CSCI 2670
Introduction to Theory of
Computing
August 23, 2005
August 23, 2005
Agenda
• Last class
– Reviewed syllabus
– Reviewed material in Chapter 0 of Sipser
– Assigned pages Chapter 0 of Sipser
• Questions?
• This class
– Begin Chapter 1
• Goal for the week
– Section 1.1
• Read Section 1.1 (pages 29 – 47) this week
August
2005
August 23,
23, 2005
2
Announcements
• Website is up
• Quiz tomorrow (8/24)
– Material: Chapter 0
• Homework due next Tuesday (8/30)
– Chapter 0 numbers 0.3 all, 0.7b, 0.12
– Chapter 1 numbers 1.3, 1.4c, 1.5f,
1.6(a, c, h)
• Unregistered students
• Please give me your student id after
class
August
2005
August 23,
23, 2005
3
Relation on A
• Function R:A×A×…×A{true, false}
– Often described as a set of elements
for which the relation is true
• Example
– A={1,2,3,4,5}
– R:A×A×A{true, false}
• R is true if the three-tuple is increasing
– {(1,2,3),(1,2,4),(2,3,4),(3,4,5)}  R
– (1,1,5)  R
August
2005
August 23,
23, 2005
4
Graphical representation
(binary relations only)
• Directed arrow with edge (a,b) if
(a,b)R
• Example: A={a,b,c,d}, R=“earlier in
alphabet”
– R={(a,b),(a,c),(a,d),(b,c),(b,d),(c,d)}
a
b
c
d
August
2005
August 23,
23, 2005
5
Equivalence Relation
• Symmetric
– {(a,a) | a  A}  R
• Reflexive
– (a,b)  R  (b,a)  R
• Transitive
– (a,b)  R  (b,c)  R  (a,c)  R
• Examples
– Equality
– “Has the same eye color”
August
2005
August 23,
23, 2005
6
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 ε)
• This week we will define a specific
class of languages – regular languages
August
2005
August 23,
23, 2005
7
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
• Today’s goal
– Formally describe DFA’s
– Interpret DFA’s
August
2005
August 23,
23, 2005
8
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)
August
2005
August 23,
23, 2005
9
Example
• State table
Hits
Wall
Paddle
Left
Paddle
Right
Grab
Left
Right
Left
Right
Stop
Right
Left
Left
Right
Stop
Stop
Impossi
ble
Left
Right
Stop
Event
State
Impossible Impossible Impossible Impossible Impossible
August
2005
August 23,
23, 2005
10
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)
Left
Right
Stop
August
2005
August 23,
23, 2005
Impossible
11
Finite automaton (formal definition)
•
A finite automaton is a 5-tuple
(Q,,,q0,F), where
1. Q is a finite set called the states
2.  is a finite set called the alphabet
3.  : Q ×   Q is the transition function
•
 corresponds to the event function from
previous example
4. q0 is the start state, and
5. F  Q is the set of accept states (also
called final states).
August
2005
August 23,
23, 2005
12
Example
• From previous example
–
–
–
–
–
Q = {Left, Right, Stop, Impossible}
 = {Hit wall, Paddle left, Paddle right, Grab}
 = The state table we constructed
q0 = Stop
F = {Left, Right, Stop}
• What if we accept any set of events that
ends with the ball in motion?
– F = {Left, Right}
August
2005
August 23,
23, 2005
13
0,1
q4
 = {0,1}
Another example
0
1
1
q1
q2
0
q3
1
0
•
•
•
•
•
Q = {q1, q2, q3, q4}
 = {0, 1}
 (next slide)
q0 = q1
F = {q3}
August
2005
August 23,
23, 2005
14
0,1
 = {0,1}
Another example
q4
0
1
1
q1
0
q2
q3
1
0
State table
0
1
q1
q2
q4
q2
q2
q3
q3
q2
q3
q4
q4
q4
August
2005
August 23,
23, 2005
15
0,1
q4
 = {0,1}
Another example
0
1
1
q1
0
q2
q3
1
0
•
Informal description of the strings
accepted by this DFA
•
All strings of 0’s and 1’s beginning with a 0
and ending with a 1
August
2005
August 23,
23, 2005
16
Group problem
•
Formally describe the DFA
(deterministic finite automaton)
illustrated in your group’s sheet
August
2005
August 23,
23, 2005
17
Group problem
 = {0, 1} for all groups
1.
2.
3.
4.
5.
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
August
2005
August 23,
23, 2005
18
Group 1
0,1
q1
0
1
q2
q3
1
0
0
q4
1
q5
0,1
Hint: What string doesn’t this DFA accept?
August
2005
August 23,
23, 2005
19
Group 2
0,1
0
q1
q2
0,1
q3
0,1
1
q4
0,1
q5
Hint: String length counts.
August
2005
August 23,
23, 2005
20
Group 3
0,1
1
q1
0
q2
q3
0,1
Hint: Symbol position counts.
August
2005
August 23,
23, 2005
21
Group 4
q1
0
q2
1
0,1
q3
0,1
0,1
q4
Hint: Can you simplify this DFA?
August
2005
August 23,
23, 2005
22
q7
Group 5
q1
0
1
1
q3
0
0
q2
0,1
1
0
0
q4
1
q5
1
0
1
q6
Hint: For each state, what do you know about
how many times each symbol has appeared?
August
2005
August 23,
23, 2005
23
Group 6
0
q1
1
q2
0
1
q3
0, 1
Hint: What happens when you get to q3?
August
2005
August 23,
23, 2005
24