fun even(n : Int) = if (n = 0) then true else odd(n - 1)
fun odd(n : Int) : Bool = if (n = 0) then false else <Bool>even(n - 1)
24
Space leaks
fun even(n : Int) = if (n = 0) then true else odd(n - 1)
fun odd(n : Int) : Bool = if (n = 0) then false else <Bool>even(n - 1)
non-tail call!
25
Space leaks
even(n)→* odd(n - 1)→* <Bool>even(n - 2)→* <Bool>odd(n - 3)→* <Bool><Bool>even(n - 4)→* <Bool><Bool>odd(n - 5)→* <Bool><Bool><Bool>even(n - 6)→* …
26
Naïve Function Casts
27
Casts in functional languages<Int>n → n<Int>v → error: “failed cast” (if v∉Int)
<σ→τ>λx:?.e → …
28
Casts in functional languages<Int>n → n<Int>v → error: “failed cast” (if v∉Int)
<σ→τ>λx:?.e → λz:σ.<τ>((λx:?.e) z)
Very useful, very popular… unsafe for space.fresh, typed proxy
cast result
29
More space leaks
fun evenk(n : Int, k : ? → ?) = if (n = 0) then k(true) else oddk(n – 1, k)
fun oddk(n : Int, k : Bool → Bool) = if (n = 0) then k(false) else evenk(n – 1, k)
30
More space leaks
fun evenk(n : Int, k : ? → ?) = if (n = 0) then k(true) else oddk(n – 1, <Bool→Bool>k)
fun oddk(n : Int, k : Bool → Bool) = if (n = 0) then k(false) else evenk(n – 1, <?→?>k)
31
More space leaks
evenk(n, k0)
→* oddk(n - 1, <Bool→Bool>k0)
→* oddk(n - 1, λz:Bool.<Bool>k0(z))→* evenk(n - 2, <?→?>λz:Bool.<Bool>k0(z))
→* evenk(n - 2, λy:?.(λz:Bool.<Bool>k0(z))(y))→* oddk(n - 3, <Bool→Bool>λy:?.(λz:Bool.<Bool>k0(z))(y))
→* oddk(n – 3, λx:Bool.(λy:?.(λz:Bool.<Bool>k0(z))(y))(x))
→* evenk(n - 4, <?→?>λx:Bool.(λy:?.(λz:Bool.<Bool>k0(z))(y))(x))
→* evenk(n - 4, λw:?.(λx:Bool.(λy:?.(λz:Bool.<Bool>k0(z))(y))(x))(w))→* oddk(n - 5, <Bool→Bool>λw:?.(λx:Bool.(λy:?.(λz:Bool.<Bool>k0(z))(y))(x))(w))
→* oddk(n - 5, λv:Bool.<Bool>(λw:?.(λx:Bool.(λy:?.(λz:Bool.<Bool>k0(z))(y))(x))(w))(v))
→* …
(…without even using k0!)
32
Space-Efficient Gradual Typing
33
Intuition
Casts are like function restrictions(Findler and Blume, 2006)
Can their representation exploit the properties of restrictions?
34
Exploiting algebraic properties
Closure under composition:
<Bool>(<Bool> v) = (<Bool>◦<Bool>) v
35
Exploiting algebraic properties
Idempotence:
<Bool>(<Bool> v) = (<Bool>◦<Bool>) v
= <Bool> v
36
Exploiting algebraic properties
Distributivity:
(<?→?>◦<Bool→Bool>) v = <(Bool◦?)→(?◦Bool)> v
37
Space-efficient gradual typing Generalize casts to coercions
(Henglein, 1994) Change representation of casts
from <τ> to <c> Merge casts at runtime:
<c>(<d> e) → <c◦d>e
merged before evaluating e
This coercion can be
simplified!
38
Space-efficient gradual typing Generalize casts to coercions
(Henglein, 1994) Change representation of casts
from <τ> to <c> Merge casts at runtime:
<c>(<d> e) → <c◦d>e
→ <c′>e
39
Tail recursion
even(n)→* odd(n - 1)→* <Bool>even(n - 2)→* <Bool>odd(n - 3)→* <Bool><Bool>even(n - 4)→* <Bool>even(n - 4)→* <Bool>odd(n - 5)→* <Bool><Bool>even(n - 6)→* <Bool>even(n - 6)→* …
40
Bounded proxies
evenk(n, k0)
→* oddk(n - 1, <Bool→Bool>k0)
→* evenk(n - 2, <?→?><Bool→Bool>k0)
→* evenk(n - 2, <Bool→Bool>k0)
→* oddk(n - 3, <Bool→Bool>k0)
→* evenk(n - 4, <?→?><Bool→Bool>k0)
→* evenk(n - 4, <Bool→Bool>k0)
→* oddk(n - 5, <Bool→Bool>k0)→* …
41
Guaranteed.
Theorem: any program state S during evaluation of a program P is bounded by
kP · sizeOR(S)
sizeOR(S) = size of S without any casts
42
Related work
Gradual typing Siek and Taha (2006, 2007)
Function proxies Findler and Felleisen (1998, 2006): Software contracts Gronski, Knowles, Tomb, Freund, Flanagan (2006):
Hybrid typing, Sage Tobin-Hochstadt and Felleisen (2006):
Interlanguage migration Coercions
Henglein (1994): Dynamic typing Space efficiency
Clinger (1998): Proper tail recursion
43
Contributions
Space-safe representation and semantics of casts for functional languages
Supports function casts and tail recursion Three implementation strategies
(coercion-passing style, trampoline, continuation marks)
Earlier error detection (see paper) Proof of space efficiency
44
The point, again
Naïve type conversions in functional programming languages are not safe for space.
But they can and should be.
Thank [email protected]
Recommended
