Understanding Parenthesis Tokenization in Haskell

Context

Before building a full arithmetic calculator in Haskell, there was one key challenge I needed to solve first: understanding and tokenizing parentheses.

Parenthesis tokenization may sound like a small detail, but it’s actually a necessary step for building my own calculator interpreter — the engine that later evaluates nested expressions like (3+(2*(5-1))) correctly, respecting operator precedence and nesting depth.

1. Why tokenization matters

In an arithmetic expression, parentheses determine the order in which subexpressions are evaluated. To process them properly, a calculator must:

• Recognize where every ( and ) occurs
• Track how deeply nested each one is
• Know which pairs of parentheses match together

Once we have this information, we can safely evaluate the innermost expressions first and work our way out — exactly how human math logic works.

2. The main entry point

To start, we define parserPar, a simple wrapper that calls the recursive worker function. It returns two lists:

• ids — positions of each parenthesis
• nums — the nesting depth at that position

parserPar :: [Char] -> ([Int], [Int])
parserPar xs = subParserPar xs [] [] [] 0 0

This setup initializes everything: the list of indices, the nesting counters, and starts recursion with both index counters (n and n2) at zero.

3. The recursive core: subParserPar

Here’s the heart of the tokenizer — the recursive system that traverses the expression and records every parenthesis and its depth:

subParserPar :: [Char] -> [Int] -> [Int] -> [Int] -> Int -> Int -> ([Int], [Int])
subParserPar [] ids nums _ _ _ = (ids, nums)
subParserPar (x:xs) ids nums valxs n n2
    | x == '(' = 
        let newids = ids ++ [n]
            newnums = nums ++ [n2]
            newvalxs = map (+1) valxs
            newvalxs2 = newvalxs ++ [1]
        in subParserPar xs newids newnums newvalxs2 (n + 1) (n2 + 1)
    | x == ')' = 
        let newvalxs = map (\x -> x - 1) valxs 
            idx = findFirstZero (reverse newvalxs) 0
            idx2 = (length valxs) - idx - 1
            newids = ids ++ [n]
            newnums = nums ++ [(nums !! idx2)]
        in subParserPar xs newids newnums (newvalxs ++ [0]) (n + 1) n2
    | otherwise = subParserPar xs ids nums valxs (n + 1) n2

Let’s decode what happens step by step:

• n — current character index in the string.
• n2 — tracks how many nested levels we’ve entered so far.
• valxs — a stack-like list representing active parentheses’ depth states.

Whenever we hit:

• An opening parenthesis ( — we add its index to ids, record the current depth in nums, and increment all ongoing nesting levels.
• A closing parenthesis ) — we decrement nesting levels, find the matching opening index using findFirstZero, and store the corresponding depth.

This creates a perfect “map” of how parentheses are nested throughout the expression.

4. Matching parentheses with findFirstZero

The helper findFirstZero is tiny but crucial — it looks backward in the list of nesting levels to locate which ( matches a given ):

findFirstZero :: [Int] -> Int -> Int
findFirstZero (xi:xsi) n
    | xi == 0 = n
    | otherwise = findFirstZero xsi (n + 1)

By scanning from the end of the nesting state, it identifies the first “zeroed” value — meaning the most recently closed parenthesis — and returns its position. This gives us a clear mapping between opening and closing brackets.

5. Why this system was essential for the calculator

Without this parenthesis tokenization system, the calculator interpreter couldn’t have existed. It’s the foundation that allows nested subexpressions to be extracted and evaluated in the correct order. Once I had (ids, nums), I could:

• Identify the deepest expression to evaluate first
• Replace it in the string with its computed result
• Re-parse and repeat until no parentheses remain

In short, this parser is the brain that makes the whole calculator interpreter work — a recursive mechanism that gives structure and hierarchy to raw characters.

6. What this teaches about Haskell

• Recursion replaces loops — the entire traversal is recursive and state-free.
• Pattern matching allows simple, readable branching for (, ), and other cases.
• Purity ensures no mutable state — every step returns new lists.

Writing this tokenizer was an essential milestone. Once the parentheses could be correctly identified and paired, everything else — arithmetic evaluation, precedence, and final interpretation — naturally built upon it.

In short: Parenthesis tokenization wasn’t just a side utility — it was the first real piece of the calculator’s interpreter logic.