Monadic parser combinators
A Parser combinator, as wikipedia describes it, is a higherorder function that accepts several parsers as input and returns a new parser as its output.
They can be very powerful when you want to build modular parsers and leave them open for further extension. But it can be tricky to get the error reporting right when using a 3^{rd} party combinator library, and they tend to be slower in imperative languages. Nonetheless, it is an interesting cornerstone in functional programming and PLT, so it shouldn't hurt to learn about them by building one on our own.
We're going to start by writing a library that describes several tiny parsers and functions that operate on those parsers. Then, we're going to build some parsers to demonstrate the usefulness of our work.
To give you a small flash forward, here is a parser that accepts Cstyle identifiers, written with the help of our handy combinators:
 matches strings that satisfy [azAZ][azAZ09]+
ident :: Parser String
 One letter or '_', followed by zero of more '_', letters or digits
= alpha_ `thenList` many' (alpha_ <> digit)
ident where
= letter <> char '_' alpha_
It is recommended for you to have some basic understanding of:
 Parsers
 Monads in functional programming
 Haskell
This post is a derivative of two papers I had read recently (1, 2). If you like a denser reading, you can go through the papers instead. The full code for this blog can be found here on GitHub.
The Parser type
Before we begin to define combinators that act on parsers, we must choose a representation for a parser first.
A parser takes a string and produces an output that can be just about anything.
A list parser will produce a list as it's output,
an integer parser will produce Int
s,
a JSON parser might return a custom ADT representing a JSON.
Therefore, it makes sense to make Parser a polymorphic type. It also makes sense to return a list of results instead of a single result since grammar can be ambiguous, and there may be several ways to parse the same input string.
An empty list, then, implies the parser failed to parse the provided input. (1)
newtype Parser a = Parser { parse :: String > [(a, String)] }
You might wonder why we return the tuple (a, String)
, not just a
.
Well, a parser might not be able to parse the entire input string.
Often, a parser is only intended to parse some prefix of the input, and let another parser do the rest of the parsing.
Thus, we return a pair containing the parse result a
and the unconsumed string subsequent parsers can use.
We could have used the type
keyword let Parser
be an alias for String > [(a, String)]
,
but having a unique data type lends us the ability to instantiate it as a typeclass,
which is something we'll do later on.
Baby parsers
We can start by describing some basic parsers that do very little work.
A result
parser always succeeds in parsing without consuming the input string.
result :: a > Parser a
= Parser $ \inp > [(val, inp)] result val
The parser zero
will always fail by returning an empty list.
zero :: Parser a
= Parser $ const [] zero
item
unconditionally accepts the first character of any input string.
item :: Parser Char
= Parser parseItem
item where
= []
parseItem [] :xs) = [(x, xs)] parseItem (x
Let's try some of these parsers in GHCi:
*Main> parse (result 42) "abc"
42, "abc")]
[(*Main> parse item "abc"
'a', "bc")] [(
Building parsers on demand
The basic parsers we defined above are of very little use.
Ideally, we would want parsers that accept input strings that satisfy certain constraints.
Say we want a parser that consumes a string if its first character satisfies a predicate.
We can generalize this idea by writing a function that takes a (Char > Bool)
predicate and returns a parser that only consumes an input string if its first character returns True
when supplied to the predicate.
The simplest solution for this would be:
sat :: (Char > Bool) > Parser Char
= Parser parseIfSat
sat p where
: xs) = if p x then [(x, xs)] else []
parseIfSat (x = [] parseIfSat []
However, since we already have an item
parser that unconditionally extracts
the first character from a string, we could use this as an opportunity to create a basic parser combinator.
Before writing a combinator, we must first instantiate Parser
as a Monad
. (2)
instance Monad Parser where
>>= f = Parser $ \inp >
p concat [parse (f v) inp'  (v, inp') < parse p inp]
 a > Parser a
return = result
The bind
operation takes a Parser a
(p) and a function a > Parser b
(f), and returns a Parser b
.
The idea is to apply p
, if it fails, then we have an empty list which results in concat [[]]
= []
.
If p
successfully parses inp
into one or more possible parse results,
we apply f
to each of the results to get corresponding Parser b
s and then apply those to the rest of the input.
With this new extension, our sat
parser can be rewritten as:
=
sat p  Apply `item`, if it fails on an empty string, we simply short circuit and get `[]`.
>>= \x >
item if p x
then result x
else zero
Now we can use the sat
combinator to describe several useful parsers.
