MC 306 Theory of Computation Thursday, 92503

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MC 306 Theory of ComputationThursday, 9/25/03

• Todays Class
• Outline of Exam 1
• More on combining regular (DFA/NFA) languages
• Regular Expressions

• Exam 1 next Thursday
• Bring questions on Tuesday!
• Exercises
• p. 61 2.19ab

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Exam 1 Outline

• Material Exam 1 covers all material through
  todays class
• Chapter 1, Math Preliminaries Material covered
  in class and assignments (sets, relations,
  equivalence relations)
• Chapter 2, Finite Automata 2.1-2.4 DFAs, NFAs,
  converting NFAs to DFAs, combining regular
  languages, regular expressions, equivalence of FA
  and regular expression languages, converting
  regular expressions to NFAs, e
• Kinds of questions Stating definitions and
  theorems carefully, giving examples and
  counterexamples, doing problems similar to class
  examples, exercises, and hand-in problems
• Reference Sheet You may bring one 8½ x 11
  sheet with any notes you wish to write on it.

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Regular Expressions versus DFAs/NFas

• Every DFA/NFA M determines a language L(M)
• Every regular expression r determines a language
  L(r)
• Theorem. The set of DFA/NFA languages exactly
  equals the set of regular expression languages.
• Hence we call these languages regular, which can
  be interpreted either way by this theorem

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Proving Equivalence of DFA and regular expression
languages.

• An If and Only If Theorem is really 2 theorems
• Theorem 1 If L L(r) is a language
  corresponding to a regular expression r, then
  there is a DFA/NFA Mr that accepts L, i.e., L
  L(Mr)
• Proof plan Start with a regular expression r
  that corresponds to L. Show how to use r to
  design a DFA/NFA that accepts L.
• Theorem 2 If L L(M) is a language
  corresponding to a DFA/NFA M, then there is a
  regular expression rM corresponding to L, i.e., L
  L(rM).
• Proof plan Start with a DFA/NFA M that accepts
  L. Show how to use M to design a regular
  expression that corresponds to L.

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Theorem 1 Convert from regular expression to NFA

• Since definition of regular expression is
  inductive, we do a base case, and then an
  inductive case (work done on board).
• Base case If the regular expression is a single
  symbol ?, ?, x??
• Inductive case If r1 and r2 are regular
  expressions that weve built NFAs for, how do we
  build an NFA for (r1 ? r2), (r1r2), and r1?

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Example

• Find an NFA that accepts the language described
  by the regular expression
• (a ? b) aa
• Note Well prove other half of Theorem, and do
  an example, next time.