Type Systems
execution er- rors
Handbook of Computer Science and Engineering, Chapter 103. CRC Press, 1997. 1 Copyright 1996 by Digital Equipment Corporation. Revised Friday, March 28, 1997, 4:45 pm.
type typing polymor- phism data abstraction subtyping type inference
nil
x Boolean x Boolean not x
2
safe
bad behavior ill behaved
3
all
4
weakly typed
5
Economy of execution nil Economy of small-scale development Economy of compilation interfaces Economy of large-scale development Economy of language features n n
6
decidably verifiable transparent enforceable
Formal type systems type soundness well-typed programs are well behaved
7
terms scoping static lexical scoping dynamic scoping free variables substitution has-type M A M A subtype-of A B equal-type A B static typing environ- ments M A M A ∫ M A M A has- value results
8
equal type type Bool type Bool structural equivalence by-name equivalence branding
9
∫ entails static typing environment x A xn An x xn dom assertion typing judgment M A M ∫ M A M A ∫ true Bool true Bool x Nat ∫ x Nat x Nat x Nat ∫ Q valid ∫ true Bool invalid ∫ true Nat
∫ Q
10
∫ n ∫ n ∫
premise i ∫ i conclusion ∫ Nat M N M N M N M N M N
n n ∫ Q ∫ M Nat ∫ N Nat ∫ n Nat ∫ M N Nat
∫ Q
(formal) type system formal proof systems
11 ∫ Nat
∫ Q ∫ Q ∫ Nat n ∫ Nat n ∫ Nat
M A ∫ M A M n inferred Nat ∫ Q ∫ true Bool true true true not typeable ∫ M Nat ∫ N Bool ∫ M N Nat true true
12
. ∫ M A [M] Ï [A] M A ∫ M A M M’ ∫ M’ A
x M x M x A M x A M x M x A M A B A B Basic ÏBasic
13 Bool Nat x x y x x y y syntactically identical
A B ÏBasic A B M,N x x A M M N
∫ A A ∫ A
∫ Q ∫ A A ∫ M A M A
axiom x x A A
14 ∫ A ∫ A ∫ Q x x A ∫ M B x dom
x ∫ A xÌdom ∫ Q x A ∫ Q
∫ Q ÏBasic ∫ A ∫ B ∫ ∫ A B
x ’ x A ∫ Q x A ∫ M B ∫ M A B ∫ N A ’ x A ∫ x A ∫ x A M A B ∫ M N B
x ’ x A x A A B B A A
∫ Q ∫ Q ∫ Q ∫ Q ∫ ∫ ∫ ∫ ∫ ∫ y ∫ Q x y ∫ Q x y ∫ y ∫ y z ∫ Q x y z ∫ Q x y z ∫ y x y z ∫ z x y z ∫ y z y ∫ z y z
15 Unit unit Bool true false Nat Unit Void Null Unit Unit unit Unit ∫ Q ∫ Q ∫ Unit ∫ unit Unit
Bool A N N A N N ifA A N N A ifA ∫ Q ∫ Q ∫ Q ∫ Bool ∫ true Bool ∫ false Bool
∫ M Bool ∫ N A ∫ N A
∫ ifA M then N else N A
16 Nat succ Nat Nat pred is ero ∫ Q ∫ Q ∫ M Nat ∫ Nat ∫ Nat ∫ succ M Nat
∫ M Nat ∫ M Nat ∫ pred M Nat ∫ is ero M Bool
product types A A A A first second with M x x N with with tuple types A ... An with
∫ A ∫ A ∫ M A ∫ M A
∫ A A ∫ ÜM M á A A
∫ M A A ∫ M A A
∫ first M A ∫ second M A
∫ M A A x A x A ∫ N B
∫ with x A x A M do N B
A A A left inLeft A right inRight isLeft isRight
17 asLeft asRight asLeft asLeft A A isLeft isRight asLeft asRight Bool case M M x x N N
∫ A ∫ A ∫ M A ∫ A ∫ A ∫ M A
∫ A A ∫ inLeftA M A A ∫ inRightA M A A
∫ M A A ∫ M A A ∫ isLeft M Bool ∫ isRight M Bool
∫ M A A ∫ M A A
∫ asLeft M A ∫ asRight M A
∫ M A A x A ∫ N B x A ∫ N B
∫ caseB M of x A then N x A then N B
Bool Unit Unit case Int Nat Nat Nat errorA asRight inLeftA unit A labeled record types variant types
18 with M l ln x xn N with A A Record first A second A2
li li
∫ A ∫ An ∫ M A ∫ Mn An
∫ Record l A ln An ∫ record l M ln Mn Record l A ln An
∫ M Record l A ln An jÏ n
∫ M lj Aj
∫ M Record l A ln An x A xn An ∫ N B
∫ with l x A ln xn An M do N B
is l isLeft is- Right as l asLeft asRight case A A Variant left A right A2 red green blue Variant red Unit green Unit blue Unit
