# My Weblog

## Proving the correctness of code using SMT sovler.

SMT solvers are great tool in computer science. One use of these solvers to prove the correctness of code. I wrote Haskell code to compute the inverse of polynomial using extended euclidean algorithm in Galois Field GF ( 2n ). If we try to analyze the code of inversePolynomial then every thing is normal except passing the depth parameter. It is unnecessary because we know that recursion is going to terminate. Initially I wrote the code without depth and the program was not terminating so I wrote mail to Levent Erkok and according to him

Your code suffers from what’s known as the “symbolic termination”. See Section 7.4 of: http://goo.gl/0E7wkm for a discussion of this issue in the context of Cryptol; but the same comments apply here as well. Briefly, when you recurse in inversePoly, SBV does not know when to stop the recursion of both branches in the ite: Thus it keeps expanding both branches ad infinitum.

Luckily, symbolic termination is usually easy to deal with once you identify what the problem is. Typically, one needs to introduce a recursion depth counter, which would stop recursion when it reaches a pre-determined depth that is determined to be safe. In fact, SBV comes with the regular GCD example that talks about this issue in detail:

I’d recommend reading the source code of the above; it should clarify the problem. You’ll need to find out a recursion-depth that’ll work for your example (try running it on concrete values and coming up with a safe number); and then prove that the bound is actually correct as explained in the above. The recursion depth bound just needs to be an overapproximation: If the algorithm needs 20 iterations but you use 30 as your recursion depth, nothing will go wrong; it’ll just produce bigger code and thus be less efficient and maybe more complicated for the backend prover. Of course, if you put a depth of 18 when 20 is needed, then it will be wrong and you won’t be able to establish correctness.

My verification condition is inverse of inverse of polynomial is same as polynomial. Right now I have no idea about proving the upper bound on depth so I just took 20. If you have any suggestion/comments then please let me know.

```
import Data.SBV.Bridge.Z3
import Data.Maybe

inversePolynomial ::  [ Int ] -> [ Int ] ->  SWord16
inversePolynomial poly reducer =  inversePoly  reducer' rem' first' second' depth'  where
poly' =  polynomial poly :: SWord16
reducer' =  polynomial reducer :: SWord16
first' =  0 :: SWord16
second' = 1 :: SWord16
depth' = 20 :: SWord16
( quot', rem' ) = pDivMod poly' reducer'

inversePoly :: SWord16 -> SWord16 -> SWord16 -> SWord16  ->  SWord16 -> SWord16
inversePoly  reducer'' poly'' first'' second'' depth'' =
ite ( depth'' .== 0 ||| rem'' .== ( 0 :: SWord16 ) ) (  second'' )  ( inversePoly  poly'' rem'' second'' ret ( depth'' - 1 ) )  where
( quot'', rem'' ) = pDivMod reducer'' poly''
ret =  pAdd ( pMult ( quot'', second'', [] ) ) first''

isInversePolyCorrect :: SWord16 -> SWord16 -> SBool
isInversePolyCorrect poly reducer = inversePoly reducer ( inversePoly reducer rem first second depth ) first second depth .== rem where
( quot, rem ) = pDivMod  poly reducer
first = 0 :: SWord16
second = 1 :: SWord16
depth = 20 :: SWord16

*Main> showPoly ( inversePolynomial  [ 6,4,1,0] [ 8,4,3,1,0] )
"x^7 + x^6 + x^3 + x"
*Main> proveWith z3 { verbose=False }  \$ forAll ["x"] \$ \x ->  pMod x ( 283 :: SWord16 ) ./=0 ==> isInversePolyCorrect x ( 283 :: SWord16 )
Q.E.D.
*Main> proveWith cvc4 { verbose=False }  \$ forAll ["x"] \$ \x ->  pMod x ( 283 :: SWord16 ) ./=0 ==> isInversePolyCorrect x ( 283 :: SWord16 )
*** Exception: fd:10: hFlush: resource vanished (Broken pipe)
*Main> proveWith yices { verbose=False }  \$ forAll ["x"] \$ \x ->  pMod x ( 283 :: SWord16 ) ./=0 ==> isInversePolyCorrect x ( 283 :: SWord16 )
Q.E.D.
```

September 21, 2013

## Counting Inversions

First week of algorithm course programming assignment is to compute the number of inversion count of given array. We have a given array A[0...n - 1] of n distinct positive integers and we have to count all the pairs ( i , j ) such that i < j and A[i] > A[j].
1. Brute Force
Brute force idea is very simple. Run two loops and keep counting the respective pairs for which i < j and A[i] > A[j]. The complexity of brute force is O( n2 ).

```bruteForce :: [ Int ] -> Int
bruteForce [] = 0
bruteForce [_] = 0
bruteForce ( x : xs ) = ( length . filter ( x > ) \$ xs ) + bruteForce xs

*Main> bruteForce [ 3 , 1 , 2]
2
*Main> bruteForce [ 2,3,8,6,1]
5
```

