Sunday, August 10, 2008

RJson update

I've been meaning to write a little more about RJson for a while. The library has come quite a long way since the post describing the original version. One feature suggested to me by Jeremy Shaw, which should go in the next release, is automatic boxing and unboxing of types with single unary constructors. For example, suppose we have the following type:

data Counter = Counter Int deriving Show

We derive instances of ‘ToJson’ and ‘FromJson’ for this type using the following template Haskell code:


Now, if we don't define custom ‘ToJson’ or ‘FromJson’ instances, the default behavior is this:

toJsonString (Counter 7)
--> "7"

fromJsonString (undefined :: Counter) "7"
--> Counter 7

This is a big improvement over the default behavior in older versions, which would have converted (Counter 7) to a single-element JSON array.

Thursday, April 24, 2008

Pattern matching and generic functions

I've been working some more on Shlisp. The interpreter is becoming reasonably complete in terms of implementing the core language functionality. It now uses Boehm GC to implement garbage collection, and setjmp, longjmp and stack copying to implement continuations (effectively by adding continuations to C++ itself, using the technique described here).

Now that the interpreter is fairly reliable, I've been writing some actual Shlisp code. This post is about the pattern-matching and generic function features. Shlisp makes use of the same pattern matcher for three purposes: ordinary pattern matching, lambda argument lists and generic function dispatch. Although this means that the pattern-matching syntax isn't perfectly optimized for any one of these domains, I think the reduction in complexity is adequate compensation. The basic pattern matcher does more or less what you would expect. Here's an example of destructuring a sequence:

(match '(1 2 3 4)
(() ())
((x y | zs) zs))

=> (3 4)

The ‘|’ is used to introduce a pattern which must match each element of the rest of a sequence. This pattern can itself be complex, so it is easy to define a function which accepts (say) a single argument followed by a list of pairs:

(defun foo args
(match args
((x | (fst snd)) (some-more-code ...))))

By convention, pattern-matching derivatives of definition macros have a colon appended to their names. For example the previous function can be defined more concisely as follows:

(defun: foo
((x | (fst snd)) (some-more-code ...)))

The ‘match’ macro returns no value if none of the specified patterns match, but an error will be raised if none of the patterns for a ‘defun:’ match. The following syntax is used to match on the type of a value:

(defun: as-integer
(((x : integer) x))
(((x : double) (floor x))))

This syntax extends to the definition of simple methods for generic functions. A generic function is defined like this:

(defgeneric generic-add)

And methods are added like this:

(defmethod generic-add ((x : number) (y : number))
(+ x y))
(defmethod generic-add ((x : string) (y : string))
(concat x y))

The ‘number’ type is a supertype of all numeric types. As befits a generic function system, we can add an additional method for a subtype of the ‘number’ type:

(defmethod generic-add ((x : integer) (y : integer))
(+ 10 x y)) ; Silly (but different) behavior.

In the case of generic functions with overlapping argument types, the dispatch algorithm selects the method with the most specific argument list, going from left to right along the arguments. The ‘defmethod’ form is only able to dispatch on fixed numbers of arguments, but there is also a ‘defmethod:’ form which integrates pattern matching with generic dispatch. This form accepts an arbitrary pattern, but still ensures that methods with overlapping argument types are dispatched appropriately. In the case of ‘generic-add’, we can allow each method to receive any number of arguments:

(defmethod: generic-add ((| (xs : number)))
(apply + xs))

As I mentioned in the previous post, it is possible to add generic dispatch to a builtin, non-generic function. For example, if we wanted to extend ‘+’ so that it could be used to concatenate symbols, the following code would do the trick:

(add-generic +)

(defmethod: + ((| (xs : symbol)))
(apply concat xs))

The generic dispatch mechanism is activated only if the builtin addition operation receives bad arguments, so the performance of ordinary arithmetic operations needn't suffer too much.

Thursday, March 27, 2008


I've been thinking recently about how to go about writing a simple self-hosting lisp compiler targeting LLVM. The basic plan of action is this:

  • Write a quick and dirty interpreter for the language (“shlisp”).

  • Write a compiler for shlisp in shlisp itself.

  • Use the interpreter to bootstrap the compiler.

