GATE CSE 2002 | Question: 1.7

Question

The language accepted by a Pushdown Automaton in which the stack is limited to $10$ items is best described as

• A. Context free
• B. Regular
• C. Deterministic Context free
• D. Recursive

Comments

FINITE CONFIGURATION THAT IS FINITE STACK OR FINITE TAPE SIMPLY MEANS REGULAR LANGUAGES!

---

since its a finite stack which means no of states is finite so its a regular language!

Answer 1

Correct Option: B

With only finite positions in stack, we can have only finite configurations and these can also be modeled as states in a finite automata.

Comments

@avraw

We can construct DFA for a^nb^n for n<10 as it is regular

---

As there is a finite number of items mentioned that is 10 items. So, a FA is always possible. That’s why regular language is best described.

Can we say conclude this answer with the above statement?

---

@Sachin Mittal 1, 

using this approach, we will have at most 10 states, Finite State automata which generates Regular Grammar. please verify.

Answer 2

The language is best described as Regular.

Here is the reasoning:

1. A standard Pushdown Automaton (PDA) gets its power to recognize context-free languages from its unbounded (infinitely deep) stack. This stack allows it to handle an arbitrary depth of nesting, like in the language $\{a^n b^n \mid n \ge 0\}$.

2. When you limit the stack to a fixed, finite size (in this case, 10 items), you are essentially removing its unbounded memory.

3. The machine now has a finite number of states (from the automaton part) and a finite number of possible stack configurations. The total "memory" of the machine is the combination of its current state and the entire content of its 10-item stack.

4. Since the number of states is finite and the number of stack configurations is finite, the total number of possible machine configurations is also finite.

5. A machine with a finite number of configurations is, by definition, a Finite Automaton (FA). This constrained PDA can be perfectly simulated by a standard FA, where each state in the FA corresponds to a (state, stack content) pair from the PDA.

6. Since the machine is equivalent in power to a Finite Automaton, the language it accepts must be regular.