The document discusses finite automata and provides an example of using a finite state machine to model the scoring system in tennis. Key points: - Finite automata are formal systems that can have a finite number of states and transition between states based on inputs. They are commonly represented as graphs. - An example finite automata is presented to model the scoring in a tennis game, with states like "Love-15" and transitions labeled with "s" for server point or "o" for opponent point. - Strings of points can be processed by starting in the start state and following the transitions between states for each point in the string. If the final state is reached, the input string is accepted.

1. 1
Finite Automata
What Are They?
An Example: Scoring in Tennis
Courtesy: Jeff Ullman

2. 2
What is a Finite Automaton?
A formal system.
Remembers only a finite amount of
information.
Information represented by its state.
State changes in response to inputs.
Rules that tell how the state changes in
response to inputs are called transitions.
Courtesy: Jeff Ullman

3. 3
Informal Explanation
Finite automata are finite collections of
states with transition rules that take
you from one state to another.
Original application was sequential
switching circuits, where the “state”
was the settings of internal bits.
Today, several kinds of software can be
modeled by FA.
Courtesy: Jeff Ullman

4. 4
Representing FA
Simplest representation is often a
graph.
 Nodes = states.
 Arcs indicate state transitions.
 Labels on arcs tell what causes the
transition.
Courtesy: Jeff Ullman

5. 5
Tennis
 Like ping-pong, except you are very
tiny and stand on the table.
 Match = 3-5 sets.
 Set = 6 or more games.
Courtesy: Jeff Ullman

6. 6
Scoring a Game
One person serves throughout.
To win, you must score at least 4
points.
You also must win by at least 2 points.
Inputs are s = “server wins point” and
o = “opponent wins point.”
Courtesy: Jeff Ullman

7. 7
Love
Start
Love-15
15-Love
s
o
Love-30
15-all
30-Love
s
s
o
o
Love-40
15-30
30-15
40-Love
s
s
s
o
o
o
Server
Wins
Opp’nt
Wins
s
o
40-15
15-40
30-all
s
s
s
o
o
o
30-40
40-30
s
s
s
o
o
o
deuce
s
s
o
o
Ad-out
Ad-in
s
o
s
o
s
o
Courtesy: Jeff Ullman

8. 8
Acceptance of Inputs
Given a sequence of inputs (input
string ), start in the start state and
follow the transition from each symbol
in turn.
Input is accepted if you wind up in a
final (accepting) state after all inputs
have been read.
Courtesy: Jeff Ullman

9. 9
Example: Processing a String
Love
Start
Love-15
15-Love
s
o
Love-30
15-all
30-Love
s
s
o
o
Love-40
15-30
30-15
40-Love
s
s
s
o
o
o
Server
Wins
Opp’nt
Wins
s
o
40-15
15-40
30-all
s
s
s
o
o
o
30-40
40-30
s
s
s
o
o
o
deuce
s
s
o
o
Ad-out
Ad-in
s
o
s
o
s
o
s o s o s o s o s o s s
*
Courtesy: Jeff Ullman

10. 10
Example: Processing a String
Love
Start
Love-15
15-Love
s
o
Love-30
15-all
30-Love
s
s
o
o
Love-40
15-30
30-15
40-Love
s
s
s
o
o
o
Server
Wins
Opp’nt
Wins
s
o
40-15
15-40
30-all
s
s
s
o
o
o
30-40
40-30
s
s
s
o
o
o
deuce
s
s
o
o
Ad-out
Ad-in
s
o
s
o
s
o
s o s o s o s o s o s s
*
Courtesy: Jeff Ullman

11. 11
Example: Processing a String
Love
Start
Love-15
15-Love
s
o
Love-30
15-all
30-Love
s
s
o
o
Love-40
15-30
30-15
40-Love
s
s
s
o
o
o
Server
Wins
Opp’nt
Wins
s
o
40-15
15-40
30-all
s
s
s
o
o
o
30-40
40-30
s
s
s
o
o
o
deuce
s
s
o
o
Ad-out
Ad-in
s
o
s
o
s
o
s o s o s o s o s o s s
*
Courtesy: Jeff Ullman

12. 12
Example: Processing a String
Love
Start
Love-15
15-Love
s
o
Love-30
15-all
30-Love
s
s
o
o
Love-40
15-30
30-15
40-Love
s
s
s
o
o
o
Server
Wins
Opp’nt
Wins
s
o
40-15
15-40
30-all
s
s
s
o
o
o
30-40
40-30
s
s
s
o
o
o
deuce
s
s
o
o
Ad-out
Ad-in
s
o
s
o
s
o
s o s o s o s o s o s s
*
Courtesy: Jeff Ullman

13. 13
Example: Processing a String
Love
Start
Love-15
15-Love
s
o
Love-30
15-all
30-Love
s
s
o
o
Love-40
15-30
30-15
40-Love
s
s
s
o
o
o
Server
Wins
Opp’nt
Wins
s
o
40-15
15-40
30-all
s
s
s
o
o
o
30-40
40-30
s
s
s
o
o
o
deuce
s
s
o
o
Ad-out
Ad-in
s
o
s
o
s
o
s o s o s o s o s o s s
*
Courtesy: Jeff Ullman

