Type Reconstruction - uni-saarland.de · Type Reconstruction ... Polymorphism Let-Polymorphism ......

Type Reconstruction

Sven Woop

wooppsuni-sbde

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Seminar Types and Programming Languages

Goal Calculating a principle type of a not type-annotated term

More Formally Given a pair (t) compute the mostgeneral type T such that gt t T es well typed

Example

2 Steps Derive a set of contraints find the principal unifier for these constraints

We compute principal types not principal typings

XXXfxfxxf )(

XXxx




Index

Basics Standard Unification Nonstandard Unification Typing-rules for simply typed -calculus

Type Reconstruction Constraint typing rules for -calculus CT-Rules and Recursive types

Polymorphism Let-Polymorphism

Overview



Unification

Unification [Robinson 1965]

Unification in linear space complexity[Martelli Montanary 1984]




Standard Unification

More precisely syntactic equational unification We define the set of terms as

st = x | f(t1tn) with x Var f FuncSymbols Given an equation

s twe search a substitution such that

s = t is called a unifier for s t





We call a unifier 1 more general than a unifier 2 iffthere is a substitution such that 1 = 2We write 1 2

A principal unifier of s t is a unifier such that for allunifiers lsquo of s t we have lsquo

Unification Theorem Each equations t has a principal unifier if it isunifiable




Example

f(xy) f(ay)

1 = x = a y = b is a unifier as1 f(xy) = 1 f(ay)f(ab) = f(ab)

2 = x = a is a principal unifier2 f(xy) = 2 f(ay)f(ay) = f(ay)

y = b 2 = 1




Example

f(x) g(a) is not unifiable

x f(x) is not unifiable by standard unification




Unification by MartelliMontanari

t t R | MM R |

f() g() R | MM if f g or Arity(f) Arity(g)

f(s1sn) f(t1tn) R | MM s1 t1 sn tn R |

x t R | MM [x= t] R | [x=t] if x var(t)(Self Occurence Check)

x t R | MM if x var(t)

t x R | MM x t R |

| MM




Motivation

x f(x) is not unifiable with a finite term But thefollowing regular tree is an infinite solution

f

f

f

finite representation

fx =




Equivalence Test

s =

fun eq(nm) =

if nm s then

true

else if Label(n) Label(m) or Arity(n) Arity(m)

false

else

s = s nm

Arity(n)

eq(nimi)

i = 1




Nonstandard Unification

See the unification problem t1 t2 as a graph unificationproblem Let eq be a function that computes theequivalence between two nodes in a graph

While not eq(t1t2) do

let (nm) be a pair of nodes with Label(n) Label(m)

or Arity(n) Arity(m)

if Label(n) = f and Label(m) = g and f g or

Arity(n) Arity(m) then return

else if Label(n) = x then subst(xm)

else if Label(m) = x then subst(xn)Note No occurencecheck Solutions are infiniteregular trees




Example f(g(x)) f(x)

f

g

x

f

x

a b

[ x= a ]

f

g

f

a



Typing rules for simplytyped lambda calculus




Typing Rules

Var)-(Ty

Tx

Tx

Abs)-(Ty

2121

221

TTtTxTtTx

App)-(Ty

T

321

22321

TttTtTt

If)-(Tytelsethen if

321

321

TttTtTtBoolt

Rec)-(Ty

2211

2211

TtTxTtTx

Note Abstractionsand recursions aretype annotated




Example

Supposition It exists aprincipal type annotation

x Bool Bool y Bool x y (Bool Bool) Bool Bool

x Nat Bool y Nat x y (Nat Bool) Nat Bool

x Bool Y y Bool x y (Bool Y) Bool Y

x X Y y X x y (X Y) X Y

x y x y



Constraint typing rules

Principal Types Curry and Feys [1958]

Algorithm to compute principal types Hindley [1969]

Type reconstruction Algorithm W Damas and Milner[1982]






Example



XXXfxfxxf )(

XXxx




CT-Rules by PierceVisage eXPert PDF Copyright copy 19982003 Visage Software

This document was created with free TRIAL version of Visage eXPert PDFThis watermark will be removedafter purchasing the licensed full version of Visage eXPert PDF Please visit httpwwwvisagesoftcom for more details



Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt

If)-(CT|telsethen if

|||

3213212321

333222111

TTBoolTCCCTttCTtCTtCTt

Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx

Let all Xi be fresh type variables




Idea

Do just the same as the standard typing rules

Introduce fresh type variables each time a type canlsquot becomputed directly

Construct constraints consisting of the conditions thetyping rules check




CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221

XXXCXxfxXxXfCXXxfxXxXf

CXXXCXxfxXxXfCXXXXxfXxXf

XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types

CT-Rules can be maintained

Use the Nonstandard Unification Algorithm




Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC

xxxVisage eXPert PDF Copyright copy 19982003 Visage Software



Unification

1X

2X

A




Unification

][ 1 AX

2X

A

Most general type of

is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy

CXXXyxxXyXxCXXXCXyxxXyXx

CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C

52121 )(y XXXyxxXyXx




Correctness

typedwellisCsatisfies σ t

4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C

Note It exists a principaltype annotation



Let-Polymorphism

Let-polymorphism Milner [1978]




Polymorphism

Polymorphism is a language mechanism thatallow a single part of a program to be used withdifferent types in different contexts




Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule




Solution type scheme

Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn

Let X1Xn be the free type variables of Tthat do not occur in We define




Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool

Now the program is well typed




Problem side effects

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref

r (X11X11)Ref X11=Nat

r (X12X12)Ref X12=Bool

no type clash




Solution

Only if t1 is a value

Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Note The type scheme is introduced afterthe typechecking of the term x=t1 Thismeans that you can not use xpolymorphically in the term t1 itself (nopolymorphic recursion)




Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool

no value no type scheme

type clash




Restriction

You can not compute a polymorphic function

Eg

val f = let val i = ref true

in

fn x =gt fn y =gt

(if i then x else y) before i = not(i)

end

f is no polymorphic function




Runtime is Exponential

let val f0 = fun x =gt (xx) in

let val f1 = fun y =gt f0 (f0 y) in




f4 (fun z =gt z)

end end end end end

The following program is well typed but takes a long time to typecheck




Runtime Analysis

let val f0 =

fun x =gt (xx) in

let val f1 = fun y =gt

f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()

Program Derived Type Constraints0

2

4

8

16

Type Size20

22

24

28

216




Overview

Unification Nonstandard Unification Constraint typing rules for -calculus (similar to standard

typing rules) It exists a principal type annotation as the solution of a

set of constraints (Unification Theorem) Constraint typing rules and recursive types Let-Polymorphism




Historical Context Unification [Robinson 1965] Unification in linear space complexity

[Martelli Montanary 1984] Nonstandard Unification




Type Reconstruction with Recursive Types[Huet 1975 1976]





QuestionsVisage eXPert PDF Copyright copy 19982003 Visage Software





Example



XXXfxfxxf )(

XXxx




Index




Overview



Unification





















Example

f(xy) f(ay)



y = b 2 = 1




Example







t t R | MM R |





t x R | MM x t R |

| MM




Motivation


f

f

f


fx =




Equivalence Test

s =

fun eq(nm) =

if nm s then

true


false

else

s = s nm

Arity(n)

eq(nimi)

i = 1

















f

g

x

f

x

a b

[ x= a ]

f

g

f

a







Typing Rules

Var)-(Ty

Tx

Tx

Abs)-(Ty

2121

221

TTtTxTtTx

App)-(Ty

T

321

22321

TttTtTt

If)-(Tytelsethen if

321

321

TttTtTtBoolt

Rec)-(Ty

2211

2211

TtTxTtTx





Example






x y x y












Example



XXXfxfxxf )(

XXxx








Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt


|||

3213212321

333222111


Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx





Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Index




Overview



Unification





















Example

f(xy) f(ay)



y = b 2 = 1




Example







t t R | MM R |





t x R | MM x t R |

| MM




Motivation


f

f

f


fx =




Equivalence Test

s =

fun eq(nm) =

if nm s then

true


false

else

s = s nm

Arity(n)

eq(nimi)

i = 1

















f

g

x

f

x

a b

[ x= a ]

