1. State Machines
EECS 20
Lecture 8 (February 2, 2001)
Tom Henzinger

2. Finite-Memory Systems
Discrete-Time Delay: remember last input value
Discrete-Time Moving Average: remember last 2 input values
Parity: remember if number of past inputs “true” is even
Infinite-Memory Systems
Count: remember if number of past inputs “true” is greater than number of past inputs “false”

3. t, t, t, f, f, t, t, …
Parity
t, f, t, f, f, f, t, …

4. { The Parity System Parity : [ Nats0  Bools ]  [ Nats0  Bools ]
such that  x  [ Nats0  Bools ] ,  y  Nats0 ,
( Parity (x) ) (y) =
true if | trueValues (x,y) | is even
false if | trueValues (x,y) | is odd {
where trueValues (x,y) = { z  Nats0 | z < y  x (z) = true }

5. t, t, f, f, f, t, t, …
Count
t, t, t, t, t, f, t, …

6. { The Count System Count : [ Nats0  Bools ]  [ Nats0  Bools ]
such that  x  [ Nats0  Bools ] ,  y  Nats0 ,
( Count (x) ) (y) =
true if | trueValues (x,y) |  | falseValues (x,y) |
false if | trueValues (x,y) | < | falseValues (x,y) | {
where falseValues (x,y) = { z  Nats0 | z < y  x (z) = false }

7. An Implementation of the Parity System
Bools  Dt
Bools

8. An Implementation of the Parity System
t, f, t, …  Dt t

9. An Implementation of the Parity System
t, f, t, …  Dt
t, f

10. An Implementation of the Parity System
t, f, t, …  Dt
t, f, f

11. An Implementation of the Parity System
t, f, t, …  Dt
t, f, f, t

12. An Implementation of the Parity System
t, f, t, …  Dt
t, f, f, t
Memory = “state” of the system

13. Systems with finite memory are naturally implemented as
finite state machines
( or finite transition systems ) .

14. An Implementation of the Count System
Ints
Ints
Bools
pos D0
Bools

15. { { ± : Bools  Ints  Ints such that  x  Bools,  y  Ints,
± (x,y) =
y if x = true
y if x = false {
pos : Ints  Bools
such that  x  Ints,
pos (x) =
true if x  0
false if x < 0 {

16. An Implementation of the Count System3
t, t, t, …
pos D0
t, t, t, t

17. An Implementation of the Count System3
t, t, t, …
pos D0
t, t, t, t
infinite state

18. Systems with countable ( integer ) memory are naturally implemented as
infinite state machines
( or infinite transition systems ) .

19. ( Systems with uncountable ( real ) memory,
such as continuous-time delay,
are not naturally viewed naturally as transition systems. )

20. A Discrete-Time Reactive SystemF
Nats0  Inputs
Nats0  Outputs
F : [ Nats0  Inputs ]  [ Nats0  Outputs ]

21. State-Based ImplementationF 2 2
Update
Nats0  Inputs
Nats0  Outputs 1 1 Dc
Nats0  States
Nats0  States
Update: memory-free
Memory = States

22. update : States  Inputs  States  Outputs
State Implementation F 2 2
Update
Nats0  Inputs
Nats0  Outputs 1 1 Dc
Nats0  States
Nats0  States
update : States  Inputs  States  Outputs
c  States ( “initial state” )

23. State Implementation of the Parity System
Inputs = Bools
Outputs = Bools
States = Bools
initialState = true
update : States  Inputs  States  Outputs
such that  q  States,  x  Inputs,
update (q,x) 1 = ( q  x )
update (q,x) 2 = q

24. State Implementation of the Count System
Inputs = Bools
Outputs = Bools
States = Ints
initialState = 0
update : States  Inputs  States  Outputs
such that  q  States,  x  Inputs,
update (q,x) 1 = ± (x,q)
update (q,x) 2 = pos (q)

25. A State Machine
Inputs ( set of possible input values )
Outputs ( set of possible output values )
States ( set of states )
initialState  States
update : States  Inputs  States  Outputs

26. A state machine is
finite
iff
States is a finite set.
Parity : can be implemented by a two-state machine.
Count : cannot be implemented by finite-state machine.

27. Discrete-Time Reactive Systems
Every memory-free system can be implemented by a one-state machine.
Every causal system can be implemented by a state machine ( take as state the entire history of inputs ), and every system that can be implemented by a state machine is causal.

28. The Block Diagram of a State Machine2 2
Update
Nats0  Inputs
Nats0  Outputs 1 1
DinitialState
Nats0 States
Nats0 States

29. The Transition Table of a Finite-State Machine ( Parity )
Current state Input Next state Output
true true false true
true false true true
false true true false
false false false false

30. The Transition Diagram of a State Machine ( Parity )
true / true
true
false
true / false
false / false
false / true
States = Bools Inputs = Bools Outputs = Bools