How to Convert Infix to Postfix Manually?

Converting an infix expression (the standard mathematical notation we use daily, like A + B) into postfix notation (also known as Reverse Polish Notation or RPN, where operators follow their operands, like A B +) is a fundamental concept in computer science. Manually performing this conversion helps in understanding how compilers and interpreters process expressions. There are primarily two common manual methods: one involving full parenthesization and operator movement, and another mimicking a stack-based algorithm.

Understanding Key Concepts

Before diving into the conversion methods, it's crucial to grasp these fundamental concepts:

• Infix Notation: The most common way to write expressions, with operators placed between their operands (e.g., 2 + 3).
• Postfix Notation (RPN): Operators are placed after their operands (e.g., 2 3 +). This notation doesn't require parentheses to define the order of operations, making it simpler for computer evaluation.
• Operator Precedence: Determines the order in which operators are evaluated. For example, multiplication and division generally have higher precedence than addition and subtraction.
• Operator Associativity: Defines the order of evaluation for operators with the same precedence. Most common arithmetic operators (+, -, *, /) are left-associative, meaning operations are performed from left to right. Exponentiation (^) is typically right-associative.

Here's a standard precedence table (higher number means higher precedence):

Parentheses () override these rules, forcing a specific order of evaluation.

Method 1: The Parenthesization and Operator Movement Method

This technique clearly illustrates the relationship between operators and their operands in postfix form. It's an excellent way to conceptualize the structural change from infix to postfix.

1. Understand Operator Precedence and Associativity: Familiarize yourself with the rules mentioned above.
2. Fully Parenthesize the Infix Expression: Explicitly add parentheses around every operation, respecting the precedence and associativity rules. Each operator, along with its two operands, should be enclosed in its own set of parentheses. For example, A + B * C should first be understood as A + (B * C) due to precedence, then fully parenthesized as (A + (B * C)).
3. Move Operators: For each innermost set of parentheses (Operand1 Operator Operand2), move the operator to the immediate right of its corresponding right parenthesis. The structure inside becomes (Operand1 Operand2 Operator). Repeat this process for all parenthesized sub-expressions, working outwards from the innermost.
4. Remove All Parentheses: Once all operators have been moved to their new positions, remove all remaining parentheses. The resulting expression will be in postfix notation.

Example for Method 1: Converting A + B * C - D

Let's convert the infix expression A + B * C - D to postfix using this method:

1. Precedence & Associativity Analysis:
   • * has higher precedence than + and -.
   • + and - have equal precedence and are left-associative.
2. Fully Parenthesize:
   • Identify the highest precedence operation: B * C. Parenthesize it: A + (B * C) - D.
   • For A + (B * C) - D, + and - have equal precedence. Due to left associativity, A + (B * C) is evaluated before subtracting D. So, parenthesize A + (B * C): (A + (B * C)) - D.
   • Finally, the entire expression forms one operation: ((A + (B * C)) - D).
3. Move Operators (Innermost to Outermost):
   • Start with (B * C). Move * to the right of its ): (B C *). The expression becomes ((A + (B C *)) - D).
   • Next, consider (A + (B C *)). Move + to the right of its ): (A (B C *) +). The expression becomes ((A B C * +) - D).
   • Finally, for ((A B C * +) - D), move - to the right of its ): (A B C * + D -).
4. Remove Parentheses:
   • The final result is: A B C * + D -

Method 2: The Scan and Process (Mental Stack) Method

This method simulates the logic of an algorithm that uses a stack, and it's generally more efficient for manual conversion, especially for longer expressions.

1. Initialize: Create an empty "output" string/list and an empty "operator stack."
2. Scan from Left to Right: Process each character (token) of the infix expression one by one.
   • Operand: If the token is an operand (a letter, number, or variable), immediately append it to the output string.
   • Left Parenthesis (: Push it onto the operator stack.
   • Right Parenthesis ): Pop operators from the stack and append them to the output until a left parenthesis ( is encountered. Discard both parentheses.
   • Operator: If the token is an operator (+, -, *, /, ^):
     • Pop operators from the stack and append them to the output as long as the stack is not empty, the top of the stack is not a left parenthesis, and the operator at the top of the stack has higher or equal precedence (and is left-associative) compared to the current operator.
     • Once the condition is no longer met, push the current operator onto the stack.
3. End of Expression: After scanning the entire infix expression, pop any remaining operators from the stack and append them to the output.

Example for Method 2: Converting A + B * C - D

Let's use the same example A + B * C - D.

The final postfix expression is A B C * + D -.

Why Postfix?

Postfix notation is beneficial because it eliminates the need for parentheses and simplifies expression evaluation for computers. Expressions in postfix can be evaluated using a simple stack-based algorithm, making it a crucial intermediate step in compilers and interpreters for converting source code into executable instructions. For more details on expression conversion and evaluation, you can explore resources on data structures and algorithms.