2. Divide and Conquer
We can modify the merge sort to compute the inversion count. The idea is to compute the inversion count of during the merge phase. Inversion count of array C resulting from merging two sorted array A and B is sum of inversion count of A, inversion count of B and their cross inversion. The inversion count of single element is 0 ( base case ). The only trick is compute the cross inversion. What will be the inversion count of following array if we merge them ?
A = [ 1 , 2 , 3 ]
B = [ 4 , 5 , 6 ]
Clearly 0 because all element of A is less than the first element of B. What is the inversion count of C = [ 4 , 5 , 6 ] and D = [ 1 , 2 , 3 ] ? First element of D ( 1 ) is less than first element of C ( 4 ) so we have total three pairs ( 4 , 1 ) ( 5 , 1 ) ( 6 , 1). Similarly for 2 and 3 ( total 9 inversion count ). It reveals that if we have any element y in D is less than certain element x in C then all the element of A after x will also be greater than y and we will have those many cross inversion counts. More clear explanation. Wrote a quick Haskell code and got the answer.

```import Data.List
import Data.Maybe ( fromJust )
import qualified Data.ByteString.Lazy.Char8 as BS

inversionCnt :: [ Int ] -> [ Int ] -> [ Int ] -> Int -> ( Int , [ Int ] )
inversionCnt [] ys  ret  cnt = ( cnt , reverse ret ++ ys )
inversionCnt xs []  ret  cnt = ( cnt , reverse ret ++ xs )
inversionCnt u@( x : xs ) v@( y : ys )  ret  cnt
| x <= y = inversionCnt xs v  (  x : ret   )   cnt
| otherwise = inversionCnt u ys  ( y : ret )  ( cnt + length u )

merge ::  ( Int , [ Int ] ) -> ( Int , [ Int ] ) -> ( Int , [ Int ] )
merge ( cnt_1 , xs ) ( cnt_2 , ys ) = ( cnt_1 + cnt_2 + cnt , ret ) where
( cnt , ret ) = inversionCnt xs ys []   0

mergeSort ::  [ Int ] -> ( Int , [ Int ] )
mergeSort [ x ] = ( 0 , [ x ] )
mergeSort  xs   = merge ( mergeSort xs' ) ( mergeSort ys' ) where
( xs' , ys' ) = splitAt ( div ( length xs ) 2 ) xs

main =  BS.interact \$ ( \x -> BS.pack \$ show x ++ "\n" ) . fst
.  mergeSort . map ( fst . fromJust .  BS.readInt :: BS.ByteString -> Int  )
. BS.lines

Mukeshs-MacBook-Pro:Puzzles mukeshtiwari\$ ghc  -fforce-recomp -O2  Inversion.hs
[1 of 1] Compiling Main             ( Inversion.hs, Inversion.o )
Mukeshs-MacBook-Pro:Puzzles mukeshtiwari\$ time ./Inversion <  IntegerArray.txt
2407905288

real	0m8.002s
user	0m7.881s
sys	0m0.038s
```

I tried to compare this solution with GeeksforGeeks‘s solution written in C and it was a complete surprise to me.

```Mukeshs-MacBook-Pro:Puzzles mukeshtiwari\$ time ./a.out <  IntegerArray.txt
Number of inversions are 2407905288

real	0m0.039s
user	0m0.036s
sys	0m0.002s
```

Almost 200 times faster! My Haskell intuition told me that I am doing certainly some wrong. I started analyzing the code and got the point. In inversionCnt function, I am doing extra work for computing the length of list u ( O ( n ) ) so rather than doing this extra work, I can pass the length.

```import Data.List
import Data.Maybe ( fromJust )
import qualified Data.ByteString.Lazy.Char8 as BS

inversionCnt :: [ Int ] -> [ Int ] -> [ Int ] -> Int -> Int -> ( Int , [ Int ] )
inversionCnt [] ys  ret _ cnt = ( cnt , reverse ret ++ ys )
inversionCnt xs []  ret _ cnt = ( cnt , reverse ret ++ xs )
inversionCnt u@( x : xs ) v@( y : ys )  ret n cnt
| x <= y = inversionCnt xs v  (  x : ret   )  ( pred n ) cnt
| otherwise = inversionCnt u ys  ( y : ret ) n ( cnt + n )

merge ::  ( Int , [ Int ] ) -> ( Int , [ Int ] ) -> ( Int , [ Int ] )
merge ( cnt_1 , xs ) ( cnt_2 , ys ) = ( cnt_1 + cnt_2 + cnt , ret ) where
( cnt , ret ) = inversionCnt xs ys [] ( length xs )  0

mergeSort ::  [ Int ] -> ( Int , [ Int ] )
mergeSort [ x ] = ( 0 , [ x ] )
mergeSort  xs   = merge ( mergeSort xs' ) ( mergeSort ys' ) where
( xs' , ys' ) = splitAt ( div ( length xs ) 2 ) xs

main =  BS.interact \$ ( \x -> BS.pack \$ show x ++ "\n" ) . fst
.  mergeSort . map ( fst . fromJust .  BS.readInt :: BS.ByteString -> Int  )
. BS.lines

Mukeshs-MacBook-Pro:Puzzles mukeshtiwari\$ time ./Inversion <  IntegerArray.txt
2407905288

real	0m0.539s
user	0m0.428s
sys	0m0.031s
```

Now 10 times slower and still lot of room for improvement but for now I am happy with this :). You can also try to solve INVCNT. Run this code on Ideone.

July 4, 2013

## Computing a^n using binary representation of natural number in Agda

While following the Agda tutorial, I wrote this code to compute an. Not very elegant example probably because I am not expert and still learning about dependent types. Maybe after getting some more experience and knowledge, I will try to prove the correctness and complexity of this algorithm. See the post of Twan van Laarhoven about proving correctness and complexity of merge sort in Agda.

```
module PowerFunction where

data ℕ⁺ : Set where
one : ℕ⁺
double : ℕ⁺ → ℕ⁺
double⁺¹ : ℕ⁺ → ℕ⁺

add : ℕ⁺ → ℕ⁺ → ℕ⁺
add one one = double one
add one ( double x ) = double⁺¹ x
add one ( double⁺¹ x ) = double ( add x one )
add ( double x ) one =  double⁺¹ x
add ( double x ) ( double y ) = double ( add x y )
add ( double x ) (  double⁺¹ y ) =  double⁺¹ (  add x y )
add (  double⁺¹ x ) one =   double (  add x one )
add (  double⁺¹ x ) (  double y ) =  double⁺¹ (  add x y  )
add (  double⁺¹ x ) (  double⁺¹ y ) = double ( add (  add x y ) one )

mult : ℕ⁺ → ℕ⁺ → ℕ⁺
mult x one = x
mult x ( double y ) = double ( mult x y )
mult x ( double⁺¹ y ) = add x ( double ( mult x y ) )

pow : ℕ⁺ → ℕ⁺ → ℕ⁺
pow one _ = one
pow x  one =  x
pow x ( double one ) = mult x x
pow x ( double y ) =  pow ( pow x y ) ( double one )
pow x ( double⁺¹ y ) =  mult x ( pow ( pow x y ) ( double one ) )

```

Computing 2 ^ 6
pow ( double one ) ( double ( double⁺¹ one ))
double (double (double (double (double (double one)))))
If you have suggestion then please let me know.

