We define closure as a set of things. We describe closure properties of regular languages as the operations implemented on regular languages which ensure that a new regular language will be produced.
All regular languages are closed under the mentioned operations. These operations are as follows:
Let's go over a detailed explanation of the closure properties of regular languages.
If a language
To implement Kleen star closure on the language
We can illustrate this process in the following way:
Let's assume there are two regular languages,
We can represent union as
Let's assume there are two languages,
The final states remain the same as for the two automata.
If there are two languages,
The intersection is checked by De Morgan's law, which states the following:
The process for the intersection is slightly different from other properties. The steps to implement intersection are as follows:
We can see this in the illustration below:
For concatenation, let's assume there are two regular languages,
Assume that there are two languages,
The final state of
There is one step for the complement process:
We can see this in the illustration below:
Let's assume a regular language,
To reverse a language, we perform the following the steps:
If there are
Consider the two languages,
This difference is
The process to take the difference between the two languages is as follows:
Homomorphism is allowing a string for each input symbol of a language.
For a regular language,
Homomorphism is closed under regular languages. The algorithm is defined for the regular expression of the language.
Suppose that
Let a regular expression of a regular language be
Then,
By equating, we see that
Consider
To prove inverse homomorphism, we perform the following steps:
The intuition behind this is that on the input,
Let
Let the language
Let the string be
Let another string be
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