CSE 230 W11 - Homework #4, Due Wednesday, March 2nd



Preliminaries

To 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 “HW4”; you will receive a confirmation email after submitting. Please note that this address is unmonitored; if you have any questions about the assignment, email Pat at prondon@cs.ucsd.edu.

> {-# LANGUAGE TypeSynonymInstances, FlexibleContexts, NoMonomorphismRestriction, OverlappingInstances, FlexibleInstances #-}
> import Data.Map hiding (map)
> import Control.Monad.State
> import Control.Monad.Error
> import Control.Monad.Writer
> import Test.QuickCheck 
> import Control.Monad (forM, forM_)
> import Data.List (transpose, intercalate)
> quickCheckN n = quickCheckWith $ stdArgs { maxSuccess = n}

Problem 1: An Interpreter for WHILE++

Previously, you wrote a simple interpreter for WHILE. For this problem, you will use monad transformers to build an evaluator for WHILE++ which, adds exceptions and I/O to the original language.

As before, we have variables, and expressions.

> type Variable = String
> type Store = Map Variable Value
>
> data Value =
> IntVal Int
> | BoolVal Bool
> deriving (Show)
>
> data Expression =
> Var Variable
> | Val Value
> | Op Bop Expression Expression
> deriving (Show)
>
> data Bop =
> Plus
> | Minus
> | Times
> | Divide
> | Gt
> | Ge
> | Lt
> | Le
> deriving (Show)

Programs in the language are simply values of the type

> data Statement =
> Assign Variable Expression
> | If Expression Statement Statement
> | While Expression Statement
> | Sequence Statement Statement
> | Skip
> | Print String Expression
> | Throw Expression
> | Try Statement Variable Statement
> deriving (Show)

The only new constructs are the Print, Throw and the Try statements.

We will use the State [monad][2] to represent the world-transformer. Intuitively, State s a is equivalent to the world-transformer s -> (a, s). See the above documentation for more details. You can ignore the bits about StateT for now.

Use monad transformers to write a function

> evalS :: (MonadState Store m, MonadError Value m, MonadWriter String m) => Statement -> m ()
> evalS = error "TODO"

and use the above function to implement a second function

> execute :: Store -> Statement -> (Store, Maybe Value, String)
> execute = error "TODO"

such that execute st s returns a triple (st', exn, log) where

Requirements

In the case of exceptional termination, the st' should be the state *at the point where the last exception was thrown, and log should include all the messages upto that point — make sure you stack your transformers appropriately!

Example 1

If st is the empty state (all variables undefined) and s is the program

X := 0 ;
Y := 1 ;
print "hello world: " X;
if X < Y then
throw (X+Y)
else
skip
endif;
Z := 3

then execute st s should return the triple

(fromList [("X", IntVal 0), ("Y",  IntVal 1)], Just (IntVal 1), "hello world: IntVal 0\n")

The program is provided as a Haskell value below:

> mksequence = foldr Sequence Skip
> testprog1 = mksequence [Assign "X" $ Val $ IntVal 0,
> Assign "Y" $ Val $ IntVal 1,
> Print "hello world: " $ Var "X",
> If (Op Lt (Var "X") (Var "Y")) (Throw (Op Plus (Var "X") (Var "Y")))
> Skip,
> Assign "Z" $ Val $ IntVal 3]

Example 2

If st is the empty state (all variables undefined) and s is the program

X := 0 ;
Y := 1 ;
try
if X < Y then
A := 100;
throw (X+Y);
B := 200;
else
skip
endif;
catch E with
Z := E + A
endwith

then execute st s should return the triple

( fromList [("A", IntVal 100), ("E", IntVal 1)
,("X", IntVal 0), ("Y", IntVal 1)
,("Z", IntVal 101)]
, Nothing
, "")

Again, the program as a Haskell value:

> testprog2 = mksequence [Assign "X" $ Val $ IntVal 0,
> Assign "Y" $ Val $ IntVal 1,
> Try (If (Op Lt (Var "X") (Var "Y"))
> (mksequence [Assign "A" $ Val $ IntVal 100,
> Throw (Op Plus (Var "X") (Var "Y")),
> Assign "B" $ Val $ IntVal 200])
> Skip)
> "E"
> (Assign "Z" $ Op Plus (Var "E") (Var "A"))]

Problem 2: [Circuit Testing]

Credit: UPenn CIS552

For this problem, you will look at a model of circuits in Haskell.

Signals

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 zs

Simulation

Next, 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 20

Testing support (QuickCheck helpers)

Next, 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] $ \_ -> arbitrary

To 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) 0

And to probe signals at specific points.

> sampleAt n (Sig b) = b !! n
> sampleAtN n signals = map (sampleAt n) signals
> sample1 = sampleAt 0
> sampleN = sampleAtN 0

Basic Gates

The 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)

Basic Signals

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 : xs

Combinational circuits

NOTE 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)

Problem: Subtraction

  1. Using 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"
  1. Using the 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"

Problem: Multiplication

  1. Using 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"
  1. Define a 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"