June 12, 2013

## ZeroMQ for distributed computing.

This post is influenced by ØMQ – The Guide By Pieter Hintjens and translation of codes in Haskell. I suggest you to read The Guide By Pieter Hintjens and if you are interested in Haskell code then you see these codes.

The client sends “Accept hello from Client. ” to the server, which replies with “I got you.”. This Haskell server opens a ØMQ socket on port 5555, reads requests on it, and replies with “I got you” to each request.The REQ-REP socket pair is in lockstep. The client issues send and then receive, in a loop (or once if that’s all it needs). Doing any other sequence (e.g., sending two messages in a row) will result in a return code of -1 from the send or receive call.

```
import System.ZMQ3
import qualified Data.ByteString.Char8 as BS
import Data.ByteString.Lazy.Internal as BL
import Data.IORef
main = do
c <- context
s <- socket c Rep
bind s "tcp://127.0.0.1:5555"
counter <- newIORef 0
forever \$ do
print res
send' s [] ( BL.packChars \$ "I got you. " ++ show t )
modifyIORef counter ( +1 )
close s
destroy c
return ()

```

Client

```import System.ZMQ3
import qualified Data.ByteString.Char8 as BS
import qualified Data.ByteString.Lazy.Internal as BL
import Data.IORef

main = do
c <- context
s <- socket c Req
connect s "tcp://127.0.0.1:5555"
counter <- newIORef 0
forever \$ do
send' s [] ( BL.packChars \$ "Accept hello from Client. " ++ show t )
print msg
modifyIORef counter ( +1 )
return ()

```

Running a server with two clients.

```
Mukeshs-MacBook-Pro:ZMQ mukeshtiwari\$ ./ZeroMqServer
"Accept hello from Client. 0"
"Accept hello from Client. 1"
"Accept hello from Client. 2"
"Accept hello from Client. 3"
"Accept hello from Client. 4"
"Accept hello from Client. 5"
"Accept hello from Client. 6"
"Accept hello from Client. 7"
"Accept hello from Client. 8"
"Accept hello from Client. 9"
"Accept hello from Client. 10"
"Accept hello from Client. 11"
"Accept hello from Client. 12"
"Accept hello from Client. 13"
"Accept hello from Client. 14"
"Accept hello from Client. 15"

"Accept hello from Client. 0"
"Accept hello from Client. 228"
"Accept hello from Client. 1"
"Accept hello from Client. 229"
"Accept hello from Client. 2"
"Accept hello from Client. 230"
"Accept hello from Client. 3"
"Accept hello from Client. 231"
"Accept hello from Client. 4"
"Accept hello from Client. 232"

First client.
Mukeshs-MacBook-Pro:ZMQ mukeshtiwari\$ ./ZeroMqClient
"I got you. 0"
"I got you. 1"
"I got you. 2"
"I got you. 3"
"I got you. 4"
"I got you. 5"
"I got you. 6"
"I got you. 7"
"I got you. 8"
"I got you. 9"
"I got you. 10"
"I got you. 11"
"I got you. 12"

Second client
Mukeshs-MacBook-Pro:ZMQ mukeshtiwari\$ ./ZeroMqClient
"I got you. 228"
"I got you. 230"
"I got you. 232"
"I got you. 234"
"I got you. 236"
"I got you. 238"
"I got you. 240"
"I got you. 242"
"I got you. 244"
"I got you. 246"
"I got you. 248"
"I got you. 250"
"I got you. 252"
"I got you. 254"
"I got you. 256"
"I got you. 258"
"I got you. 260"
"I got you. 262"
"I got you. 264"
"I got you. 266"
```

### Publish-Subscribe

Data publishing server which publishes weather data for zip codes in range 500 and 2000.

```import System.ZMQ3
import qualified Data.ByteString.Char8 as BS hiding ( putStrLn )
import qualified Data.ByteString.Lazy.Internal as BL
import System.Random

main = do
c <- context
publisher <- socket c Pub
bind publisher "tcp://127.0.0.1:5556"
bind publisher "ipc://weather.ipc"
forever \$ do
zipcode <- randomRIO ( ( 500 , 2000 ) ::  ( Int , Int ) )
temp <- randomRIO ( 10 , 45 ) :: IO Int
relhum <- randomRIO ( 0 , 100 ) :: IO Int
putStrLn \$ show zipcode ++ " " ++ show temp ++ " " ++ show relhum
send' publisher [] ( BL.packChars \$ show zipcode ++ " " ++ show temp ++
" " ++  show relhum )
close publisher
destroy c
return ()
```

Client who is only interested in two zip codes.

```import System.ZMQ3
import qualified Data.ByteString.Char8 as BS
import qualified Data.ByteString.Lazy.Internal as BL
import System.Random

main = do
c <- context
subscriber <- socket c Sub
connect subscriber "tcp://127.0.0.1:5556"
subscribe subscriber ( BS.pack "1000" )
subscribe subscriber ( BS.pack "1010" )
forever \$ do
print update

close subscriber
destroy c

