Express

Express is a library for manipulating dynamically typed Haskell expressions. It’s like Data.Dynamic but with support for encoding applications and variables.

It provides the Expr type and over a hundred functions for building, evaluating, comparing, folding, canonicalizing and matching Exprs. See Express’s Haddock documentation for more details.

This library has been used in the implementation of Speculate and Extrapolate.

Installing

To install the latest Express version from Hackage, just run:

$ cabal update
$ cabal install express

Starting from Cabal v3.0, you need to pass --lib as an argument to cabal install:

$ cabal install express --lib

Basics

To import Express just:

> import Data.Express

For types that are Show instances, we can use val to encode values as Exprs.

> let false = val False
> :t false
false :: Expr
> print false
False :: Bool

> let one = val (1 :: Int)
> :t one
one :: Expr
> print one
1 :: Int

As seen above, the Show instance for Expr produces a string with the encoded value and it’s type.

For types that aren’t Show instances, like functions, we can use value to encode values as Exprs.

> let notE = value "not" not
> :t notE
notE :: Expr
> print notE
not :: Bool -> Bool

Using :$ we can apply function valued Exprs, to other Exprs.

> let notFalse = notE :$ false
> :t notFalse
notFalse :: Expr
> notFalse
not False :: Bool

Using evaluate and eval we can evaluate Exprs back into a regular Haskell value.

> evaluate notFalse :: Maybe Bool
Just True
> evaluate notFalse :: Maybe Int
Nothing
> eval False notFalse
True
> eval (0::Int) notFalse
0

Example 1: heterogeneous lists

Like with Data.Dynamic, we can use Express to create heterogeneous lists.

Here, we use applications of val to create a heterogeneous list:

> let xs = [val False, val True, val (1::Int), val (2::Int), val (3::Integer), val "123"]
> :t xs
xs :: [Expr]
> xs
[ False :: Bool
, True :: Bool
, 1 :: Int
, 2 :: Int
, 3 :: Integer
, "123" :: [Char]
]

We can then apply evaluate to select values of different types:

> import Data.Maybe
> mapMaybe evaluate xs :: [Bool]
[False,True]
> mapMaybe evaluate xs :: [Int]
[1,2]
> mapMaybe evaluate xs :: [Integer]
[3]
> mapMaybe evaluate xs :: [String]
["123"]

Example 2: listing applications

Carrying on from Example 1, we define an heterogeneous list of functions encoded as Exprs:

> let fs = [value "not" not, value "&&" (&&), value "abs" (abs :: Int -> Int)]
> :t fs
fs :: [Expr]

Using $$ we list the type correct applications of functions in fs to values in xs.

> catMaybes [f $$ x | f <- fs, x <- xs]
[ not False :: Bool
, not True :: Bool
, (False &&) :: Bool -> Bool
, (True &&) :: Bool -> Bool
, abs 1 :: Int
, abs 2 :: Int
]

Example 3: u-Extrapolate

u-Extrapolate is a property-based testing library capable of generalizing counter-examples. It’s implementation has under 40 lines of code. Besides, using Express to encode expressions, it uses LeanCheck for generating test values.

import Data.Express
import Test.LeanCheck hiding (counterExample, check)

Given a maximum number of tests and a property, the following counterExample function returns either Nothing when tests pass or Just a counterexample encoded as an Expr.

counterExample :: (Listable a, Express a) => Int -> (a -> Bool) -> Maybe Expr
counterExample maxTests prop  =  listToMaybe
  [expr x | x <- take maxTests list, not (prop x)]

Examples (REPL):

> counterExample 100 (\(x,y) -> x + y == y + x)
Nothing
> counterExample 100 (\x -> x == x + x)
Just (1 :: Integer)
> counterExample 100 (\xs -> nub xs == (xs :: [Int]))
Just ([0,0] :: [Int])

