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 ‘⊢’


(q0, aw, z0) ⊢ (q1, w, yz0)

is possible if and only if

δ(q0, a, z0) ϵ (q1, yz0).


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.


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.