```

Our server is publishing lot of data but client is only interested in two zip codes.

```

Mukeshs-MacBook-Pro:ZMQ mukeshtiwari\$ ./ZeroMqWeatherPubServer
1568 27 85
924 41 46
1461 15 46
1867 44 28
1013 23 100
1052 13 6
1720 45 6
1480 12 94
1852 33 4
1295 20 6
925 18 77
935 37 94
1670 11 6
1285 39 38
1613 44 99
1888 26 62
1011 21 45
993 45 45
1402 26 86
1639 13 65
1285 18 40
1960 38 18
1160 27 39
1374 16 59
665 25 22

Mukeshs-MacBook-Pro:ZMQ mukeshtiwari\$ ./ZeroMqWeatherClient
"1000 34 89"
"1010 19 70"
"1010 15 86"
"1000 38 57"
"1010 12 42"
"1000 25 1"
"1000 28 78"
"1000 25 16"
"1000 28 82"
"1010 28 98"
"1000 12 77"
"1010 11 16"
"1010 44 14"
"1010 12 89"
"1000 32 8"
"1010 37 55"
"1010 17 21"
"1000 13 96"
"1000 18 51"
"1010 16 38"
"1000 18 21"
"1010 37 60"
"1010 17 25"
"1000 33 43"
"1000 34 44"
"1010 33 78"
"1010 35 63"
"1000 39 50"
"1000 45 70"
"1000 26 3"
"1010 34 12"
"1010 26 3"
"1000 32 75"
"1010 14 68"
"1000 44 75"
"1010 27 54"
"1000 21 39"
"1010 12 65"
"1010 43 29"
"1010 25 60"
```

### Divide and Conquer

For this problem we will calculate the number of primes less than $10^{7}$. Our ventilator will push the task to workers and they will perform the task. After finishing the job, they will send the result back to sink.

Ventilator

```import System.ZMQ3
import qualified Data.ByteString.Char8 as BS
import Data.ByteString.Lazy.Internal as BL
import Data.IORef

main = do
c <- context
sender <- socket c Push
bind sender "tcp://127.0.0.1:5557"

sink <- socket c Push
connect sink "tcp://127.0.0.1:5558"

putStrLn "Press enter when workers are ready."
_ <- getChar
putStrLn "Sending the task to workers.\n"

send' sink [] ( BL.packChars . show \$ 0 )

forM_ [ 0..9999 ] \$ \ i -> do
putStrLn \$ "Sending the range [ " ++ show ( 1000 * i + 1 ) ++ " .. " ++ show  ( 1000 * i + 1 + 999 ) ++ "] to worker for primality testing."
send' sender [] ( BL.packChars . show \$ 1000 * i + 1 )

close sink
close sender
destroy c

```

Worker for computing the prime number.

```import System.ZMQ3
import qualified Data.ByteString.Char8 as BS
import qualified Data.ByteString.Lazy.Internal as BL
import Data.IORef
import Data.Bits

powM :: Integer -> Integer -> Integer ->  Integer
powM a d n = powM' a d n where
powM' a d n
| d == 0 = 1
| d == 1 = mod a n
| otherwise = mod q n  where
p = powM'  ( mod ( a^2 ) n ) ( shiftR d 1 ) n
q = if (.&.) d 1 == 1 then mod ( a * p ) n else p

calSd :: Integer -> ( Integer , Integer )
calSd n = ( s , d ) where
s = until ( \x -> testBit ( n - 1) ( fromIntegral x ) )  ( +1 ) 0
d = div ( n - 1 ) (  shiftL 1 ( fromIntegral s )  )

rabinMiller::Integer->Integer->Integer->Integer-> Bool
rabinMiller  n s d a
| n == a = True
| otherwise =  case x == 1 of
True -> True
_ ->   any ( == pred n ) . take ( fromIntegral s )
. iterate (\e -> mod ( e^2 ) n ) \$ x
where
x = powM a d n

isPrime::Integer-> Bool
isPrime n
| n <= 1 = False
| n == 2 = True
| even n = False
| otherwise  = all ( == True ) . map ( rabinMiller n s d ) \$  [ 2 , 3 , 5 , 7 , 11 , 13 , 17 ] where
( s , d ) = calSd n

primeRange :: Integer -> Integer -> [ Bool ]
primeRange m n = map isPrime [ m .. n ]

main = do
c <- context

sender <- socket c Push
connect sender "tcp://127.0.0.1:5558"

forever \$ do
let n = read . BS.unpack \$ num  :: Integer
let len = length . filter ( == True ) . primeRange  n  \$  n + 999
putStrLn \$ "Received range [ " ++  show n ++ " .. " ++ show ( n + 999 ) ++ "  and number of primes in this range is " ++ show len
send' sender [] ( BL.packChars . show \$ len  )

close sender
destroy c

```

Sink for collecting the results.

```import System.ZMQ3
import qualified Data.ByteString.Char8 as BS
import qualified Data.ByteString.Lazy.Internal as BL
import Data.IORef

main = do
c <- context

print m
sum <- newIORef  0
forM_ [ 1.. 10000 ] \$ \i -> do
let n = read . BS.unpack \$ num :: Integer
putStrLn \$ "Received a number " ++ show n ++ " from one of worker"
modifyIORef sum ( + n )

putStrLn \$ "Total number of primes in the range is " ++ show t
destroy c
```

