CSCI 1302

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CSCI 1302

• Deterministic Finite-state Automata
• Turing Machines

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Finite-state Automata

• Deterministic Nondeterministic
• Useful for defining simple machine
• No memory (storage)
• Defined as algorithm input, and returns T/F

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What is Automata?

• Good for designing computation models with
  extremely limited memory
• Regular expressions
• Determinisitic Finite state machines (DFAs)
• Non-deterministic finite state machines (NFAs)

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Traditional Desperate Example
Good
Bad
Single
Dating
Bad
Good
Bad
Good
Good/bad
Engaged
Married
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Formal Definition

• A finite automaton is a 5-tuple
• (Q, S, d, q0, F), where
• Q is a finite set called states
• S is a finite set called the alphabet
• d Q x S -gt the transition function
• q0 is a member of Q and is the start state
• F is a subset of Q and is the set of accept states

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Our Example

• Q single, dating, engaged, married
• S good, bad
• d is described
• Single is the start state
• F married

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Another ExampleMachine M
1
0
1
q2
q1
0
L(M) w w ends in a 1
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Yet Another ExampleMachine M
0
1
1
0
q2
q3
q1
0,1
L(M) w w contains at least one 1 and
an even number of 0s follow the last 1
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Another ExampleMachine M
1
0
1
q2
q1
0
L(M) w w is empty, or ends in a 0
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Another ExampleMachine M
0
1
s
0
1
q1
q3
0
0
1
q2
q4
0
1
L(M) w w begins and ends with the same
number
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Fun at your next party

• L(M) w w contains the substring 10 and ends
  in a 1
• L(M) w w does not contain three 0s
• L(M) w w starts with a 1 and ends with a 0
• L(M) w every other character is a 1 in w
• L(M) w w has an even number of 0s or even
  number of 1s
• L(M) w w contains the same number of 0s and
  1s

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Non-deterministic Finite Automata (NFAs)

• Can explore all paths simultaneously by
  multiple transitions for a single input
• An oracle always picks the correct path
• Can also have e transitions (arbitrary)

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Example

• L(M) w w has an even number of 0s or even
  number of 1s

0
1
s
e
e
q1
q3
0
1
1
0
q2
q4
1
0
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Pushdown Automata (PDAs)

• More powerful than DFAs
• Similar to DFAs, but have a stack
• Can use the top of the stack to decide transition
• Can manipulate the stack
• For example
• L(M) w w contains the same number of 0s and
  1s
• Why is a stack necessary for this?

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Example L(M) 0n1n n gt 0
1A/e
0 Z/AZ
0 A/AA
F
q
p
e
eZ/Z
0 Z/AZ (to humans if you see a zero and a Z is
on the stack, replace it with AZ
Note stack has a Z on it by default
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Computational History

• Study limits of computational processes
• What can computers accomplish
• What is not doable by computers
• 1930s
• Turing Machine defined by Alan M. Turing in 1936

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Turing Machine Defined

• Control unit
• Read/Write head
• Infinitely-long tape
• Finite states (start, halt, others)
• Deterministic transition function

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Turing Machine Visualized



1
0
0
1
0

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The Church-Turing Thesis

• Turing Computable
• Input value on tape (binary)
• Execute program
• Read output value from tape
• Computational power of Turing machine limit of
  computational power of any algorithmic system
• i.e. If you cant do it with a Turing machine,
  you cant do it computationally

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FIN