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# hw4-sol - CSC236 Homework 4 Question 1 Part(a The automaton...

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CSC236: Homework 4 March 17, 2010 Question 1 Part (a) The automaton accepts words over { a, b } * where the number of a s is congurent to the number of b s modulo 3. Part (b) A B C Part (c) Prove by induction that: The new DFA is in state A iff the old DFA is in one of the states 1 , 5 , 9; The new DFA is in state B iff the old DFA is in one of the states 2 , 6 , 7; The new DFA is in state C iff the old DFA is in one of the states 3 , 4 , 8. The induction base is true since the initial states are A and 1. The induction step follows by considering all the possible transitions. Finally, note that the new DFA accepts iff it is in state A , which happens iff the old DFA is is one of the states 1 , 5 , 9, which happens iff the old DFA accepts. Thus the two DFAs recognize the same language. 1

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Question 2 Here is the DFA: O A 0 A 1 R 0 R 1 0 1 1 0 0 1 0 1 1 0 Here are the semantics of the different states: 1. State O captures the empty string. 2. States A x capture strings starting with x and containing an equal num- ber of occurrences of 01 and 10.
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