Match List I with List II: Choose the correct answer from the options given below:List I List II (A) Type 0 (I) Finite automata (B) Type 1 (II) Tuning machine (C) Type 2 (III) Linear bound automata (D) Type 3 (IV) Pushdown automata
(A) - (II), (B) - (III), (C) - (IV), (D) - (I)
Understanding Chomsky Hierarchy and Automata Matching
The Chomsky Hierarchy is a classification of formal grammars. It groups grammars into four types (Type 0, Type 1, Type 2, and Type 3) based on their generative power. Each type of grammar corresponds to a specific class of formal languages, and these language classes, in turn, are recognized by specific types of automata.
Let's break down the correspondence between the grammar types listed in List I and the automata models listed in List II:
- Type 0 Grammars: These are also known as Unrestricted Grammars. They can generate all recursively enumerable languages. The computational model capable of recognizing recursively enumerable languages is the Turing Machine.
- Type 1 Grammars: These are known as Context-Sensitive Grammars. They generate context-sensitive languages. These languages are recognized by Linear Bounded Automata (LBA). An LBA is essentially a Turing Machine with a tape restricted to the size of the input.
- Type 2 Grammars: These are known as Context-Free Grammars. They generate context-free languages. Context-free languages are recognized by Pushdown Automata (PDA). A PDA is a finite automaton with an additional stack memory.
- Type 3 Grammars: These are known as Regular Grammars. They generate regular languages. Regular languages are the simplest in the hierarchy and are recognized by Finite Automata (FA). Finite automata have no additional memory beyond their finite states.
Matching Grammar Types to Automata
Based on the standard Chomsky Hierarchy correspondence, we can create the following matches:
- (A) Type 0 matches with (II) Turing Machine.
- (B) Type 1 matches with (III) Linear Bound Automata.
- (C) Type 2 matches with (IV) Pushdown Automata.
- (D) Type 3 matches with (I) Finite Automata.
| List I (Grammar Type) | List II (Automata) | Match |
|---|---|---|
| (A) Type 0 | (II) Turing Machine | (A) - (II) |
| (B) Type 1 | (III) Linear Bound Automata | (B) - (III) |
| (C) Type 2 | (IV) Pushdown Automata | (C) - (IV) |
| (D) Type 3 | (I) Finite Automata | (D) - (I) |
Putting these matches together, we get the combination: (A) - (II), (B) - (III), (C) - (IV), (D) - (I).
This arrangement correctly pairs each type of grammar from the Chomsky hierarchy with the corresponding abstract machine (automaton) that recognizes the languages generated by that type of grammar.
Revision Table: Chomsky Hierarchy Summary
| Type | Grammar Type | Language Type | Automaton | Production Rule Form (\(\alpha \rightarrow \beta\)) |
|---|---|---|---|---|
| 0 | Unrestricted | Recursively Enumerable | Turing Machine | No restrictions on \(\alpha\) and \(\beta\) (except \(\alpha \neq \epsilon\)) |
| 1 | Context-Sensitive | Context-Sensitive | Linear Bounded Automaton (LBA) | \(|\alpha| \le |\beta|\) (except possible \(\epsilon\) from start symbol S under certain conditions) |
| 2 | Context-Free | Context-Free | Pushdown Automaton (PDA) | \(A \rightarrow \beta\), where \(A\) is a single non-terminal |
| 3 | Regular | Regular | Finite Automaton (FA) | \(A \rightarrow aB\) or \(A \rightarrow a\) (right-linear) or \(A \rightarrow Ba\) or \(A \rightarrow a\) (left-linear) |
Additional Information on Automata Power
The Chomsky Hierarchy not only classifies grammars and languages but also establishes a hierarchy of computational power among the automata. As we move from Type 3 to Type 0, the expressive power of the grammars and languages increases, and consequently, the complexity and capabilities of the required automata also increase.
- Finite Automata (FA): The least powerful. Can only recognize regular languages. No memory beyond current state.
- Pushdown Automata (PDA): More powerful than FAs. Can recognize context-free languages. Have a stack for memory, allowing them to handle nested structures.
- Linear Bounded Automata (LBA): More powerful than PDAs. Can recognize context-sensitive languages. Have a tape like a Turing Machine, but restricted to the length of the input string, providing bounded memory.
- Turing Machines (TM): The most powerful in this hierarchy. Can recognize recursively enumerable languages. Have an infinite tape, representing unbounded memory and computational power equivalent to modern computers for decidable/recognizable problems.
This hierarchy demonstrates that adding memory or computational capability to an automaton allows it to recognize a wider class of languages.
Important Questions from Context Free Grammars - Teaching
The reduced grammar equivalent to the grammar, whose production rules are given below, is
S → AB | CA
B → BC | AB
A → a
C → a B | b
Consider the following grammars:
G 1: S → aSb|bSa|aa
G 2: S → aSb|bSa|SS|λ
G 3: S → aSb|bSa|SS|a
G 4: S → aSb|bSa|SS|SSS|λ
Which of the following is correct w.r.t. the above grammars?Consider L = L 1 ∩ L 2
Where L 1 = {0 m1 m20 n1 n |m, n >= 0}
L 2 = {0 m1 n2 k | m, n, k ≥ 0}
Then, the language L is
Consider the language \(L = \left\{ {{a^n}{b^{n - 3}}\ | n > 2} \right\}\) on ∑ = {a, b}. Which one of the following grammars generates the language L?
Consider the following grammar:
S → XY
X → YaY | a and y → bbXWhich of the following statements is/are true about the above grammar?
(a) Strings produced by the grammar can have consecutive three a's
(b) Every string produced by the grammar have alternate a and b
(c) Every string produced by the grammar have at least two a's
(d) Every string produced by the grammar have b's in multiple of 2.