f

g

f

a







Typing Rules

Var)-(Ty

Tx

Tx

Abs)-(Ty

2121

221

TTtTxTtTx

App)-(Ty

T

321

22321

TttTtTt

If)-(Tytelsethen if

321

321

TttTtTtBoolt

Rec)-(Ty

2211

2211

TtTxTtTx





Example






x y x y












Example



XXXfxfxxf )(

XXxx








Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt


|||

3213212321

333222111


Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx





Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview



















Unification





















Example

f(xy) f(ay)



y = b 2 = 1




Example







t t R | MM R |





t x R | MM x t R |

| MM




Motivation


f

f

f


fx =




Equivalence Test

s =

fun eq(nm) =

if nm s then

true


false

else

s = s nm

Arity(n)

eq(nimi)

i = 1

















f

g

x

f

x

a b

[ x= a ]

f

g

f

a







Typing Rules

Var)-(Ty

Tx

Tx

Abs)-(Ty

2121

221

TTtTxTtTx

App)-(Ty

T

321

22321

TttTtTt

If)-(Tytelsethen if

321

321

TttTtTtBoolt

Rec)-(Ty

2211

2211

TtTxTtTx





Example






x y x y












Example



XXXfxfxxf )(

XXxx








Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt


|||

3213212321

333222111


Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx





Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview



































Example

f(xy) f(ay)



y = b 2 = 1




Example







t t R | MM R |





t x R | MM x t R |

| MM




Motivation


f

f

f


fx =




Equivalence Test

s =

fun eq(nm) =

if nm s then

true


false

else

s = s nm

Arity(n)

eq(nimi)

i = 1

















f

g

x

f

x

a b

[ x= a ]

f

g

f

a







Typing Rules

Var)-(Ty

Tx

Tx

Abs)-(Ty

2121

221

TTtTxTtTx

App)-(Ty

T

321

22321

TttTtTt

If)-(Tytelsethen if

321

321

TttTtTtBoolt

Rec)-(Ty

2211

2211

TtTxTtTx





Example






x y x y












Example



XXXfxfxxf )(

XXxx








Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt


|||

3213212321

333222111


Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx





Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview



























Example

f(xy) f(ay)



y = b 2 = 1




Example







t t R | MM R |





t x R | MM x t R |

| MM




Motivation


f

f

f


fx =




Equivalence Test

s =

fun eq(nm) =

if nm s then

true


false

else

s = s nm

Arity(n)

eq(nimi)

i = 1

















f

g

x

f

x

a b

[ x= a ]

f

g

f

a







Typing Rules

Var)-(Ty

Tx

Tx

Abs)-(Ty

2121

221

TTtTxTtTx

App)-(Ty

T

321

22321

TttTtTt

If)-(Tytelsethen if

321

321

TttTtTtBoolt

Rec)-(Ty

2211

2211

TtTxTtTx





Example






x y x y












Example



XXXfxfxxf )(

XXxx








Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt


|||

3213212321

333222111


Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx





Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Example

f(xy) f(ay)



y = b 2 = 1




Example







t t R | MM R |





t x R | MM x t R |

| MM




Motivation


f

f

f


fx =




Equivalence Test

s =

fun eq(nm) =

if nm s then

true


false

else

s = s nm

Arity(n)

eq(nimi)

i = 1

















f

g

x

f

x

a b

[ x= a ]

f

g

f

a







Typing Rules

Var)-(Ty

Tx

Tx

Abs)-(Ty

2121

221

TTtTxTtTx

App)-(Ty

T

321

22321

TttTtTt

If)-(Tytelsethen if

321

321

TttTtTtBoolt

Rec)-(Ty

2211

2211

TtTxTtTx





Example






x y x y












Example



XXXfxfxxf )(

XXxx








Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt


|||

3213212321

333222111


Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx





Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Example







t t R | MM R |





t x R | MM x t R |

| MM




Motivation


f

f

f


fx =




Equivalence Test

s =

fun eq(nm) =

if nm s then

true


false

else

s = s nm

Arity(n)

eq(nimi)

i = 1

















f

g

x

f

x

a b

[ x= a ]

f

g

f

a







Typing Rules

Var)-(Ty

Tx

Tx

Abs)-(Ty

2121

221

TTtTxTtTx

App)-(Ty

T

321

22321

TttTtTt

If)-(Tytelsethen if

321

321

TttTtTtBoolt

Rec)-(Ty

2211

2211

TtTxTtTx





Example






x y x y












Example



XXXfxfxxf )(

XXxx








Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt


|||

3213212321

333222111


Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx





Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview





















t t R | MM R |





t x R | MM x t R |

| MM




Motivation


f

f

f


fx =




Equivalence Test

s =

fun eq(nm) =

if nm s then

true


false

else

s = s nm

Arity(n)

