Role of Pushdown Automata in Compilers

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In syntax analysis, the most commonly used grammars are context-free grammars. In lexical analysis, we use finite state machines (FSMs) because they work well for simpler tasks like token recognition. However, FSMs are not sufficient for parsing context-free languages, hence pushdown automata are used for this purpose.

Pushdown Automata are not physical machines but theoretical computational models with greater computational power than FSMs. Their power comes from an additional component—a stack—which enables them to handle more complex languages that FSMs cannot process. These complex languages are those defined by context-free grammars.

In this chapter, we will explore the fundamentals of pushdown automata, their structure, and their role in compiler design, along with examples for better understanding.

What is a Pushdown Automaton?

A pushdown automaton is an abstract computational model designed to process languages that require additional memory. Unlike finite state machines that rely solely on states and transitions, pushdown automata use a stack to store and retrieve symbols.

This stack operates on a manner called last-in, first-out (LIFO). It means the last item added to the stack is the first one removed. This mechanism helps pushdown machines to keep track of nested structures and dependencies. And this is making them ideal for parsing tasks.

Components of a Pushdown Automaton

To understand how a pushdown automaton works, we must understand its components. A pushdown automaton consists of the following components −

• Finite Set of States − Includes one starting state and possibly one or more accepting states.
• Input Alphabet − The set of symbols the machine can read from the input string.
• Stack Alphabet − The set of symbols that can be pushed onto or popped from the stack.
• Stack − An infinite memory structure that supports push and pop operations.
• State Transition Function − Defines how the machine moves between states based on the current input symbol, stack symbol, and current state.

How Pushdown Machines Work?

The operation of a pushdown machine is done with the following steps −

• Input Processing − The machine reads one symbol at a time from the input string.
• Stack Operations − Then based on the input symbol and the current state, the machine can do the following tasks (but one at a time)
  • Push a symbol onto the stack.
  • Pop a symbol from the stack.
  • Replace the top stack symbol with another symbol or sequence of symbols.
• State Transition − The machine moves to a new state or stays in the current state.
• Acceptance or Rejection − When the input string is fully processed then the machine either accepts or rejects. This is done based on its final state and the condition of the stack.

Example: A Pushdown Machine for {aⁿbⁿ}

This is the typical example of context-free grammar that can be accepted with pushdown automata.

Let us consider a pushdown automaton that accepts the strings of the form {aⁿbⁿ | n ≥ 0}. This language consists of strings where the number of a’s is equal to the number of b’s.

Structure of the Pushdown Machine

• States − Two states: S1 (starting state) and S2.
• Input Symbols − {a, b}.
• Stack Symbols − {X, ∇}, where X tracks the count of as and ∇ marks the bottom of the stack.

Operation

• State S1 − Push an X onto the stack for each a read from the input.
• State S2 − Pop an X from the stack for each b read from the input.

Acceptance

The machine accepts the string if the input is fully read. And the stack is empty (except for ∇), and the machine reaches an accepting state.

Transition Table

The transition table for this machine is as follows −

Example Run: Input aabb

Initial Configuration − Input = aabb, Stack = ∇

• Step 1 − Read a. Push X → Stack = X∇
• Step 2 − Read a. Push X → Stack = XX∇
• Step 3 − Read b. Pop X → Stack = X∇
• Step 4 − Read b. Pop X → Stack = ∇

Final Step − Input is fully processed, then the stack becomes empty. And machine is in an accepting state → Accept.

Applications of Pushdown Automata in Compiler Design

Let us see the applications of pushdown automata in compiler design paradigm. Pushdown machines are widely used in the syntax analysis phase of compilers. They provide the foundation for parsers that process context-free grammars. Here are some key applications −

• Parsing Nested Structures − Pushdown machines can handle nested constructs like parentheses, brackets, and function calls. For example, checking if ((a+b)*c) has balanced parentheses or not
• Recognizing Context-Free Languages − They are useful for parsing languages where it is defined by context-free grammars. For example arithmetic expressions.
• Translation − Extended pushdown machines can translate infix expressions like (2 + 3 * 5) into postfix notation (2 3 5 * +).

Advantages of Pushdown Automata

Pushdown automata can process languages that finite state machines cannot. The stack allows for handling complex dependencies and nested structures. Features like replace operations or output functions enhance their capabilities.

Challenges and Limitations in Pushdown Automata

It can be challenging to design a pushdown automaton for certain languages can be challenging. The reliance on a stack makes them unsuitable for languages where advanced memory structures are needed. Pushdown machines cannot process all computational problems; they are limited to context-free languages.

Real-World Examples of Pushdown Automata

A pushdown automaton can verify if a string of parentheses, like {()()}, is correctly balanced. Pushdown automata are used for translating infix expressions into postfix form for easier evaluation by machines. They are also used for parsing the syntax of languages like C, Java, and Python during compilation.

Conclusion

Pushdown automata are a vital part of the syntax analysis phase in compiler design. They are useful in compilers to process and validate context-free languages efficiently.

In this chapter, we explained the concept of pushdown automata and their role in compiler design. We also understood their components, such as states, input alphabets, and stacks. We analyzed their operation through examples like {aⁿbⁿ}. We also discussed their applications in parsing context-free grammars and handling nested structures.