- Reverse Polish Notation, or postfix notation, places the operators after the operands, eliminating the need to use parentheses.
- It was derived from the Polish prefix notation created by Jan Łukasiewicz to simplify propositional logic.
- It was massively popularized by Hewlett-Packard in iconic calculators such as the HP-9100A and the HP-35.
- In computing, it is implemented through the use of stacks, being fundamental in systems such as Unix and stack-oriented languages.
You've probably wondered how those old scientific calculators worked inside, or why some programming languages seem to write math backwards. Even though we've been taught since school that addition and subtraction symbols always go in the middle of numbers, there's a fascinating world called Reverse Polish Notation that breaks with everything we know.
This system is not a modern invention; it has its roots in pure logic and the desire to make algebraic expressions more efficient. Unlike what we usually do, here the order of the factors does alter the way they are written, allowing us to completely disregard parentheses and simplify data entry into computer systems.
The origin: From Jan Łukasiewicz to computing
It all started back in 1924 thanks to a brilliant Polish mathematician, logician, and philosopher named Jan Łukasiewicz . He set out to simplify propositional logic and created what is known as prefix notation (or Polish notation), where the operator is placed before the operands. For example, if we want to add 3 and 4, instead of writing 3 + 4, we would write + 3 4.
Although it initially seemed like a curiosity for logicians and philosophers, and was even mentioned by Alonzo Church in his texts on mathematical logic, the idea eventually found its way into the world of computer science. In fact, the LISP programming language bases its syntax precisely on this prefix model, demonstrating that what originated in logic ended up being a powerful tool for coding.
What exactly is Reverse Polish Notation?
While prefix notation puts the operator before the operand, Reverse Polish Notation (also known as RPN or postfix notation) does the opposite: it places the operators after the operands. Thus, our usual sum of 2 and 5 becomes 2 5 +.
The most powerful aspect of this system is that, as long as the operators have a fixed number of operands, the parentheses can be eliminated . Take the operation 5 * (12 + 4) as an example. In the traditional system, we need the parentheses to force the addition first; however, in RPN this is simply written as 5 12 4 + * . This is much cleaner and more straightforward for a machine.
The golden age of Hewlett-Packard and calculators
In the mid-60s, Australian scientist Charles Leonard Hamblin perfected the algorithms for implementing this system in electronic devices. It was then that Hewlett-Packard decided to invest in it, launching the HP-9100A in 1968, a desktop calculator that changed the game. Shortly after, the legendary HP-35 arrived , the first pocket-sized scientific calculator, which put RPN in the hands of thousands of engineers and students between 1972 and 1975.
Using these machines required a little extra mental effort to interpret the result, but it was much more efficient in terms of keystrokes . Instead of typing a long equation and pressing equals, you entered numbers and operators that were processed instantly, making the workflow much more efficient.
Technical implementation: Stacks and pipes
From a programming perspective, RPN is wonderful because it is extremely easy to implement using a data structure called a stack . The system stores numbers on the stack and, as soon as an operator appears, it extracts the last values entered, performs the calculation, and pushes the result back onto the stack.
This efficiency is the reason why, although we no longer see as many physical calculators using this system, RPN remains alive in the inner workings of modern technology. It is widely used in stack-oriented languages and in operating systems based on pipelines or data flow, a classic example being its presence in the Unix environment.
Curiosities about Łukasiewicz and logic
Jan Łukasiewicz not only left us this way of writing math. He was a pioneer in multivalued logic and modal logic. He even tackled intellectual challenges like the liar paradox (the phrase "this is false"). His solution was brilliant: he argued that logical principles only apply to objects that can be values of variables, and since that contradictory phrase cannot be the value of a variable, it is simply outside of logic and cannot be analyzed using traditional logical methods.
This fascinating journey from the mind of a Polish logician to the circuits of HP calculators and the Unix kernel shows us how an alternative way of organizing information can optimize data processing. By eliminating the need for parentheses and relying on the use of stacks, Reverse Polish Notation remains an invisible but fundamental pillar in the architecture of many modern computer systems.