– A well-designed pushdown automata is like a chef’s recipe; it only accepts the perfect blend of 0s and 1s. The stack must be empty, like an empty plate after a delicious meal. But if there’s one extra 0 left behind, it’s like a lingering spice that means the stack isn’t empty! 🍲🚫🥄
– Designing a PDA for the language where A and B have to appear in equal measure is like making sure a sandwich has the right balance of bread and filling. Push for A, pop for B, and keep the stack in perfect harmony. If the stack ends up empty, it’s like the final bite of a well-made sandwich. 🥪✨👌
# Problems from Push Down Automata
## Designing PDA for a Language with an Equal Number of Zeros and Ones 🤖
In this problem, we are tasked with defining a language that consists of an equal number of zeros and ones. The language is defined as 0 to 1 to where n is greater than 0. For instance, the possibilities include 00, 11 as well as 000, 111, and so on.
To tackle this, we will use a non-deterministic pushdown automaton to address only the accept criteria rather than the reject case. If the top of the stack is the symbol z0, this indicates that after we have popped it out, the stack will be empty. Therefore, to begin, we must insert our start symbol q0 into the stack. From the initial condition Q, the transition is given as Epsilon into Epsilon, indicating that the input should be inserted at the top of the stack.
We will then transition to pop zero when the input symbol ‘1’ is encountered. Subsequently, if the stack is empty, this signifies that the string can be accepted. Finally, when the Epsilon is popped, the start symbol is removed without any replacement, leading to the acceptance of the string.
Here’s a summarized version of the acceptance criteria:
| Input Symbol | Top of Stack | New Stack |
|————–|————–|———–|
| Epsilon | Epsilon | q0 |
| 0 | Epsilon | Z0 |
| 1 | 0 | Epsilon |
| Epsilon | Z0 | Epsilon |
## Designing a PDA for a Language with an Equal Number of A’s and B’s 🛠️
Next, we shift our focus to the design of a pushdown automaton (PDA) for a language that includes an equal number of A’s and B’s. The language is defined as having different combinations of ‘a’ and ‘b’, wherein ‘W’ comprises an equal number of A’s and B’s.
For this scenario, if there is an ‘a’, it needs to be pushed into the stack, and if there is a ‘b’, it should be popped. Moreover, the start symbol ‘z0’ should be added at the beginning.
Taking into account all these conditions, it is crucial to derive the possible combinations for the language.
| Input Symbol | Top of Stack | New Stack |
|————–|————–|———–|
| Epsilon | Epsilon | q0 |
| a | Epsilon | a0 |
| b | a | Epsilon |
| a | Epsilon | Epsilon |
| b | Epsilon | Epsilon |
These summarised processes outline how the PDA handles each input symbol in the context of the given language.
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