The PDA can be defined by a 6-tuple and by a 7-tuple, adding top of the stack element as 7th member of tuple. Which definition is more correct?
In the field of computational complexity theory, specifically in the study of pushdown automata (PDAs), the definition of a PDA can vary depending on the context and the specific sources being referenced. It is important to note that both the 6-tuple and 7-tuple definitions are valid and widely accepted in the field. However, the 7-tuple definition, which includes the addition of the top of the stack element as the 7th member of the tuple, is more commonly used and provides a more complete representation of the PDA's behavior.
To understand the rationale behind the inclusion of the top of the stack element in the 7-tuple definition, let's first review the basic components of a PDA. A PDA is a theoretical model of computation that extends the capabilities of a finite automaton by incorporating a stack. It consists of the following components:
1. Q: a finite set of states.
2. Σ: a finite set of input symbols.
3. Γ: a finite set of stack symbols.
4. δ: a transition function that maps the current state, input symbol, and top of the stack symbol to the next state, symbol to be pushed onto the stack, and direction to move the stack head (left or right).
5. q0: the initial state.
6. Z: the initial stack symbol.
7. F: a set of accepting states.
The 6-tuple definition of a PDA includes all of the above components except for the top of the stack element. This means that the transition function, δ, does not take into account the current symbol at the top of the stack when determining the next state and stack operation. This can limit the expressive power of the PDA, as it cannot make decisions based on the contents of the stack.
On the other hand, the 7-tuple definition includes the top of the stack element as an additional component. This allows the transition function, δ, to consider the current state, input symbol, and top of the stack symbol when determining the next state and stack operation. By including the top of the stack element in the definition, the PDA gains the ability to make decisions based on the contents of the stack, enabling it to recognize more complex languages.
To illustrate the difference between the two definitions, let's consider an example. Suppose we have a PDA that recognizes the language L = {a^n b^n | n ≥ 0}, where the number of 'a's is equal to the number of 'b's. In the 6-tuple definition, the PDA would not be able to keep track of the number of 'a's and 'b's on the stack, as it does not consider the top of the stack symbol during transitions. However, in the 7-tuple definition, the PDA can use the top of the stack symbol to keep track of the number of 'a's and 'b's, allowing it to recognize the language L.
While both the 6-tuple and 7-tuple definitions of a PDA are valid, the 7-tuple definition, which includes the top of the stack element, is more commonly used and provides a more comprehensive representation of the PDA's behavior. By considering the top of the stack symbol during transitions, the PDA gains additional expressive power and can recognize more complex languages.
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