li li
∫ A ∫ An ∫ A ∫ An ∫ M A jÏ n
∫ Variant l A ln An ∫ variant l A ln An l M Variant l A ln An
∫ M Variant l A ln An jÏ n
∫ M is lj Bool
∫ M Variant l A ln An jÏ n
∫ M as lj Aj
19
∫ M Variant l A ln An x A ∫ N B xn An ∫ Nn B
∫ caseB M of l x A then N ln xn An then Nn B
Ref A A unit ∫ A ∫ M A ∫ Ref A ∫ ref M Ref A
∫ M Ref A ∫ M Ref A ∫ N A ∫ deref M A ∫ M N Unit
Ref Ref Array A @ Nat Nat Ref A
arrayA N M @ N M let cell Ref A ref M and and cellN Ref A ref M in ÜN x Nat if x then cell else if else if x N then cellN else errorRef A á bound M @ first M
M N A @ if N < first M then deref second M N
else errorA M N P @ if N < first M then second M N P
else errorUnit
20 Array A A let ∫ A ∫ Array A
∫ N Nat ∫ M A ∫ M Array A ∫ array N M Array A ∫ bound M : Nat
∫ N Nat ∫ M Array A ∫ N Nat ∫ M Array A ∫ P A ∫ M N A ∫ M N P Unit
A A A unfold fold A A A B A B A un- fold fold M M fold unfold M’ M’ ∫ Q Ìdom ’ ∫ Q ∫ A ∫ Q ’ ∫ ∫ A
21 ∫ M A A ∫ M A
∫ fold A M A ∫ unfold A M A A
ListA A nil cons listCase
ListA @ Unit A
nilA ListA @ fold inLeft unit
consA A ListA ListA @ hd A tl ListA fold inRight Ühd tlá
listCaseA B ListA B A ListA B B @ l ListA n B c A ListA B case unfold l of unit Unit then n p A ListA then c p
case with A ®A íA
®A A @ x B unfoldB x x foldB x B unfoldB x x
íA A A A @ f A A x B f unfoldB x x foldB x B f unfoldB x x B 7 A A
V @ äxã @ x ä-ã V
ä x Mã @ foldV x V äMã äM Nã @ unfoldV äMã äNã
22 fold unfold
Unit A Bool Nat Proc D I C I A E C I E
C C D C I E C
23 E I N
E E E E
∫ C ∫ E A ∫ M A C E ∫ D a S signature S declaration D x Nat x Nat ∫ Q ∫ A A ∫ C C ∫ E A E A ∫ D a S D S
I ∫ A IÌdom ∫ Q I A ∫ Q
∫ Q ∫ Q ∫ Q ∫ Bool ∫ Nat ∫ Proc
∫ C ∫ E A AÏ Bool Nat ∫ I C a I Proc ∫ I A E a I A
∫ I A ∫ E A ∫ C ∫ C
∫ I E ∫ C C
24 ∫ D a I A I A ∫ C ∫ I Proc ∫ E Bool ∫ C ∫ D C ∫ I ∫ E C
I A ∫ Q ∫ Q
I A ∫ I A ∫ N Nat
∫ E Nat ∫ E Nat ∫ E Nat ∫ E Nat
∫ E E Nat ∫ E E Bool
type parameters M, M A x A x A id @ x x A type instantiation id A x A x M universally quantified M Ó A forall M A M A id Ó
25
A B A B Ó A M,N x x A M M N M M A
Ó A A M M
∫ Q ∫ A A ∫ M A M A
x ∫ A xÌdom ∫ Q Ìdom ∫ Q x A ∫ Q ∫ Q
’ ∫ Q ∫ A ∫ B ∫ A ’ ∫ ∫ A B ∫ Ó A
x ’ x A ∫ Q x A ∫ M B ∫ M A B ∫ N A ’ x A ∫ x A ∫ x A M A B ∫ M N B
26 ∫ M A ∫ M Ó A ∫ B ∫ M Ó A ∫ M B B A
Ó A A B A B A id Ó A id A A 7 A A id Ó id existentially quantified ∫ A ∫ B M B A
∫ Ô A ∫ packÔ A B with M Ô A
∫ M Ô A x A ∫ N B ∫ B
∫ openB M as x A in N B
Unit Unit true false cond BoolInterface @ ÔBool Record true Bool false Bool cond Ó Bool Bool true false cond Bool Unit Unit pack
27 boolModule BoolInterface @
packBoolInterface Bool Unit Unit with record true inLeft unit false inRight unit
cond x Bool y y
case x of x Unit then y x Unit then y Bool boolOp in if bool- Op true then else
openNat boolModule as Bool boolOp Record true Bool false Bool cond Ó Bool in boolOp cond Nat boolOp true boolOp true open Bool open
∫ A B A B A B A B Top
28 ∫ Q ∫ A A ∫ A B A B ∫ M A M A
A A B B Top A B A’ B’ A’ A B B’ M A B A A’ A M B B’ B M A B A’ B’ A’ B’ A B A’ B’ Unit Unit Unit Unit Unit
∫ A ∫ A B ∫ B C ∫ a A ∫ A B ∫ A A ∫ A C ∫ a B
∫ Q ∫ A ∫ A’ A ∫ B B’ ∫ Top ∫ A Top ∫ A B A’ B’
Nat Int