Running Ventilator, 3 workers and sink

```Mukeshs-MacBook-Pro:ZMQ mukeshtiwari\$ ./TaskVent
Sending the range [ 9992001 .. 9993000] to worker for primality testing.
Sending the range [ 9993001 .. 9994000] to worker for primality testing.
Sending the range [ 9994001 .. 9995000] to worker for primality testing.
Sending the range [ 9995001 .. 9996000] to worker for primality testing.
Sending the range [ 9996001 .. 9997000] to worker for primality testing.
Sending the range [ 9997001 .. 9998000] to worker for primality testing.
Sending the range [ 9998001 .. 9999000] to worker for primality testing.
Sending the range [ 9999001 .. 10000000] to worker for primality testing.

Received range [ 9978001 .. 9979000  and number of primes in this range is 60
Received range [ 9981001 .. 9982000  and number of primes in this range is 49
Received range [ 9984001 .. 9985000  and number of primes in this range is 69
Received range [ 9987001 .. 9988000  and number of primes in this range is 57
Received range [ 9990001 .. 9991000  and number of primes in this range is 60
Received range [ 9993001 .. 9994000  and number of primes in this range is 71
Received range [ 9996001 .. 9997000  and number of primes in this range is 58
Received range [ 9999001 .. 10000000  and number of primes in this range is 53

Received range [ 9982001 .. 9983000  and number of primes in this range is 69
Received range [ 9985001 .. 9986000  and number of primes in this range is 62
Received range [ 9988001 .. 9989000  and number of primes in this range is 58
Received range [ 9991001 .. 9992000  and number of primes in this range is 57
Received range [ 9994001 .. 9995000  and number of primes in this range is 64
Received range [ 9997001 .. 9998000  and number of primes in this range is 67
Received range [ 9986001 .. 9987000  and number of primes in this range is 59
Received range [ 9989001 .. 9990000  and number of primes in this range is 58
Received range [ 9992001 .. 9993000  and number of primes in this range is 58
Received range [ 9995001 .. 9996000  and number of primes in this range is 62
Received range [ 9998001 .. 9999000  and number of primes in this range is 64

Sink
Received a number 53 from one of worker
Received a number 67 from one of worker
Received a number 64 from one of worker
Received a number 52 from one of worker
Received a number 59 from one of worker
Received a number 58 from one of worker
Received a number 58 from one of worker
Received a number 62 from one of worker
Received a number 64 from one of worker
Total number of primes in the range is 664579
```

First start all the workers and sink and then run ventilator. See more about distributed computing in Haskell Eden and distributed-process. If you have any suggestion then please let me know. Some of the contents and images are taken from Pieter Hintjens‘s tutorial by his permission.

May 13, 2013

## Missionaries and cannibals problem

In the missionaries and cannibals problem, three missionaries and three cannibals must cross a river using a boat which can carry at most two people, under the constraint that, for both banks, if there are missionaries present on the bank, they cannot be outnumbered by cannibals (if they were, the cannibals would eat the missionaries.) The boat cannot cross the river by itself with no people on board. We can solve this problem using depth first search. Representing the states as number of missionaries, cannibals and boat location in following way.

```
data Side = Side { missionaries :: Int , cannibals :: Int } deriving ( Show , Eq  )
data Loc = L | R deriving ( Show , Eq  )
data State = State { left :: Side , right :: Side , bloc :: Loc } deriving ( Show , Eq  )

```

Target is to move every missionaries and cannibals from left to right so our initial state is initialState and final state is finalState.

```
initialState = State { left =  Side 3 3  , right =  Side 0 0  , bloc = L }
finalState = State  { left =  Side 0 0  , right = Side 3 3  , bloc = R }

```

Possible movements
1. move ( 2 , 0 )
2. move ( 1 , 0 )
3. move ( 1 , 1 )
4. move ( 0 , 1 )
5. move ( 0 , 2 )
where move ( M , C ) is movement of M missionaries and C cannibals from one side to other side depending on boat location. If boat location is left then move ( M , C ) is moving M missionaries and C cannibals from left to right and vice versa. Using depth first search, we can explore every possible movement

```path :: [ State ] -> [ State ] -> [ State ]
path []   vis = reverse vis
path  ( x : xs )  vis
| x  == finalState  = reverse ( x : vis )
| elem x vis = path xs vis
| otherwise = path xs'  vis'  where
vis'  = x : vis
xs' =   ( filter  isLegal  ( move x ) ) ++ xs
```

Encoding the moves and testing the legality

```
move :: State -> [ State ]
move ( State ( Side ml cl ) ( Side mr cr ) L ) =
[ State ( Side ( ml - 2 ) cl ) ( Side ( mr + 2 ) cr ) R
, State ( Side ( ml - 1 ) cl ) ( Side ( mr + 1 ) cr ) R
, State ( Side ( ml - 1 ) ( cl - 1 ) ) ( Side ( mr + 1 ) ( cr + 1 ) ) R
, State ( Side ml ( cl - 2 ) ) ( Side mr ( cr + 2 ) ) R
, State ( Side ml ( cl - 1 ) ) ( Side mr ( cr + 1 ) ) R
]
move  ( State ( Side ml cl ) ( Side mr cr ) R ) =
[ State ( Side ( ml + 2 ) cl ) ( Side ( mr - 2 ) cr ) L
, State ( Side ( ml + 1 ) cl ) ( Side ( mr - 1 ) cr ) L
, State ( Side ( ml + 1 ) ( cl + 1 ) ) ( Side ( mr - 1 ) ( cr - 1 ) ) L
, State ( Side ml ( cl + 2 ) ) ( Side mr ( cr - 2 ) ) L
, State ( Side ml ( cl + 1 ) ) ( Side mr ( cr - 1 ) ) L
]

isLegal :: State -> Bool
isLegal ( State ( Side ml cl ) ( Side mr cr ) y )
| ml == 0 = mr >= cr  && cr >= 0
| mr == 0 = ml >= cl  && cl >= 0
| otherwise = ml >= cl && mr >= cr && cl >= 0 && cr >= 0

