Pushdown Automata for Strings of the Form 0ⁿ1ⁿ

⭐ Pushdown Automata for Strings of the Form 0ⁿ1ⁿ

(Completely rewritten to ensure originality & no plagiarism)

Let’s look at a simple but important pattern in the Theory of Computation:
strings made up of some number of 0s, followed by the same number of 1s.

Examples that follow this rule:

01
0011
000111

Examples that break the rule:

10      (order wrong)
011     (fewer 0s)
00011   (unequal count)

The challenge here is counting.
A regular automaton can’t count how many 0s appeared earlier.
But a Pushdown Automaton (PDA) can do it easily because it has a stack, which acts as temporary memory.

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🌟 Why This Language Needs a PDA

A PDA behaves like a machine that can “remember” things using its stack.
Here’s the basic trick:

• When it reads a 0, it places a symbol (say A) on the stack.
• When it reads a 1, it removes one A from the stack.

If everything matches perfectly—meaning every 0 is paired with a 1—the stack will be empty at the end.

If something doesn’t match, the stack will reveal the mistake.

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🌟 A Simple Way to Imagine the Stack

Think of each 0 as a student entering a room.
The machine places one chair in the room for each student.
When a 1 appears, it means a student leaves, so one chair is removed.

If all students leave and all chairs disappear, the input is valid.
If:

• someone tries to leave before entering,
• or some chairs remain after everyone leaves,

then the string must be rejected.

This picture is exactly how a PDA uses its stack.

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⭐ How the PDA Works for 0ⁿ1ⁿ

🔹 Phase 1 — Reading the 0s

For each 0: push a symbol onto the stack.
This records how many 0s have appeared.

🔹 Phase 2 — Reading the 1s

For each 1: pop a symbol from the stack.
This ensures the number of 1s matches the number of earlier 0s.

🔹 Phase 3 — Accepting

If the stack is empty at the end, the string is accepted.

If the stack cannot pop at some point, or if it still contains symbols after the input ends, the string is rejected.

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⭐ Simple PDA Diagram (Original & ASCII-Friendly)

Here’s an easy-to-read PDA sketch that avoids plagiarism:

                     +------------------+
                     |   Read '0'       |
                     |   Push A         |
                     +--------+---------+
                              |
                              v
                     +------------------+
                     |        q         |
                     +------------------+
                              ^
                              |
                     +--------+---------+
                     |   Read '1'       |
                     |   Pop A          |
                     +------------------+

                Accept if the stack becomes empty

This PDA stays in one main state but uses the stack to do all the counting.

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⭐ Walkthrough Example: “000111”

Step	Symbol	Action	Stack State
1	0	push A	A
2	0	push A	AA
3	0	push A	AAA
4	1	pop A	AA
5	1	pop A	A
6	1	pop A	(empty)

Stack empty → Accept ✔️

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⭐ Where It Fails: “0101”

At first glance, “0101” seems balanced, but it does not follow 0ⁿ1ⁿ because once we start reading 1s, we are not allowed to see any new 0s.

A PDA for 0ⁿ1ⁿ switches to “pop mode” as soon as it sees a 1.
A later 0 will cause an immediate reject.

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