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:
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:
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
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:
Now, summarize the complete PDA for given L is: