I have the following Haskell functions:
expM :: Integer -> Integer -> Integer -> Integer
expM x y = rem (x^y)
And
exMME :: Integer -> Integer -> Integer -> Integer
exMME b 0 m = 1
exMME b e m = exMME' b e m 1 0 where
exMME' b e m c e'
| e' < e = exMME' b e m ((b * c) `mod` m) (e'+1)
| otherwise = c
What I want to do is use quickCheck to compare these two functions so i can see that they produce the same answer and which one is the fastest.
To test if they have the same answers I want to let QuickCheck create random positive integers with the exception of 0. So I created a Gen:
positives :: Gen Integer
positives =
do -- Pick an arbitrary integer:
x <- arbitrary
if (x == 0)
then return 1
else if (x < 0)
then return (-x)
else
return x
This works from the command line (ghci) but I have a prop:
prop_CompareAnswerExMFM :: Integer -> Integer -> Integer -> Bool
prop_CompareAnswerExMFM b e m =exMFM b e m == exM b e m
And each time i call this with QuickCheck prop_CompareAnswerExMFM it doesn't us my gen. After reading some stuff i toughed that i needed to define a instance:
instance Arbitrary Integer where
arbitrary = positives
This doesn't work because a arbitrary instance of Integer already exist. Again after some googling I say that the standard way to solve this is to use a wrapper:
newtype Positives = Positives Integer
deriving (Eq, Ord, Show)
instance Arbitrary Positives where
arbitrary = positives
positives :: Gen Positives
positives =
do -- Pick an arbitrary integer:
x <- arbitrary
if (x == 0)
then return 1
else if (x < 0)
then return (-x)
else
return x
But after playing around i keep getting errors like can't resolve this, No instance for (Num Positives) arising from the literal '0' or Can't make a derived instance of 'Num Positives'.
I think i'm making it way to complex for what i want but I just can't figure it out. I hope someone can help me or send me in the right direction.
Thanks
The problem with your code is that in positives the variable x has type Integer, so your return statements need to include the Positives constructor:
positives :: Gen Positives
positives =
do -- Pick an arbitrary integer:
x <- arbitrary
if (x == 0)
then return $ Positives 1
else if (x < 0)
then return $ Positives (-x)
else
return $ Positives x
If it helps, here is another way to write (a similarly working) positives function:
positives' :: Gen Positives
positives' = fmap (\x -> Positives (1 + abs x)) arbitrary
Here the arbitrary call is a Gen Integer, so the function argument to fmap has type Integer -> Positives.
To use your Positives newtype with QuickCheck you would use the Positives (de-)constructor to get at the Integer values:
prop_addition :: Positives -> Positives -> Bool
prop_addition (Positives a) (Positives b) = a + b >= 2
ghci> quickCheck prop_addtion
As #Carsten mentions in the comments, QuickCheck as a Positive a type which has an arbitrary instance for numeric and ordered types a.
Here's a quick way that doesn't require much understanding of QuickCheck but is a bit of a hack:
prop_CompareAnswerExMFM :: Integer -> Integer -> Integer -> Bool
prop_CompareAnswerExMFM b e m =
exMFM absB absE absM == exM absB absE absM
where -- following guarantees args are positive integers > 0
absB = 1 + abs b
absE = 1 + abs e
absM = 1 + abs m
and then you can just use
quickCheck prop_factored
Related
I wrote my first program in Haskell today. It compiles and runs successfully. And since it is not a typical "Hello World" program, it in fact does much more than that, so please congrats me :D
Anyway, I've few doubts regarding my code, and the syntax in Haskell.
Problem:
My program reads an integer N from the standard input and then, for each integer i in the range [1,N], it prints whether i is a prime number or not. Currently it doesn't check for input error. :-)
Solution: (also doubts/questions)
To solve the problem, I wrote this function to test primality of an integer:
is_prime :: Integer -> Bool
is_prime n = helper n 2
where
helper :: Integer -> Integer -> Bool
helper n i
| n < 2 * i = True
| mod n i > 0 = helper n (i+1)
| otherwise = False
It works great. But my doubt is that the first line is a result of many hit-and-trials, as what I read in this tutorial didn't work, and gave this error (I suppose this is an error, though it doesn't say so):
prime.hs:9:13:
Type constructor `Integer' used as a class
In the type signature for `is_prime':
is_prime :: Integer a => a -> Bool
According to the tutorial (which is a nicely-written tutorial, by the way), the first line should be: (the tutorial says (Integral a) => a -> String, so I thought (Integer a) => a -> Bool should work as well.)
is_prime :: (Integer a) => a -> Bool
which doesn't work, and gives the above posted error (?).
