News

• PA 6 is out, due Fri 12/7

• NO PA 7

• We will release it, but will not be graded ...

• ... but will be on the final.

Brief Digression Into PA6

• Decorators e.g. Profiling

• ScalaCheck e.g. BST Bug

• Case Classes e.g. IntList

Today's Plan: Subtyping

When is an object like another?

• When does one object behave like another?

• When can we substitute one widget with another?

Example

• What kinds of Point objects can we send to screenSaver?

• What is a Point?

Essential for safe code reuse

Recap: Objects and Types

Objects

• Maps from names to properties

• Properties = Data + Functions (+ Objects)

Recap: Types

• Short descriptions of Objects

• Maps from names to types

Description of object properties

• The actions one can perform on object

• Field and Methods and their respective types

Recap: Types

• Short descriptions of Objects

• Maps from names to types

Example: ticker

{ def next(): Any
  def hasNext: Boolean }

Example: Int

{ def .+(other: Int) : Int // add       method
  def .-(other: Int) : Int // subtract  method
  ...//etc }

Recap: Types

• Short descriptions of Objects

• Maps from names to types

Example: Point

class Point {
  val name : String                 // immutable field
  var x    : Int                    // mutable field
  var y    : Int                    // mutable field
  def render(g: Graphics): Unit     // method 
  def move(d: Dimension, del: Int)  // method
} 

Recall ScreenSaver

def screenSaver(p: Point) {

  // Only uses properties described in Point 

}

A Type is a Contract

• Method (ScreenSaver) promises to only use attributes of Point

• Caller (main) promises to only send Point values

• Compiler must verify that both sides stick to contract

• Key: what exactly is a Point value?

What is a Point ?

What objects can we really call ScreenSaver with?

• Only those created by new Point(...) ?

• That seems too restrictive. What about ?
  • Dot
  • ColorDot
  • Box
  • ColorBox

• Why are those also allowed?

Subtyping

What objects can we really call ScreenSaver with?

• Intuitively, any object with Point attributes

• If object has more attributes, no problem!

• This notion is called subtyping
  • Dot
  • ColorDot
  • Box
  • ColorBox
  • .. are subtypes of Point

Notation: S <: T means type S is a subtype of type T

If S<:T then we can use S object in place of T object

If S<:T then S object IS A T object

Subtype Polymorphism

• If S <: T then we can use S object in place of T object

• If a function expects T objects, it can be called with S objects

Subtype Polymorphism

If S <: T then we can use S objects in place of T objects

• Dot <: Point
• ColorDot <: Dot <: Point
• Box <: Dot <: Point
• ColorBox <: Dot <: Point

def screen(p: Point) {
  // Only uses properties described in Point 
}

So screen can be called with Dot, ColorDot, Box, ColorBox

Sounds great in theory, but when is S <: T

Sounds great in theory, but when is S <: T

Intuition

When every object of type S "IS A" object of type T

• When S has all attributes of T

• (... and maybe extra ones.)

Sounds great in theory, but when is S <: T

Example: S <: T if S has all attributes of T

class ColorDot {
  val color : Color  // Extra Field
  val name  : String
  var x     : Int
  var y     : Int
  def render(g: Graphics): Unit
  def move(d: Dimension, delta: Int) 
}

is a (subtype of)

class Dot {
  val name : String
  var x    : Int
  var y    : Int
  def render(g: Graphics): Unit
  def move(d: Dimension, delta: Int) 
}

Subtyping Rule S <: T

Width Subtyping: when S has more attributes than T

If

• S = { val f0: S0; val f1: S1; ... ; val fn: Sn }

• T = { val f1: S1; ... ; val fn: Sn }

Then

• S <: T

Hold on a minute!

We know

• Dot <: Point (extra field: name)

• ColorDot <: Dot (extra field: color)

But how do we get

• ColorDot <: Point ?

Subtyping Rule S <: T

Transitive Subtyping: The Is-A relation is transitive

If

• S <: Thing

• Thing <: T

Then

• S <: T

But intuition can be a tricky thing...

Given types S and T, how can we know if S <: T ?