So far, I've made good progress in writing an interpreter in C++ for shlisp. It differs in some interesting respects from Common Lisp and Scheme:

No cons

The basic data structure is the ‘sequence’, which is supposed to be implemented as some sort of general-purpose extensible array, of the kind you find in Perl, Python, etc. Having linked lists as a default data structure is arguably a bad idea. Lisp code is littered with explicit manipulation of lists, which makes it hard to reuse list-manipulating code on other kinds of sequences. Furthermore, the use of cons pairs to create lists allows for “improper lists,” creating all sorts of complications regarding which functions will and will not accept improper lists. Eventually, I want to implement some sort of STL-like generic sequence library. For now, I've made the interface to sequences relatively neutral w.r.t. their underlying implementation (most of the sequence-manipulating functions could be efficiently implemented for linked lists or other sequential structures). I have found that it is very easy to work with sequences given a decent library of utility functions — possibly even easier than with linked lists. The one exception is that it is not possible to construct sequences recursively using recursive function calls. However, most code of this sort can be written just as easily and declaratively using unfold-style combinators. For example, here's the code for the ‘range’ function, which generates a list of consecutive integers in a given range:

(defun range (start end)
(\(x stop yield)
(if (>= x end)
(yield (+ 1 x))))

That's just as concise as the directly recursive version. The ‘generate’ combinator has a simple definition (though the lack of tail-call elimination in the interpreter makes this version hugely inefficient):

(defun generate (f start)
(let ((accum (seq start)))
(letf loop ()
(last accum)
(\() accum)
(push accum x)

Shlisp uses ‘\’ instead of ‘lambda’. ‘letf’ is the equivalent of Scheme's named let.

A final advantage of sequences over cons-based lists is that many natural imperative operations (e.g. pushing an item onto a sequence) can be implemented as functions instead of macros.

No distinction between strings and symbols

There doesn't seem to be much point in making this distinction these days (and indeed, there are many historic lisps that didn't make it). Interning strings at runtime isn't a sufficiently useful operation to be built into the language, and constant strings can always be interned by an optimizing compiler.

No boolean values

Zero and the empty sequence are considered false; everything else is true. I don't see much point in having boolean values in a dynamically typed language. In Scheme, the main use for #t and #f in practice seemed to be to provide return values for exceptional cases (e.g. a function that normally returned an integer might return #f on failure). I prefer to pass success and failure continuations to the function in these cases. For example:

(parse-integer "123"
(\() (write "Parsing failed."))
(\(n) (write "Success!")))

Another benefit to eliminating #t and #f is that (come the compiler) there is more room for tagged representations of other simple types.

A pragmatic approach to macro hygiene

Shlisp is a lisp-1 (i.e. functions and variables share the same namespace), so there is potentially an issue with unhygienic macros. Hygienic macro systems are nice, but I don't feel up to implementing one at the moment. Instead, I've added the ‘global’ special form for explicitly accessing symbols in the global environment. The following expressions illustrate how it is used:

(let ((+ -)) (+ 1 2)) => -1
(let ((+ -)) ((global +) 1 2)) => 3

‘global’ works for functions and macros, but not special forms. The upshot of all this is that well-written macros can be guaranteed to work so long as none of the special forms they use have been shadowed by other bindings. This is an improvement over Common Lisp, where well-written macros can only be guaranteed to work so long as none of the special forms or functions or variables they use have been shadowed. Shlisp is lexically scoped, so I do not think there should be any problem implementing ‘global’ efficiently in a compiler.

Very basic support for generic functions

I wanted to find some sort of middle ground between Scheme (no support for generic functions) and Common Lisp (very elaborate support). Although I'd like to add a proper generic function system at some point, I don't want to have to have it built into the language (the amount of C++ code needs to be kept to a minimum). Currently, I've settled on the following solution. Any function may raise a BADARGS error if given an argument list which it can't handle. Functions also have a delegates property, which is a list of functions. If a BADARGS error is raised, the argument list is passed to each of the delegate functions in turn, until one of the functions executes without raising BADARGS. This alone is enough to implement simple generic functions. I expect that in the future, the delegates list for a generic function will contain a single delegate that implements a more sophisticated generic dispatch algorithm.

...and that's it

The interpreter's currently about 2500 lines of C++. I'm hoping that once all of the features I want have been implemented it won't go over 4000 or so. With luck, it won't be necessary to write a garbage collector (since as long as the interpreter can bootstrap the compiler without running out of memory, there's no need).