And why does it not work? What is the difference between this line (which doesn't work) and the line (which works)?
Also, what is the idiomatic way to loop through 1 to N? I'm not completely satisfied with the loop in my code. Please suggest improvements. Here is my code:
--read_int function
read_int :: IO Integer
read_int = do
line <- getLine
readIO line
--is_prime function
is_prime :: Integer -> Bool
is_prime n = helper n 2
where
helper :: Integer -> Integer -> Bool
helper n i
| n < 2 * i = True
| mod n i > 0 = helper n (i+1)
| otherwise = False
main = do
n <- read_int
dump 1 n
where
dump i x = do
putStrLn ( show (i) ++ " is a prime? " ++ show (is_prime i) )
if i >= x
then putStrLn ("")
else do
dump (i+1) x
You are misreading the tutorial. It would say the type signature should be
is_prime :: (Integral a) => a -> Bool
-- NOT Integer a
These are different types:
Integer -> Bool
This is a function that takes a value of type Integer and gives back a value of type Bool.
Integral a => a -> Bool
This is a function that takes a value of type a and gives back a value of type Bool.
What is a? It can be any type of the caller's choice that implements the Integral type class, such as Integer or Int.
(And the difference between Int and Integer? The latter can represent an integer of any magnitude, the former wraps around eventually, similar to ints in C/Java/etc.)
The idiomatic way to loop depends on what your loop does: it will either be a map, a fold, or a filter.
Your loop in main is a map, and because you're doing i/o in your loop, you need to use mapM_.
let dump i = putStrLn ( show (i) ++ " is a prime? " ++ show (is_prime i) )
in mapM_ dump [1..n]
Meanwhile, your loop in is_prime is a fold (specifically all in this case):
is_prime :: Integer -> Bool
is_prime n = all nondivisor [2 .. n `div` 2]
where
nondivisor :: Integer -> Bool
nondivisor i = mod n i > 0
(And on a minor point of style, it's conventional in Haskell to use names like isPrime instead of names like is_prime.)
Part 1: If you look at the tutorial again, you'll notice that it actually gives type signatures in the following forms:
isPrime :: Integer -> Bool
-- or
isPrime :: Integral a => a -> Bool
isPrime :: (Integral a) => a -> Bool -- equivalent
Here, Integer is the name of a concrete type (has an actual representation) and Integral is the name of a class of types. The Integer type is a member of the Integral class.
The constraint Integral a means that whatever type a happens to be, a has to be a member of the Integral class.
Part 2: There are plenty of ways to write such a function. Your recursive definition looks fine (although you might want to use n < i * i instead of n < 2 * i, since it's faster).
If you're learning Haskell, you'll probably want to try writing it using higher-order functions or list comprehensions. Something like:
module Main (main) where
import Control.Monad (forM_)
isPrime :: Integer -> Bool
isPrime n = all (\i -> (n `rem` i) /= 0) $ takeWhile (\i -> i^2 <= n) [2..]
main :: IO ()
main = do n <- readLn
forM_ [1..n] $ \i ->
putStrLn (show (i) ++ " is a prime? " ++ show (isPrime i))
It is Integral a, not Integer a. See http://www.haskell.org/haskellwiki/Converting_numbers.
map and friends is how you loop in Haskell. This is how I would re-write the loop:
main :: IO ()
main = do
n <- read_int
mapM_ tell_prime [1..n]
where tell_prime i = putStrLn (show i ++ " is a prime? " ++ show (is_prime i))
I am writing some code to work with arbitrary radix numbers in haskell. They will be stored as lists of integers representing the digits.
I almost managed to get it working, but I have run into the problem of converting a list of tuples [(a_1,b_1),...,(a_n,b_n)] into a single list which is defined as follows:
for all i, L(a_i) = b_i.
if there is no i such that a_i = k, a(k)=0
In other words, this is a list of (position,value) pairs for values in an array. If a position does not have a corresponding value, it should be set to zero.