For example, a char
parser that only consumes a string beginning with a specific character.
char :: Char > Parser Char
= sat (== x) char x
*Main> parse (char 'a') "abc"
'a', "bc")] [(
A parser for ASCII digits:
 import Data.Char (isDigit, isLower, isUpper)
digit :: Parser Char
= sat isDigit digit
And similarly small but useful parsers:
lower :: Parser Char
= sat isLower
lower
upper :: Parser Char
= sat isUpper upper
$ ghci i main.hs
*Main> parse lower "aQuickBrownFox"
'a',"QuickBrownFox")] [(
Now that we have upper
, lower
and digit
this opens up new possibilities for combinations:
 An
alphabet
parser that accepts a char that is consumable by eitherupper
orlower
.  An
alphanumeric
parser that accepts a char, eitheralphabet
ordigit
.
Clearly, an or'
combinator that captures this recurring pattern will come in handy.
Let us begin by describing a plus
combinator that concatenates the result returned by two parsers:
 Applies two parsers to the same input, then returns a list
 containing results returned by both of them.
plus :: Parser a > Parser a > Parser a
`plus` q = Parser $ \inp > parse p inp ++ parse q inp p
Haskell has a MonadPlus typeclass defined in the prelude like so:
class (Monad m) => MonadPlus m where
mzero :: m a
mplus :: m a > m a > m a
mzero
represents failure, and mplus
represents combination of two monads.
Since Parser
is already a monad, we can instantiate the MonadPlus
typeclass to enforce
this idea:
 Add this to the list of imports:
 import Control.Monad (MonadPlus (..))
instance MonadPlus Parser where
= zero
mzero = plus mplus
The or'
combinator can then be:
or' :: Parser a > Parser a > Parser a
`or'` q = Parser $ \inp > case parse (p `plus` q) inp of
p > []
[] :xs) > [x] (x
In fact, the Alternative
typeclass already defines this functionality with the choice (<>
) operator:
instance Alternative Parser where
= zero
empty <>) = or' (
Finally, we can return to the letter
and alphanum
parsers:
letter :: Parser Char
= lower <> upper
letter
alphanum :: Parser Char
= letter <> digit alphanum
We can now take them for a spin in GHCi:
*Main> parse letter "p0p3y3"
'p',"0p3y3")]
[(*Main> parse letter "30p3y3"
[]*Main> parse alphanum "foobar"
'f',"oobar")] [(
As an aside, we can use the
sequencing (>>) operator to write more concise code at times.
Consider the function string
for example, where string "foo"
returns a parser that only accepts strings which begin with "foo".
string :: String > Parser String
"" = result ""
string :xs) =
string (x>> string xs >> result (x:xs) char x
Using >>=
notation, we would have had to write:
:xs) =
string (x
char x>>= const string xs  same as \_ > string xs
>>= const result (x:xs)  same as \_ > result (x:xs)
*Main> parse (string "prefix") "prefixxxxx"
"prefix", "xxxx")] [(
Using the do notation
Haskell provides a handy do notation for readably sequencing monadic computations. This is useful when composing monadic actions becomes a bit gnarly looking. Consider this example that composes the outputs of several parsers:
= parser1 >>= \x1 >  1. apply parser1
parser >>= \x2 >  2. use parser1's output to make parser2
make_parser2 x1 >>= \x3 >  3. Use parser2's output to make parser3
make_parser3 x2 return (f x1 x2 x3)  4. Combine all parse results to form the final result
Using the do
notation, the above code snippet becomes:
= do
parser < parser1
x1 < make_parser2 x1
x2 < make_parser3 x2
x3 return (f x1 x2 x3)
Moving forward, we will prefer the do
notation over >>=
wherever it improves readability.
Combinators for repetition
You may be familiar with the regex matchers +
and *
.
a*
matches zero or more occurrences of the letter 'a', whereas a+
expects one or more occurrences of the letter 'a'.
We can represent the *
matcher as a combinator like so:
many' :: Parser a > Parser [a]
= do
many' p < p  apply p once
x < many' p  recursively apply `p` as many times as possible
xs return (x:xs)
Looks decent, but when run in GHCi, it fails to produce the expected result:
*Main> parse (many' $ char 'x') "xx"
[]
If you try to work out the application of this parser by hand, you'll notice a flaw in our base case:
In the final recursive call, when the input string is ""
, x < p
fails, and we short circuit to return []
.