Tuesday, January 29, 2008

Deserializing JSON

I've improved the code from the last post so that it is now capable of deserializing as well as serializing. The improved library is on hackage. I'll post some example code soon.

Thursday, December 27, 2007

Pushing Haskell's type system to the limits: A reflective JSON serializer

Since it looks like Haskell web development might start to take off soon, I've been thinking about how to write a good JSON serializer library. The obvious way to start is to define a simple type to represent JSON data structures:

import Data.Ratio
import qualified Data.Map as M

data JsonData = JDString String |
JDNumber Rational |
JDList [JsonData] |
JDObject (M.Map String JsonData)

instance Show JsonData where

It's easy enough to convert other data structures to the JsonData type, but serializing more complex types takes a lot of boilerplate code. So, maybe the the solution is to create a new class containing a ‘toJson’ function:

class Json a where
toJson :: a -> JsonData

instance Json Integer where
toJson i = show i

instance Json [a] where
toJson [] = []
toJson l@(x:xs) = "[" ++ (join "," l) ++ "]"
where join _ [] = []
join _ l@[x] = l
join k (x:ys) = x : k : (join k ys)

Once instances have been added for other basic datatypes, this works nicely enough. However, there are two significant problems. First, since Haskell's String type is just a synonym for [Char], there is no way to handle Strings any differently from normal lists. Second, if you use this approach for serializing simple record types, you end up with a lot of boilerplate instance declarations like the following:

import qualified Data.Map as M

data MyRecord1 = MyRecord1 { _x :: Int,
_y :: Int,
_z :: Int }
data MyRecord2 = MyRecord2 { _rec :: MyRecord1,
_a :: Int,
_b :: Int,
_c :: Int }

instance Json MyRecord1 where
toJson r = JDObject (M.fromList [("x", _x r),
("y", _y r),
("z", _z r)])

instance Json MyRecord2 where
toJson r = JDObject (M.fromList [
("rec", toJson (_rec r)),
("a", _a r),
("b", _b r),
("c", _c r)

Ideally, we would like to be able to give some sort of default procedure for serializing a record, so that unless we specify otherwise, the record will be converted into a JSON object with appropriate field names. The default should work correctly for nested records, and using it should require at most a single line of code per record type. This ideal can certainly be achieved in a dynamic, reflective language such as Common Lisp, Python or Ruby, but it is absolutely out of the question in Haskell 98. Surprisingly, with the addition of a number of Generic programming extensions to GHC, it is now possible to implement a generic JSON serializer for Haskell too. As you might expect, this involves some pretty hairy manipulation of the type system, so I'd like to set the bait before getting into the details. Here goes. Once we're done, the following code works:

import Json -- Generic Json serialization library (to be written)

data OuterRecord { x :: Int,
y :: String,
r :: InnerRecord }

data InnerRecord { z :: Int }


> toJson (OuterRecord { x=1,
r=(InnerRecord { z=7 })})

=> {"x":1,"y":"foo","r":{"z":7}} :: JsonData

If you have the luxury of designing your record types with JSON in mind, you just need to add one such line of code for every type. If you do need to customize serialization of a particular type, it's just a matter of creating an instance:

data CustomRecord { dontSerializeMe :: Int,
serializeMe :: Int }

instance Json CustomRecord where
toJson r = show (serializeMe r)

> toJson (CustomRecord { dontSerializeMe=1,
serializeMe=2 })

=> 2 :: JsonData

(In fact, the finished implementation of the serializer provides an easier way of excluding fields from serialization.) The code uses the Scrap Your Boilerplate approach to generics. Although GHC comes with the Data.Generics module, which implements the system described in the first SYB paper, my code is based on the third SYB paper (Scrap Your Boilerplate With Class). For this reason, you will need to install the syb-with-class package from Hackage in order to compile it. The current version (0.3) does not compile on ghc-6.8 out of the box, but only because of a trivial build config issue that's easily fixed (see below).

