Do you think IIT Guwahati certified course can help you in your career?
Introduction
In this blog, we will learn about the closure properties of a regular language. But before discussing closure and its properties, we should first understand what a regular language is.
A regular language may be expressed using regular expressions, non- deterministic or deterministic finite automata, or state machines. A language is a collection of strings composed of characters from a specific alphabet or set of symbols. Regular languages are a subset of the entire string set.
Regular languages are one of the first ideas taught in computability classes and are utilized in parsing and developing programming languages. These are important for assisting computer scientists in seeing patterns in data and grouping specific computational issues together; once this is done, they may utilize comparable methodologies to address the problems that have been grouped. In computability theory, regular languages are an essential issue.
When we talk about groups of objects, we use "Closure." If we have two regular languages, L1 and L2, and L is generated by performing specific operations on L1 and L2, then L is also regular.
Let us consider an example; let us take a confectionery set. Each component of the set contains a different type of candy. What happens if we take a candy from the set candy and drop it on the clean ground? We can still eat it. Thus it qualifies as candy. The set candy is closed using the "drop" method.
Closure properties used in Regular languages are as follows:
Union
Concatenation
Intersection
Reversal
Difference
Complementation
Homomorphism
Inverse Homomorphism
Union
Statement: If L1 and L2 are regular languages, then their union L1 U L2 is also a regular language.
Proof: Let M1 and M2 be two finite Automata that accept the L1 and L2 regular languages, respectively. If we wish to demonstrate that the union of L1 and L2 is likewise a regular language, we may do the following:
Make a new starting point.
Make ?-transitions from the new state to each of M1 and M2's original states.
Concatenation
Statement: Concatenation of two regular languages is also regular.
Proof: Let M1 and M2 be two finite automata, and L1 and L2 be the languages that M1 and M2 accept, respectively. We aim to demonstrate that L1L2 = L, i.e., their concatenation yields regular language. Let M be a finite automaton composed of M1 and M2.
Proof: Assume L1 and L2 are regular languages, and we wish to demonstrate that the intersection of L1 and L2 is likewise a regular language. De Morgen's Law may be used to calculate the intersection of languages L1 and L2.
Reversal
Statement: Under reversal, the set of regular languages is closed.
Proof: Let M be a deterministic finite automaton that accepts L; we will create M' from M so that M and M' states are the same. Make the final state of M the accepting state of M' and the final state of M the beginning state of M'. The direction of the edges in M' has been reversed. It denotes that the string has been written backward, i.e., the reverse of abbc is cbba.
Difference
Statement: If L1 and L2 are regular languages, then L1-L2 is as well, which implies all the strings in L1 but not in L2.
Proof: Since L2 is regular, its complement L2' is also regular, and L1? L2' too is regular; this proves that L1-L2 is also regular.
Complementation
Statement: According to the theorem, the complement of two regular languages is also regular.
Proof: If M is a deterministic finite automaton that accepts L, we may write L= L(M), then L' = L. (M1). The DFA M1 is similar to M, except that M's accepting states are now M1's non-accepting states and vice versa.
Homomorphism
A function that returns a string for each symbol in an alphabet is a homomorphism.
Statement: If L is a regular language and h is an alphabet homomorphism, then h(L) = h(w) | w is in L is also a regular language.
Proof: Assume E is a regular expression for L. Apply h for each symbol in E. The language of the resultant RE is h (L).
Inverse Homomorphism
Statement: If h is a homomorphism from the alphabet? To alphabet A and L is a regular language on A, then h' (L) is likewise regular.
Closure properties define how a language remains closed under operations like union, concatenation, intersection, complement, and Kleene star, ensuring the result is still in the same language class.
What is the closure of a language?
The closure of a language refers to the smallest superset containing all strings derived from applying specific operations while preserving the language's properties.
What are the properties of RL (Regular Languages)?
Regular languages (RL) are closed under union, intersection, concatenation, complement, difference, reversal, and Kleene star, making them stable under various operations in automata theory.
Conclusion
Closure properties of regular languages play a crucial role in automata theory and formal language processing. These properties ensure that regular languages remain closed under various operations such as union, intersection, complement, concatenation, and Kleene star, making them highly predictable and easy to manipulate. Understanding these properties helps in simplifying complex problems, constructing new languages, and proving language regularity.
519+ registered 