I have read this (https://wiki.haskell.org/How_to_work_on_lists) but I don't think any of these methods are suitable for this task.
baseN :: Integer -> Integer -> [Integer]
baseN n b = convert_digits (baseN_digits n b)
chunk :: (Integer, Integer) -> [Integer]
chunk (e,m) = m : (take (fromIntegral e) (repeat 0))
-- This is broken because the exponents don't count for each other's zeroes
convert_digits :: [(Integer,Integer)] -> [Integer]
convert_digits ((e,m):rest) = m : (take (fromIntegral (e)) (repeat 0))
convert_digits [] = []
-- Converts n to base b array form, where a tuple represents (exponent,digit).
-- This works, except it ignores digits which are zero. thus, I converted it to return (exponent, digit) pairs.
baseN_digits :: Integer -> Integer -> [(Integer,Integer)]
baseN_digits n b | n <= 0 = [] -- we're done.
| b <= 0 = [] -- garbage input.
| True = (e,m) : (baseN_digits (n-((b^e)*m)) b)
where e = (greedy n b 0) -- Exponent of highest digit
m = (get_coef n b e 1) -- the highest digit
-- Returns the exponent of the highest digit.
greedy :: Integer -> Integer -> Integer -> Integer
greedy n b e | n-(b^e) < 0 = (e-1) -- We have overshot so decrement.
| n-(b^e) == 0 = e -- We nailed it. No need to decrement.
| n-(b^e) > 0 = (greedy n b (e+1)) -- Not there yet.
-- Finds the multiplicity of the highest digit
get_coef :: Integer -> Integer -> Integer -> Integer -> Integer
get_coef n b e m | n - ((b^e)*m) < 0 = (m-1) -- We overshot so decrement.
| n - ((b^e)*m) == 0 = m -- Nailed it, no need to decrement.
| n - ((b^e)*m) > 0 = get_coef n b e (m+1) -- Not there yet.
You can call "baseN_digits n base" and it will give you the corresponding array of tuples which needs to be converted to the correct output
Here's something I threw together.
f = snd . foldr (\(e,n) (i,l') -> ( e , (n : replicate (e-i-1) 0) ++ l')) (-1,[])
f . map (fromIntegral *** fromIntegral) $ baseN_digits 50301020 10 = [5,0,3,0,1,0,2,0]
I think I understood your requirements (?)
EDIT:
Perhaps more naturally,
f xs = foldr (\(e,n) fl' i -> (replicate (i-e) 0) ++ (n : fl' (e-1))) (\i -> replicate (i+1) 0) xs 0
The following data structure can be tested with the Tasty-SmallCheck related code that follows. There is a relation that has to hold with the constructor ShB: the second and the third positive integers should be at most as large as the first one.
data Shape = ShA Int Int Bool
| ShB Int Int Int Bool Bool
deriving (Show,Read,Eq)
The constructor ShA should have positive Int's but otherwise there is no relation between the parameters.
auxShA :: (Positive Int, Positive Int, Bool) -> Shape
auxShA (i,j,b) = ShA (fromIntegral i) (fromIntegral j) b
auxShB :: (Positive Int, Positive Int, Positive Int) -> Bool -> Bool -> Shape
auxShB (a1,a2,a3) = ShB i u d
where
(i,u,d) = auxTriplet (a1,a2,a3)
auxTriplet :: (Positive Int, Positive Int, Positive Int) -> (Int,Int,Int)
auxTriplet (a,b,c)
| a >= b && a >= c = (fromIntegral a, fromIntegral b, fromIntegral c)
| b >= a && b >= c = (fromIntegral b, fromIntegral a, fromIntegral c)
| otherwise = (fromIntegral c, fromIntegral a, fromIntegral b)
consB :: (Serial m a1, Serial m a2, Serial m a3, Serial m b, Serial m c) =>
((a1,a2,a3) -> b -> c -> e) -> Series m e
consB f = decDepth $
f <$> series
<~> series
<~> series
instance Monad m => Serial m Shape where
series = cons1 auxShA \/ consB auxShB
The generated cases are otherwise ok but there are duplicates that can be seen e.g. with
list 4 series :: [Shape]
The question is, how to generate the test cases with SmallCheck (tasty) when the following holds?
there are properties that has to hold, e.g. the first parameter has to Positive
what if the first parameter should be larger than 10::Int?
And continuing, what if the second parameter should between the first - 5 and the first, and the third should be between the second - 5 and the second?
Or, how to generate test cases that dynamically can depend on the previous generated values?
