Prefix Notation

Overview

Prefix notation, or Polish notation, is a mathematical and logical notation where every operator precedes all of its operands. This contrasts with the more common infix notation (e.g., 2 + 3) and postfix notation (e.g., 2 3 +). The primary advantage of prefix notation is its inherent lack of ambiguity, meaning it does not require parentheses or other grouping symbols to define the order of operations.

Key Concepts

In prefix notation, the structure is straightforward:

• An operator comes first.
• Followed by its required operands.

For example, the infix expression (2 + 3) * 4 becomes * + 2 3 4 in prefix notation.

Deep Dive

Evaluating prefix expressions is typically done using a stack-based approach or recursively. When parsing an expression:

• If a token is an operand, push it onto a stack.
• If a token is an operator, pop the required number of operands from the stack, perform the operation, and push the result back onto the stack.

Consider the expression + 2 * 3 4:

1. Read +.
2. Read 2.
3. Read *.
4. Read 3.
5. Read 4. Since * needs two operands, it takes 3 and 4. Result: 12.
6. Now, + needs two operands. It takes 2 and the result of * (which is 12). Result: 14.

This systematic approach guarantees correct evaluation without any need for parentheses.

Applications

Prefix notation is widely used in computer science, particularly in:

• Compiler design for parsing and evaluating expressions.
• Functional programming languages where it aligns with function application syntax.
• Command-line interfaces and scripting languages.
• Abstract Syntax Trees (ASTs) can be represented in prefix form.

Challenges & Misconceptions

A common misconception is that prefix notation is difficult to read or write. While it may seem unusual at first, its unambiguous nature simplifies parsing and reduces errors compared to infix notation, especially for complex expressions. The lack of parentheses is a feature, not a bug.

FAQs

What is an example of prefix notation?

The infix expression 5 - 1 is represented as - 5 1 in prefix notation.

How is prefix notation evaluated?

It’s typically evaluated using a stack or recursively. Operators are applied to the operands that immediately follow them.