In the given situation, the input string will be rejected by the PDA. When the PDA has reached the final state and there are still elements remaining in the input buffer, it means that there is no defined transition for the next input element in the final state. Since the PDA cannot make a transition for the next input element, it will not be able to accept the input string.
It is important to note that a PDA can only accept an input string if it can consume the entire input string and reach a final state, following a series of transitions defined by the PDA's transition function. If the PDA cannot consume the entire input string or if it cannot reach a final state, then it will reject the input string.
Empty stack acceptance is a separate concept that is unrelated to the acceptance or rejection of the input string. Empty stack acceptance allows a PDA to accept an input string even if it has not completely emptied its stack, as long as it has reached a final state. This is done by adding an additional condition to the acceptance criteria of the PDA, such that the PDA will also be considered to have accepted the input string if it has reached a final state and the stack is empty or has a certain specific configuration
