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README
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
Expr
s. 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
Basics
To import Express
just:
> import Data.Express
For types that are Show
instances,
we can use val
to encode values as Expr
s.
> 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 Expr
s.
> let notE = value "not" not
> :t notE
notE :: Expr
> print notE
not :: Bool > Bool
Using :$
we can apply function valued Expr
s, to other Exprs.
> let notFalse = notE :$ false
> :t notFalse
notFalse :: Expr
> notFalse
not False :: Bool
Using evaluate
and eval
we can evaluate Expr
s 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 Expr
s:
> 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: uExtrapolate
[uExtrapolate] is a propertybased testing library capable of generalizing counterexamples. 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 `asTypeOf` [[undefined :: Int]])
"Bool" > mapT val (tiers `asTypeOf` [[undefined :: Bool]])
"[Int]" > mapT val (tiers `asTypeOf` [[undefined :: [Int]]])
"[Bool]" > mapT val (tiers `asTypeOf` [[undefined :: [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 counterexample, 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 :$) (gen e2)
gen e
 isVar e = []
 otherwise = [holeAsTypeOf e  isListable e]
isListable = not . null . tiersFor
The need for isListable
above makes sure we only replace by variables what we
can enumerate. Our candidate generalizations are listed in nonincreasing
order of generality:
> candidateGeneralizations (value "not" not :$ 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 counterexample, the following function returns a counterexample 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)
[uExtrapolate] 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 fullfeatured version without the above limitations and with support for conditional generalizations.
Example 4: uSpeculate
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 [uSpeculate] example in the [eg](eg) folder for the full code
of speculateAbout
.
[uSpeculate] 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 fullfeatured version without the above limitations.
Further reading
For a detailed documentation, please see Express's Haddock documentation.
For more examples, see the [eg](eg) and [bench](bench) folders.