First thought was to write constructors to Shape that check that inputs are valid (e.g. the bullet points above), but the problem of duplicate test case generation would remain with that approach.
The above code uses similar solution as in
SmallCheck invariant -answer.
We have two functions that compare two different power functions, and return true if they return the same value (on the same input).
Then we have two other functions that test these functions against two lists to see if there is any value that doesn't return true.
But instead of using lists that use a range [1..100], we would like to use QuickCheck.
Is it possible to make QuickCheck only return positive integers?
Code:
comparePower1 :: Integer -> Integer -> Bool
comparePower1 n k = power n k == power1 n k
comparePower2 :: Integer -> Integer -> Bool
comparePower2 n k = power n k == power2 n k
testing1 = and [comparePower1 n k | n <- [0..100], k <- [0..100]]
testing2 = and [comparePower2 n k | n <- [0..100], k <- [0..100]]
QuickCheck has support for Positive numbers, but for the sake of this tutorial I will show you how to create your own Generator. One of the main features of QuickCheck is that you can design your own generator to output just what you need. For instance
genPos :: Gen Int
genPos = abs `fmap` (arbitrary :: Gen Int) `suchThat` (> 0)
Then you can create your own list genereator
genListOfPos :: Gen [Int]
genListOfPos = listOf genPos
Finally you can use forAll, pass the generator and profit.
main :: IO ()
main = do
quickCheck $ forAll genPos $ \x -> x > 0
quickCheck $ forAll genListOfPos $ all (> 0)
I have this theorem (not sure if that's the right word), and I want to get all the solutions.
pairCube limit = do
m <- natural exists "m"
n <- natural exists "n"
a <- natural exists "a"
constrain $ m^3 .== n^2
constrain $ m .< limit
return $ m + n .== a^2
res <- allSat (pairCube 1000)
-- Run from ghci
extractModels res :: [[Integer]]
This is trying to solve the problem:
There are infinite pairs of integers (m, n) such that m^3 = n^2 and m + n is a perfect square. What is the pair with the greatest m less than 1000?
I know the actual answer, just through brute forcing, but I want to do with SBV.
However, when I run the code it gives only the following values (in the form [m, n, a]):
[[9,27,6],[64,512,24],[]]
However, there are several other solutions with an m value less than 1000 that aren't included.
It's always good to give a full program:
{-# LANGUAGE ScopedTypeVariables #-}
import Data.SBV
pairCube :: SInteger -> Symbolic SBool
pairCube limit = do
(m :: SInteger) <- exists "m"
(n :: SInteger) <- exists "n"
(a :: SInteger) <- exists "a"
constrain $ m^(3::Integer) .== n^(2::Integer)
constrain $ m .< limit
return $ m + n .== a^(2::Integer)
main :: IO ()
main = print =<< allSat (pairCube 1000)
When I run it, I get:
Main> main
Solution #1:
m = 0 :: Integer
n = 0 :: Integer
a = 0 :: Integer
Solution #2:
m = 9 :: Integer
n = 27 :: Integer
a = -6 :: Integer
Solution #3:
m = 1 :: Integer
n = -1 :: Integer
a = 0 :: Integer
Solution #4:
m = 9 :: Integer
n = 27 :: Integer
a = 6 :: Integer
Solution #5:
m = 64 :: Integer
n = 512 :: Integer
a = -24 :: Integer
Solution #6:
m = 64 :: Integer
n = 512 :: Integer
a = 24 :: Integer
Unknown
Note the final Unknown.
Essentially, SBV queried Z3, and got 6 solutions; when SBV asked for the 7th, Z3 said "I don't know if there's any other solution." With non-linear arithmetic, this behavior is expected.
To answer the original question (i.e., find the max m), I changed the constraint to read:
constrain $ m .== limit
and coupled it with the following "driver:"
main :: IO ()
main = loop 1000
where loop (-1) = putStrLn "Can't find the largest m!"
loop m = do putStrLn $ "Trying: " ++ show m
mbModel <- extractModel `fmap` sat (pairCube m)
case mbModel of
Nothing -> loop (m-1)
Just r -> print (r :: (Integer, Integer, Integer))
After running about 50 minutes on my machine, Z3 produced:
(576,13824,-120)
So, clearly the allSat based approach is causing Z3 to give-up way earlier than what it can actually achieve if we fix m and iterate ourself. With a non-linear problem, expecting anything faster/better would be too much to ask of a general purpose SMT solver..