This document discusses closure properties and decidability problems for formal languages. It contains tables listing whether operations like union, intersection and concatenation preserve regularity, context-freeness, etc. It also lists which problems, like membership, emptiness and equivalence are decidable for different language classes like regular, context-free and recursive languages. Some example problems are given to test understanding of these concepts.

1. THEORY OF
COMPUTATION
Tricks to remember Closure Properties of
Formal Languages

2. Operation Regular DCFL CFL CSL Recursi
ve
REL
Union ✔ ✘ ✔ ✔ ✔ ✔
Intersection ✔ ✘ ✘ ✔ ✔ ✔
Difference ✔ ✘ ✘ ✔ ✔ ✘
Complement ✔ ✔ ✘ ✔ ✔ ✘
Concatenation ✔ ✘ ✔ ✔ ✔ ✔
Kleene closure ✔ ✘ ✔ ✔ ✔ ✔
+ve closure ✔ ✘ ✔ ✔ ✔ ✔
Homomorphism ✔ ✘ ✔ ✘ ✘ ✔
I-
Homomorphism
✔ ✔ ✔ ✔ ✔ ✔
Closure Properties of Formal Languages

3. Operation Regular DCFL CFL CSL Recursi
ve
REL
Substitution ✔ ✘ ✔ ✔ ✘ ✔
Reverse ✔ ✘ ✔ ✔ ✔ ✔
L ⋃ regular ✔ ✔ ✔ ✔ ✔ ✔
L ⋂ regular ✔ ✔ ✔ ✔ ✔ ✔
L – regular ✔ ✔ ✔ ✔ ✔ ✔
Regular - R ✔ ✔ ✘ ✔ ✔ ✘
L / reg ✔ ✘ ✔ ✘ ✔ ✔
Reg* / R ✔ ✘ ✔ ✘ ✔ ✔
Subset ✘ ✘ ✘ ✘ ✘ ✘
Init ✔ ✔ ✔ ✔ ✔ ✔

4. Decidability & Undecidability of Formal Languages
Problem Regula
r
DCFL CFL Rec. REL
Is W in L? (membership
problem)
✔ ✔ ✔ ✔ ✘
Is L= ϕ ? (emptiness problem) ✔ ✔ ✔ ✘ ✘
Is L= finite? (finiteness problem) ✔ ✔ ✔ ✘ ✘
Is L1=L2 ? (equivalence
problem)
✔ ✔ ✘ ✘ ✘
Is L1⊆L2 ? (subset problem) ✔ ✘ ✘ ✘ ✘
Is ‘L’ regular? (regularity
problem)
✔ ✔ ✘ ✘ ✘
Is L ambiguous? (ambiguity
pro.)
✔ ✘ ✘ ✘ ✘
Is L=Σ* ? (universality problem) ✔ ✔ ✘ ✘ ✘
Is L1⋂L2 = ϕ ? (disjoint
problem)
✔ ✘ ✘ ✘ ✘

5. Related Questions
O Let L1 – regular language, L2 – CFL, L3 – recursive
language, L4 – recursive enumerable but not
recursive. Then which of the following is false.
a) 𝐿3 𝑐
⋃ 𝐿4 is REL
b) 𝐿2 ⋃ 𝐿3 is recursive language
c) L1*⋂ 𝐿2 is CFL
d) L1 ⋃ 𝐿2 is CFL

6. O L1- regular, L2- DCFL, L3- REL but not recursive.
Then which of the following is false?
a) L1⋂L2 is DCFL
b) L1⋂L2⋂L3 is REL
c) L1⋃L2 is CFL
d) L1⋂L3 recursive language

7. O L1- recursive language, L2- REL but not recursive.
Then which of the following is true?
a) 𝐿1 is recursive and 𝐿2 is REL
b) 𝐿1 is recursive and 𝐿2 is not REL
c) 𝐿1 and 𝐿2 are REL
d) 𝐿1 is REL and 𝐿2 is recursive

8. O which of the following is undecidable?
a) Is L(DFA)=L(NFA) ?
b) G is a CFG is W in G?
c) G is CFG is L(G)=Σ* ?
d) G is CFG is L(G)=ϕ ?