To handle this scenario, we can use our or'
combinator:
many' :: Parser a > Parser [a]
=
many' p do
< p  apply p once
x < many' p  recursively apply `p` as many times as possible
xs return (x : xs)
<> return []
 In case `p` fails either in the initial call, or in one of the
 recursive calls to itself, we return an empty list as the parse result.
And we're golden:
*Main> parse (many' $ char 'x') "xxx123"
"xxx","123")] [(
If the use of <>
is still confusing to you, try working it out on paper.
Analogous to the regex +
matcher, we can write a many1
combinator that accepts one or more occurrences of an input sequence.
Piggybacking off of many'
, this can be simply written as:
many1 :: Parser a > Parser [a]
= do
many1 p < p
x < many' p
xs return (x:xs)
Parsing a list of identifiers
If you haven't realized by now, we've built some combinators capable of parsing regular languages. Circling back to the beginning of this post, here is a combinator that parses a valid Cstyle identifier:
ident :: Parser String
= do
ident < alpha_
x < many' (alpha_ <> digit)
xs return (x : xs)
where
= letter <> char '_' alpha_
*Main> parse ident "hello_123_ = 5"
"hello_123_"," = 5")] [(
To make it even more concise, we can define a then'
combinator which combines
the result produced by two parsers using a caller provided function.
then' :: (a > b > c) > Parser a > Parser b > Parser c
=
then' combine p q >>= \x >
p >>= \xs >
q $ combine x xs result
A thenList
combinator can then combine to parse results of type a
and [a]
using (:)
.
thenList :: Parser a > Parser[a] > Parser[a]
= then' (:) thenList
Now our identifier parser becomes even shorter:
ident :: Parser String
= alpha_ `thenList` many' (alpha_ <> digit)
ident where
= letter <> char '_' alpha_
Now, lets take our combinations a step further. Say we want to parse a list of commaseparated identifiers, Here is one way to do that:
idList :: Parser [String]
= do
idList < ident
firstId < many' (char ',' >> ident)
restIds return (firstId : restIds)
A tokenseparated list of items is a commonly occurring pattern in language grammar.
As such, we can abstract away this idea with a sepBy
combinator:
 Accept a list of sequences forming an `a`, separated by sequences forming a `b`.
sepBy :: Parser a > Parser b > Parser [a]
`sepBy` sep = do
p < p
x < many' (sep >> p)
xs return (x : xs)
= ident `sepBy` char ',' idList
Now, what if the list of identifiers was enclosed in braces like in an array?
We can define another combinator, bracket
, to parse strings enclosed within specific sequences.
bracket :: Parser a > Parser b > Parser c > Parser b
= do
bracket open p close < open
_ < p
x < close
_ return x
Sequencing operators can be used to write bracket
in a slightly more elegant manner:
= open >> p <* close bracket open p close
Using this, our parser for a list of items can be written as:
= bracket (char '[') ids (char ']')
idList where
= ident `sepBy` char ',' ids
Let's test this implementation in GHCi:
*Main> parse idList "[foo,bar,baz]" [(["foo","bar","baz"],"")]
Perfect!
Parsing natural numbers
Since our parsers are polymorphic, we can return a parse result containing an input string's evaluated value. Here is a parser that consumes and evaluates the value of a natural number:
 Add this to the list of imports:
 import Data.Text.Internal.Read (digitToInt)
nat :: Parser Int
=
nat >>= eval
many1 digit where
= result $ foldl1 op [digitToInt x  x < xs]
eval xs `op` n = 10 * m + n m
A natural number is one or more decimal digits, which we then fold to produce a base 10 value.
Alternatively, we can use the builtin read
to implement nat
:
= read <$> many1 digit nat
Handling whitespace
As you may already have noticed, the parsers we've written so far aren't great at dealing with whitespace.
*Main> parse idList "[a, b, c]"
[]
Ideally, we should ignore any whitespace before or after tokens. Generally, it is a tokenizer's job to handle whitespaces and return a list of tokens that the parser can then use. However, it is possible to skip a tokenizer completely when using combinators.
We can define a token
combinator that takes care of all trailing whitespace:
 Add 'void' and 'isSpace' to import lists.
 import Control.Monad (MonadPlus (..), void)
 import Data.Char (isDigit, isLower, isUpper, isSpace)
spaces :: Parser ()
= void $ many' $ sat isSpace
spaces
token :: Parser a > Parser a
= p <* spaces token p
And a parse'
combinator that removes all leading whitespace:
parse' :: Parser a > Parser a
= spaces >> p parse' p
The parse'
combinator is applied to the final parser once to ensure there is no leading whitespace.
