Space-Efficient Manifest Contracts. Michael Greenberg Princeton University POPL 2015
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1 Space-Efficient Manifest Contracts Michael Greenberg Princeton University POPL 2015
2 (First-order) contracts Specifications Written in code Checked at runtime 2
3 (First-order) contracts Specifications Written in code Checked at runtime assert(n 0) 2
4 (First-order) contracts Specifications Written in code Checked at runtime assert(n 0) sqrt : {x:float x 0} Float 2
5 Higher-order contracts ({x:int x 0} {x:int x 0}) {y:int y 0} You give a function f on Nats, I return a Nat 3 even-odd rule Findler and Felleisen 2002
6 Higher-order contracts ({x:int x 0} {x:int x 0}) {y:int y 0} You give a function f on Nats, I return a Nat 3 even-odd rule Findler and Felleisen 2002
7 Higher-order contracts ({x:int x 0} {x:int x 0}) {y:int y 0} You give a function f on Nats, I return a Nat If you don t get a Nat, oops you blame me 3 even-odd rule Findler and Felleisen 2002
8 Higher-order contracts ({x:int x 0} {x:int x 0}) {y:int y 0} You give a function f on Nats, I return a Nat If you don t get a Nat, oops you blame me If f is called with a negative number, oops you blame me 3 even-odd rule Findler and Felleisen 2002
9 Higher-order contracts ({x:int x 0} {x:int x 0}) {y:int y 0} You give a function f on Nats, I return a Nat If you don t get a Nat, oops you blame me If f is called with a negative number, oops you blame me If f returns a negative, oops I blame you 3 even-odd rule Findler and Felleisen 2002
10 Checking contracts at runtime Nat 7 4
11 Checking contracts at runtime Nat 7 7 4
12 Checking contracts at runtime Nat 7 7 Nat -1 4
13 Checking contracts at runtime Nat 7 7 Nat -1 blame 4
14 Checking contracts at runtime Nat 7 7 Nat -1 blame Pos Pos pred 4
15 Checking contracts at runtime Nat 7 7 Nat -1 blame ( Pos ) Pos pred 1 4
16 Checking contracts at runtime Nat 7 7 Nat -1 blame ( ) Pos pred 1 Pos Pos ( ( 1)) pred Pos 4
17 Checking contracts at runtime Nat 7 7 Nat -1 blame ( ) Pos pred 1 Pos Pos ( ( 1)) pred Pos blame 4
18 Bad space behavior ( ) Nat Nat f v Nat (f( Nat v)) 5
19 Bad space behavior ( ) Nat Nat f v Nat (f( Nat v)) My paper: a solution! 5
20 Function proxies Set... min : {l:α set not (empty) l}α List... head : {l:α list not (null l)}α 6
21 Function proxies Set... min : {l:α set not (empty) l}α List... head : {l:α list not (null l)}α 6
22 Function proxies Set empty = null min = ( not empty α ) head List... head = ( not null α ) fun x. 7
23 Tail calls let odd = let even = (λn:int. if (n==0) then false else even (n-1)) (λn:int. if (n==0) then true else odd (n-1)) 8
24 Tail calls let odd = let even = (λn:int. if (n==0) then false else even (n-1)) (λn:int. if (n==0) then true else odd (n-1)) 8
25 Tail calls let odd = let even = (λn:int. if (n==0) then false else even (n-1)) (λn:int. if (n==0) then true else odd (n-1)) 8
26 What tail calls? let odd = let even = (λn:int. even (n-1)) (λn:int. odd (n-1)) even ( 3) 9
27 What tail calls? let odd = let even = (λn:int. even (n-1)) (λn:int. odd (n-1)) even ( 3) odd ( 2) 9