SYBWC makes it possible to define extensible generic functions. An ordinary generic function operates over instances of GHC's Data.Generics.Basics.Data class. For example, the ‘gshow’ function has the following type:

gshow :: (Data a) => a -> String

‘gshow’ works for the most common Haskell datatypes, but there is no way to customize it for your own datatypes. With SYBWC, generic functions can easily be extended using instance declarations (hence “with class”). Suppose we want to define our own generic show function using SYBWC. We begin, ironically enough, with a small amount of boilerplate code:

import Data.Generics.SYB.WithClass.Basic

class MyShow a where
myShow :: a -> String

-- Note that the type of the myShowD field is identical
-- to the type of 'myShow' in the MyShow class.
data MyShowD a = MyShowD { myShowD :: a -> String }

myShowProxy :: Proxy myShowD
myShowProxy = error "This value should never be evaluated!"

instance MyShow a => Sat (MyShowD a) where
dict = MyShowD { myShowD = myShow }

This is all rather involved, and I refer you to the SYBWC paper for the gorey details. The good thing is that this code follows exactly the same template whatever the type signature of your generic function, so you don't really need to understand any of it.

With the boilerplate out of the way, the rest is pretty straightforward. First, we need to define a generic implementation of myShow that will be called if there isn't a more specific implementation for a given type. Focusing on the instance declaration alone, we need the following:

instance Data JsonD a => MyShow a where

Note that the ‘Data’ class in Data.Generics.SYB.WithClass takes two type arguments. This is crucial to the increased flexibility of the SYBWC implementation of generics. In effect, the declaration above adds a new superclass to the Data class, telling GHC that every value that is an instance of ‘Data’ is also an instance of ‘Json’. This is very powerful technique, since it lets us treat any field of a record passed to ‘toJson’ as an instance of ‘Json’ — that's what's going to help us to recursively serialize record types. Defining a good generic show function is a project in itself, so I'll move on from this example and get straight to the JSON serializer. Here's the JSON boilerplate:

class Json a where
toJson :: a -> JsonData
exclude :: a -> [String]

data JsonD a = JsonD { toJsonD :: a -> JsonData }

jsonProxy :: Proxy JsonD
jsonProxy = error "This value should never be evaluated!"

instance Json t => Sat (JsonD t) where
dict = JsonD { toJsonD = toJson }

Now we have to decide how our generic serialization function is going to work. For record types, it should recursively serialize each value in the record and then create a JSON object with the appropriate field names. For other types, it's less obvious what should be done; I've chosen the following behavior. For algebraic types, the serializer outputs a list containing the name of the type followed by all of the arguments to the type constructor (if any). For primitive types (e.g. ‘Int’), a runtime error is signaled, since no sensible generic behavior can be defined. Here's the definition for the generic version of ‘toJson’:

instance Data JsonD t => Json t where
toJson x
| isAlgType (dataTypeOf jsonProxy x) =
case (map stripInitialUnderscores (getFields x)) of
[] -> JDList (JDString (dataTypeName (dataTypeOf jsonProxy x)) :
(gmapQ jsonProxy (toJsonD dict) x))
fs -> JDObject (M.fromList (zip fs (gmapQ jsonProxy (toJsonD dict) x)))
| True =
error ("Unable to serialize the primitive type '" ++
dataTypeName (dataTypeOf jsonProxy x) ++ "'")

getFields :: Data JsonD a => a -> [String]
getFields = constrFields . (toConstr jsonProxy)

The ‘dataTypeOf’ function returns a DataType value representing a Haskell data type. Its first argument is a “proxy”, which fixes the superclass of the Data class that we're dealing with (in our case, the ‘Json’ class). Once the DataType of the value has been obtained, we check that it is an algebraic type, signaling an error if it is not. Now we call the ‘getFields’ function, which returns [] for ordinary algebraic data types and a list of field names for record types. In either case, we use the ‘gmapQ’ function (which also takes a proxy argument) to map ‘toJson’ over all the constructor arguments. Crucially, however, there is no direct recursive call to ‘toJson’. Instead, the following expression is used to obtain a suitable function:

(toJsonD dict)

This allows the most specialized instance of ‘toJson’ to be selected for each field of the record. If ‘toJson’ were called directly, the code would still type-check and compile, but the specialized instances of ‘toJson’ would not be called, leading to the wrong behavior at runtime. Again, the SYBWC paper gives the details of how this works.

