1. MATH 224 – Discrete Mathematics
Deterministic Finite Automata Finite State Machines
Set of States Q
Start State q0  Q
Accepting States qf  Q
Alphabet  e.g, {0, 1}
Transitions  (, qi)  qj q1 q2 q3 q0
qf1
qf2  qi qj
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2. MATH 224 – Discrete Mathematics
DFA Represented by a Directed Graph
Multitasking in an operating System
Time slice expired
Start
Ready Queue
Running
Dequeued
Finished with resource and ready to run
Blocked waiting for a resource, e.g. data from a disk
Blocked Waiting
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3. MATH 224 – Discrete Mathematics
DFA for Multiples of 3 in binary
Start q0
Example 1
46 remainder 1
string
States
Example 2
51 remainder 0
string
States
qf-3 1 1 1 1 q1 q2
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4. MATH 224 – Discrete Mathematics
DFA to Recognize Floating Point
0 - 9
Start
0 - 9 q0 . q2
qf-0
+, –
0 - 9
e, E
0 - 9 q5 q1 q3
+, –
0 - 9 q4
0 - 9
Qf-1
0 - 9
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5. MATH 224 – Discrete Mathematics
Crossing the Bridge in Minimum Time 1, 2, 5 and 10 minutes
5 &10 q6
Start
Only 2 of the final states are shown here, one of the two shortest and one of the next shortest. q0
2 &10 q5
1 & 2
2 & 5
1&10
What would the longest time be?
2 minutes
1 & 5 q1 q4 q3 1 2 q2
5 minutes
3 minutes 1
19 minutes q7 q8 5
5 &10
q10
6 minutes
qf17 q9
1 &10
q11
q12
q15
13 minutes
16 minutes
1 & 2 1 2
q16
1 & 2
17 minutes
q13
qf14
15 minutes
17 minutes
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6. MATH 224 – Discrete Mathematics
Other DFAs
Construct a DFA that recognizes legal identifier names in C++ using α to represent letters and δ to represent the digits 0 through 9.
Construct a DFA to recognize even numbers in binary.
Construct a DFA to recognize multiples of four in binary.
Construct a DFA to recognize multiples of 5 in binary.
Construct a DFA to recognize multiples of 5 in decimal.
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7. MATH 224 – Discrete Mathematics
Representing a DFA in a Computer Program
Probably the simplest way to represent a DFA in a program is to use a 2-dimensional array, where each column corresponds to a character in the alphabet and each row corresponds to a state. So for example here is the DFA that recognizes multiples of 3 in binary. 1
Blank 3
Halt 2
qf-3
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8. MATH 224 – Discrete Mathematics
DFA for Multiples of 6 in binary
Start q0
qf-6 q3 1 1 1 1 1 q1 q2
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