Before starting this part of the assignment,
$ cabal install quickcheckTo complete this homework, download this file as plain text and answer each question, filling in code where it says "TODO". Your code must typecheck against the given type signatures. Feel free to add your own tests to this file to exercise the functions you write. Submit your homework by sending this file, filled in appropriately, to cse230@goto.ucsd.edu with the subject “HW3”; you will receive a confirmation email after submitting. Please note that this address is unmonitored; if you have any questions about the assignment, post to Piazza.
> {-# LANGUAGE TypeSynonymInstances, FlexibleContexts, NoMonomorphismRestriction, OverlappingInstances, FlexibleInstances #-}> module Hw3 where> import qualified Data.Map as Map> import Control.Monad.State
> import Control.Monad.Error
> import Control.Monad.Writer> import Test.QuickCheck hiding ((===))
> import Control.Monad (forM, forM_)
> import Data.List (transpose, intercalate)> quickCheckN n = quickCheckWith $ stdArgs { maxSuccess = n}Tell us your name, email and student ID, by replacing the respective strings below
> myName = "Write Your Name Here"
> myEmail = "Write Your Email Here"
> mySID = "Write Your SID Here"Recall the old type of binary search trees from HW2.
> data BST k v = Emp
> | Bind k v (BST k v) (BST k v)
> deriving (Show)
>
> toBinds :: BST t t1 -> [(t, t1)]
> toBinds Emp = []
> toBinds (Bind k v l r) = toBinds l ++ [(k,v)] ++ toBinds rThe following function tests whether a tree satisfies the binary-search-order invariant.
> isBSO :: Ord a => BST a b -> Bool
> isBSO Emp = True
> isBSO (Bind k v l r) = all (< k) lks && all (k <) rks && isBSO l && isBSO r
> where lks = map fst $ toBinds l
> rks = map fst $ toBinds rFinally, to test your implementation, we will define a type of operations over trees
> data BSTop k v = BSTadd k v | BSTdel k
> deriving (Eq, Show)and a function that constructs a tree from a sequence of operations
> ofBSTops :: Ord k => [BSTop k v] -> BST k v
> ofBSTops = foldr doOp Emp
> where doOp (BSTadd k v) = bstInsert k v
> doOp (BSTdel k) = bstDelete k and that constructs a reference Map from a sequence of operations
> mapOfBSTops :: Ord k => [BSTop k a] -> Map.Map k a
> mapOfBSTops = foldr doOp Map.empty
> where doOp (BSTadd k v) = Map.insert k v
> doOp (BSTdel k) = Map.delete kand functions that generate an arbitrary BST operations
> keys :: [Int]
> keys = [0..10]
>
> genBSTadd, genBSTdel, genBSTop :: Gen (BSTop Int Char)
> genBSTadd = liftM2 BSTadd (elements keys) (elements ['a'..'z'])
> genBSTdel = liftM BSTdel (elements keys)
> genBSTop = frequency [(5, genBSTadd), (1, genBSTdel)] Write an insertion function
> bstInsert :: (Ord k) => k -> v -> BST k v -> BST k v
> bstInsert = error "TBD"such that bstInsert k v t inserts a key k with value v into the tree t. If k already exists in the input tree, then its value should be replaced with v. When you are done, your code should satisfy the following QC properties.
> prop_insert_bso :: Property
> prop_insert_bso = forAll (listOf genBSTadd) $ \ops ->
> isBSO (ofBSTops ops)
>
> prop_insert_map = forAll (listOf genBSTadd) $ \ops ->
> toBinds (ofBSTops ops) == Map.toAscList (mapOfBSTops ops)Write a deletion function for BSTs of this type:
> bstDelete :: (Ord k) => k -> BST k v -> BST k v
> bstDelete k t = error "TBD"such that bstDelete k t removes the key k from the tree t. If k is absent from the input tree, then the tree is returned unchanged as the output. When you are done, your code should satisfy the following QC properties.
> prop_delete_bso :: Property
> prop_delete_bso = forAll (listOf genBSTop) $ \ops ->
> isBSO (ofBSTops ops)
>
> prop_delete_map = forAll (listOf genBSTop) $ \ops ->
> toBinds (ofBSTops ops) == Map.toAscList (mapOfBSTops ops)The following function determines the height of a BST
> height (Bind _ _ l r) = 1 + max (height l) (height r)
> height Emp = 0We say that a tree is balanced if
> isBal (Bind _ _ l r) = isBal l && isBal r && abs (height l - height r) <= 2
> isBal Emp = TrueWrite a balanced tree generator
> genBal :: Gen (BST Int Char)
> genBal = error "TBD"such that
> prop_genBal = forAll genBal isBalRig it so that your insert and delete functions also create balanced trees. That is, they satisfy the properties
> prop_insert_bal :: Property
> prop_insert_bal = forAll (listOf genBSTadd) $ isBal . ofBSTops
>
> prop_delete_bal :: Property
> prop_delete_bal = forAll (listOf genBSTop) $ isBal . ofBSTopsCredit: UPenn CIS552
For this problem, you will look at a model of circuits in Haskell.
