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# What is the Instantaneous description of PDA?

The Instantaneous description is called as an informal notation, and explains how a Push down automata (PDA) computes the given input string and makes a decision that the given string is accepted or rejected.

The PDA involves both state and content of the stack.

Stack is often one of the important parts of PDA at any time.

So, we make a convenient notation for describing the successive configurations of PDA for string processing.

The factors of PDA notation by triple (q, w, γ) were

- q is the current state.
- w is the remaining input alphabet.
- γ is the current contents of the PDA stack.

Generally, the leftmost symbol indicates the top of the stack γ and the bottom at the right end. This type of triple notation is called an instantaneous description or ID of the pushdown automata.

A move from one instantaneous description to another is denoted by the symbol ‘⊢’

Therefore,

(q0, aw, z0) ⊢ (q1, w, yz0)

is possible if and only if

δ(q0, a, z0) ϵ (q1, yz0).

## Example

Consider an example as given below −

Show the IDs or moves for input string w = “aabb” of PDA where,

**M = ({q0, q1, q2}, {a, b}, {a, b, Z0}, δ, q0, Z0, {q2}),**

Where δ is defined as follows −

δ(q0, a, Z0) = {(q0, aZ0)} Rule (1) δ(q0, a, a) = {(q0, aa)} Rule (2) δ(q0, b, a) = {(q1, λ)} Rule (3) δ(q1, b, a) = {(q1, λ)} Rule (4) δ(q1, λ, Z0) = {(q2, λ)} Rule (5) δ(q0, λ, Z0) = {(q2, λ)} Rule (6)

And we need to find out whether string w is accepted by PDA or not.

## Solution

The Instantaneous Description for the string w = “aabb”. It is explained below −

(q0, aabb, Z0) |- (q0, abb, aZ0) based on Rule (1) |- (q0, bb, aaZ0) based on Rule (2) |- (q1, b, aZ0 based on Rule (3) |- (q1, λ, Z0) based on Rule (3) |- (q2, λ, λ) based on Rule (5)

Therefore, PDA reaches a configuration of (q2, λ, λ) i.e. PDA stack is empty and it has reached a final state. So the string ‘w’ is accepted.

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