At this point, we have a working generic serialization function for record types. All we need to do now is add instances of the ‘Json’ class for common Haskell types. For example:

instance Json Int where
toJson i = show i

With this definition, we can now serialize nested records containing integers:

data OuterRecord = { _x :: Int,
_y :: Int,
_rec :: InnerRecord }
data InnerRecord = { _a :: Int,
_b :: Int }


> toJson (OuterRecord { _x=1,
_rec=(InnerRecord { _a=3, _b=4 }) })

=> {"x":1,"y":2,"rec":{"a":3,"b":4}} :: JsonData

The ‘$(derive ...)’ lines invoke some Template Haskell code that automatically derives an instance of SYBWC's ‘Data’ class for the given datatype. That's basically it. We have a generic, extensible JSON serializer, and for the most part, we haven't sacrificed any type safety.1 The only problem that remains is that of treating strings differently from other lists. This is easily accomplished in the instance declaration for lists with the help of GHC's ‘cast’ function:

instance (Typeable a, Json a) => Json [a] where
toJson l =
case (cast l) :: Maybe String of
Just s -> JDString s
Nothing -> JDList (map toJson l)

[It has been pointed out to me that the use of a cast here is not required. An alternative is to use the same trick that the Prelude uses to ensure that ‘show’ treats strings properly (namely, addition of a ‘showList’ method to the Show class).]

The finished library adds an ‘exclude’ member to the ‘Json’ class. This can be used to exclude fields of a record from serialization without having to write a full custom instances. For example:

data CustomRecord { _dontSerializeMe :: Int,
_serializeMe :: Int }

instance Json CustomRecord where
excluded _ = ["_dontSerializeMe"]
toJson = genericToJson

> toJson (CustomRecord { _dontSerializeMe=1, _serializeMe=2 })

=> {"serializeMe":2} :: JsonData

I have not found a way of providing a default implementation for ‘toJson’ which is compatible with Haskell's type system, so if you want the default serialization behavior, you must manually set ‘toJson’ to ‘genericToJson’ (a function provided by the library).

Installing syb-with-class

In order to compile syb-with-class successfully for ghc-6.8, you need to make the following changes to the file ‘syb-with-class.cabal’. (Disclaimer: this is just a quick hack, not a proper fix.)

  • In the line beginning ‘Build-Depends:’, add ‘array’ to the dependencies.

  • In the line beginning ‘GHC-Options:’, add the option ‘-fglasgow-exts’.

You can now use the following commands to build and install the package:

runghc Setup.hs configure
runghc Setup.hs build
runghc Setup.hs install



module Json (Json, toJson, genericToJson, JsonData(..)) where

import Data.Generics.SYB.WithClass.Basics
import Data.Generics.SYB.WithClass.Instances
import Data.Generics.SYB.WithClass.Context
import Data.Generics.SYB.WithClass.Derive
import Language.Haskell.TH.Syntax
import qualified Data.Map as M
import qualified Text.Printf as P
import Data.Char
import Data.Ratio
import Data.Array

-- | The 'JsonData' type is a Haskell representation of a JSON
-- data structure.
data JsonData = JDString String |
JDNumber Rational |
JDList [JsonData] |
JDObject (M.Map String JsonData)

join :: a -> [a] -> [a]
join _ [] = []
join _ l@[x] = l
join k (x:ys) = x : k : (join k ys)

concatJoin :: String -> [String] -> String
concatJoin k l = concat (join k l)

alistToJsonDict :: [(String, String)] -> String
alistToJsonDict l =
"{" ++
concatJoin "," (map (\(k,v) -> (escapeString k) ++ ":" ++ v) l)
++ "}"

-- Special characters which will be pretty printed.
escapeMap :: M.Map Char String
escapeMap = M.fromList [
('\\', "\\"), ('"', "\""), ('\'', "'"), ('\n', "n"),
('\r', "r"), ('\f', "f"), ('\t', "t"), ('\b', "\b")]
escape :: Char -> Maybe String
escape c = M.lookup c escapeMap