```

Complete source code ( In case if you are lazy )

```import Data.List

data Side = Side { missionaries :: Int , cannibals :: Int } deriving ( Show , Eq  )
data Loc = L | R deriving ( Show , Eq  )
data State = State { left :: Side , right :: Side , bloc :: Loc } deriving ( Show , Eq  )

initialState = State { left =  Side 3 3  , right =  Side 0 0  , bloc = L }
finalState = State  { left =  Side 0 0  , right = Side 3 3  , bloc = R }

path :: [ State ] -> [ State ] -> [ State ]
path []   vis = reverse vis
path  ( x : xs )  vis
| x  == finalState  = reverse ( x : vis )
| elem x vis = path xs vis
| otherwise = path xs'  vis'  where
vis'  = x : vis
xs' =   ( filter  isLegal  ( move x ) ) ++ xs

move :: State -> [ State ]
move ( State ( Side ml cl ) ( Side mr cr ) L ) =
[ State ( Side ( ml - 2 ) cl ) ( Side ( mr + 2 ) cr ) R
, State ( Side ( ml - 1 ) cl ) ( Side ( mr + 1 ) cr ) R
, State ( Side ( ml - 1 ) ( cl - 1 ) ) ( Side ( mr + 1 ) ( cr + 1 ) ) R
, State ( Side ml ( cl - 2 ) ) ( Side mr ( cr + 2 ) ) R
, State ( Side ml ( cl - 1 ) ) ( Side mr ( cr + 1 ) ) R
]
move  ( State ( Side ml cl ) ( Side mr cr ) R ) =
[ State ( Side ( ml + 2 ) cl ) ( Side ( mr - 2 ) cr ) L
, State ( Side ( ml + 1 ) cl ) ( Side ( mr - 1 ) cr ) L
, State ( Side ( ml + 1 ) ( cl + 1 ) ) ( Side ( mr - 1 ) ( cr - 1 ) ) L
, State ( Side ml ( cl + 2 ) ) ( Side mr ( cr - 2 ) ) L
, State ( Side ml ( cl + 1 ) ) ( Side mr ( cr - 1 ) ) L
]

isLegal :: State -> Bool
isLegal ( State ( Side ml cl ) ( Side mr cr ) y )
| ml == 0 = mr >= cr  && cr >= 0
| mr == 0 = ml >= cl  && cl >= 0
| otherwise = ml >= cl && mr >= cr && cl >= 0 && cr >= 0

```

``` Main> path [ initialState ] [] [State {left = Side {missionaries = 3, cannibals = 3}, right = Side {missionaries = 0, cannibals = 0}, bloc = L},State {left = Side {missionaries = 2, cannibals = 2}, right = Side {missionaries = 1, cannibals = 1}, bloc = R},State {left = Side {missionaries = 3, cannibals = 2}, right = Side {missionaries = 0, cannibals = 1}, bloc = L},State {left = Side {missionaries = 3, cannibals = 0}, right = Side {missionaries = 0, cannibals = 3}, bloc = R},State {left = Side {missionaries = 3, cannibals = 1}, right = Side {missionaries = 0, cannibals = 2}, bloc = L},State {left = Side {missionaries = 1, cannibals = 1}, right = Side {missionaries = 2, cannibals = 2}, bloc = R},State {left = Side {missionaries = 2, cannibals = 2}, right = Side {missionaries = 1, cannibals = 1}, bloc = L},State {left = Side {missionaries = 0, cannibals = 2}, right = Side {missionaries = 3, cannibals = 1}, bloc = R},State {left = Side {missionaries = 0, cannibals = 3}, right = Side {missionaries = 3, cannibals = 0}, bloc = L},State {left = Side {missionaries = 0, cannibals = 1}, right = Side {missionaries = 3, cannibals = 2}, bloc = R},State {left = Side {missionaries = 1, cannibals = 1}, right = Side {missionaries = 2, cannibals = 2}, bloc = L},State {left = Side {missionaries = 0, cannibals = 0}, right = Side {missionaries = 3, cannibals = 3}, bloc = R}] *Main> map bloc . path [ initialState ] \$ [] [L,R,L,R,L,R,L,R,L,R,L,R] ```
If you have any suggestion or comment then please let me know.

April 19, 2013

## SPOJ Pell (Mid pelling)

Pell (Mid pelling) is related to Pell’s equation. It is similar to Project Euler 66 and SPOJ EQU2. I just wrote the quick solution from mathworld but I found Chakravala method very interesting. Accepted Haskell code.

```import qualified Data.ByteString.Char8 as BS
import Data.List
import Data.Maybe ( fromJust )

continuedFraction :: Integer -> [ Integer ]
continuedFraction n = map ( \ ( a , _ , _ ) -> a ) . iterate fun \$ ( d , 0 , 1 )
where
d = truncate . sqrt . fromIntegral \$ n
fun ( a0 , p0 , q0 ) = ( a1 , p1 , q1 ) where
p1 = a0 * q0 - p0
q1 = div ( n - p1 ^ 2 ) q0
a1 = div ( d + p1 ) q1

pellSolver :: Integer -> BS.ByteString
pellSolver n
| perfectSqr n = BS.pack. show \$ ( -1 )
| otherwise =  ( BS.pack . show \$ p ) `BS.append` ( BS.pack " " )
`BS.append` ( BS.pack.show \$ q ) where
d = truncate . sqrt . fromIntegral \$ n
lst = takeWhile ( /= 2 * d ) . continuedFraction \$ n
len = length lst
r@( x : y : xs ) = take ( if even len then len else 2 * len ) .
continuedFraction \$ n
( p0 , p1 , q0 , q1 ) = ( x , x * y + 1 , 1 , y )
( p , _ ) = foldl' compute ( p1 , p0 ) \$ xs
( q , _ ) = foldl' compute ( q1 , q0 ) \$ xs
compute :: ( Integer , Integer ) -> Integer -> ( Integer , Integer )
compute ( p1 , p0 ) a = ( a * p1 + p0 , p1 )

perfectSqr :: Integer -> Bool
perfectSqr n = d * d == n where
d = truncate . sqrt . fromIntegral \$ n

main = BS.interact \$  BS.unlines . map ( pellSolver . readI ) . tail . BS.lines
```

March 23, 2013

## SPOJ EMAIL ID

Finally solved EMAIL ID using python though it was very hard for me to switch from Haskell to python. While doing this problem, I learned quite a lot about regular expressions, found some cool site like pythontutor and problem solving with python . Accepted code in python.