28 What tail calls? let odd = let even = (λn:int. even (n-1)) (λn:int. odd (n-1)) even ( 3) odd ( 2) even ( 1) 9
29 What tail calls? let odd = let even = (λn:int. even (n-1)) (λn:int. odd (n-1)) even ( 3) odd ( 2) even ( 1) odd ( 0) 9
30 What tail calls? let odd = let even = (λn:int. even (n-1)) (λn:int. odd (n-1)) even ( 3) odd ( 2) even ( 1) odd ( 0) false 9
31 What tail calls? let odd = let even = (λn:int. even (n-1)) (λn:int. odd (n-1)) even ( 3) Contracts break tail calls! odd ( 2) even ( 1) odd ( 0) false 9
32 Bad space behavior Functional Programming - Tail Calls = Bad News Contracts change asymptotic space behavior Big barrier to adoption 10
33 Space-efficient manifest contracts a semantics for manifest contracts checks consume constant space behave just like classic contracts 11
34 Westward the Course of Empire Takes Its Way Emanuel Leutze
35 Contracts Made Manifest Greenberg, Pierce, and Weirich POPL 2010 no Are contracts types? yes Latent Manifest 13
36 Casts <T1 T2> l e I know e has type T1 Treat it as type T2 If I m wrong, blame l 14
37 Casts <T1 T2> l e B ::= Bool... T ::= {x:b e} T1T2 15
38 Casts between refinements <{x:int true} {x:int x 0}> l 7 * 7 16
39 Casts between refinements <{x:int true} {x:int x 0}> l 7 * 7 <{x:int true} {x:int x 0}> l -1 * blame l 16
40 Casts between functions <T1 T2 U1 U2> l f...is a value a/k/a function proxy. 17
41 Casts between functions (<T1 T2 U1 U2> l f) v <T2 U2> l (f (<U1 T1> l v)) 18
42 19
43 (<{x:int true}{x:int true} {x:int x 0}{x:Int x 0}> l λx:{x:int true}. x-1) 0 19
44 (<{x:int true}{x:int true} {x:int x 0}{x:Int x 0}> l λx:{x:int true}. x-1) 0 <{x:int true} {x:int x 0}> l (λx:{x:int true}. x-1 (<{x:int x 0} {x:int true}> l 0)) * 19
45 (<{x:int true}{x:int true} {x:int x 0}{x:Int x 0}> l λx:{x:int true}. x-1) 0 <{x:int true} {x:int x 0}> l (λx:{x:int true}. x-1 (<{x:int x 0} {x:int true}> l 0)) * <{x:int true} {x:int x 0}> l (λx:{x:int true}. x-1 0) * 19
46 (<{x:int true}{x:int true} {x:int x 0}{x:Int x 0}> l λx:{x:int true}. x-1) 0 <{x:int true} {x:int x 0}> l (λx:{x:int true}. x-1 (<{x:int x 0} {x:int true}> l 0)) * <{x:int true} {x:int x 0}> l (λx:{x:int true}. x-1 0) * <{x:int true} {x:int x 0}> l -1 * blame l 19
47 (<{x:int true}{x:int true} {x:int x 0}{x:Int x 0}> l λx:{x:int true}. x-1) 0 <{x:int true} {x:int x 0}> l (λx:{x:int true}. x-1 (<{x:int x 0} {x:int true}> l 0)) * <{x:int true} {x:int x 0}> l (λx:{x:int true}. x-1 0) * <{x:int true} {x:int x 0}> l -1 * blame l 19
48 Pop quiz When we execute <(NatNat)Nat (PosPos)Pos> l will we check Nat or Pos in the domain s domain? 20
49 Insight #1: use coercions <T1 T2> l 21
50 Coercions between predicates <{x:int true} {x:int x 0}> l 7 * 7 <{x:int true} {x:int x 0}> l -1 * blame l 22
51 Coercions between predicates <{x:int true} {x:int x 0}> l 7 * 7 <{x:int true} {x:int x 0}> l -1 * blame l Totally ignored! 22
52 Coercions between predicates <{x:int true} {x:int x 0}> l 7 * 7 <{x:int true} {x:int x 0}> l -1 * blame l Nat 23
53 Coercions between functions (<{x:int true}{x:int true} {x:int x 0}{x:Int x 0}> l f) v <{x:int true} {x:int x 0}> l (f (<{x:int x 0} {x:int true}> l v)) 24
