Application of Stack: Conversion of Infix to Postfix Notation Using Stack

Definition

Infix to postfix conversion is the process of transforming an arithmetic or logical expression written in infix form, where operators appear between operands, into postfix form, where operators appear after operands, using a stack to temporarily store operators and parentheses.

A stack is a linear data structure that follows the LIFO (Last In, First Out) principle, meaning the last inserted element is the first one to be removed. During conversion, the stack helps keep track of operators that must be delayed until the correct order of output is determined.

Main Content

1. Infix Notation

• In infix notation, the operator is placed between two operands, such as A + B, X * Y, or (P + Q) / R.
• It is the most familiar notation used in algebra and everyday calculations, but it can become ambiguous without rules for precedence, associativity, and parentheses.

Infix expressions are easy for humans to read but difficult for machines to process directly because the order of evaluation is not always obvious. For example:

• A + B * C does not mean (A + B) * C
• Instead, multiplication has higher precedence, so the result is A + (B * C)

This means a computer must analyze the expression carefully before evaluation or translation.

2. Postfix Notation

• In postfix notation, the operator is placed after the operands, such as AB+, ABC*+, or PQ+R/.
• Postfix expressions do not require parentheses because the order of evaluation is implicit in the structure of the expression.

Postfix notation is highly useful in computers because it is unambiguous and easy to evaluate using a stack. For example:

• Infix: A + B * C
• Postfix: A B C * +

Evaluation becomes straightforward:

1. Read operands and push them onto a stack.
2. When an operator appears, pop the required operands.
3. Perform the operation and push the result back.

This simplicity is why postfix notation is widely used in compilers, interpreters, and calculators.

3. Stack-Based Conversion Rules

• Operators are stored temporarily in a stack until their correct position in the postfix expression is known.
• The conversion process depends on operator precedence and associativity.

Key rules include:

Operands

• are directly added to the postfix output.

Left parenthesis (

• is pushed onto the stack.

Right parenthesis )

• causes operators to be popped from the stack into the output until ( is found.
• When an operator is encountered, it is compared with the operator on the top of the stack:
• If the stack top has higher precedence, pop it to output.
• If the stack top has equal precedence and the operator is left-associative, pop it.
• Otherwise, push the current operator onto the stack.

Typical precedence order:

1. ^ exponentiation
2. *, /, %
3. +, -

Associativity:

• +, -, *, / are usually left-associative
• ^ is often right-associative

This rule-based use of a stack ensures correct conversion without needing to repeatedly scan the entire expression.

Working / Process

1. Scan the infix expression from left to right

• Read one symbol at a time: operand, operator, or parenthesis.
• Initialize an empty stack and an empty postfix output string.

2. Apply conversion rules while scanning

• If the symbol is an operand, append it directly to the postfix output.
• If the symbol is (, push it onto the stack.
• If the symbol is ), pop and append all operators until ( is encountered, then discard (.
• If the symbol is an operator, compare it with the top of the stack and pop higher-precedence or equal-precedence operators according to associativity, then push the current operator.

3. Pop remaining operators after the scan ends

• After the entire infix expression has been processed, pop all remaining operators from the stack and append them to the postfix output.
• No parentheses should remain in the stack at the end.

Example conversion of A + B * C:

Step-by-step processing:

Final postfix expression: ABC*+

ASCII diagram for stack-based conversion of A + B * C:

Expression: A + B * C

Read A   -> output: A
Read +   -> stack: [+]
Read B   -> output: AB
Read *   -> stack: [+, *]
Read C   -> output: ABC
End      -> pop * then +

Example with parentheses: (A + B) * C

• ( is pushed
• A and B go to output
• + stays inside parentheses
• ) causes + to be popped
• * waits until the end

Final postfix: AB+C*

Advantages / Applications

Removes ambiguity in expressions

Postfix notation eliminates the need for parentheses and makes the evaluation order explicit.

Efficient evaluation in computers

Postfix expressions can be evaluated easily with a stack, making them highly suitable for calculators, compilers, and expression parsers.

Important in compiler design and parsing

Infix-to-postfix conversion is a foundational technique used in syntax analysis, expression evaluation, and intermediate code generation.

Summary

• Infix expressions place operators between operands, while postfix expressions place operators after operands.
• A stack is used to manage operators based on precedence, associativity, and parentheses.
• The conversion process scans the expression from left to right and builds the postfix form systematically.
• Important terms to remember: infix notation, postfix notation, stack, precedence, associativity, parentheses