14. 14
Example: Processing a String
Love
Start
Love-15
15-Love
s
o
Love-30
15-all
30-Love
s
s
o
o
Love-40
15-30
30-15
40-Love
s
s
s
o
o
o
Server
Wins
Opp’nt
Wins
s
o
40-15
15-40
30-all
s
s
s
o
o
o
30-40
40-30
s
s
s
o
o
o
deuce
s
s
o
o
Ad-out
Ad-in
s
o
s
o
s
o
s o s o s o s o s o s s
*
Courtesy: Jeff Ullman

15. 15
Example: Processing a String
Love
Start
Love-15
15-Love
s
o
Love-30
15-all
30-Love
s
s
o
o
Love-40
15-30
30-15
40-Love
s
s
s
o
o
o
Server
Wins
Opp’nt
Wins
s
o
40-15
15-40
30-all
s
s
s
o
o
o
30-40
40-30
s
s
s
o
o
o
deuce
s
s
o
o
Ad-out
Ad-in
s
o
s
o
s
o
s o s o s o s o s o s s
*
Courtesy: Jeff Ullman

16. 16
Example: Processing a String
Love
Start
Love-15
15-Love
s
o
Love-30
15-all
30-Love
s
s
o
o
Love-40
15-30
30-15
40-Love
s
s
s
o
o
o
Server
Wins
Opp’nt
Wins
s
o
40-15
15-40
30-all
s
s
s
o
o
o
30-40
40-30
s
s
s
o
o
o
deuce
s
s
o
o
Ad-out
Ad-in
s
o
s
o
s
o
s o s o s o s o s o s s
*
Courtesy: Jeff Ullman

17. 17
Example: Processing a String
Love
Start
Love-15
15-Love
s
o
Love-30
15-all
30-Love
s
s
o
o
Love-40
15-30
30-15
40-Love
s
s
s
o
o
o
Server
Wins
Opp’nt
Wins
s
o
40-15
15-40
30-all
s
s
s
o
o
o
30-40
40-30
s
s
s
o
o
o
deuce
s
s
o
o
Ad-out
Ad-in
s
o
s
o
s
o
s o s o s o s o s o s s
*
Courtesy: Jeff Ullman

18. 18
Example: Processing a String
Love
Start
Love-15
15-Love
s
o
Love-30
15-all
30-Love
s
s
o
o
Love-40
15-30
30-15
40-Love
s
s
s
o
o
o
Server
Wins
Opp’nt
Wins
s
o
40-15
15-40
30-all
s
s
s
o
o
o
30-40
40-30
s
s
s
o
o
o
deuce
s
s
o
o
Ad-out
Ad-in
s
o
s
o
s
o
s o s o s o s o s o s s
*
Courtesy: Jeff Ullman

19. 19
Example: Processing a String
Love
Start
Love-15
15-Love
s
o
Love-30
15-all
30-Love
s
s
o
o
Love-40
15-30
30-15
40-Love
s
s
s
o
o
o
Server
Wins
Opp’nt
Wins
s
o
40-15
15-40
30-all
s
s
s
o
o
o
30-40
40-30
s
s
s
o
o
o
deuce
s
s
o
o
Ad-out
Ad-in
s
o
s
o
s
o
s o s o s o s o s o s s
*
Courtesy: Jeff Ullman

20. 20
Example: Processing a String
Love
Start
Love-15
15-Love
s
o
Love-30
15-all
30-Love
s
s
o
o
Love-40
15-30
30-15
40-Love
s
s
s
o
o
o
Server
Wins
Opp’nt
Wins
s
o
40-15
15-40
30-all
s
s
s
o
o
o
30-40
40-30
s
s
s
o
o
o
deuce
s
s
o
o
Ad-out
Ad-in
s
o
s
o
s
o
s o s o s o s o s o s s
*
Courtesy: Jeff Ullman

21. 21
Example: Processing a String
Love
Start
Love-15
15-Love
s
o
Love-30
15-all
30-Love
s
s
o
o
Love-40
15-30
30-15
40-Love
s
s
s
o
o
o
Server
Wins
Opp’nt
Wins
s
o
40-15
15-40
30-all
s
s
s
o
o
o
30-40
40-30
s
s
s
o
o
o
deuce
s
s
o
o
Ad-out
Ad-in
s
o
s
o
s
o
s o s o s o s o s o s s *
Courtesy: Jeff Ullman

22. 22
Example: Recognizing Strings
Ending in “ing”
nothing Saw i
i
Not i
Saw ing
g
i
Not i or g
Saw in
n
i
Not i or n
Start
Courtesy: Jeff Ullman

23. 23
Automata to Code
 In C/C++, make a piece of code for
each state. This code:
1. Reads the next input.
2. Decides on the next state.
3. Jumps to the beginning of the code for
that state.
Courtesy: Jeff Ullman

24. 24
Example: Automata to Code
2: /* i seen */
c = getNextInput();
if (c == ’n’) goto 3;
else if (c == ’i’) goto 2;
else goto 1;
3: /* ”in” seen */
. . .
Courtesy: Jeff Ullman

25. 25
Language of an Automaton
The set of strings accepted by an
automaton A is the language of A.
Denoted L(A).
Different sets of final states -> different
languages.
Example: As designed, L(Tennis) =
strings that determine the winner.
Courtesy: Jeff Ullman