Before moving on to generalize counterexamples, we need a way to compute ground expressions from an expression with variables. For that, we will use grounds and tiersFor:

grounds :: Expr -> [Expr]
grounds e  =  map (e //-)
           .  concat
           $  products [mapT ((,) v) (tiersFor v) | v <- nubVars e]

tiersFor :: Expr -> [[Expr]]
tiersFor e  =  case show (typ e) of
  "Int"    ->  mapT val (tiers :: [[Int]])
  "Bool"   ->  mapT val (tiers :: [[Bool]])
  "[Int]"  ->  mapT val (tiers :: [[ [Int] ]])
  "[Bool]" ->  mapT val (tiers :: [[ [Bool] ]])
  _        ->  []

Above, we restrict ourselves to Int, Bool, [Int] and [Bool] as test types. So we can now compute the grounds of an expression with variables:

> grounds (value "not" not :$ var "p" (undefined :: Bool))
[ not False :: Bool
, not True :: Bool
]
> grounds (value "&&" (&&) :$ var "p" (undefined :: Bool) :$ var "q" (undefined :: Bool))
[ False && False :: Bool
, False && True :: Bool
, True && False :: Bool
, True && True :: Bool
]

To compute candidate generalizations from a given counter-example, we use the following function:

candidateGeneralizations :: Expr -> [Expr]
candidateGeneralizations  =  map canonicalize
                          .  concatMap canonicalVariations
                          .  gen
  where
  gen e@(e1 :$ e2)  =
    [holeAsTypeOf e | isListable e]
    ++ [g1 :$ g2 | g1 <- gen e1, g2 <- gen e2]
    ++ map (:$ e2) (gen e1)
    ++ map (e1 :$) (gen e2)
  gen e
    | isVar e    =  []
    | otherwise  =  [holeAsTypeOf e | isListable e]
  isListable  =  not . null . tiersFor

The need for isListable above makes sure we only replace by variables what we can enumerate. Our candidate generalizations are listed in non-increasing order of generality:

> candidateGeneralizations (value "not" not :$ val False)
[ p :: Bool
, not p :: Bool
]
Prelude> candidateGeneralizations (value "||" (||) :$ val False :$ val True)
[ p :: Bool
, p || q :: Bool
, p || p :: Bool
, p || True :: Bool
, False || p :: Bool
]

For a given maximum number of tests, property and counter-example, the following function returns a counter-example generalization if one is found. It goes through the list of candidate generalizations and returns the first for which all tests fail.

counterExampleGeneralization :: Express a => Int -> (a -> Bool) -> Expr -> Maybe Expr
counterExampleGeneralization maxTests prop e  =  listToMaybe
  [g | g <- candidateGeneralizations e
     , all (not . prop . evl) (take maxTests $ grounds g)]

We can finally define our check function, that will test a property and report a counterexample and a generalization when either are found.

check :: (Listable a, Express a) => (a -> Bool) -> IO ()
check prop  =  putStrLn $ case counterExample 500 prop of
  Nothing -> "+++ Tests passed.\n"
  Just ce -> "*** Falsified, counterexample:  " ++ show ce
          ++ case counterExampleGeneralization 500 prop ce of
             Nothing -> ""
             Just g -> "\n               generalization:  " ++ show g
          ++ "\n"

Now we can find counterexamples and their generalizations:

> check $ \xs -> sort (sort xs :: [Int]) == sort xs
+++ Tests passed.

> check $ \xs -> length (nub xs :: [Int]) == length xs
*** Falsified, counterexample:  [0,0] :: [Int]
               generalization:  x:x:xs :: [Int]

> check $ \x -> x == x + (1 :: Int)
*** Falsified, counterexample:  0 :: Int
               generalization:  x :: Int

> check $ \(x,y) -> x /= (y :: Int)
*** Falsified, counterexample:  (0,0) :: (Int,Int)
               generalization:  (x,x) :: (Int,Int)

u-Extrapolate has some limitations:

• it only supports properties with one argument (uncurried);
• it only supports generalization of Int, Bool, [Int] and [Bool] values;
• there is no way to configure the number of test arguments.

Please see Extrapolate for a full-featured version without the above limitations and with support for conditional generalizations.