54 Coercions between functions (<{x:int true}{x:int true} {x:int x 0}{x:Int x 0}> l f) v <{x:int true} {x:int x 0}> l (f (<{x:int x 0} {x:int true}> l v)) Nat 24
55 Coercions between functions (<{x:int true}{x:int true} {x:int x 0}{x:Int x 0}> l f) v <{x:int true} {x:int x 0}> l (f (<{x:int x 0} {x:int true}> l v)) Nat Int 24
56 Coercions between functions (<{x:int true}{x:int true} {x:int x 0}{x:Int x 0}> l f) v <{x:int true} {x:int x 0}> l (f (<{x:int x 0} {x:int true}> l v)) Int Nat 25
57 Coercions between functions <T1 T2 U1 U2> l <U1 T1> l <T2 U2> l 26
58 Coercions between functions <T1 T2 U1 U2> l <U1 T1> l <T2 U2> l 26
59 Coercions between functions <T1 T2 U1 U2> l <U1 T1> l <T2 U2> l 26
60 Makeup exam When we execute <(NatNat)Nat (PosPos)Pos> l will we check Nat or Pos in the domain s domain? 27
61 Makeup exam When we execute <(NatNat)Nat (PosPos)Pos> l ( ) Nat Pos Pos will we check Nat or Pos in the domain s domain? 27
62 Bodies in Urban Spaces Willi Dorner / Studio 70
63 Insight #2: avoid redundant checks Nat Even Nat 6 29
64 Insight #2: avoid redundant checks Nat Even Nat 6 Nat Even 6 29
65 Insight #2: avoid redundant checks Nat Even Nat 6 Nat Even 6 Nat 6 29
66 Insight #2: avoid redundant checks Nat Even Nat 6 Nat Even 6 Nat
67 Insight #2: avoid redundant checks Nat Even Nat 6 Nat Even 6 Nat 6 6 Nat Even Nat 7 29
68 Insight #2: avoid redundant checks Nat Even Nat 6 Nat Even 6 Nat 6 6 Nat Even Nat 7 Nat Even 7 29
69 Insight #2: avoid redundant checks Nat Even Nat 6 Nat Even 6 Nat 6 6 Nat Even Nat 7 Nat Even 7 blame 29
70 Insight #2: avoid redundant checks Nat Even Nat 6 Nat Even 6 Nat 6 6 Nat Even Nat 7 Nat Even 7 blame Nat Even Nat -1 29
71 Insight #2: avoid redundant checks Nat Even Nat 6 Nat Even 6 Nat 6 6 Nat Even Nat 7 Nat Even 7 blame Nat Even Nat -1 blame 29
72 Insight #2: avoid redundant checks Never fails! Nat Even Nat 6 Nat Even 6 Nat 6 6 Nat Even Nat 7 Nat Even 7 blame Nat Even Nat -1 blame 29
73 Eliminating redundant checks let odd = let even = (λn:int. even (n-1)) (λn:int. odd (n-1)) even ( 3) 30
74 Eliminating redundant checks let odd = let even = (λn:int. even (n-1)) (λn:int. odd (n-1)) even ( 3) odd ( 2) 30
75 Eliminating redundant checks let odd = let even = (λn:int. even (n-1)) (λn:int. odd (n-1)) even ( 3) odd ( 2) even ( 1) 30
76 Eliminating redundant checks let odd = let even = (λn:int. even (n-1)) (λn:int. odd (n-1)) even ( 3) odd ( 2) even ( 1) odd ( 0) 30
77 Eliminating redundant checks let odd = let even = (λn:int. even (n-1)) (λn:int. odd (n-1)) even ( 3) odd ( 2) Redundant! even ( 1) odd ( 0) 30
78 Eliminating redundant checks let odd = let even = (λn:int. even (n-1)) (λn:int. odd (n-1)) even ( 3) odd ( 2) even ( 1) odd ( 0) 30
79 Eliminating redundant checks let odd = let even = (λn:int. even (n-1)) (λn:int. odd (n-1)) even ( 3) odd ( 2) even ( 1) odd ( 0) false 30
80 Eliminating redundant checks let odd = let even = (λn:int. even (n-1)) (λn:int. odd (n-1)) even ( 3) odd ( 2) even ( 1) Redundant! odd ( 0) false 30
81 Eliminating redundant checks let odd = let even = (λn:int. even (n-1)) (λn:int. odd (n-1)) even ( 3) odd ( 2) even ( 1) odd ( 0) false 30
