1. Representation Application and Examples
Finite Automata 1
Representation
Application and Examples

2. What are FA?
Finite automata are finite collections of states with transition rules that take you from one state to another.

3. Representing FA Simplest representation is a graph. Nodes = states.
Arcs indicate state transitions.
Labels on arcs tell what causes the transition.

4. FA model of on-off switch
push on
Start

5. Protocol for Sending Data
Ready
Sending
data in
done
timeout
Start

6. FA Recognizing Strings Ending in “ing”
Get next character in any state except the finish state
Where we go depends on which character
empty
Saw i i
Not i
Saw ing g
Not i or g
Saw in n
Not i or n
Start
Double circle denotes
“finish” or “accepting”
state
String recognized may not be the entire input

7. Automata to Code
In C/C++, make a piece of code for each state that does the following
Reads the next input.
Decides on the next state.
Jumps to the beginning of the code for that state.

8. Code for state “saw i” 2: /* i seen */ c = getNextInput();
if (c == ’n’) goto 3;
else if (c == ’i’) goto 2;
else goto 1;
3: /* ”in” seen */
. . .

9. E-commerce example E-commerce requires protocols to prevent fraud
Protocols can be modeled by finite automata
Example: protocol to ensure that e-money is transferred from customer to merchant on shipped purchases

10. E-commerce 2
pay
cancel
cancel 2 1 3 4
start
redeem
transfer
bank
customer
Pay does not change state of customer. Free to cancel order
Bank can go down either cancel path or path that leads to e-money transfer following a redeem request from e-store

11. E-commerce 3 Store preferred path a->b->c->f->g
redeem
start b c f e g
pay
ship
transfer
store
Store preferred path a->b->c->f->g
Alternate paths allow ship before transfer complete

12. This representation is not complete
pay
cancel a
redeem
start b c f e g
pay
ship
transfer
cancel 2 1 3 4
start
redeem
transfer
All parts must have a
defined response to every
input even if the response
is no change of state.

13. Each loop labeled by actions that do not change statesR
start b c f e g P S T C
P,C
P,C,R,S,T C 2 1 3 4
start R T
P,S
P,C,R,S
P=pay
C=cancel
R=redeem
S=ship
T=transfer

14. Customer is in a “free” state
start b c f e g P S T C
P,C
P,C,S,R,T C 2 1 3 4
start R T
P,S
P,C,R,S
No interactions between
store and bank can
change customer’s state

15. Analysis of Store-Bank interactions reveals paths to failure of e-protocols
Analysis requires “product” FA with all possible combinations of (store, bank) states and paths that connect them.
Will discuss construction of product FAs later in course

16. Store-Bank product FA
Paths exist to state (c,2) that has no exit. Customer cancelled and store shipped before transfer of payment from bank.

18. Analysis of protocol using FA
start b c f e g P S T C
P,C
P,C,S,R,T C 2 1 3 4
start R T
P,S
P,C,R,S
If customer state is C,
then interactions between
store and bank can
lead to failure

19. Store-Bank product automaton 3
start b c f e g P S T C
P,C
Product automaton
reveals problem with
the protocol. State
(c,2) is accessible
from (a,1) but only
loops back. C 2 1 3 4
start R T
P,S
P,C,R,S
State (c,2) is case where
store has shipped before
redeeming payment from
bank and customer has
canceled

20. Far out example
On a distant planet, there are three species, a, b, and c.
Any two different species can mate. If they do:
The participants die.
Two children of the third species are born.

21. Strange Planet – (2)
Observation: the number of individuals never changes.
The planet fails if at some point all individuals are of the same species.
Then, no more breeding can take place.
State = sequence of three integers – the numbers of individuals of species a, b, and c.

22. Strange Planet – Questions
In a given state, must the planet eventually fail?
In a given state, is it possible for the planet to fail, if the wrong breeding choices are made?

23. Relationship to Computers
These questions mirror real ones about protocols.
Planet “failure” is like protocol “termination”
We want protocols to terminate
“Must the planet fail” is like asking whether a protocol, in its current state, is guaranteed to terminate.

24. Strange Planet – Transitions
An a-event occurs when individuals of species b and c breed and are replaced by two a’s.
Analogously: b-events and c-events.
Represent these by symbols a, b, and c, respectively.

25. Strange Planet with 2 Individuals
200
002
020
110
101
011 a c b
all states are “must-fail”
all possible events lead to states where have all same species
analogous to “termination” states automata

26. “Meta-states” are populations of different size
200
020
002 a b c
011
101
110
This analysis of a “meta-state” consist of “exhaustive
enumeration”.
List all possible species distributions.
For each distribution, list all possible transitions.

27. Strange Planet with 3 Individuals
“must-fail” state
111 a c b
300
030
003 a a
210
102
021
201
120
012 c b b c
“can’t-fail” states are parts of cycles.

28. Strange Planet with 4 Individuals
400
022
130
103
211 a c b
040
202
013
310
121 b a c
004
220
301
031
112 c b a
“might-fail” states appear
Some breeding choices lead to termination
Others lead to a cycle

29. Taking Advantage of Symmetry
In exhaustive enumeration of meta-states, we see repeated patterns.
Nodes and edges have different labels but topology is same.
200
002
020
110
101
011 a c b

30. Taking Advantage of Symmetry 2
Outcome (termination) is the same regardless of which species is extinct and which 2 have a single individual.
200
002
020
110
101
011 a c b

31. Taking Advantage of Symmetry 3
Develop a “prototype” that illustrates states and transitions allowed for populations of various size
List species-specific populations in decreasing order
Replace specific type of transition (a, b, c) with x
prototype
200
002
020
110
101
011 a c b
110
200 x

32. Prototypes for 2, 3, 4 x
110
200 x
211
400
111 x x x
300
310 x
210
220 x
The “might-fail” state of prototype 4 is a nondeterminism.
Different transitions are possible from 211 on the same input.

33. Prototype for meta-state 5
Drop the unnecessary edge label “x”
Prototype has standard form of a directed graph
Basic graph algorithms can find cycles
410
500
320
311
221
1 must-fail state; 1 can’t fail cycle; 1 might fail state

34. Strange planet example illustrates
3 types of states you could find in finite automata: must terminate, might terminate and can’t terminate
Methods of analysis to find those states: exhaustive enumeration and graph theory