CSCI%202670%20Introduction%20to%20Theory%20of%20Computing

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CSCI 2670Introduction to Theory of Computing

• September 4, 2007

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Languages

• Alphabet
• Finite collection of objects (denoted ?)?
• String
• Concatenation of 0 or more elements of an
  alphabet
• Language
• Collection of strings
• ? is the set of all strings over ? (including
  e)?
• This week we will define a specific class of
  languages regular languages

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Deterministic finite automata (DFA)?

• Method for modeling computers with limited memory
• Language recognizer
• Idea
• Keep track of current state
• Events can move you from one state to another
• Todays goal
• Formally describe DFAs
• Interpret DFAs

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Example

• Ball in frictionless room
• Moves left, right or not at all
• Three possible states left, right, stop
• One other state impossible
• Start at rest (in stop state)?
• State changes under four conditions
• Ball hits a wall (reverse direction)?
• Paddle hits left (ball moves left)?
• Paddle hits right (ball moves right)?
• Hand grabs ball (stop moving)?

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Example

• State table

Event
State
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Example

• Ball in frictionless room
• Ball hits a wall (reverse direction)?
• Paddle hits left (ball moves left)?
• Paddle hits right (ball moves right)?
• Hand grabs ball (stop moving)?

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Finite automaton (formal definition)?

• A finite automaton is a 5-tuple (Q,?,?,q0,F),
  where
• Q is a finite set called the states
• ? is a finite set called the alphabet
• ? Q ? ? Q is the transition function
• ? corresponds to the event function from previous
  example
• q0 is the start state, and
• F ? Q is the set of accept states (also called
  final states).

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Example

• From previous example
• Q
• ?
• ?
• q0
• F

• Left, Right, Stop, Impossible
• Hit wall, Paddle left, Paddle right, Grab
• The state table we constructed
• Stop
• Left, Right, Stop
• What if we accept any set of events that ends
  with the ball in motion?
• F Left, Right

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Another example
0,1
? 0,1
q4
0
1
1
q1
q2
q3
0
1
0

• Q
• ?
• ?
• q0
• F

q1, q2, q3, q4 0, 1 (next slide)? q1 q3
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Another example
0,1
? 0,1
q4
0
1
1
q1
q2
q3
0
1
0
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Another example
0,1
? 0,1
q4
0
1
1
q1
q2
q3
0
1
0

• Informal description of the strings accepted by
  this DFA
• All strings of 0s and 1s beginning with a 0 and
  ending with a 1

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Group problem

• Formally describe the DFA (deterministic finite
  automaton) illustrated in your groups sheet

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Group problem

• 0, 1 for all groups
• Q is a finite set called the states
• ? is a finite set called the alphabet
• ? Q ? ? Q is the transition function
• q0 is the start state, and
• F ? Q is the set of accept states (also called
  final states).
• Include informal description

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Group 1
0,1
q1
q2
q4
1
0
q3
0
1
0
1
q5
0,1
Hint What string doesnt this DFA accept?
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Group 2
0,1
q2
0,1
q3
0
q1
0,1
1
q4
0,1
q5
Hint String length counts.
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Group 3
0,1
1
q1
q2
0
q3
0,1
Hint Symbol position counts.
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Group 4
0
q1
q2
1
0,1
0,1
0,1
q3
q4
Hint Can you simplify this DFA?
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Group 5
0,1
q7
1
q1
1
q3
q5
1
0
0
0
0
0
0
q4
q2
1
q6
1

Hint For each state, what do you know about how
many times each symbol has appeared?
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Group 6
0
q1
1
q2
0
1
q3
0, 1
Hint What happens when you get to q3?