eq(nimi)

i = 1

















f

g

x

f

x

a b

[ x= a ]

f

g

f

a







Typing Rules

Var)-(Ty

Tx

Tx

Abs)-(Ty

2121

221

TTtTxTtTx

App)-(Ty

T

321

22321

TttTtTt

If)-(Tytelsethen if

321

321

TttTtTtBoolt

Rec)-(Ty

2211

2211

TtTxTtTx





Example






x y x y












Example



XXXfxfxxf )(

XXxx








Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt


|||

3213212321

333222111


Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx





Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Motivation


f

f

f


fx =




Equivalence Test

s =

fun eq(nm) =

if nm s then

true


false

else

s = s nm

Arity(n)

eq(nimi)

i = 1

















f

g

x

f

x

a b

[ x= a ]

f

g

f

a







Typing Rules

Var)-(Ty

Tx

Tx

Abs)-(Ty

2121

221

TTtTxTtTx

App)-(Ty

T

321

22321

TttTtTt

If)-(Tytelsethen if

321

321

TttTtTtBoolt

Rec)-(Ty

2211

2211

TtTxTtTx





Example






x y x y












Example



XXXfxfxxf )(

XXxx








Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt


|||

3213212321

333222111


Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx





Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Equivalence Test

s =

fun eq(nm) =

if nm s then

true


false

else

s = s nm

Arity(n)

eq(nimi)

i = 1

















f

g

x

f

x

a b

[ x= a ]

f

g

f

a







Typing Rules

Var)-(Ty

Tx

Tx

Abs)-(Ty

2121

221

TTtTxTtTx

App)-(Ty

T

321

22321

TttTtTt

If)-(Tytelsethen if

321

321

TttTtTtBoolt

Rec)-(Ty

2211

2211

TtTxTtTx





Example






x y x y












Example



XXXfxfxxf )(

XXxx








Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt


|||

3213212321

333222111


Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx





Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview

































f

g

x

f

x

a b

[ x= a ]

f

g

f

a







Typing Rules

Var)-(Ty

Tx

Tx

Abs)-(Ty

2121

221

TTtTxTtTx

App)-(Ty

T

321

22321

TttTtTt

If)-(Tytelsethen if

321

321

TttTtTtBoolt

Rec)-(Ty

2211

2211

TtTxTtTx





Example






x y x y












Example



XXXfxfxxf )(

XXxx








Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt


|||

3213212321

333222111


Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx





Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview





















f

g

x

f

x

a b

[ x= a ]

