PDA Acceptance

A language can be accepted by Pushdown automata using two approaches:

1. Acceptance by Final State: The PDA is said to accept its input by the final state if it enters any final state in zero or more moves after reading the entire input.

Let P =(Q, ∑, Γ, δ, q0, Z, F) be a PDA. The language acceptable by the final state can be defined as:

L(PDA) = {w | (q0, w, Z) ⊢* (p, ε, ε), q ∈ F} 

2. Acceptance by Empty Stack: On reading the input string from the initial configuration for some PDA, the stack of PDA gets empty.

Let P =(Q, ∑, Γ, δ, q0, Z, F) be a PDA. The language acceptable by empty stack can be defined as:

N(PDA) = {w | (q0, w, Z) ⊢* (p, ε, ε), q ∈ Q}  

Equivalence of Acceptance by Final State and Empty Stack

• If L = N(P1) for some PDA P1, then there is a PDA P2 such that L = L(P2). That means the language accepted by empty stack PDA will also be accepted by final state PDA.
• If there is a language L = L (P1) for some PDA P1 then there is a PDA P2 such that L = N(P2). That means language accepted by final state PDA is also acceptable by empty stack PDA.

Example:

Construct a PDA that accepts the language L over {0, 1} by empty stack which accepts all the string of 0's and 1's in which a number of 0's are twice of number of 1's.

Solution:

There are two parts for designing this PDA:

• If 1 comes before any 0's
• If 0 comes before any 1's.

We are going to design the first part i.e. 1 comes before 0's. The logic is that read single 1 and push two 1's onto the stack. Thereafter on reading two 0's, POP two 1's from the stack. The δ can be

δ(q0, 1, Z) = (q0, 11, Z)        Here Z represents that stack is empty  

δ(q0, 0, 1) = (q0, ε)  

Now, consider the second part i.e. if 0 comes before 1's. The logic is that read first 0, push it onto the stack and change state from q0 to q1. [Note that state q1 indicates that first 0 is read and still second 0 has yet to read].

Being in q1, if 1 is encountered then POP 0. Being in q1, if 0 is read then simply read that second 0 and move ahead. The δ will be:

δ(q0, 0, Z) = (q1, 0Z)  

 δ(q1, 0, 0) = (q1, 0)  

δ(q1, 0, Z) = (q0, ε)        (indicate that one 0 and one 1 is already read, so simply read the second 0)  

δ(q1, 1, 0) =     (q1, ε)  

Now, summarize the complete PDA for given L is:

δ(q0, 1, Z) = (q0, 11Z)  

δ(q0, 0, 1) = (q1, ε)  

δ(q0, 0, Z) = (q1, 0Z)  

δ(q1, 0, 0) = (q1, 0)  

δ(q1, 0, Z) = (q0, ε)  

δ(q0, ε, Z) = (q0, ε)      ACCEPT state