```
import re

if __name__ == "__main__":
n = int ( raw_input() )
c = 1
while c <= n :
email =  re.findall ( "[a-zA-Z0-9][a-zA-Z0-9._]{4,}@[a-zA-Z0-9]+\.(?:com|edu|org|co\.in)", raw_input() )
t = len ( email )
print 'Case #' + str ( c ) + ': ' + str ( t )
for i in xrange ( t ) : print email[i]
c += 1

```

March 11, 2013

## My experience with online learning

I am one big confused soul who is always curious for learning. I found Coursera and Udacity very helpful. I try to take as many courses as possible but recently I took the Heterogeneous Parallel Programming by Professor Wen-mei W. Hwu and it was fun course. The video lectures were excellent and assignments were not very hard. They were designed to make us familiar with basics of cuda programming and apart from that I got certificate signed by Professor Wen-mei W. Hwu :). While doing some more research about heterogeneous computing which involves CPU, GPU and FPGA all together in single system, today I found FPGA Programming for the Masses. There are two promising technologies, one from Altera and other is Liquid Metal from IBM.

Altera OpenCL-to-FPGA Framework.
Altera proposed a framework for synthesizing bitstreams from OpenCL. The framework consists of a kernel compiler, a host library, and a system integration component. The kernel compiler, based on the open source LLVM compiler infrastructure, synthesizes OpenCL kernel code into hardware. The host library provides APIs and bindings for establishing communication between the host part of the application running on the processor and the kernel part of the application running on the FPGA. The system integration component wraps the kernel code with memory controllers and a communication interface (such as PCIe)
Liquid Metal.
The Liquid Metal project at IBM aims to provide not only high-level synthesis for FPGAs,1 but also, more broadly, a single unified language for programming heterogeneous architectures. The project offers a language called Lime, which can be used to program hardware (FPGAs), as well as software running on conventional CPUs and more exotic architectures such as GPUs. The Lime language was founded on principles that eschew the complicated program analysis that plagues C-based frameworks, while also offering powerful programming models that espouse the benefits of functional and stream-oriented programming.
Heterogeneous parallel programming looks very promising but still long way to go. My certificate

.

March 9, 2013

## Parsing Email ID

Now a days I am exploring one of the most awesome feature of Haskell, parsing. There are lot of parsing libraries but Parsec is the popular one. This parsing code is written for SPOJ EMAIL ID problem ( Unfortunately it’s getting time limit exceed. I have seen couple of python solution accepted so I am hopeful that there must be another algorithm probably using regular expressions to solve the problem ) but you can build more sophisticated email-id parser by adding more functionality. Tony Morris excellent parsing tutorial is must read.

```import Data.List
import qualified Text.Parsec.ByteString as PB
import Text.Parsec.Prim
import Text.Parsec.Char
import Text.Parsec.Combinator
import qualified Data.ByteString.Char8 as BS
import Control.Applicative hiding ( ( <|> ) , many )

validChars :: PB.Parser Char
validChars  = alphaNum <|> oneOf "._"

dontCare :: PB.Parser Char
dontCare = oneOf "~!@#\$%^&*()<>?,."

{--
_ <- many dontCare
fi <- alphaNum
se <- validChars
th <- validChars
fo <- validChars
ft <- validChars
let addr = fi : se : th : fo : ft : restAddr
char '@'
dom <- many1 alphaNum
rest <- try ( string ".com" <|> string ".org"
<|>  string ".edu" ) <|> try ( string ".co.in" )
_ <- many dontCare
return \$  addr ++ (  '@': dom ++ rest )

--}

emailAddress = conCatfun <\$> ( many dontCare *> alphaNum ) <*> validChars <*>
validChars <*> validChars <*> validChars <*> many alphaNum  <*>
( char '@' *> many1 alphaNum ) <*> ( try ( string ".com" <|>
string ".org" <|>  string ".edu" ) <|> try ( string ".co.in" )
<* many dontCare ) where
conCatfun a b c d e f dom rest =
( a : b : c : d : e : f ) ++ ( '@' : dom ) ++ rest

collectEmail :: BS.ByteString -> String
collectEmail email = case  parse emailAddress "" email of
Left err ->  ""

process :: ( Int , [ String ] ) -> BS.ByteString
process ( k , xs ) = ( BS.pack "Case " ) `BS.append` ( BS.pack . show \$ k )
`BS.append` ( BS.pack ": " ) `BS.append` ( BS.pack . show . length \$ xs )
`BS.append` ( BS.pack "\n" ) `BS.append` ( BS.pack
(  unlines . filter ( not . null ) \$  xs ) )

main = BS.interact \$ BS.concat .  map process . zip [ 1.. ] . map ( map collectEmail .
BS.words ) . tail . BS.lines

```
```Mukeshs-MacBook-Pro:Haskell mukeshtiwari\$ cat t.txt
2
svm11@gmail.com
svm11@gmail.com svm12@gmail.co.in  ~!@#\$%^&*()<>?svm12@gmail.co.in~!@#\$%^&*()
Case 1: 1
svm11@gmail.com
Case 2: 2
svm11@gmail.com
svm12@gmail.co.in
svm12@gmail.co.in
```

#### Update

I tried again to solve this problem using regular expression. Following the tutorial, I wrote this code but I got compiler error because of old version of ghc on SPOJ ( GHC-6.10.4 ). It’s working fine on my system but I still have to test if it is correct and fast enough to get accepted.