f

g

f

a







Typing Rules

Var)-(Ty

Tx

Tx

Abs)-(Ty

2121

221

TTtTxTtTx

App)-(Ty

T

321

22321

TttTtTt

If)-(Tytelsethen if

321

321

TttTtTtBoolt

Rec)-(Ty

2211

2211

TtTxTtTx





Example






x y x y












Example



XXXfxfxxf )(

XXxx








Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt


|||

3213212321

333222111


Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx





Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview























Typing Rules

Var)-(Ty

Tx

Tx

Abs)-(Ty

2121

221

TTtTxTtTx

App)-(Ty

T

321

22321

TttTtTt

If)-(Tytelsethen if

321

321

TttTtTtBoolt

Rec)-(Ty

2211

2211

TtTxTtTx





Example






x y x y












Example



XXXfxfxxf )(

XXxx








Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt


|||

3213212321

333222111


Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx





Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Typing Rules

Var)-(Ty

Tx

Tx

Abs)-(Ty

2121

221

TTtTxTtTx

App)-(Ty

T

321

22321

TttTtTt

If)-(Tytelsethen if

321

321

TttTtTtBoolt

Rec)-(Ty

2211

2211

TtTxTtTx





Example






x y x y












Example



XXXfxfxxf )(

XXxx








Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt


|||

3213212321

333222111


Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx





Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Example






x y x y












Example



XXXfxfxxf )(

XXxx








Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt


|||

3213212321

333222111


Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx





Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




























Example



XXXfxfxxf )(

XXxx








Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt


|||

3213212321

333222111


Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx





Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview






















Example



XXXfxfxxf )(

XXxx








Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt


|||

3213212321

333222111


Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx





Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview
























Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt


|||

3213212321

333222111


Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx





Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview





















Var)-(CT|

Tx

Tx

Abs)-(CT|

|

2121

221

CTXtXxCTtXx

App)-(CT|

||

212121

222111

XTTCCXttCTtCTt


|||

3213212321

333222111


Rec)-(CT|

|

212211

2211

TXCTtXxCTtXx





Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Idea







CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















CT-Var

TxTx

|

TxTx

Ty-Var

CT-Var




CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















CT-Rec

||

212211

2211

TXCTtXxCTtXx

2211

212211

TtTxTTTtTx

Ty-Rec




CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















CT-Abs

2121

221

TTtTxTtTx

CTXtXxCTtXx

||

2121

221

Ty-Abs

CT-Abs




CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















CT-App

321

22321

TttTtTTt

|||

32121321

222111

XTTCCXttCTtCTt

Ty-App




Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Example

|)(|)(

|)(|

|||

4212121

24221

24321421

1321321

221121

221



XxXxXfXfXxXfXxXxXf

421

432

321

3

XXXXXXXXX

C

)( xfxxf




Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Example

421

432

321

3

XXXXXXXXX

C

][ 3211 XXX

4232

432

3232

31

XXXXXXX

XXXXC

][ 434212 XXXXX

4242

442

4242

32

XXXXXXX

XXXXC

])([ 4424344413 XXXXXXXXX

444444

4444

444444

33

)()(

)()(

XXXXXXXXXX

XXXXXXC




Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Recursive Types






Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Simple Example

|||Xx|Xx

211211

21121

1111

XXXXXxxXxXXXXxxXx

XxXx

211 XXXC




Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Unification

1X

2X

A




Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Unification

][ 1 AX

2X

A


is

xxx

2211 X XXX




Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Advanced Example

352121

35422521

242314

213113

11

2

|)(y |)(y

|||

||

|Xy


CXXXCXyxxXyCXXXXxx

XxXx

542

423

311

3

XXXXXXXXX

C

)(y y yxxxF FFfix



1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview



















1X

1X

3X

3X

2X

4X

2X

4X

5X

A



3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview



















3X

3X

2X

4X

2X

4X

5X

][ 1 AX

A B



2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview



















2X

4X

2X

4X

5X

][ 31 BXAX

A B C



4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview



















4X 4X

5X

][ 231 CXBXAX

A B C





Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Correctness


4321 )( XXxfxXxXf

421

432

321

XXXXXXXXX

C




Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview



















Let-Polymorphism





Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Polymorphism





Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Let-Polymorphism

let

id = xx

in

id 1

id true

end

id X1 X1

X1=Nat

X1=Bool

type clash

Naive Let-Rule





Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview





















Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n

Var2)-(CT|

][ 111

CTxXXXXCTXXx nnn





Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Example

let

id = xx

in

id 1

id true

end

id X1X1 X1

id X11 X11 X11=Nat

id X12 X12 X12=Bool






let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview





















let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r X1(X1X1)Ref



no type clash




Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Solution


Let)-(CT|endin let

|)(|

21

211

CTttxCTtCTXXxCTtx n





Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Example

let

r = ref(xx)

in

r= xNatsucc x

(r) true

end

r (X1X1)Ref

r (X1X1)Ref X1=Nat

r (X1X1)Ref X1=Bool


type clash




Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Restriction


Eg


in

fn x =gt fn y =gt


end











f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview


























f4 (fun z =gt z)

end end end end end





Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Runtime Analysis

let val f0 =

fun x =gt (xx) in


f0 (f0 y) in


f1 (f1 y) in


f2 (f2 y) in


f3 (f3 y) in

f4 (fun z =gt z)

end end end end end

X0X0X0X0

X1X1 (X1X1)(X1X1)

X2X2((((X2X2)(X2X2))

((X2X2)(X2X2)))

(((X2X2)(X2X2))

((X2X2)(X2X2))))

()


2

4

8

16

Type Size20

22

24

28

216




Overview




















Overview



































Date post:	11-May-2018
Category:	Documents
Upload:	phamquynh
View:	217 times
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