To enable compound expressions to be written without parentheses, the Polish mathematician Jan Lukasiewicz designed a notation that lists the operators after the numbers and variables (and not in between). The notation is called Reverse Polish Notation1.

Instead of 3 + 4, one writes 3 4 +. This allows compound expressions to be written without parentheses and interpreted unambiguously. Another example: For (3+4)∗(6-2) one writes 3 4 + 6 2 - ∗. So first 3+4 is calculated, then 6 - 2 and finally the two factors are multiplied together. The result of the calculation is 28.

You can imagine that the numbers are written one after the other into the cellar or onto the stack. If an operator is read in (or comes as input), the two uppermost numbers are taken from the stack, calculated with each other and the result is put back on the stack.

Note: Z is a number, O is an operator

flowchart LR
    START[ ] -- Start --> Q0((QO))
    Q0 -- Z,$/$ --> Q1(((Q1)))
    Q1 -- Z,$/Z$ --> Q2((Q2))
    Q2 -- Z,Z/ZZ<br/>O,Z/ε --> Q2
    Q2 -- ε,$/$ --> Q1

    style START fill:#FFFFFF00, stroke:#FFFFFF00;  

Note: Z is a number, O is an operator

flowchart LR
    START[ ] -- Start --> Q0(((Q0)))
    Q0 -- Z,$/#Z$ --> Q0
    Q0 -- Z,#/Z# --> Q1((Q1))
    Q1 -- Z,Z/ZZ --> Q1
    Q1 -- O,Z/ε --> Q2((Q2))
    Q2 -- ε,Z/Z --> Q1
    Q2 -- ε,#/ε --> Q3
    Q3((Q3)) -- ε,Z/#Z --> Q0

    style START fill:#FFFFFF00, stroke:#FFFFFF00;