Are the languages of the following regular expressions the same?

a(bc)*

ab*c*

Question

Are the languages of the following regular expressions the same?

a(bc)*

ab*c*

BlackTom AI · Accepted Answer

The question asks whether the languages of the two given regular expressions are the same, and provides two answer choices.
Option 1: True. This would claim that a(bc)* and ab*c* denote the same set of strings. However, a(bc)* generates strings that start with a and then have zero or more occurrences of the exact block bc, producing strings like a, abc, abcbc, abcbcbc, and so on. In contrast, ab*c* allows any number of b's after the initial a (including zero), followed by any number of c's (including zero), which yields strings such as a, ab, abb, aaaa? Wait, not that last one—only b's after a and then c's. More concretely, a, ab, abb, ..., ac, acc, accc, etc. Since the two forms permit different patterns after the initial a, they do not generate the same languages. Hence this statement is incorrect.
Option 2: False. This aligns with the actual difference: the first expression requires bc in sequence repeated, the second allows a run of b's followed by a run of c's without the mandatory pairing of bc blocks. Therefore the languages are not the same, making this option true about the question's truth value.
In summary, the claim that the languages are the same is incorrect, and the correct assessment is that the languages differ because the structural constraints—repeated bc blocks versus arbitrary b*s then c*s—lead to distinct sets of strings.

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Are the languages of the following regular expressions the same? a(bc)* abc

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