Consider the regular language L = (111 + 11111)&lowast;. The minimum number of states in any DFA accepting this languages is:Correct answer is '9'. Can you explain this answer?

Question

Answer

A regular language is a language that can be expressed using a regular expression or can be recognized by a finite automaton.

In the given language L, the regular expression can be written as (111 11111).

This regular expression represents a language that consists of a sequence of three '1's followed by a sequence of five '1's.

For example, the strings '11111111' and '1111111111' are in the language L, but the strings '1111' and '111111' are not in the language L.

A finite automaton can also be constructed to recognize the language L. The automaton will have two states, one for the initial three '1's and another for the following five '1's.

In summary, the regular language L = (111 11111) consists of strings that start with three '1's followed by five '1's.

Answer

Given language L = (111 + 11111)^∗

Strings, that belongs to the language

L = {ϵ,111,11111,111111,11111111,111111111,111111111,…

Due to concatenation with (111)^∗ if we have three consecutive string lengths in L, then all higher string lengths will be in L.

We have strings of length 8,9,10 in L and so all higher length strings are also in L.

So, required DFA will be:

So, there are 5 states are final states and 4 states are non-final states, total number of states are 9 states.
Hence, the correct answer is 9.

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