Question

What are the closure properties for CFGs? What does it mean for a CFG to be closed under one of these operations?

Answer 1

Closure properties of CFG:

The following are the closure properties of CFG.
1. union
2.concatenation
3.kleen closure
4.intersection
5.complementation

A detailed explanation of each operation.

1. union:

A1 and A2 are considered as the two context-free languages.

the union of A1 and A2 is denoted as A1UA2.

the union of A1 and A2 (A1UA2) is context-free language if A1 and A2 are context-free languages.

2. concatenation:

A1 and A2 are considered as the two context-free languages.

the concatenation of A1 and A2 is denoted as A1.A2.

the concatenation of A1 and A2 (A1.A2) is context-free language if A1 and A2 are context-free languages.

3.kleen closure:

kleen closure of a language A is denoted as A*.

If language A is context-free, then A* also context-free.

4.Intersection:

A1 and A2 are considered as the two context-free languages.

the intersection of A1 and A2 is denoted as A1∩A2.

If A1 and A2 are context-free languages, then there is no need for the intersection of A1 and A2 (A1∩A2) to be context-free language.

Complementation:

A is considered a context-free language.

Its complement is derived by subtracting A from the whole set.
the whole set is ∑*.
the complement of A is ∑*-A.

If a CFG is closed under one of the above operations, it means that it satisfies that one particular operation only.
Example:
If a language L satisfy the union operation, there is no need for it to satisfy the addition operation.
Each operation is unique.