A signal is a list of booleans.
> newtype Signal = Sig [Bool]By convention, all signals are infinite. We write a bunch of lifting functions that lift boolean operators over signals.
> lift0 :: Bool -> Signal
> lift0 a = Sig $ repeat a
>
> lift1 :: (Bool -> Bool) -> Signal -> Signal
> lift1 f (Sig s) = Sig $ map f s
>
> lift2 :: (Bool -> Bool -> Bool) -> (Signal, Signal) -> Signal
> lift2 f (Sig xs, Sig ys) = Sig $ zipWith f xs ys
>
> lift22 :: (Bool -> Bool -> (Bool, Bool)) -> (Signal, Signal) -> (Signal,Signal)
> lift22 f (Sig xs, Sig ys) =
> let (zs1,zs2) = unzip (zipWith f xs ys)
> in (Sig zs1, Sig zs2)
>
> lift3 :: (Bool->Bool->Bool->Bool) -> (Signal, Signal, Signal) -> Signal
> lift3 f (Sig xs, Sig ys, Sig zs) = Sig $ zipWith3 f xs ys zsNext, we have some helpers that can help us simulate a circuit by showing how it behaves over time. For testing or printing, we truncate a signal to a short prefix
> truncatedSignalSize = 20
> truncateSig bs = take truncatedSignalSize bs
>
> instance Show Signal where
> show (Sig s) = show (truncateSig s) ++ "..."
>
> trace :: [(String, Signal)] -> Int -> IO ()
> trace desc count = do
> putStrLn $ intercalate " " names
> forM_ rows $ putStrLn . intercalate " " . rowS
> where (names, wires) = unzip desc
> rows = take count . transpose . map (\ (Sig w) -> w) $ wires
> rowS bs = zipWith (\n b -> replicate (length n - 1) ' ' ++ (show (binary b))) names bs
>
> probe :: [(String,Signal)] -> IO ()
> probe desc = trace desc 1
>
> simulate :: [(String, Signal)] -> IO ()
> simulate desc = trace desc 20Next, we have a few functions that help to generate random tests
> instance Arbitrary Signal where
> arbitrary = do
> x <- arbitrary
> Sig xs <- arbitrary
> return $ Sig (x : xs)
>
> arbitraryListOfSize n = forM [1..n] $ \_ -> arbitraryTo check whether two values are equivalent
> class Agreeable a where
> (===) :: a -> a -> Bool
>
> instance Agreeable Signal where
> (Sig as) === (Sig bs) =
> all (\x->x) (zipWith (==) (truncateSig as) (truncateSig bs))
>
> instance (Agreeable a, Agreeable b) => Agreeable (a,b) where
> (a1,b1) === (a2,b2) = (a1 === a2) && (b1 === b2)
>
> instance Agreeable a => Agreeable [a] where
> as === bs = all (\x->x) (zipWith (===) as bs)To convert values from boolean to higher-level integers
> class Binary a where
> binary :: a -> Integer
>
> instance Binary Bool where
> binary b = if b then 1 else 0
>
> instance Binary [Bool] where
> binary = foldr (\x r -> (binary x) + 2 *r) 0And to probe signals at specific points.
> sampleAt n (Sig b) = b !! n
> sampleAtN n signals = map (sampleAt n) signals
> sample1 = sampleAt 0
> sampleN = sampleAtN 0The basic gates from which we will fashion circuits can now be described.
> or2 :: (Signal, Signal) -> Signal
> or2 = lift2 $ \x y -> x || y
>
> xor2 :: (Signal, Signal) -> Signal
> xor2 = lift2 $ \x y -> (x && not y) || (not x && y)
>
> and2 :: (Signal, Signal) -> Signal
> and2 = lift2 $ \x y -> x && y
>
> imp2 :: (Signal, Signal) -> Signal
> imp2 = lift2 $ \x y -> (not x) || y
>
> mux :: (Signal, Signal, Signal) -> Signal
> mux = lift3 (\b1 b2 select -> if select then b1 else b2)
>
> demux :: (Signal, Signal) -> (Signal, Signal)
> demux args = lift22 (\i select -> if select then (i, False) else (False, i)) args
>
> muxN :: ([Signal], [Signal], Signal) -> [Signal]
> muxN (b1,b2,sel) = map (\ (bb1,bb2) -> mux (bb1,bb2,sel)) (zip b1 b2)
>
> demuxN :: ([Signal], Signal) -> ([Signal], [Signal])
> demuxN (b,sel) = unzip (map (\bb -> demux (bb,sel)) b)Similarly, here are some basic signals
> high = lift0 True
> low = lift0 False
>
> str :: String -> Signal
> str cs = Sig $ (map (== '1') cs) ++ (repeat False)
>
> delay :: Bool -> Signal -> Signal
> delay init (Sig xs) = Sig $ init : xsNOTE When you are asked to implement a circuit, you must ONLY use the above gates or smaller circuits built from the gates.
