Various Properties of context free languages (CFL)

A Context-Free Language (CFL) is a formal language generated by a Context-Free Grammar (CFG) or Type 2 grammar (according to Chomsky classification). CFLs are accepted by Pushdown Automata and are essential in creating programming languages and building compilers.

Key Points:

• Made with Grammar Rules: A Context-Free Grammar (CFG) has rules that say how you can build valid sentences or expressions. Each rule replaces one symbol with a combination of other symbols.
• Recognized by a Pushdown Automaton (PDA): A CFL can be checked or accepted by a special machine called a Pushdown Automaton. It's like a basic computer that uses a stack (like a pile of plates) to keep track of what it’s doing.
• Can Handle Nested Structures: CFLs are more powerful than regular languages. They can describe things like matching parentheses (( )), which regular languages can't handle properly.
• Regularity: context-free languages are Non-Regular PDA language. 

Properties of CFL

1. Closure properties

The context-free languages are closed under some specific operation, closed means after doing that operation on a context-free language the resultant language will also be a context-free language. Some such operation are: 

• Union Operation
• Concatenation
• Kleene closure
• Reversal operation
• Homomorphism
• Inverse Homomorphism
• Substitution
• prefix operation
• Quotient with regular language
• Cycle operation
• Union with regular language
• Intersection with regular language
• Difference with regular language

Context free language is not closed under some specific operation, not-closed means after doing that operation on a context-free language the resultant language not remains to be a context-free language anymore. 
Some such operation are: 

• Intersection
• Complement
• Subset
• Superset
• Infinite Union
• Difference, Symmetric difference (XOR, NAND, NOR or any other operation which get reduced to intersection and complement)

Read more about Closure Properties of Context Free Languages

2. Decision Properties

• Emptiness: We can decide if a CFL is empty (i.e., it has no strings).
• Finiteness: We can decide if a CFL has only a finite number of strings.
• Membership: We can decide if a string belongs to a CFL (using parsing techniques like CYK algorithm).
• Equivalence: Undecidable. We cannot always determine if two CFLs are exactly the same.
• Universality (L = Σ*) : Undecidable.
• Inclusion (L₁ ⊆ L₂) : Undecidable.

3. Deterministic property

The context-free language can be: 

• DCFL-Deterministic : These are the languages which can be recognized by deterministic pushdown automata.
• NDCFL-Non-deterministic: These are the context free languages are that which can not be recognized by DPDA but can be recognized by NPDA context free language.

Read more about Difference Between NPDA and DPDA