# 24 days of Hackage, 2015: day 10: s-cargot: using S-expression syntax

## Table of contents for the whole series

A table of contents is at the top of the article for day 1.

## Day 10

There are times when I’m jealous of the Lisp world.

One of those times is when defining some domain-specific language, because in the Lisp world, the natural thing to do is to represent it using S-expressions as the concrete syntax, and not fuss with defining yet another special syntax, along with writing a parser for that syntax into the abstract syntax as well as a pretty-printer from the abstract syntax back to the concrete syntax. Maybe in the long run users might want a special syntax that is not just S-expressions, but for quick initial prototyping, at least, it seems worthwhile to not commit to any special syntax and just use S-expressions. Although there is nothing magical about S-expressions (or XML or JSON or any other generic representation of a tree data structure), they are particularly concise and flexible.

S-expressions are so useful that in my first job as a software engineer in the 1990s, we actually had a C++ S-expression library for input and output of a format that amounted to a domain-specific language (this was before XML was invented) that was processed by many tools (including a validator that I wrote in Standard ML).

A library on Hackage for working with S-expressions in Haskell is
`s-cargot`

. There have
been many others, but most of them have gone sadly unmaintained,
whereas this one is new and comes with bells and whistles.

Today I’ll give an example of how to use this library, in the context of a problem domain in which having a concrete syntax is important.

## Installation

We need to add `s-cargot`

to `stack.yaml`

:

```
- s-cargot-0.1.0.0
```

## The task: symbolic differentiation

The task we solve here is, appropriately enough, a translation from the Lisp (Scheme, to be precise) code for a symbolic differentiator of mathematical expressions in the classic computer science textbook “Structure and Interpretation of Computer Programs”. I won’t be walking through the solution here, but just focusing on some syntax issues.

Example: given a mathematical expression such as the linear function
`5x + 7`

, we want to find the symbolic derivative with respect to `x`

,
to get `5`

.

## Simplest syntax for the expression type

How do we model an expression and a function to compute a derivative
of an expression? Let’s start with the most vanilla possible way,
which is to define a data type for expression, `Exp`

, along with
ordinary alphanumeric constructors `N`

(for number), `V`

(for
variable), `Plus`

(for sum of subexpressions), `Times`

(for product of
subexpressions). A sample subset of an appropriate HSpec/QuickCheck
spec to define what we need:

```
module SymbolicDifferentiation.AlphaSyntaxSpec where
import SymbolicDifferentiation.AlphaSyntax (Exp(N, V, Plus, Times), deriv)
import Test.Hspec (Spec, hspec, describe, it, shouldBe)
import Test.Hspec.QuickCheck (prop)
import Test.QuickCheck ((==>))
spec :: Spec
spec =
describe "symbolic differentiation" $ do
prop "d/dx (x + n) == 1" $ \x n ->
deriv (Plus (V x) (N n)) x `shouldBe` N 1
prop "d/dx (x + y) == x, if x /= y" $ \x y ->
x /= y ==>
deriv (Times (V x) (V y)) x `shouldBe` V y
prop "d/dx (a * x + b) == x" $ \a x b ->
deriv (Plus (Times (N a) (V x)) (N b)) x `shouldBe` N a
it "d/dx (x * y * (x + 3)) == (x * y) + y * (x + 3)" $ do
deriv (Times (Times (V "x") (V "y"))
(Plus (V "x") (N 3))) "x" `shouldBe`
(Plus (Times (V "x") (V "y"))
(Times (V "y") (Plus (V "x") (N 3))))
```

The syntax looks OK for simple expressions, but ugly when you have lots of nested subexpressions, with parentheses for grouping, as in the final artificial example.

Here is the code for a still-naive symbolic differentiator, having
only a few heuristics built in for some simplification through
rewriting (for example, adding `0`

to an expression results in that
expression rather than construction of a superfluous `N 0`

subexpression):

```
module SymbolicDifferentiation.AlphaSyntax where
-- | Variable in an expression.
type Var = String
-- | Expression.
data Exp
= N Int -- ^ number
| V Var -- ^ variable
| Plus Exp Exp -- ^ sum
| Times Exp Exp -- ^ product
deriving (Show, Eq)
-- | Derivative of expression with respect to a variable.
deriv :: Exp -> Var -> Exp
deriv (N _) _ = N 0
deriv (V v') v = N (if v' == v then 1 else 0)
deriv (Plus e1 e2) v = plus (deriv e1 v) (deriv e2 v)
deriv (Times e1 e2) v = plus (times e1 (deriv e2 v))
(times (deriv e1 v) e2)
-- | Smart constructor that simplifies while combining subexpressions.
plus :: Exp -> Exp -> Exp
plus (N 0) e = e
plus e (N 0) = e
plus (N n1) (N n2) = N (n1 + n2)
plus e1 e2 = Plus e1 e2
-- | Smart constructor that simplifies while combining subexpressions.
times :: Exp -> Exp -> Exp
times (N 0) _ = N 0
times _ (N 0) = N 0
times (N 1) e = e
times e (N 1) = e
times (N n1) (N n2) = N (n1 * n2)
times e1 e2 = Times e1 e2
```