29 WorkingAge @ Variant student Unit, adult Unit Age @ Variant child Unit, student Unit, adult Unit, senior Unit Worker @ Record name String age WorkingAge profession String Person @ Record name String age Age WorkingAge Age Worker Person Ref A Ref B A B Ref A Ref B
∫ A B ∫ A B ∫ A B ∫ A B
∫ A A B B ∫ A A B B
li
∫ A B ∫ An Bn ∫ An ∫ An m
∫ Record l A ln m An m Record l B ln Bn
li
∫ A B ∫ An Bn ∫ Bn ∫ Bn m
∫ Variant l A ln An Variant l B ln m Bn m
bounded variables Top A B A B A B A B A B
30 <: ∫ A Ìdom ’ A ∫ Q ’ A ∫ Q A ∫ Q ’ A ∫ ’ A ∫ A
Top ∫ A ∫ A ∫ B Top ∫ A B ∫ A ∫ A B
A M M A M Top M A M Ó A B
A B A B Ó A B M,N x x A M M N A M M A
Ó A B B A A M M A x x
31 A ∫ B ∫ A’ A A’ ∫ B B’ ∫ Ó A B ∫ Ó A B Ó A’ B’
A ∫ M B ∫ M Ó A B ∫ A’ A ∫ A M Ó A B ∫ M A’ A’ B
Ô A B partially abstract type A A ∫ B ∫ A A A ∫ B B’ ∫ Ô A B ∫ Ô A B Ô A’ B’
∫ C A ∫ C M C B
∫ packÔ A B A C with M Ô A B
∫ M Ô A B ∫ D A x B ∫ N D
∫ openD M as A x B in N D
type equivalence ∫ A B ∫ A
32 A A ∫ A A ∫ A B A B retyping rule ∫ M A ∫ A B ∫ M B term equivalence ∫ M N A ∫ Int N N M
typability M M’ M A A M’ M’ M
M A ∫ M A
M 7 x x ∫ M
M 7 x y x 7 ’ y ∫ M
x B x x B
33 ∫ x Top B x x Top B B B x Top
∫ x B unfoldB x x B B B 7 unfoldB x B B
∫ x B x B x B B B 7 Ó x B B B Type M M M fail x A M x Ì dom x
Type x @ if x A Ï for some A then A else fail Type x A M @ A Type x A) M Type M N @ if Type M 7 Type N B for some B then B else fail
z y z y , Type y ) z y z Type y z ) y z if Type y z ) y 7 Type y z ) z B for some B then B else fail
34 if 7 B for some B then B else fail B7
Good A M A
Good @ Ï dom Good A B @ Good A and Good B Good Ó A @ Good A Type x @ if x A Ï for some A then A else fail Type x A M @ if Good A then A Type x A) M else fail Type M N @ if Type M 7 Type N B for some B then B else fail Type M @ Ó Type ) M Type M A @ if Type M 7 Ó B for some ,B and Good A then A B else fail
Subtype A, B A B Subtype A B A B
Good @ Ï dom Good Top @ true Good A B @ Good A and Good B Good Ó A B @ Good A and Good A B
35 Subtype A Top @ true Subtype @ true Subtype A @ for A= Top if B Ï for some B then Subtype B A else false Subtype A B A’ B’ @ Subtype A’ A and Subtype B B’ Subtype Ó A B Ó ’ A’. B’ @ Subtype , A’, A and Subtype , ’ A’ , ’ B, B’ Subtype A B @ false otherwise Expose @ if A Ï for some A then Expose A else fail Expose A @ A otherwise Type x @ if x A Ï for some A then A else fail Type x A M @ if Good A then A Type x A) M else fail Type M N @ if Expose Type M 7 A B for some A,B and Subtype Type N A then B else fail Type A M @ if Good A then Ó A Type A) M else fail Type M A @ if Expose Type M 7 Ó A’ B for some ,A’,B and Good A and Subtype A A’ then A B else fail
36
37
A B A B C A C B C C A B A B D D A D B D not
38 A B B
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40 System Sciences Proc. 11th Annual ACM Symposium on Principles of Programming Languages Handbook of Theoretical Computer Science, Proc. 12th Annual ACM Symposium on Principles of Programming Languages Proc. 9th Annual IEEE Symposium on Logic in Computer Science (To appear in Information and Computation.) Proc. 19th Annual ACM Sympo- sium on Principles of Programming Languages Proc. Colloquium sur la programma- tion Information Pro- cessing, . . Information and Computation Proc. 9th Annual IEEE Symposium on Logic in Computer Science . Information and Computation
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