Match List-I with List-II: Where L1: Regular language L2: Context-free language L3: Recursive language L4: Recursively enumerable language List-1 List-2 (a) L' 3 U L 4 (i) Context-free language (b) L' 2 U L 3 (ii) Recursively enumerable language (c) L 1* ∩ L 2 (iii) Recursive language Choose the correct from those given below:
(a)-(ii); (b)-(iii); (c)-(i)
Understanding Formal Language Classes and Closure Properties
This question asks us to match combinations of operations on different types of formal languages with the resulting language classes. We are given four main language classes:
- L1: Regular language
- L2: Context-free language (CFL)
- L3: Recursive language
- L4: Recursively enumerable language (RE)
These classes form a hierarchy, where Regular ⊆ CFL ⊆ Recursive ⊆ RE. Each class has specific closure properties under various set operations like union, intersection, complement, and Kleene star.
Closure Properties of Language Classes
Let's quickly review the relevant closure properties for these language classes:
| Operation | Regular (L1) | CFL (L2) | Recursive (L3) | RE (L4) |
|---|---|---|---|---|
| Union (∪) | Closed | Closed | Closed | Closed |
| Intersection (∩) | Closed | Not Closed | Closed | Closed |
| Complement (') | Closed | Not Closed (Complement of CFL is Recursive) | Closed | Not Closed |
| Kleene Star (*) | Closed | Closed | Closed | Closed |
Now let's analyze each expression given in List-I.
Analyzing List-I Expressions
(a) L'₃ ∪ L₄
- L₃ is a Recursive language.
- The complement of a Recursive language (L'₃) is always a Recursive language. This is because the class of Recursive languages is closed under complement.
- L₄ is a Recursively enumerable language (RE).
- We need to find the class of L'₃ ∪ L₄, which is the union of a Recursive language and an RE language.
- The union of any two Recursively enumerable languages is Recursively enumerable. Since Recursive languages are a subclass of RE languages, the union of a Recursive language and an RE language is always a Recursively enumerable language.
Thus, L'₃ ∪ L₄ is a Recursively enumerable language.
This matches with (ii) Recursively enumerable language.
(b) L'₂ ∪ L₃
- L₂ is a Context-free language (CFL).
- The complement of a Context-free language (L'₂) is not necessarily Context-free. A known property is that the complement of a Context-free language is always a Recursive language.
- L₃ is a Recursive language.
- We need to find the class of L'₂ ∪ L₃, which is the union of a Recursive language (L'₂) and a Recursive language (L₃).
- The union of any two Recursive languages is always a Recursive language. This is because the class of Recursive languages is closed under union.
Thus, L'₂ ∪ L₃ is a Recursive language.
This matches with (iii) Recursive language.
(c) L₁* ∩ L₂
- L₁ is a Regular language.
- The Kleene star of a Regular language (L₁*) is always a Regular language. This is because the class of Regular languages is closed under Kleene star.
- L₂ is a Context-free language (CFL).
- We need to find the class of L₁* ∩ L₂, which is the intersection of a Regular language (L₁*) and a Context-free language (L₂).
- The intersection of a Regular language and a Context-free language is always a Context-free language.
Thus, L₁* ∩ L₂ is a Context-free language.
This matches with (i) Context-free language.
Summary of Matchings
- (a) L'₃ ∪ L₄ corresponds to (ii) Recursively enumerable language.
- (b) L'₂ ∪ L₃ corresponds to (iii) Recursive language.
- (c) L₁* ∩ L₂ corresponds to (i) Context-free language.
| List-I | Resulting Language Class | List-II |
|---|---|---|
| (a) L'₃ ∪ L₄ | Recursively enumerable | (ii) |
| (b) L'₂ ∪ L₃ | Recursive | (iii) |
| (c) L₁* ∩ L₂ | Context-free | (i) |
Comparing our matchings with the given options, the correct choice is (a)-(ii); (b)-(iii); (c)-(i).
Revision Table: Formal Language Properties
| Property | Regular | CFL | Recursive | RE |
|---|---|---|---|---|
| Recognized by | Finite Automata (DFA/NFA) | Pushdown Automata (PDA) | Turing Machine (decides) | Turing Machine (accepts) |
| Decidable? (Membership) | Yes | Yes | Yes | No (partially) |
| Closed under Union | Yes | Yes | Yes | Yes |
| Closed under Intersection | Yes | No | Yes | Yes |
| Closed under Complement | Yes | No (complement is Recursive) | Yes | No |
| Closed under Kleene Star | Yes | Yes | Yes | Yes |
Additional Information: Hierarchy and Properties
The hierarchy of these language classes is strict: Regular ⊊ CFL ⊊ Recursive ⊊ RE ⊊ All Languages.
- Regular Languages: Simplest class, recognized by finite automata. Can be described by regular expressions.
- Context-Free Languages: More powerful, recognized by pushdown automata. Described by context-free grammars. Used widely in parsing programming languages.
- Recursive Languages: Decidable languages. For any string, a Turing Machine can halt and tell whether the string is in the language or not.
- Recursively Enumerable Languages: Recognizable languages. For strings in the language, a Turing Machine will halt and accept. For strings not in the language, the Turing Machine might halt and reject, or it might loop forever.
Understanding the closure properties is crucial for determining the class of a language resulting from operations on known language classes. For example, while CFLs are not closed under intersection, the intersection of a CFL and a Regular language *is* always a CFL. Similarly, while CFLs are not closed under complement, their complements fall into the Recursive language class.
Important Questions from Context Free Languages - Teaching
Consider the following statements about Context Free Language (CFL):
Statement I: CFL is closed under homomorphism.
Statement II: CFL is closed under complement.
Which of the following is correct?
Given below are two statements
Statement I : The family of context free languages is closed under homomorphism
Statement II : The family of context free languages is closed under reversal
In light of the above statements, choose the correct answer from the options given below
Which of the following statements is true ?
Consider the following statements with respect to the language L = {a nb n|n≥ 0}
S 1: L 2is context free language
S 2: L kis context-free language for any given k ≥ 1
S 3: L̅ and L* are context free languages
Which one of the following is correct?
The language L = {a ib c i | i ≥ 0} over the alphabet {a, b, c} is :