The syntax of the code, using pattern matching, looks reasonably good
to me. It’s the `Exp`

data construction that looks a bit ugly,
although not terrible. So we have a situation in which the implementor
of this little expression language is fairly happy, but the user is
not. In fact, we haven’t even provided a way for a user outside the
system to create expressions: right now, we have an API but no parser
from a string into an expression.

## Parsing from what format?

One way to go is to write a custom parser from a custom syntax, for
example maybe write a function `fromString :: String -> Exp`

such that

```
fromString "5x + 7y + 20" ==
Plus (Plus (Times (N 5) (V "x"))
(Times (N 7) (V "y")))
(N 20)
```

We would also want to write a pretty-printer ```
toString :: Exp ->
String
```

such that maybe it’s smart enough to go

```
toString (Plus (Plus (Times (N 5) (V "x"))
(Times (N 7) (V "y")))
(N 20)) == "5x + 7y + 20"
```

It is straightforward to write such a parser using
`parsec`

or the
like, but you can imagine that sometimes it might be annoying to
design the syntax for a much more complex language (including special
infix operators, precedences, scoping keywords, different kinds of
braces, etc.) and also make users learn it.

So let’s assume for the purpose of this article that we have a reason to prefer prefix-only S-expressions, just as the Lisp community does in order to avoid all the hassles of a custom syntax.

### (Update of 2015-12-14) Providing a custom syntax using `Earley`

On
day 14,
I created a custom syntax and a parser using the `Earley`

parser library.

### (Update of 2015-12-17) Providing a custom pretty-printer using `ansi-wl-pprint`

The inverse of parsing is pretty-printing, covered on day 17.

### QuickCheck tests

Here are some sample QuickCheck tests to show what it is we want to be
able to do. (Note that for convenience, we are using string
interpolation through the
`here`

package as
introduced yesterday,
day 9.)

```
{-# LANGUAGE QuasiQuotes #-}
{-# LANGUAGE OverloadedStrings #-}
module SymbolicDifferentiation.SExpSpec where
import SymbolicDifferentiation.AlphaSyntax (Exp(N, V, Plus, Times))
import qualified SymbolicDifferentiation.SExp as SExp
import Test.Hspec (Spec, hspec, describe, shouldBe)
import Test.Hspec.QuickCheck (prop)
import Data.String.Here (i)
-- | Required for auto-discovery.
spec :: Spec
spec =
describe "S-expression syntax for expression" $ do
prop "(+ x a)" $ \a ->
SExp.parse [i|(+ x ${a})|] `shouldBe`
Right (Plus (V "x") (N a))
prop "(* (+ x a) (+ y b))" $ \a b ->
SExp.parse [i|
(* (+ x ${a})
(+ y ${b}))
|] `shouldBe`
Right (Times (Plus (V "x") (N a))
(Plus (V "y") (N b)))
it "(!? x y)" $
SExp.parse "(!? x y)" `shouldBe`
Left "\"!?\" is not a valid operator"
```

I’ve included a little test to indicate that we want some kind of error handling also. Nothing is more annoying to a user than terrible parse error messages. We won’t provide great messages here, but at least will give an idea of how one could.

## The S-expression parser

`s-cargot`

provides very flexible ways of constructing S-expression
parsers, based on what kind of syntax you want to allow (full Scheme
or not, for example), and allows hooks on many levels to support
readers as well as specifying the desired atom parser.

Some imports first:

```
{-# LANGUAGE QuasiQuotes #-}
{-# LANGUAGE OverloadedStrings #-}
module SymbolicDifferentiation.SExp where
import SymbolicDifferentiation.AlphaSyntax (Exp(N, V, Plus, Times))
import qualified Data.SCargot as S
import Data.SCargot.Language.Basic (basicParser)
import Data.SCargot.Repr.WellFormed
(WellFormedSExpr(WFSList, WFSAtom), fromWellFormed)
import qualified Data.Text as Text
import Data.Text (Text)
import Data.Text.Read (signed, decimal)
import Data.String.Here (i)
-- | Error when parsing.
type Error = String
```

The main driver is a pipeline of calling an `s-cargot`

parser and then
calling our own parser of an S-expression into our `Exp`

type:

```
-- | For simplicity, we use 'basicParser' which just treats every atom
-- as 'Text', which we parse later rather than up front.
parse :: Text -> Either Error Exp
parse text = parseOneSexp text >>= toExp
parseOneSexp :: Text -> Either Error (WellFormedSExpr Text)
parseOneSexp = S.decodeOne (S.asWellFormed basicParser)
```

Once we have a well-formed S-expression, we can pick it apart, while catching errors if we encounter one.

```
toExp :: WellFormedSExpr Text -> Either Error Exp
toExp (WFSAtom text) = fromAtom text
toExp (WFSList [WFSAtom operatorText, sexp1, sexp2]) = do
operator <- fromOperator operatorText
e1 <- toExp sexp1
e2 <- toExp sexp2
return (operator e1 e2)
toExp list@(WFSList _) = Left [i|${list} should have exactly 3 elements|]
fromOperator :: Text -> Either Error (Exp -> Exp -> Exp)
fromOperator "+" = return Plus
fromOperator "*" = return Times
fromOperator text = Left [i|${text} is not a valid operator|]
-- | Either an integer or a variable.
fromAtom :: Text -> Either Error Exp
fromAtom text =
case signed decimal text of
Right (n, "") ->
return (N n)
Right (_, _) ->
Left [i|extra garbage after numeric in ${text}|]
Left _ ->
return (V (Text.unpack text))
```

## Pretty-printing

And we get pretty-printing for free from `s-cargot`

if we turn our
`Exp`

into an S-expression first. I won’t show the details, but you
can use the basic S-expression printer or customize it with a lot
options including indentation strategy. Let’s just use the basic.

A sample test for our `SExp.prettyPrint`

:

```
prop "pretty-printing" $ \a b ->
SExp.prettyPrint (Times (Plus (V "x") (N a))
(Plus (V "y") (N b))) `shouldBe`
[i|(* (+ x ${a}) (+ y ${b}))|]
```

The code:

```
fromExp :: Exp -> WellFormedSExpr Text
fromExp (N n) = WFSAtom (Text.pack (show n))
fromExp (V x) = WFSAtom (Text.pack x)
fromExp (Plus e1 e2) = WFSList [WFSAtom "+", fromExp e1, fromExp e2]
fromExp (Times e1 e2) = WFSList [WFSAtom "*", fromExp e1, fromExp e2]
prettyPrint :: Exp -> Text
prettyPrint =
S.encodeOne (S.setFromCarrier fromWellFormed (S.basicPrint id))
. fromExp
```

## Summary of S-expression parsing and pretty-printing

So there you have it: with a little bit of boilerplate you can get an experience similar to that of working in Lisp. Note that with even clever Template Haskell work with quasiquotation you could go further than the pure text templates we’ve used for convenience, and create pattern templates as well.

We didn’t have to write a traditional parser, and we were able to separate well-formedness from further processing. This is really useful in many contexts: in my experience, the multi-level error checking makes good error messages easier to create. Also, not discussed here is how S-expressions can also help with prediction and completion.

## Optional further notes on syntax for domain-specific languages

I wanted to point out that for some problem domains, such as this one that happens to be mathematical, it is sometimes popular to make things look mathematical. I’ve deliberately presented it without that attempt first, but now I’ll show some syntactic variants.

### Alphanumeric identifiers as operators

First, one can use operator syntax with backticks and precedence levels:

