Table of Contents
Definition
A pushdown automaton (PDA) is a type of automaton that extends the capabilities of a finite automaton by incorporating a stack. It is also known as a pushdown machine or a stack automaton.
A PDA is a theoretical model of computation that can recognize context-free languages. It consists of:
- An input tape: This tape contains the input symbols that the PDA reads sequentially.
- A control unit: It controls the behavior of the PDA and determines the state transitions based on the input and the current state.
- A stack: It is a last-in, first-out (LIFO) data structure that allows the PDA to store and retrieve symbols. The stack provides the additional memory needed to recognize context-free languages.
The operation of a PDA involves the following steps:
- The PDA starts in an initial state with an empty stack.
- It reads the input symbols one by one from the input tape.
- Based on the current state, the input symbol, and the symbol at the top of the stack, the PDA determines the next state and the action to be performed.
- The PDA can perform three types of actions:
- Push: It adds a symbol to the top of the stack.
- Pop: It removes the symbol from the top of the stack.
- Stay: It does not modify the stack.
- The PDA repeats steps 2-4 until it reaches a final state or the input is completely processed.
Difference between a Pushdown Automaton and a finite automaton is the stack
The stack allows the PDA to remember information about the previously seen symbols, enabling it to recognize context-free languages that cannot be handled by finite automata
Pushdown automata have various applications in computer science, particularly in the field of compiler design and parsing. They are used to analyze and process programming languages by recognizing the syntactic structure described by context-free grammars.
Type of Pushdown Automata
There are two main types of pushdown automata (PDA) based on their acceptance criteria: deterministic pushdown automata (DPDA) and nondeterministic pushdown automata (NPDA)
Deterministic Pushdown Automaton (DPDA)
- A deterministic pushdown automaton is a PDA that operates deterministically, meaning that for every combination of current state, input symbol, and stack symbol, there is at most one possible next move. In other words, there is no ambiguity in the transition from one state to another.
DPDAs are defined by a 7-tuple (Q, Σ, Γ, δ, q0, Z, F), where:
- Q is a finite set of states.
- Σ is the input alphabet.
- Γ is the stack alphabet.
- δ is the transition function: Q × (Σ ∪ {ε}) × Γ → Q × Γ*, where ε represents an empty string and Γ* denotes zero or more stack symbols.
- q0 is the initial state.
- Z is the initial stack symbol.
- F is the set of final/accepting states.
Nondeterministic Pushdown Automaton (NPDA):
- A nondeterministic pushdown automaton is a PDA that can have multiple possible next moves for a given combination of current state, input symbol, and stack symbol. It can guess the correct transition among the possible choices.
NPDAs are defined by a 7-tuple (Q, Σ, Γ, δ, q0, Z, F), where the components have the same meanings as in DPDA.
Nondeterminism in NPDA allows them to have more expressive power than DPDAs. However, this also means that their behavior can be more complex, and they may require backtracking or exploring multiple paths to determine the correct acceptance of an input.