What is Polish Notation?
Polish notation, invented by Polish logician Jan Εukasiewicz in 1924, is a mathematical notation where operators precede their operands. In computer science, this is called prefix notation.
Comparison: Infix vs Prefix
| Infix (Standard) | Prefix (Polish) | Result |
|---|---|---|
5 + 6 |
+ 5 6 |
11 |
3 * 4 |
* 3 4 |
12 |
(2 + 3) * 4 |
* + 2 3 4 |
20 |
2 + 3 * 4 |
+ 2 * 3 4 |
14 |
Key Advantages
1. No Parentheses Needed
The structure is unambiguous without grouping symbols.
Infix:
((5 + 6) * (7 - 2))
Prefix:
* + 5 6 - 7 2
2. No Operator Precedence Rules
You don't need to remember that multiplication comes before addition.
3. Easy to Parse
Left-to-right evaluation with simple recursive descent.
Reading Polish Notation
The key is understanding arity - how many arguments each operator takes:
Expression: + 5 6
Result: 11
Complex Example Walkthrough
Let's evaluate: multiply add 2 3 subtract 10 5
Step-by-Step Evaluation
Initial State
multiply needs 2 arguments, so we evaluate the next two expressions.
Step 1: Evaluate first argument (add 2 3)
add 2 3 β 5
Step 2: Evaluate second argument (subtract 10 5)
subtract 10 5 β 5
Step 3: Final multiplication
multiply 5 5 β 25
Parse Tree Visualization
Polish notation creates a natural tree structure:
Expression: multiply add 2 3 subtract 10 5
Polish Notation in Pangea
Pangea extends basic Polish notation with named operations:
// Instead of: + 5 6
add 5 6
// Instead of: * + 2 3 - 10 5
multiply add 2 3 subtract 10 5
// With variables and control flow
set x 10
if greater x 5
print " x is large "
print " x is small "Why Use Polish Notation?
β Consistency
Every expression follows the same pattern: operation arg1 arg2 ... argN
β Extensibility
Easy to add new operations without worrying about precedence.
β Simplicity
No need for complex parsing rules or ambiguity resolution.
β Natural for Functional Programming
Functions as first-class operators fit naturally.
β Trade-off: Readability
While unambiguous and consistent, Polish notation can be less intuitive for humans accustomed to mathematical infix notation. However, with practice, it becomes natural.
Practice Examples
Try to evaluate these in your head:
1. add multiply 2 3 4
Show answer
multiply 2 3 = 6
add 6 4 = 10
2. multiply add 1 2 add 3 4
Show answer
add 1 2 = 3
add 3 4 = 7
multiply 3 7 = 21
3. subtract multiply 5 5 multiply 3 3
Show answer
multiply 5 5 = 25
multiply 3 3 = 9
subtract 25 9 = 16
π Next Step
Now that you understand Polish notation, learn how Pangea evaluates expressions using recursive descent and phrase length calculation.