```
module SymbolicDifferentiation.AlphaOperatorSyntax where
-- | Variable in an expression.
type Var = String
-- | Precedences for our expression constructors.
infixl 6 `Plus`
infixl 7 `Times`
-- | Expression.
data Exp
= N Int -- ^ number
| V Var -- ^ variable
| Plus Exp Exp -- ^ sum
| Times Exp Exp -- ^ product
deriving (Show, Eq)
-- | Derivative of expression with respect to a variable.
deriv :: Exp -> Var -> Exp
deriv (N _) _ = N 0
deriv (V v') v = N (if v' == v then 1 else 0)
deriv (Plus e1 e2) v = deriv e1 v `plus` deriv e2 v
deriv (Times e1 e2) v = e1 `times` deriv e2 v
`plus`
deriv e1 v `times` e2
-- | Precedences for our expression "smart" constructors.
infixl 6 `plus`
infixl 7 `times`
-- | Smart constructor that simplifies while combining subexpressions.
plus :: Exp -> Exp -> Exp
N 0 `plus` e = e
e `plus` N 0 = e
N n1 `plus` N n2 = N (n1 + n2)
e1 `plus` e2 = e1 `Plus` e2
-- | Smart constructor that simplifies while combining subexpressions.
times :: Exp -> Exp -> Exp
N 0 `times` _ = N 0
_ `times` N 0 = N 0
N 1 `times` e = e
e `times` N 1 = e
N n1 `times` N n2 = N (n1 * n2)
e1 `times` e2 = e1 `Times` e2
```

Note that nothing has really changed, except the use of infix function definitions and calls, and use of precedence to remove parentheses, such as in the big expression for the derivative of a product of expressions. But clarity is starting to be lost, for those not already familiar with the problem domain and conventions.

### Symbolic identifiers as operators

One can go further and use symbolic identifiers in place of the backticked alphanumeric operators. This is where many of us start wondering what is going on:

```
module SymbolicDifferentiation.OperatorSyntax where
-- | Variable in an expression.
type Var = String
-- | Precedences for our expression constructors.
infixl 6 :+:
infixl 7 :*:
-- | Expression.
data Exp
= N Int -- ^ number
| V Var -- ^ variable
| Exp :+: Exp -- ^ sum
| Exp :*: Exp -- ^ product
deriving (Show, Eq)
-- | Derivative of expression with respect to a variable.
deriv :: Exp -> Var -> Exp
deriv (N _) _ = N 0
deriv (V x ) y = N (if x == y then 1 else 0)
deriv (e1 :+: e2) v = deriv e1 v .+. deriv e2 v
deriv (e1 :*: e2) v = e1 .*. deriv e2 v
.+.
deriv e1 v .*. e2
-- | Precedences for our expression "smart" constructors.
infixl 6 .+.
infixl 7 .*.
-- | Smart constructor that simplifies while combining subexpressions.
(.+.) :: Exp -> Exp -> Exp
N 0 .+. e = e
e .+. N 0 = e
N n1 .+. N n2 = N (n1 + n2)
e1 .+. e2 = e1 :+: e2
-- | Smart constructor that simplifies while combining subexpressions.
(.*.) :: Exp -> Exp -> Exp
N 0 .*. _ = N 0
_ .*. N 0 = N 0
N 1 .*. e = e
e .*. N 1 = e
N n1 .*. N n2 = N (n1 * n2)
e1 .*. e2 = e1 :*: e2
```

Depending on your taste, you might find that this funny syntax makes the tests look nicer:

```
spec :: Spec
spec =
describe "symbolic differentiation" $ do
prop "d/dx (x + n) == 1" $ \x n ->
deriv (V x :+: N n) x `shouldBe` N 1
prop "d/dx (x + y) == x, if x /= y" $ \x y ->
x /= y ==>
deriv (V x :*: V y) x `shouldBe` V y
prop "d/dx (a * x + b) == x" $ \a x b ->
deriv (N a :*: V x :+: N b) x `shouldBe` N a
it "d/dx (x * y * (x + 3)) == (x * y) + y * (x + 3)" $ do
deriv (V "x" :*: V "y" :*: (V "x" :+: N 3)) "x" `shouldBe`
(V "x" :*: V "y") :+: (V "y" :*: (V "x" :+: N 3))
```

All this looks kind of cute if you’re used to it, but I think it can look really weird otherwise; and I know that many non-Haskellers seeing this before seeing the vanilla way get the wrong idea that you have to write this way in Haskell and turn away in disgust and confusion. This is why I showed the vanilla way first, and show this only to illustrate that there are libraries out there that do try to be very suggestive in symbolic operator usage, and also that there is nothing special going on here: it’s just a different way of saying exactly the same thing as the first version, with the second version (backticked alphanumerics as operators) being a transitional step toward this third version. It’s good to know about all three variants, regardless of which one you prefer to read and write.

But what about the S-expression route, which is to decouple the user (represented by the tests) side of things from the abstract syntax and the Haskell syntax? My intuition is that there are real benefits in at least providing an alternate S-expression syntax for whatever other concrete syntax is made available.

## Conclusion

S-expressions are a time-honored way of representing data. The
`s-cargot`

library comes with ways to build custom S-expression
parsers and pretty-printers and also comes with useful defaults.

## All the code

All my code for my article series are at this GitHub repo.

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