We know ColorDot <: Dot. What about ColorLine and Line?

class Line {
  val p1: Dot
  val p2: Dot
}

and

class ColorLine {
  val p1: ColorDot
  val p2: ColorDot
}

QUIZ: How can we know if S <: T ?

We know ColorDot <: Dot. What about ColorLine and Line?

class Line {
  val p1: Dot
  val p2: Dot
}

class ColorLine {
  val p1: ColorDot
  val p2: ColorDot
}

For any function def foo(l: Line), is it OK to call foo with a ColorLine ?

A. Hells, Yes!

B. OMG, NO!

Given types S and T, how can we know if S <: T ?

We know ColorDot <: Dot.

What about ColorLine and Line?

For any function

def foo(l: Line)

Is it OK to call foo with a ColorLine ? Yes

• Only thing foo can do is use p1 and p2 as Dot...

• A ColorLine has those fields with properties of ColorDot

• But a ColorDot is a Dot

• Hence, we can call foo with ColorLine objects.

Subtyping Rule S <: T

Depth Subtyping: when S fields are subtypes of T fields

If

• S = { val f1: S1; ... ; val fn: Sn }

• T = { val f1: T1; ... ; val fn: Tn }

• S1 <: T1 and S2 <: T2 and ... Sn <: Tn

Then

• S <: T

Digression: Subtyping on Function Types ...

Given types S and T, how can we know if S <: T ?

Subtyping Function Types

• First, we look at outputs (with the same input type)
  • When is (In => S) <: (In => T)?

• Next, lets look at inputs (with the same output type)
  • When is (S => Out) <: (T => Out)?

• Finally, we can combine the two...
  • When is (S1 => S2) <: (T1 => T2)?

Given types S and T, how can we know if S <: T ?

Subtyping Function Types

• First, we look at outputs (with the same input type)
  • When is (In => S) <: (In => T)?
• Next, lets look at inputs (with the same output type)
  • When is (S => Out) <: (T => Out)?
• Finally, we can combine the two...
  • When is (S1 => S2) <: (T1 => T2)?

Output Subtyping: When is In=>S <: In=>T ?

Is () => ColorDot <: () => Dot ?

Suppose I have a method whose input has type () => Dot

def screenSaver(f:() => Dot): Unit

Also, suppose I have a function that returns ColorDot <: Dot

val getColorDot: () => ColorDot

QUIZ: Is it safe to do screenSaver(getColorDot) ?

A. Yes

B. No

Output Subtyping: When is In=>S <: In=>T ?

Is () => ColorDot <: () => Dot ?

Suppose I have a method whose input has type () => Dot

def screenSaver(f:() => Dot): Unit

Also, suppose I have a function that returns ColorDot <: Dot

val getColorDot: () => ColorDot

QUIZ: Is it safe to do screenSaver(getColorDot) ? YES

• Only thing screenSaver can do is call f to get a Dot ...
• ... on which it operates using Dots properties

• When screenSaver calls getColorDot it gets a ColorDot ...
• ... which is a Dot (has all of Dots properties) ...

Subtyping Rule In => S <: In => T

Function Subtyping

Can substitute/use a function with another that produces fewer outputs ...

If

• S <: T

Then

• In => S <: In => T

Example

• As ColorDot <: Dot
• We have () => ColorDot <: () => Dot

Given types S and T, how can we know if S <: T ?

Subtyping Function Types

• First, we look at outputs (with the same input type)
  • (In => S) <: (In => T)?
  • If S <: T
• Next, lets look at inputs (with the same output type)
  • When is (S => Out) <: (T => Out)?
• Finally, we can combine the two...
  • When is (S1 => S2) <: (T1 => T2)?

Input Subtyping: When is S=>Out <: T=>Out ?

Is ColorDot=>Graphic <: Dot=>Graphic ?

If I have a function whose input must be Dot => Graphic

def plotGraphic(x: Dot => Graphic): Unit

and a function from ColorDot to Graphic

val cd2g: ColorDot => Graphic

QUIZ: Is it safe to do plotGraphic(cd2g) ?

