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CSC312 Automata Theory Lecture # 13 Chapter # 7 by Cohen Kleene’s Theorem (Part-3)

CSC312 Automata Theory Lecture # 13 Chapter # 7 by Cohen Kleene’s Theorem (Part-3). Proof of Part-3 (Cont…). Rule 3: (Concatenation of two FAs)

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CSC312 Automata Theory Lecture # 13 Chapter # 7 by Cohen Kleene’s Theorem (Part-3)

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  1. CSC312 Automata Theory Lecture # 13 Chapter # 7 by Cohen Kleene’s Theorem (Part-3)

  2. Proof of Part-3 (Cont…) • Rule 3: (Concatenation of two FAs) • If there is an FA called FA1 that accepts the language defined by the RE r1 and there is an FA called FA2 that accepts the language defined by RE r2, then there is an FA that we shall call FA3 that accepts the language defined by the concatenation r1r2, the product language. • OR • Using the FAs corresponding to r1 and r2, an FA can be build corresponding to r1.r2. • . P-143, Exercise Q. No.5)

  3. Proof of Rule 3: • We prove rule 3 by showing how to construct the new machine from the two old machines i.e. we shall prove that FA3 exists by showing how to construct it. • Let FA1 be an FA corresponding to RE r1, and FA2 be an FA corresponding to RE r2. Now let FA3 be an FA corresponding to RE (r1.r2), then the initial state of FA3 must correspond to the initial state of FA1 and

  4. the final state of FA3 must correspond to a final state of FA2. • Since the strings of language L3 corresponding to r1.r2 are obtained by concatenating the strings of L1 to those of L2, therefore the moment a final state of first FA is entered, the possibility of the initial state of second FA will be included as well.

  5. In general, FA3 will be different from both FA1 and FA2, so the labels of the states of states of FA3 may be supposed to be Z1, Z2, Z3,…., where Z1 is supposed to be initial state.

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