82 Redundant checks Same color same check Formally: decidable pre-order on refinement types Is this enough? 31
83 How many checks? Finitely many because of simple types!
84 How many checks? Types: Finitely many because of simple types!
85 Bounds Types: Finitely many types Appear once, at most What s the worst that can happen?
86 Bounds Types: Finitely many types Appear once, at most What s the worst that can happen?
87 Bounds Types: Finitely many types Appear once, at most What s the worst that can happen?
88 Bounds Types: Finitely many types Appear once, at most What s the worst that can happen?
89 Bounds Types: Finitely many types Appear once, at most What s the worst that can happen?
90 Eliminating redundant checks How do we merge lists of checks? e Invariant: the checks on the stack have no redundancy. We ll merge the new checks in, dropping redundant checks. 34
91 Eliminating redundant checks How do we merge lists of checks? ( ) e e Invariant: the checks on the stack have no redundancy. We ll merge the new checks in, dropping redundant checks. 34
92 Merging, in detail How do we merge lists of checks? e + =? 35
93 Merging, in detail How do we merge lists of checks? ( ) e e + =? 35
94 Merging, in detail How do we merge lists of checks? ( ) e e + =? new 35
95 Merging, in detail How do we merge lists of checks? ( ) e e + =? new old 35
96 Merging, in detail How do we merge lists of checks? ( ) e e + = 36
97 Merging, in detail How do we merge lists of checks? ( ) e e + = 36
98 Merging, in detail How do we merge lists of checks? ( ) e e + = 36
99 Merging, in detail How do we merge lists of checks? ( ) e e + = 36
100 Merging, in detail How do we merge lists of checks? ( ) e e + = 36
101 Merging, in detail How do we merge lists of checks? ( ) e e + = 36
102 Merging, in detail How do we merge lists of checks? ( ) e e + = 36
103 Merging, in detail How do we merge lists of checks? ( ) e e + = 36
104 Merging, in detail How do we merge lists of checks? Go from new to old Drop redundant checks ( ) e e on the old coercion new old + =
105 Merging function proxies ( ( f ))vv 38
106 Merging function proxies ( ( f ))vv 38
107 Merging function proxies ( ( f ))vv (( f) v) 38
108 Merging function proxies ( ( f ))vv (( f) v) 38
109 Merging function proxies ( ( f ))vv (( f) v) (f ( v) ) 38
110 Merging function proxies new ( ( f ))vv old (( f) v) (f ( v) ) 38
111 Merging function checks Domain: new to old Codomain: old to new new old + = 39
112 Merging function checks Domain: new to old Codomain: old to new new old + = + 39
113 Merging function checks Domain: new to old Codomain: old to new new old + =
114 Merging function checks Domain: new to old Codomain: old to new new old + = 40
115 Merging function checks new Go from right to left old Domain: new to old Codomain: old to new + = Drop redundant checks Contravariance 40
116 Proofs Soundness Congruence lemma Classic semantics and space-efficient semantics behave identically e 1 e 2 + ht 1 )T 2 i l e 1 ht 1 )T 2 i l e 2 E E result E e E 41
117 Congruence lemma e e e e e 42
118 Congruence lemma e1 e1 Our step: check ( ) ( ) e1 e1 e 43
119 Congruence lemma e1 e1 Our step: check ( ) ( ) e1 e1 e1 e1 e 44
120 Outlook Types: + = Use coercions, not casts Merge redundant checks 45
121 Outlook Can we scale to dependency? Types: + = Simple types finite number Dependent types infinite number 46
122 Outlook Can we scale to dependency and effects? keep lowest check + = Idea: partial orders/lattices 5 47 x: x
123 Space-Efficient Manifest Contracts Michael Greenberg Princeton University POPL 2015
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