A. Yes

B. No

Input Subtyping: When is S=>Out <: T=>Out ?

Is ColorDot=>Graphic <: Dot=>Graphic ?

def plotGraphic(f: Dot => Graphic) {
  val d = new Dot(0, 0) // not ColorDot
  plot(f(d))    
}

def cd2g(cd: ColorDot) = { 
  val c = cd.color
  setColor(cd.color)
  fillColorOval()
} 

What happens if we do ...

plotGraphic(cd2g)

?

Input Subtyping: When is S=>Out <: T=>Out ?

Is ColorDot=>Graphic <: Dot=>Graphic ?

def plotGraphic(f: Dot => Graphic) {
  val d = new Dot(0, 0) // not ColorDot
  plot(f(d))            // cd2g(d) crashes! 
}

def cd2g(cd: ColorDot) = { 
  val c = cd.color
  setColor(cd.color)
  fillColorOval()
} 

plotGraphic(cd2g)

Yikes! the call f(d) in plotGraphic crashes!

• As cd2g(new Dot(0, 0)) tries to access .color field ...
• ... which does not exist in Dot!

Input Subtyping: When is S=>Out <: T=>Out ?

Is ColorDot=>Graphic <: Dot=>Graphic ?

• B. No!

• Cannot use a ColorDot=>Graphic instead of Dot=>Graphic

• Cannot substitute a ColorDot=>Graphic in place of Dot=>Graphic

• !!!

Input Subtyping: When is S=>Out <: T=>Out ?

How about Dot=>Graphic <: ColorDot=>Graphic ?

Input Subtyping: When is S=>Out <: T=>Out ?

How about Dot=>Graphic <: ColorDot=>Graphic ?

Suppose we have:

def plotColorGraphic(f: ColorDot => Graphic)

def d2g(d: Dot)

Would plotGraphic(cd2g) cause any problems?

• No! Non! Nein! Nada! Nahin!

• plotColorGraphic will only call f with ColorDot values ...

• ... all of which are Dot values ...

• ... so d2g will have no problem handling them!

Input Subtyping: When is S=>Out <: T=>Out ?

How about Dot=>Graphic <: ColorDot=>Graphic ?

Suppose we have:

def plotColorGraphic(f: ColorDot => Graphic)

def d2g(d: Dot)

Would plotGraphic(cd2g) cause any problems?

Safe to use a function that accepts more inputs

Subtyping Rule T => Out <: T => Out

Function Subtyping

Can substitute/use a function with another that accepts more inputs ...

If

• T <: S

Then

• S => Out <: T => Out

Example

• As ColorDot <: Dot

• We have Dot => Graphic <: ColorDot => Graphic

Function Subtyping in Pictures

Function Subtyping in Pictures

Function Subtyping in Pictures

Function Subtyping in Pictures

Given types S and T, how can we know if S <: T ?

Subtyping Function Types

• First, we look at outputs (with the same input type)
  • (In => S) <: (In => T)?
  • If: S <: T
• Next, lets look at inputs (with the same output type)
  • When is (S => Out) <: (T => Out)?
  • If: T <: S
• Finally, we can combine the two...
  • (In1 => Out1) <: (In2 => Out2) ?

QUIZ: Function Subtyping

You tell me!

In1 => Out1 <: In2 => Out2

If:

A. In1 <: In2 and Out1 <: Out2

B. In2 <: In1 and Out1 <: Out2

C. In1 <: In2 and Out2 <: Out1

D. In2 <: In1 and Out2 <: Out1

Function Subtyping

You tell me!

In1 => Out1 <: In2 => Out2

We can use f1: In1 => Out1 in place of f2: In2 => Out2

If:

• f1 accepts more inputs
  • In2 <: In1

• f2 produces fewer outputs
  • Out1 <: Out2

• Solution: B. In2 <: In1 and Out1 <: Out2

Given types S and T, how can we know if S <: T ?

Subtyping Function Types

If

• In2 <: In1
• Out1 <: Out2

Then

• In1 => Out1 <: In2 => Out2

Given types S and T, how can we know if S <: T ?

