Finite State Machines

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Finite State Machines

• COMP 114
• Tuesday April 23

2
Announcements

• TA of the year award
• See class web page for details

3
Topics

• Last weeks grammar
• Finite State Machines

4
What About 5 7 2 ?

• expression term plusminus term
• term factor multdiv factor
• factor number ( expression )
• FAILS!

5
What About 5 7 2 ?

• expression term plusminus term
• term factor multdiv factor
• factor number ( expression )
• The problem here is that all expressions must end
  with a plusminus term

6
New Grammar

• expression term plusminus term
• term factor multdiv factor
• factor number ( expression )

7
Finite State Machines

• Also called Finite Automata
• These are machines (real or theoretical) that
  have a finite number of states
• Imagine elevator
• current floor
• direction
• no history of where it has been

8
Examples

• Text editors
• The Find in Microsoft Word
• Could think of computer as a very large finite
  state machine
• not very useful too many states
• Brains as finite state machines?
• Maybe not analog instead of digital state

9
Sample Problem

• Man with wolf, goat, and cabbage is on left side
  of river
• Has a small rowboat, just big enough for him and
  one other thing
• Cant leave goat and wolf together
• Cant leave goat and cabbage together
• Can he get them to right bank intact?

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Model

• Current state is list of what things are on which
  side of river
• All on left
• Man and goat on right
• All on right (desired state)

MWGC -
WC - MG
- MWGC
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State Transitions

• Well indicate with arrows changes between states
• Letter indicates what happened
• g man took goat c man took cabbage
• w man took wolf m man went alone

g
MWGC -
WC - MG
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Transition Both Ways
g
MWGC -
WC - MG
g
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Some States Bad!
c
MWGC -
WG - MC

• We wont draw those

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Diagram of Solutions

• Can draw a diagram of solutions
• and transitions
• that dont lead to bad states

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Want To Try?
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Getting Started
g
MWGC -
WC - MG
g
Start
- MWGC
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State Diagram
m
g
MWGC -
MWC - G
WC - MG
m
g
c
c
w
w
Start
W - MGC
C - MWG
g
g
g
g
WGM - C
MGC - W
c
c
w
w
g
m
- MWGC
G - MWC
MG - WC
m
g
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Some Differences

• Not a typical state machine
• This problem has symmetric transitions
• Often not the case
• Only one ending state
• Often many accepting states

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Finite Automaton

• Finite set of states, Q
• Start state q0
• Set of transitions from one state to another
• Occur on input from alphabet ?
• Exactly one transition out of each state for each
  symbol in input alphabet
• A subset of final states, F ? Q

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Recognizing Integers
b
b
s
B
Start
d, b, s
b
b
S
d
OK
OK
d, s
s
d
s
d
Key
d digit, b blank, s other symbol
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Example
_ 1 2
b
b
s
B
Start
d, b, s
b
b
S
d
OK
OK
d, s
s
d
s
d
Key
d digit, b blank, s other symbol
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Example
_ 1 2
b
b
s
B
Start
d, b, s
b
b
S
d
OK
OK
d, s
s
d
s
d
Key
d digit, b blank, s other symbol
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Example
_ 1 2
b
b
s
B
Start
d, b, s
b
b
S
d
OK
OK
d, s
s
d
s
d
Key
d digit, b blank, s other symbol
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Example
_ 1 2
b
b
s
B
Start
d, b, s
b
b
S
d
OK
OK
d, s
s
d
s
d
Key
d digit, b blank, s other symbol
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Example
_ 1 2
b
b
s
B
Accept!
Start
d, b, s
b
b
S
d
OK
OK
d, s
s
d
s
d
Key
d digit, b blank, s other symbol
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Example 2
_ z34
b
b
s
B
Start
d, b, s
b
b
S
d
OK
OK
d, s
s
d
s
d
Key
d digit, b blank, s other symbol
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Example 2
_ z34
b
b
s
B
Start
d, b, s
b
b
S
d
OK
OK
d, s
s
d
s
d
Key
d digit, b blank, s other symbol
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Example 2
_ z34
b
b
s
B
Start
d, b, s
b
b
S
d
OK
OK
d, s
s
d
s
d
Key
d digit, b blank, s other symbol
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Example 2
_ z34
b
b
s
B
Start
d, b, s
b
b
S
d
NG
OK
OK
d, s
s
d
s
d
Key
d digit, b blank, s other symbol
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Example 2
_ z34
b
b
s
B
FAIL!
Start
d, b, s
b
b
S
d
NG
OK
OK
d, s
s
d
s
d
Key
d digit, b blank, s other symbol
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Example 3
1___
b
b
s
B
Start
d, b, s
b
b
S
d
OK
OK
d, s
s
d
s
d
Key
d digit, b blank, s other symbol
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Example 3
1___
b
b
s
B
Start
d, b, s
b
b
S
d
OK
OK
d, s
s
d
s
d
Key
d digit, b blank, s other symbol
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Example 3
1___
b
b
s
B
Start
d, b, s
b
b
S
d
OK
OK
d, s
s
d
s
d
Key
d digit, b blank, s other symbol
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Example 3
1___
b
b
s
B
Start
d, b, s
b
b
S
d
OK
OK
d, s
s
d
s
d
Key
d digit, b blank, s other symbol
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Example 3
1___
b
b
s
B
Accept!
Start
d, b, s
b
b
S
d
OK
OK
d, s
s
d
s
d
Key
d digit, b blank, s other symbol
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How To Implement

• Case statements
• Tables

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Case Implementation

• final int START 0
• final int BLANK 1
• final int DIGIT 2
• final int AFTER 3
• final int NG 4
• int state START
• char c

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Start Case

• for(int i 0 i lt s.length() i)
• c s.charAt(i)
• switch(state)
• case START
• if(c ' ')
• state BLANK
• else
• if(Character.isDigit(c))
• state DIGIT
• else
• state NG
• break

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Digit Case

• case DIGIT
• if(c ' ')
• state AFTER
• else
• if(Character.isDigit(c))
• state DIGIT
• else
• state NG
• break

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NG Case

• case NG
• default
• state NG
• break

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Table for Example
Table is just a very small array
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Limits of FSM

• How about a string like
• ai bi
• Same number of as as bs
• FSM cant recognize because theres no notion of
  past history.
• More complex machines can recognize
• COMP 181

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Summary

• Ive told you
• How finite state machines work
• Sketched how to design one
• How to implement one
• Have not told you
• How to make one automatically, given a language
  to recognize

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References

• Introduction to Automata Theory, Languages, and
  Computation by Hopcroft and Ullman
• Introduction to Algorithms by Cormen, Leiserson,
  and Rivest

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Next Time

• 3D Computer Graphics
• Movies!