Welcome to Intro to CS Theory

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Welcome to Intro to CS Theory

• Introduction to CS Theory
• formalization of computation
• various models of computation (increasing
  difficulty/power)
• what can / cannot be done ?

• Why a theory course ?
• relevant to practice (grammars for programming
  languages, finite automata regular expressions
  for pattern matching of strings, NP-completeness
  to determine required time complexity e.g. for
  cryptography)
• problem solving skills independent of current
  technology (specific programming languages,
  etc.), ability to express ideas clearly,
  succinctly, and correctly

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Chapter 0
Introduction
Automata Theory mathematical models of
computation Computability Theory what can
be computed ? Complexity Theory which
problems are computationally hard / easy ?

• Need math background
• review Chapter 0
• discrete math quiz next class

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Chapter 0
Strings and Languages
Alphabet non-empty finite set of symbols,
typically denoted by or , e.g. 1 0,1 ,
2 a,b,c,d , ,,0,1,2
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Chapter 0
Strings and Languages
Alphabet non-empty finite set of symbols,
typically denoted by or , e.g. 1 0,1 ,
2 a,b,c,d , ,,0,1,2
String over an alphabet a finite sequence of
symbols from the alphabet, e.g. w1 00101 over
1, w2 badcab over 2
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Chapter 0
Strings and Languages
Alphabet non-empty finite set of symbols,
typically denoted by or , e.g. 1 0,1 ,
2 a,b,c,d , ,,0,1,2
String over an alphabet a finite sequence of
symbols from the alphabet, e.g. w1 00101 over
1, w2 badcab over 2 The length of a
string w over (the number of symbols in w) is
denoted w. The string with no symbols is called
the empty string and denoted e.
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Chapter 0
Strings and Languages

• Operations on strings (let w w1w2wn)
• reverse wR wnwn-1w1
• substring wiwi1wj
• concatenation of w with a string z z1z2zm
• wz w1w2wnz1z2zm
• wk means concatenation of k copies of w

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Chapter 0
Strings and Languages

• Operations on strings (let w w1w2wn)
• reverse wR wnwn-1w1
• substring wiwi1wj
• concatenation of w with a string z z1z2zm
• wz w1w2wnz1z2zm
• wk means concatenation of k copies of w
• lexicographic ordering of strings first by
  length, then alphabetically, e.g for 0,1
• e,0,1,00,01,10,11,000,

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Chapter 0
Strings and Languages
Language a set of strings over an alphabet ,
e.g. L1 a, ab, bab L2 L3 e L4
w over 0,1 w contains more 1s than 0s
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Chapter 0
Strings and Languages
throughout the book, e.g. page 44

• Operations on languages
• typical set operations , Å, etc.

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Chapter 0
Strings and Languages
throughout the book, e.g. page 44

• Operations on languages
• typical set operations , Å, etc.
• concatenation L1.L2 w1w2 w12 L1, w22 L2

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Chapter 0
Strings and Languages
throughout the book, e.g. page 44

• Operations on languages
• typical set operations , Å, etc.
• concatenation L1.L2 w1w2 w12 L1, w22 L2
• Kleenes star L k01 Lk

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Chapter 0
Strings and Languages
throughout the book, e.g. page 44

• Operations on languages
• typical set operations , Å, etc.
• concatenation L1.L2 w1w2 w12 L1, w22 L2
• Kleenes star L k01 Lk
• Note a language L µ

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Chapter 0
Strings and Languages
throughout the book, e.g. page 44

• Operations on languages
• typical set operations , Å, etc.
• concatenation L1.L2 w1w2 w12 L1, w22 L2
• Kleenes star L k01 Lk
• reverse LR wR w 2 L
• Note a language L µ