Subtyping Function Types

If

• In2 <: In1
• Out1 <: Out2

Then

• In1 => Out1 <: In2 => Out2

Jargon

• Contravariant on input type
• Covariant on output type

End Digression: Back To Objects/Classes...(but don't forget functions!)

QUIZ: How can we know if S <: T ?

We know ColorDot <: Dot. How about these two (with getter methods)

class Line {
  def getP1(): Dot
  def getP2(): Dot
}

and

class ColorLine {
  def getP1(): ColorDot
  def getP2(): ColorDot
}

Is ColorLine <: Line ? (Is it OK to call foo with a ColorLine?)

A. Yes

B. No

Given types S and T, how can we know if S <: T ?

We know ColorDot <: Dot. How about these two (with getter methods)

class Line {
  def getP1(): Dot
  def getP2(): Dot
}

and

class ColorLine {
  def getP1(): ColorDot
  def getP2(): ColorDot
}

Yes we can call foo with a ColorLine

• foo can only get and use p1 and p2 as Dot...
• A ColorLine has those fields with properties of ColorDot
• But a ColorDot is a Dot
• Hence, we can call foo with ColorLine objects.

Given types S and T, how can we know if S <: T ?

We know ColorDot <: Dot. How about these two (with getter methods)

class Line {
  def getP1: () => Dot
  def getP2: () => Dot
}

class ColorLine {
  def getP1: () => ColorDot
  def getP2: () => ColorDot
}

Lets use the rules to derive ColorLine <: Line ?

• By Function Subtyping : () => ColorDot <: () => Dot
• Hence, ColorLine.getP1 <: Line.getP1
• And, ColorLine.getP2 <: Line.getP2
• So, by Depth Subtyping: ColorLine <: Line

Given types S and T, how can we know if S <: T ?

We know ColorDot <: Dot. How about these two (with setter methods)

class Line { 
  // other fields from before ...
  def setP1(p: Dot): Unit
  def setP2(p: Dot): Unit
}

class ColorLine {
  // other fields from before ...
  def setP1(p: ColorDot) : Unit
  def setP2(p: ColorDot) : Unit
}

Is ColorLine <: Line ?

For any function foo, can we call foo(new ColorLine(...)) ?

def foo(l: Line) {...} 

Given types S and T, how can we know if S <: T ?

OK to call foo with a ColorLine? (for any def foo(l: Line))

class Line      { def setP1(p: Dot): Unit }

class ColorLine { def setP1(p: ColorDot) : Unit }

Suppose that ColorLine <: Line ?

Consider the following foo

def foo(l: Line) { l.setP1(new Dot(0, 0)) }

Now, all hell breaks loose:

val cl     = new ColorLine
foo(cl)                     // Typechecks 
val color1 = l.getP1.color  // Runtime Error! 

But the rules would have told us this!

Given types S and T, how can we know if S <: T ?

If ColorDot <: Dot then is ColorLine <: Line ?

class Line { 
  // from before ...
  def setP1: Dot => Unit
  def setP2: Dot => Unit
}

class ColorLine {
  // from before ...
  def setP1: ColorDot => Unit
  def setP2: ColorDot => Unit
}

Fields are Not Subtypes

• ColorLine.setP1 not a subtype of Line.setP1 ...
• ... accepts fewer inputs

Recap: Subtyping = Is an object like another?

Can one widget be safely substituted with another?

Subtyping Relation S <: T

Says when a value of type S is a value of type T

• Width Subtyping: when S has more attributes than T

• Depth Subtyping: when S fields are subtypes of T fields

• Transitive Subtyping: The "Is-A" relation is transitive

• Function Subtyping when S accepts more inputs, produces fewer outputs

Recap: Subtyping Enables Polymorphism and Reuse

Recap: Subtype Polymorphism

If S <: T then we can use S objects in place of T objects

• Dot <: Point
• ColorDot <: Dot <: Point
• Box <: Dot <: Point
• ColorBox <: Dot <: Point

def screen(p: Point) {
  // Only uses properties described in Point 
}

So screen can be called with Dot, ColorDot, Box, ColorBox