-- Characters which can safely be printed as literals.
allowedMap = M.fromList (zip " !@#$%^&*()_-+={[}]|:;<>,.?/~`" (repeat True))
allowed' c o
| o >= ord 'a' && o <= ord 'z' = True
| o >= ord 'A' && o <= ord 'Z' = True
| o >= ord '0' && o <= ord '9' = True
| True = M.member c allowedMap
allowed c = allowed' c (ord c)

hexEscape :: Char -> String
hexEscape c = "\\u" ++ P.printf "%04x" (ord c)

escapeString' :: String -> String
escapeString' [] = "\""
escapeString' (c:cs)
| allowed c =
c : (escapeString' cs)
| True =
(maybe (hexEscape c) (\s -> "\\" ++ s) (escape c)) ++
(escapeString' cs)

escapeString s = '"' : escapeString' s

instance Show JsonData where
show (JDString s) = escapeString s
show (JDNumber n)
-- Show as an integer if possible, otherwise as double precision float.
| denominator n == 1 =
show (numerator n)
| True =
show ((fromIntegral (numerator n) :: Double) /
(fromIntegral (denominator n) :: Double))
show (JDList l) = "[" ++ concatJoin "," (map show l) ++ "]"
show (JDObject o) = alistToJsonDict (map (\(k,v) -> (k, show v)) (M.toList o))

-- Json class plus SYB boilerplate.
-- | New instances can be added for the 'Json' class to customize
-- JSON serialization.
class Json a where
toJson :: a -> JsonData
exclude :: a -> [String]
exclude _ = []

data JsonD a = JsonD { toJsonD :: a -> JsonData }

jsonProxy :: Proxy JsonD
jsonProxy = error "This value should never be evaluated!"

instance Json t => Sat (JsonD t) where
dict = JsonD { toJsonD = toJson }

-- Implementations of toJson for different data types.
instance Json Int where
toJson i = JDNumber (fromIntegral i)
instance Json Integer where
toJson i = JDNumber (fromIntegral i)
instance Json Float where
toJson i = JDNumber (toRational i)
instance Json Double where
toJson i = JDNumber (toRational i)
instance Integral a => Json (Ratio a) where
toJson i = JDNumber ((fromIntegral (numerator i)) %
(fromIntegral (denominator i)))
instance Json Char where
toJson c = JDString [c]

instance Json a => Json (M.Map String a) where
toJson x = JDObject ( toJson x)

-- Handles lists, with special case for [Char].
instance (Typeable a, Json a) => Json [a] where
toJson l =
case (cast l) :: Maybe String of
Just s -> JDString s
Nothing -> JDList (map toJson l)

-- TODO: Add instances for the other array types supported by GHC.
instance (Typeable a, Json a, Ix i) => Json (Array i a) where
toJson a = toJson (elems a)

stripInitialUnderscores "" = ""
stripInitialUnderscores ('_':s) = stripInitialUnderscores s
stripInitialUnderscores s = s

getFields :: Data JsonD a => a -> [String]
getFields = constrFields . (toConstr jsonProxy)

-- | This is the implementation of 'toJson' for the generic instance declaration,
-- but it's useful to be able to use the same implentation for
-- other instance declarations which override the default implementation
-- of 'exclude'.
genericToJson :: (Data JsonD a, Json a) => a -> JsonData
genericToJson x
| isAlgType (dataTypeOf jsonProxy x) =
case (map stripInitialUnderscores (filter (not . (flip elem) (exclude x)) (getFields x))) of
[] -> JDList (JDString (dataTypeName (dataTypeOf jsonProxy x)) :
(gmapQ jsonProxy (toJsonD dict) x))
fs -> JDObject (M.fromList (zip fs (gmapQ jsonProxy (toJsonD dict) x)))
| True =
error ("Unable to serialize the primitive type '" ++ dataTypeName (dataTypeOf jsonProxy x) ++ "'")

instance Data JsonD t => Json t where
toJson = genericToJson

-- Instances for tuples up to n=5.
-- Tuples are converted to (hetrogenous) JSON lists.
instance (Json a, Json b) => Json (a, b) where
toJson (a,b) = JDList [toJson a, toJson b]
instance (Json a, Json b, Json c) => Json (a,b,c) where
toJson (a,b,c) = JDList [toJson a, toJson b, toJson c]
instance (Json a, Json b, Json c, Json d) => Json (a,b,c,d) where
toJson (a,b,c,d) = JDList [toJson a, toJson b, toJson c, toJson d]
instance (Json a, Json b, Json c, Json d, Json e) => Json (a,b,c,d,e) where
toJson (a,b,c,d,e) = JDList [toJson a, toJson b, toJson c, toJson d, toJson e]

1. The one exception is the possibility of a runtime error if we try to serialize a primitive type for which no instance has been defined. In practice, given instances for all the basic Haskell String/numeric types, this sort of error is very unlikely to occur. [^]