## Concept:

# Reverse_Polish_notation

## Headline

A mathematical notation where operators follow their operands

## Illustration

Consider the following expression using regular infix operators:

```
5 + ((1 + 2) * 4) − 3
```

In RPN:

```
5 1 2 + 4 * + 3 -
```

RPN can be evaluated by means of a simple iterative, stack-based algorithm which handles one input item (operator or operand) in each iteration. An operand is pushed on the stack. An operator is immediately applied to the two top-most stack entries. When the (well-formed) input has been consumed, then the result of expression evaluation is the one and only entry on the stack.

The input "5 1 2 + 4 * + 3 -" is processed as follows:

- Push 5 on the stack.
- Push 1 on the stack.
- Push 2 on the stack.
- Apply "+" to second top of stack (1) and top of stack (2); pop twice, push 3.
- Push 4 on the stack.
- Apply "*" to second top of stack (3) and top of stack (4); pop twice, push 12.
- Apply "*" to second top of stack (5) and top of stack (12); pop twice, push 17.
- Push 3 on the stack.
- Apply "-" to second top of stack (17) and top of stack (3); pop twice, push 14.

```
module RPN where
import Stack
-- RPN as lists of operators and operands
type RPN = [Entry]
data Entry = Operator Operator | Operand Operand
deriving (Show)
data Operator = Plus | Minus | Times
deriving (Show)
type Operand = Int
-- Evaluation of RPN via stack
eval :: RPN -> Int
eval = loop empty
where
-- Loop over input
loop :: Stack -> RPN -> Int
loop s i =
if null i
then if size s == 1
then top s
else rpnError
else
loop (step (head i) s) (tail i)
-- Process head of input
step :: Entry -> Stack -> Stack
step h s =
case h of
Operand x -> push x s
Operator f ->
if size s >= 2
then
let (x,s') = (top s, pop s) in
let (y,s'') = (top s', pop s') in
push (apply f y x) s''
else rpnError
-- Interpretation of operator symbols
apply :: Operator -> Int -> Int -> Int
apply Plus = (+)
apply Minus = (-)
apply Times = (*)
-- Reusable error
rpnError :: a
rpnError = error "RPN corrupted"
```

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