```import Data.List
import Text.Regex.Posix
import qualified Data.ByteString.Char8 as BS

pat :: BS.ByteString
pat = BS.pack "[^~!@#\$%^&*()<>?,.]*[a-zA-Z0-9][a-zA-Z0-9._][a-zA-Z0-9._][a-zA-Z0-9._][a-zA-Z0-9._][a-zA-Z0-9._]*@[a-zA-Z0-9]+.(com|edu|org|co.in)[^~!@#\$%^&*()<>?,.a-zA-Z0-9]*"

collectEmail :: BS.ByteString -> BS.ByteString
collectEmail email = ( =~ ) email pat

process :: ( Int , [ BS.ByteString ] ) -> BS.ByteString
process ( k , xs ) = ( BS.pack "Case " ) `BS.append` ( BS.pack . show \$ k )
`BS.append` ( BS.pack ": " ) `BS.append` ( BS.pack . show . length \$ xs )
`BS.append` ( BS.pack "\n" ) `BS.append` ( BS.unlines xs )

main = BS.interact \$ BS.concat .  map process . zip [ 1 .. ] .
map ( filter ( not . BS.null ) . map collectEmail . BS.words ) .
tail . BS.lines
```
```Mukeshs-MacBook-Pro:SPOJ mukeshtiwari\$ cat t.txt
2
svm11@gmail.com
svm11@gmail.com svm12@gmail.co.in %&^%&%&%&%&^%&^%&^%&^mukeshtiwari.iiitm@gmail.com%&%^&^%&%&%&%&^%&%&^%&^%&^% %\$%\$#%#%#%#%\$#%&%&%&tiwa@gmail.com
Mukeshs-MacBook-Pro:SPOJ mukeshtiwari\$ ./Spoj_11105 < t.txt
Case 1: 1
svm11@gmail.com
Case 2: 3
svm11@gmail.com
svm12@gmail.co.in
mukeshtiwari.iiitm@gmail.com
```

March 8, 2013

## Learning Agda

Finally I am feeling bit comfortable with Agda and feeling great :). I started reading Ulf Norell‘s Dependently Typed Programming in Agda. Tutorials on Agda-wiki , AgdaTutorial , papers which mention Agda , stackoverflow post, Introduction to Agda and Programming in Martin-Löf’s Type Theory. I wrote some of the codes from the paper. You can also try a similar language Idris ( See the stackoverflow post about difference between these languages ).

```module Basic where

data Bool : Set where
true : Bool
false : Bool

not : Bool → Bool
not true = false
not false = true

or : Bool → Bool → Bool
or false false = false
or _ _ = true

and : Bool → Bool → Bool
and true true = true
and _ _ = false

data ℕ  : Set where
zero : ℕ
suc : ℕ → ℕ

_+_ : ℕ → ℕ → ℕ
zero + m = m
suc n + m = suc ( n + m )

_⋆_ : ℕ → ℕ → ℕ
zero ⋆ m = zero
suc n ⋆ m =   ( n ⋆ m ) + m

fold-ℕ : ℕ → ( ℕ → ℕ ) → ℕ → ℕ
fold-ℕ u _ zero = u
fold-ℕ u f ( suc n ) = f ( fold-ℕ u f n )

if_then_else_ : { A : Set } → Bool → A → A → A
if true then x else y = x
if false then x else y = y

data List ( A : Set ) : Set where
[] : List A
_::_ : A → List A -> List A

identity : ( A : Set ) → A → A
identity A x = x

zero′ : ℕ
zero′ = identity ℕ zero

apply : ( A : Set ) ( B : A → Set ) → ( ( x : A ) → B x ) → ( a : A ) → B a
apply A B f a = f a

identity₂ : ( A : Set ) → A → A
identity₂ = \A x → x

identity₃ : ( A : Set ) → A → A
identity₃ = \(A : Set ) ( x : A ) → x

identity₄ : ( A : Set ) → A → A
identity₄ = \ ( A : Set ) x → x

id : { A : Set } → A → A
id x = x

true′ : Bool
true′ = id true

one : ℕ
one = identity _ ( suc zero )

_∘_ : { A : Set } { B : A → Set } { C : ( x : A ) → B x → Set }
( f : { x : A } ( y : B x ) → C x y ) ( g : ( x : A ) → B x ) ( x : A )
→ C x ( g x )
( f ∘ g ) x = f ( g x )

plus-two = suc ∘  suc
plus-three = suc ∘ ( suc ∘ suc )

map : { A B : Set } → ( A → B ) → List A → List B
map f [] = []
map f ( x :: xs ) = f x :: map f xs

_++_ : { A : Set } → List A → List A → List A
[] ++ ys = ys
( x :: xs ) ++ ys = x :: ( xs ++ ys )

foldl : { A B : Set } → ( A → B → A ) → A → List B → A
foldl f val [] = val
foldl f val ( x :: xs ) = foldl f ( f val x ) xs

foldr : { A B : Set } → ( A → B → B ) → B → List A → B
foldr f val [] = val
foldr f val ( x :: xs ) = f x ( foldr f val xs )

--type of Vec A is  ℕ → Set. This mean Vec A is family of types indexed by
-- natural numbers

data Vec ( A : Set ) : ℕ →  Set where
[] : Vec A zero
_::_ : { n : ℕ } → A → Vec A n → Vec A ( suc n )

head : { A : Set } { n : ℕ } → Vec A ( suc n )  → A
head ( x :: xs ) = x

vmap : { A B : Set } { n : ℕ } → ( A → B ) → Vec A n → Vec B n
vmap f [] = []
vmap f ( x :: xs ) = f x :: vmap f xs

data Vec₂ ( A : Set ) : ℕ → Set where
nil : Vec₂ A zero
cons : ( n : ℕ ) → A → Vec₂ A n → Vec₂ A ( suc n )
{--
The rule for when an argument should be dotted is:
if there is a unique
type correct value for the argument it should be dotted
--}

vmap₂ : { A B : Set } ( n : ℕ ) → ( A → B ) → Vec₂ A n → Vec₂ B n
vmap₂ .zero f nil = nil
vmap₂ .( suc n ) f ( cons n x xs ) = cons n  ( f x ) ( vmap₂ n f xs )

vmap₃ : { A B : Set } ( n : ℕ ) → ( A → B ) → Vec₂ A n → Vec₂ B n
vmap₃ zero f nil = nil
vmap₃ ( suc n ) f ( cons .n x xs ) = cons n ( f x ) ( vmap₃ n f xs )

pow : ℕ →  ℕ  →  ℕ
pow _  zero = suc zero
pow a ( suc n ) =  a ⋆ pow a n

t : ℕ
t = pow ( suc ( suc  zero ) ) ( suc ( suc ( suc zero ) ) )

```

To evaluate the expression , type C-c C-n. