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

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. Parityt, t, t, f, f, t, t, …t, f, t, f, f, f, t, …

4. The Parity System Parity : [ Nats 0  Bools ]  [ Nats 0  Bools ] such that  x  [ Nats 0  Bools ],  y  Nats 0, ( Parity (x) ) (y) = true if | trueValues (x,y) | is even false if | trueValues (x,y) | is odd { where trueValues (x,y) = { z  Nats 0 | z < y  x (z) = true }

5. Countt, t, f, f, f, t, t, …t, t, t, t, t, f, t, …

6. The Count System Count : [ Nats 0  Bools ]  [ Nats 0  Bools ] such that  x  [ Nats 0  Bools ],  y  Nats 0, ( Count (x) ) (y) = true if | trueValues (x,y) |  | falseValues (x,y) | false if | trueValues (x,y) | < | falseValues (x,y) | { where falseValues (x,y) = { z  Nats 0 | z < y  x (z) = false }

7. An Implementation of the Parity System  DtDt Bools

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

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

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

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

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

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

14. An Implementation of the Count System ±D0D0 Bools Ints pos Ints

15. ± : Bools  Ints  Ints such that  x  Bools,  y  Ints, ± (x,y) = y+1 if x = true y-1 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 System ± pos t, t, t, … t, t, t, t 3 D0D0

17. An Implementation of the Count System ± pos t, t, t, … t, t, t, t 3 infinite state D0D0

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 System Nats 0  InputsNats 0  Outputs F : [ Nats 0  Inputs ]  [ Nats 0  Outputs ] F

21. State-Based Implementation Nats 0  InputsNats 0  Outputs F DcDc Update 11 22 Nats 0  States Update: memory-free Memory = States

22. State Implementation Nats 0  InputsNats 0  Outputs F DcDc Update 11 22 Nats 0  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 Machine Nats 0  InputsNats 0  Outputs D initialState Update 11 22 Nats 0  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 ) truefalse true / true true / false false / true false / false States = Bools Inputs = Bools Outputs = Bools