Example 4: u-Speculate

Using Express, it takes less than 70 lines of code to define a function speculateAbout that conjectures equations about a set of functions based on the results of testing:

> speculateAbout [hole (undefined :: Bool), val False, val True, value "not" not]
[ not False == True :: Bool
, not True == False :: Bool
, not (not p) == p :: Bool
]

> speculateAbout
>   [ hole (undefined :: Int)
>   , hole (undefined :: [Int])
>   , val ([] :: [Int])
>   , value ":" ((:) :: Int -> [Int] -> [Int])
>   , value "++" ((++) :: [Int] -> [Int] -> [Int])
>   , value "sort" (sort :: [Int] -> [Int])
>   ]
[ sort [] == [] :: Bool
, xs ++ [] == xs :: Bool
, [] ++ xs == xs :: Bool
, sort (sort xs) == sort xs :: Bool
, sort [x] == [x] :: Bool
, [x] ++ xs == x:xs :: Bool
, sort (xs ++ ys) == sort (ys ++ xs) :: Bool
, sort (x:sort xs) == sort (x:xs) :: Bool
, sort (xs ++ sort ys) == sort (xs ++ ys) :: Bool
, sort (sort xs ++ ys) == sort (xs ++ ys) :: Bool
, (x:xs) ++ ys == x:(xs ++ ys) :: Bool
, (xs ++ ys) ++ zs == xs ++ (ys ++ zs) :: Bool
]

Please see the u-Speculate example in the eg folder for the full code of speculateAbout.

u-Speculate has some limitations:

• it sometimes prints redundant equations;
• although it usually runs quickly with less than 6 symbols, runtime is exponential with the number of symbols given, providing it with more than a dozen symbols can make it run for several minutes or hours;
• there is no way to configure the size limit of reported equations;
• it only supports variables of Int, Bool, [Int], and [Bool] types.

Please see Speculate for a full-featured version without the above limitations.

Example 5: u-Conjure

Using Express, it takes less than 70 lines of code to define a function conjure that generates a function from a partial function definition and a list of primitives.

Example 5.1. Given:

factorial :: Int -> Int
factorial 0  =  1
factorial 1  =  1
factorial 2  =  2
factorial 3  =  6
factorial 4  =  24

Running:

conjure "factorial" factorial
  [ val (0 :: Int)
  , val (1 :: Int)
  , value "+" ((+) :: Int -> Int -> Int)
  , value "*" ((*) :: Int -> Int -> Int)
  , value "foldr" (foldr :: (Int -> Int -> Int) -> Int -> [Int] -> Int)
  , value "enumFromTo" (enumFromTo :: Int -> Int -> [Int])
  ]

Prints:

  factorial :: Int -> Int
  factorial x  =  foldr (*) 1 (enumFromTo 1 x)

Example 5.2. Given:

(+++) :: [Int] -> [Int] -> [Int]
[x] +++ [y]  =  [x,y]
[x,y] +++ [z,w]  =  [x,y,z,w]

Running:

conjure "++" (+++)
  [ val (0 :: Int)
  , val (1 :: Int)
  , val ([] :: [Int])
  , value "head" (head :: [Int] -> Int)
  , value "tail" (tail :: [Int] -> [Int])
  , value ":" ((:) :: Int -> [Int] -> [Int])
  , value "foldr" (foldr :: (Int -> [Int] -> [Int]) -> [Int] -> [Int] -> [Int])
  ]

Prints:

(++) :: [Int] -> [Int] -> [Int]
xs ++ ys  =  foldr (:) ys xs

Please see the u-Conjure example in the eg folder for the full code.

u-Conjure has some limitations:

• the maximum function size (7) or number of tests (60) are not configurable;
• the maximum function size has to be kept small (<=7) for a reasonable runtime. Due to this, several simple functions are simply out-of-reach;
• the number of primitive functions given has to be kept small (<12) for a reasonable runtime;
• there is no support for explicitly recursive functions thought it is possible to pass foldr and similar functions as primitives.

Please see Conjure library for an experimental version that addresses some the above limitations.

Further reading

For a detailed documentation, please see Express’s Haddock documentation.

For more examples, see the eg and bench folders.

Express is subject to a paper in the Haskell Symposium 2021 titled “Express: Applications of Dynamically Typed Haskell Expressions”.