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

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CSCI 2670Introduction to Theory of Computing

• Instructor Shelby Funk

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Today

• Syllabus
• Chapter 0
• Homework due Tuesday, August 23
• Read pages Chapter 0
• You are responsible for all this material even if
  I dont cover it in class
• Send me an e-mail telling me your favorite thing
  about Athens
• If youre new to Athens, tell me your favorite
  thing about your most recent hometown

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Course goals

• Theoretically explore the capabilities and
  limitations of computers
• Complexity theory
• What makes some problems computationally hard and
  others easy?
• Computability theory
• What problems can be solved by a computer?
• Automata theory
• How can we mathematically model computation?

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Sets, multisets and sequences

• Set
• Order and repetition dont matter
• 7,4,7,3 3,4,7
• Multiset
• Order doesnt matter, repetition does
• 7,4,7,3 3,4,7,7 ? 3,4,7
• Sequence
• Order and repetition matter
• (7,4,7,3) ? (3,4,7,7)
• Finite sequence of k elements may be called a
  k-tuple

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Set notation

• Union A?B
• Intersection A?B
• Complement A
• Cartesian Product A?B
• Also called cross product
• Power set P (A)

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Example

• A 1,2, B2,3, U x?Nx lt 6
• A?B
• A?B
• A
• A?B
• P (A)

• A 1,2, B2,3, U x?Nx lt 6
• A?B 1,2,3
• A?B 2
• A 3,4,5
• A?B (1,2), (1,3), (2,2), (2,3)
• P (A) Ø, 1, 2, 1,2

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Function

• Mechanism associating each input value with
  exactly one output value
• Domain set of all possible input values
• Range set containing all possible output values
• f D ? R

n
f (n)
f 1, 2, 3, 4 ? 2, 4
1 2 3 4
2 4 2 4
f 1, 2, 3, 4 ? 1, 2, 3, 4
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Relation

• Predicate function whose output value is always
  either true or false
• Relation predicate whose domain is the set
  AAA
• If domain is all k-tuples of A, the relation is a
  k-ary relation on A

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Graphs
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Graphs
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Graphs
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Graphs
Subgraph
Binary tree
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Directed graphs
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(2,1),(3,1),(4,3),(5,2)
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4
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Alphabets and strings

• Alphabet any finite set
• ?1 1,2,3
• ?2 ?,?,?
• String finite sequence of symbols from the given
  alphabet
• 1212123
• ??????
• Empty string, e, contains no symbols of the
  alphabet
• Language a set of strings

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Boolean logic

• Conjunction (and) ?
• Disjunction (or) ?
• Negation (not) ?
• Exclusive or (xor) ?
• Equality ?
• Implication ?

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Proof techniques

• Construction
• Prove a there exists statement by finding the
  object that exists
• Contradiction
• Assume the opposite and find a contradiction
• Induction
• Show true for a base case and show that if the
  property holds for the value k, then it must also
  hold for the value k 1

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Have a great weekend!