Given an alphabet Σ, what are the languages L over Σ for which L ∗ is...

Given an alphabet Σ, what are the languages L over Σ for which L ∗ is finite? How many such languages exist?

Expert Solution

If Σ is an alphabet, and L ⊆ Σ∗, then L is a (formal) language over Σ.
Language: A (possibly infinite) set of strings all of which are chosen from some
Σ∗
A language over Σ need not include strings with all symbols of Σ
Thus, a language over Σ is also a language over any alphabet that is a superset
of Σ
Examples:

Programming language C
Legal programs are a subset of the possible strings that can be formed from the alphabet of the language (a subset of ASCII characters)
English or French
The language of all strings consisting of
n 0s followed by n 1s (n≥0): {ǫ, 01, 0011, 000111, . . .}
The set of strings of 0s and 1s with an equal number of each:{ǫ, 01, 10, 0011, 0101, 1001, . . .}
Σ∗ is a language for any alphabet Σ
∅, the empty language, is a language over any alphabet
{ǫ}, the language consisting of only the empty string, is also a language over any alphabet
{w | w consists of an equal number of 0 and 1}

venereology answered 9 months ago

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