For example, the following is a half-adder (that adds a carry-bit to a single bit).
> halfadd :: (Signal, Signal) -> (Signal, Signal)
> halfadd (x,y) = (sum,cout)
> where sum = xor2 (x, y)
> cout = and2 (x, y)Here is a simple property about the half-adder
> prop_halfadd_commut b1 b2 =
> halfadd (lift0 b1, lift0 b2) === halfadd (lift0 b2, lift0 b1) We can use the half-adder to build a full-adder
> fulladd (cin, x, y) = (sum, cout)
> where (sum1, c1) = halfadd (x,y)
> (sum, c2) = halfadd (cin, sum1)
> cout = xor2 (c1,c2)
>
> test1a = probe [("cin",cin), ("x",x), ("y",y), (" sum",sum), ("cout",cout)]
> where cin = high
> x = low
> y = high
> (sum,cout) = fulladd (cin, x, y)and then an n-bit adder
> bitAdder :: (Signal, [Signal]) -> ([Signal], Signal)
> bitAdder (cin, []) = ([], cin)
> bitAdder (cin, x:xs) = (sum:sums, cout)
> where (sum, c) = halfadd (cin,x)
> (sums, cout) = bitAdder (c,xs)
>
> test1 = probe [("cin",cin), ("in1",in1), ("in2",in2), ("in3",in3), ("in4",in4),
> (" s1",s1), ("s2",s2), ("s3",s3), ("s4",s4), ("c",c)]
> where
> cin = high
> in1 = high
> in2 = high
> in3 = low
> in4 = high
> ([s1,s2,s3,s4], c) = bitAdder (cin, [in1,in2,in3,in4])The correctness of the above circuit is described by the following property that compares the behavior of the circuit to the reference implementation which is an integer addition function
> prop_bitAdder_Correct :: Signal -> [Bool] -> Bool
> prop_bitAdder_Correct cin xs =
> binary (sampleN out ++ [sample1 cout]) == binary xs + binary (sample1 cin)
> where (out, cout) = bitAdder (cin, map lift0 xs) Finally, we can use the bit-adder to build an adder that adds two N-bit numbers
> adder :: ([Signal], [Signal]) -> [Signal]
> adder (xs, ys) =
> let (sums,cout) = adderAux (low, xs, ys)
> in sums ++ [cout]
> where
> adderAux (cin, [], []) = ([], cin)
> adderAux (cin, x:xs, y:ys) = (sum:sums, cout)
> where (sum, c) = fulladd (cin,x,y)
> (sums,cout) = adderAux (c,xs,ys)
> adderAux (cin, [], ys) = adderAux (cin, [low], ys)
> adderAux (cin, xs, []) = adderAux (cin, xs, [low])
>
> test2 = probe [ ("x1", x1), ("x2",x2), ("x3",x3), ("x4",x4),
> (" y1",y1), ("y2",y2), ("y3",y3), ("y4",y4),
> (" s1",s1), ("s2",s2), ("s3",s3), ("s4",s4), (" c",c) ]
> where xs@[x1,x2,x3,x4] = [high,high,low,low]
> ys@[y1,y2,y3,y4] = [high,low,low,low]
> [s1,s2,s3,s4,c] = adder (xs, ys)And we can specify the correctness of the adder circuit by
> prop_Adder_Correct :: [Bool] -> [Bool] -> Bool
> prop_Adder_Correct l1 l2 =
> binary (sampleN sum) == binary l1 + binary l2
> where sum = adder (map lift0 l1, map lift0 l2) prop_bitAdder_Correct as a model, write a specification for a single-bit subtraction function that takes as inputs a N-bit binary number and a single bit to be subtracted from it and yields as outputs an N-bit binary number. Subtracting one from zero should yield zero.> prop_bitSubtractor_Correct :: Signal -> [Bool] -> Bool
> prop_bitSubtractor_Correct = error "TODO"bitAdder circuit as a model, define a bitSubtractor circuit that implements this functionality and use QC to check that your behaves correctly.> bitSubtractor :: (Signal, [Signal]) -> ([Signal], Signal)
> bitSubtractor = error "TODO"prop_Adder_Correct as a model, write down a QC specification for a multiplier circuit that takes two binary numbers of arbitrary width as input and outputs their product.> prop_Multiplier_Correct :: [Bool] -> [Bool] -> Bool
> prop_Multiplier_Correct = error "TODO"multiplier circuit and check that it satisfies your specification. (Looking at how adder is defined will help with this, but you’ll need a little more wiring. To get an idea of how the recursive structure should work, think about how to multiply two binary numbers on paper.)> multiplier :: ([Signal], [Signal]) -> [Signal]
> multiplier = error "TODO"