eecs 20 lecture 4 (january 24, 2001) tom henzinger block diagrams

EECS 20

Lecture 4 (January 24, 2001)

Tom Henzinger

Block Diagrams

1 Systems are functions

2 Signals are functions

Systems as Functions

Input Output

Domain: set of possible inputs

Range: set of possible outputs

Graph: set of pairs ( input, output )

Factorial System

!Nats Nats

Factorial System

!Nats Nats

3 6

Factorial System

!Nats Nats

4 24

Factorial System

!Nats Nats

4 24

Domain: Nats

Range: Nats

Graph: { ( 1, 1 ), ( 2, 2 ), ( 3, 6), ( 4, 24 ), … }

Inverter System

Bools

Bools

Inverter System

Bools

Bools true false

Inverter System

Bools

Bools false true

Inverter System

Bools

Bools false true

Domain: Bools

Range: Bools

Graph: { ( true, false ), ( false, true ) }

Composition of Systems

Bools

Bools Bool

s true truefalse

Composition of Systems

Bools

Bools Bool

s false falsetrue

This is again a system !

Bools Bool

s

The Identity System

Bools

Bools id

Domain: Bools

Range: Bools

Graph: { ( x, y ) Bools2 | x = y }

System Composition is Function Composition

= id

because domain ( ) = domain (id)

range ( ) = range (id)

x Bools, ( x ) = id (x)

And System

Bools

And System

Bools2 Bools

And System

Bools (true, false) false

Bools2

And System

Bools

Bools

Bools

And System

Bools

Bools

Bools true

falsefalse

Exponentiation System

Nats exp

Nats

Nats 2

3?

-

graph (exp) = { ( (x,y), z ) Nats2 Nats | z = x

y }

Exponentiation System

Nats exp

Nats

Nats 2

38

2

1

graph (exp) = { ( (x,y), z ) Nats2 Nats | z = x

y }

Nats exp

Nats

Nats 2

39

1

2

Exponentiation System

graph (exp) = { ( (x,y), z ) Nats2 Nats | z = x

y }

Bools

Bools

Bools

Bools

Bools

Bools

Block Diagram

This cannot be written easily using .

Bools

Bools

Bools

Block Diagram

true

false

Bools

Bools

Bools

Block Diagram

true

false true

false

Bools

Bools

Bools

Block Diagram

true

false true

false

false

Bools

Bools

Bools

Block Diagram

true

false

true

true

false

false

Bools

Bools

Bools

Or System

domain () = Bools2

range () = Bools

graph () = { ( (x,y), z ) Bools2 Bools | z = x y }

Bools

Bools

Bools

Or System

domain () = Bools2

range () = Bools

graph () = { ( (x,y), z ) Bools2 Bools | z = x y }

-

Block Diagrams with Forks

Block Diagrams with Forks

true

false

Block Diagrams with Forks

true

false

f

Block Diagrams with Forks

true

false

f

f

t

Block Diagrams with Forks

true

false

f

f

t

true

Joins are illegal

Joins are illegal

true

false

Joins are illegal

true

false

f

t

?

Multiple Outputs

Nats

Nats Nats0

Nats0

f

f: Nats2 Nats02 such that

x,yNats, f (x,y) = { (q,r)Nats02 | x = q· y + r r <

y }

Division System

Nats

Nats

divide

divide: Nats2 Nats02 such that

x,yNats, divide (x,y) = { (q,r)Nats02 | x = q· y + r r

< y }

1

2

1

2

quotient

remainderNats0

Nats0

Division System

Nats

Nats

divide

7

3

1

2

1

2

divide: Nats2 Nats02 such that

x,yNats, divide (x,y) = { (q,r)Nats02 | x = q· y + r r

< y }

Nats0

Nats0

Division System

Nats

Nats

divide

7

3

1

2

1

2

2

1

divide: Nats2 Nats02 such that

x,yNats, divide (x,y) = { (q,r)Nats02 | x = q· y + r r

< y }

Nats0

Nats0

Division System

Nats

Nats

divide

3

7

1

2

1

2

divide: Nats2 Nats02 such that

x,yNats, divide (x,y) = { (q,r)Nats02 | x = q· y + r r

< y }

Nats0

Nats0

Division System

Nats

Nats

divide

3

7

1

2

1

2

0

3

divide: Nats2 Nats02 such that

x,yNats, divide (x,y) = { (q,r)Nats02 | x = q· y + r r

< y }

Nats0

Nats0

Division System

Nats

Nats

divide

9

3

1

2

1

2

3

0

divide: Nats2 Nats02 such that

x,yNats, divide (x,y) = { (q,r)Nats02 | x = q· y + r r

< y }

Nats0

Nats0

Many possible implementations

Nats

Nats

1

2

1

2

C program for Euclid’s algorithm

divide

divide: Nats2 Nats02 such that

x,yNats, divide (x,y) = { (q,r)Nats02 | x = q· y + r r

< y }

Nats0

Nats0

Many possible implementations

Nats

Nats Nats0

Nats01

2

1

2

circuit

divide

divide: Nats2 Nats02 such that

x,yNats, divide (x,y) = { (q,r)Nats02 | x = q· y + r r

< y }

Block diagrams can hide outputs

divide22

1 1

zerocheck

zerocheck: Nats0 Bools such that

xNats, zerocheck (x) x = 0 .

Nats

Nats

Bools

Block Diagrams can hide outputs

divide22

1 1

zerocheck

zerocheck: Nats0 Bools such that

xNats, zerocheck (x) x = 0 .

Nats

Nats

Bools

Block Diagrams can hide outputs

2

1Nats

Nats

Boolsdivisible

divisible: Nats2 Bools such that

x,yNats, divisible (x,y) ( qNats, x = q· y ) .

Hidden inputs are illegal, for now

Hidden inputs are illegal, for now

true

?

Constant functions have no inputs

true

Constant function

true

true truetrue

Cycles are illegal, for now

divide

Block Diagrams

-are nested, directed, acyclic graphs

-allow compositional, hierarchical system description

Quiz

1. set x, x P(x)

2. function f, { x domain (f) | x = f(x) }

3. n Nats, n = 2 ( n, n+1 ) { 1, 2, 3 }

2

4. f [Nats Nats], f(x) = x 2