Introduction

Reverse Polish Notation (RPN), also known as postfix notation, is a mathematical notation where every operator follows all of its operands. It is distinct from the common "infix" notation, where operators typically are written between operands. RPN eliminates the need for parentheses that are required to denote operation precedence in traditional infix expressions. This concise method is beneficial in stack-based computations, notably seen in certain calculators and computer programming languages.

Basic Principles of RPN

In RPN, operators are written after their operands. Given an expression in infix notation, `3 + 4`, its RPN equivalent would be `3 4 +`. Here, `+` is the operator applied to `3` and `4`, its operands.

Here's a breakdown of how to evaluate an RPN expression:

1. Read from left to right.
2. Push operands on a stack.
3. Upon encountering an operator, pop the required number of operands from the stack, perform the operation, and push the result back onto the stack.
4. Once the expression is fully read, the value remaining on the stack is the result.

Variable-Length Operators in RPN

In traditional infix notation, operators usually have a fixed number of operands they work with (binary operators like `+`, `-`). However, RPN can incorporate variable-length operators, allowing for operations on a flexible number of operands.

Example of Variable-Length Operators

Consider an operation like `SUM`, which takes a variable number of operands and sums them. In an infix system, this typically requires auxiliary structures like lists or loops. In RPN, using `SUM`, you can directly state:

• Operands: `3 4 5 2`
• RPN Expression: `3 4 5 2 SUM`

In this scenario, `SUM` would pop all operands from the stack until it recognizes no more operands, summing them in the process. Let us see how an RPN processor could handle this:

• Initial Stack: `[]`
• After reading operands: `[3, 4, 5, 2]`
• Apply `SUM`: Pop every element to compute sum; resulting stack: `[14]`

Technical Explanation

Stack-Based Evaluation

The stack nature of RPN excels in managing variable-length operators due to its Last In, First Out (LIFO) structure. Here's a low-level breakdown of managing such operations:

• Compute `SUM` first: `2 + 3 + 4 + 5 = 14`
• After applying `SUM`: `[14]`
• With final operation `* 6`: `14 * 6 = 84`