Explanation Given: The set of all bit strings containing an equal number of 0s and 1s. Proof: Intuitively, the reason that a finite-state automaton cannot recognize the set of bit strings containing an equal number of 0's and l's is that there is not enough "memory" in the machine to keep track of how many extra 0's or l's the machine has read so far. The natural way to prove a negative statement such as this is by contradiction. So let us suppose that we do have a finite-state automaton M that accepts precisely the set of bit strings containing an equal number of 0's and l's. We will derive a contradiction by showing that the machine must accept some illegal strings. The idea behind the proof is that since there are only finitely many states, the machine must repeat some states as it computes. In this way, it can get into arbitrarily long loops, and this will lead us to a contradiction. To be specific, suppose that M has n states...

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Chapter 13.3, Problem 57E

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There is no finite-state automaton that recognizes the set of bit strings containing an equal number of 0s and 1s.

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Chapter 13.3, Problem 56E

Chapter 13.3, Problem 58E

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There is no finite-state automaton that recognizes the set of bit strings containing an equal number of 0s and 1s.

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Chapter 13 Solutions

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There is no finite-state automaton that recognizes the set of bit strings containing an equal number of 0s and 1s.