The token
combinator consumes all trailing whitespace,
hence ensuring there is no leading whitespace left for the subsequent parsers.
We can now write parsers that disregard whitespace:
identifier :: Parser String
= token ident identifier
At this point, we have atomic parsers that can be plugged in several places. One such place can be an arithmetic expression evaluator:
An expression parser
Finally, to demonstrate the usefulness of combinators we have defined so far, we build a basic arithmetic expression parser.
We will support the binary +
and 
operators, parenthesized expressions and integer literals.
By the end, we will have an eval
function that can evaluate expressions like so:
*Main> eval "1 + 2  3  4 + 10"
6
 consume a character and discard all trailing whitespace
charToken :: Char > Parser Char
= token <$> char
charToken
 an ADT representing a parse tree for expressions
data Expr = Add Expr Expr
 Sub Expr Expr
 Par Expr
 Lit Int
deriving (Show)
eval' :: Expr > Int
Add a b) = eval' a + eval' b
eval' (Sub a b) = eval' a  eval' b
eval' (Par a) = eval' a
eval' (Lit a) = a
eval' (
eval :: String > Int
= fst . Bifunctor.first eval' . head <$> parse expr
eval
 Our expression parser expects a string of the following grammar:
 expr ::= term (op term)*
 op ::= '+'  ''
 term ::= int  '(' expr ')'
 int ::= [09]*
 The `expr` parser first consumes an atomic term  <X>, then it
 consumes a series of "<op> <operand>"s and packs them into tuples like ((+), 2)
 We then fold the list of tuples using <X> as the initial value to produce the result.
expr :: Parser Expr
= do
expr < term
x < many' parseRest
rest return $
foldl (\x (op, y) > x `op` y) x rest
where
= do
parseRest < op
f < term
y return (f, y)
 term := int  parens
term :: Parser Expr
= int <> parens
term
 parens := '(' expr ')'
parens :: Parser Expr
= bracket (char '(') expr (char ')')
parens
 int := [09]*
int :: Parser Expr
= Lit <$> token nat
int
 op := '+'  ''
op :: Parser (Expr > Expr > Expr)
= makeOp '+' Add <> makeOp '' Sub
op where makeOp x f = charToken x >> return f
Spin up GHCi, and there we have it:
*Main> eval "1 + 2  (3  1)"
3
*Main> eval "1 + 2 + 3"
6
Our parser is decent, but it can be refactored a little further.
An expression is a list of parenthesized expressions and integer literals separated by +
or 
.
As it turns out, parsing a list of token delimited items is a common pattern captured by the chainl1
combinator:
chainl1 :: Parser a > Parser (a > a > a) > Parser a
`chainl1` op = do
p < p
first < many' $ do
rest < op
f < p
term return (f, term)
return $ foldl (\x (f, y) > f x y) first rest
expr :: Parser Expr
= term `chainl1` op expr
And with that, we have a monadic expression parser composed of several tiny and modular parsers. As an exercise, you can extend this parser and add more operators such as multiplication, division, and log.
Further reading
There are already several parser combinator libraries for many languages, as you may have guessed. Parsec in particular, is the most commonly used one among Haskell programmers. Here are more resources for you to chew on:
 Monadic Parser Combinators
 Functional Pearls  Monadic Parsing in Haskell
 Microsoft Research  Direct style monadic parser combinators for the real world
The first two should feel very familiar if you've followed the post so far. The 3rd is a paper that attempts to provide a better alternative technique for parsing using monads.
At this point, parser combinators have become another tool in your functional programming arsenal. Go forth and write some killer parsers!
Backmatter

In most implementations, the parse result is a functor that can store an error message in case the parser fails.
newtype Parser a = Parser { parse :: String > ParseResult a } type ParesResult a = Either ParseError a type ParseError = String

In order to instantiate Parser as a Monad in Haskell, we also have to make it an instance of
Functor
andApplicative
:
instance Functor Parser where
fmap f p = Parser (fmap (Bifunctor.first f) . parse p)
instance Applicative Parser where
pure = result
<*> p2 = Parser $ \inp > do
p1 < parse p1 inp
(f, inp') < parse p2 inp'
(a, inp'') return (f a, inp'')