Closure Properties of Context-Free Languages (CFLs)
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What closure properties are usually covered for CFLs
How to prove closure (CFL → CFL) using grammars/automata
- 1Step 1
Model CFLs as context-free grammars (CFGs) or pushdown automata (PDAs). Use whichever makes the construction easiest.
- 2Step 2
For union/concatenation/star, combine CFGs (nonterminals + productions) in a structured way; for homomorphisms/inverse homomorphisms, transform productions or transition labels.
- 3Step 3
Create new start symbol(s) and productions that simulate the chosen operation while preserving context-freeness.
- 4Step 4
Show: (i) any string generated by the new grammar is in the target closure set; (ii) any string in the target set can be generated by the new grammar.
- 5Step 5
Since the new grammar is CFG, the resulting language is context-free.
Typical closure results checklist
Positive closures
1Union, concatenation, Kleene star, reversal, homomorphism, inverse homomorphism, substitution."
Restricted closure
2Intersection with regular languages (and related “regular-guided” operations)."
Negative closures
3Complement and general intersection are not guaranteed to stay context-free."
Closure status (high-level)
Standard course-level classification for CFLs
Knowledge Check
Which operation is CFLs generally closed under?