Union and Intersection of Regular languages with CFL

Context-Free Languages (CFLs) are an essential class of languages in the field of automata theory and formal languages. They are generated by context-free grammars (CFGs) and are recognized by pushdown automata (PDAs). Understanding the closure properties of CFLs helps in determining which operations preserve the context-free nature of a language.

CFLs are closed under some operations, meaning that applying these operations to CFLs will always result in another CFL. However, they are not closed under certain operations, which means that performing these operations on CFLs might result in a language that is not context-free.

Regular languages are the class of languages that can be represented using finite automata, regular expressions, or regular grammar. These languages have predictable patterns and are computationally efficient to recognize.

Union of Regular language with CFG

If all regular languages are context free then union of both results is also a context-free language.

Example 1

Let’s L1 = {0*1*} is a regular language and L2 = {0^m1^m | m>=0} be a context free

And let L=L1 ∪ L2 be the union of both these languages. In the problem, L = {0*1*} is a regular language.

We know that every regular language is context-free.

So, obviously we can say the union of two always results in context-free language. Because the union of two context-free languages is a context-free language.

Hence proved.

Example 2

All strings with only a’s:
RL = { a, aa, aaa, ... }

Strings with the same number of a’s and b’s:
CFL = { ab, aabb, ababab, ... }

Union for the above two will be all strings that are either all a’s OR have equal a’s and b’s

Example strings in the result:

• a → in RL
• aa → in RL
• ab → in CFL
• aabb → in CFL
• abab → in CFL

So, obviously we can say the union of two always results in context-free language

Intersection of Regular language with CFG

The intersection of a regular and a context-free language always result in a context-free language.

Example 1

L1 = {0*1*} is a regular language and

L2 = {0^m1^m | m>=0} is a CFL

The intersection of two languages is as follows −

L= L1 ∩ L2

L={ 0^m1^m | m>=0} which is context-free .

So, finally it is concluded that the intersection of regular language and context free language generates a context free language.

Example 2

All strings with only a’s:
RL = { a, aa, aaa, ... }

Strings with the same number of a’s and b’s:
CFL = { ab, aabb, ababab, ... }

Intersection contains strings that start with a AND have equal a’s and b’s

Example strings in the result:

• ab → starts with a and a = b
• aabb → starts with a and a = b
• abab → starts with a and a = b

You can take more such examples and verify that the union and intersection of a regular language and a context-free language always results in a context-free language.

For more information about Regular language and context free language you can refer to the article given in link

Regular Language and Context Free language