Question

Design a PDA to accept each of the following languages. You may design your PDA to accept either by final state or empty stack, whichever is more convenient.

(a) {aʰbᵏaᵐbⁿ:h+k=m+n}
(b) {aⁿb:n≥0) ∪ {aⁿb:n≥0)∪ {aⁿb:n≥0)}

Answer 1

To design a PDA to accept languages (a) and (b), a pushdown automaton with final state can be used. In both cases, the automaton checks the balance of 'a's and 'b's and the empty state of the stack.

Step-by-step explanation:

To design a PDA to accept the language (a) {aʰbᵏaᵐbⁿ:h+k=m+n}, we can use a pushdown automaton with final state. Here are the steps:

1. Start with an initial state.
2. For each 'a' encountered in the input, push it onto the stack.
3. For each 'b' encountered in the input, check if there is a corresponding 'a' on top of the stack. If there is, pop the 'a' from the stack and continue.
4. After processing the entire input, check if the stack is empty and if the number of 'a's and 'b's on the stack matches the number of 'a's and 'b's in the input. If both conditions are met, transition to the final state. Otherwise, reject the input.

To design a PDA to accept the language (b) {aⁿb:n≥0) ∪ {aⁿb:n≥0)∪ {aⁿb:n≥0)}, we can also use a pushdown automaton with final state. Here are the steps:

1. Start with an initial state.
2. For each 'a' encountered in the input, push it onto the stack.
3. For each 'b' encountered in the input, check if there is a corresponding 'a' on top of the stack. If there is, pop the 'a' from the stack and continue.
4. After processing the entire input, check if the stack is empty. If it is, transition to the final state. Otherwise, reject the input.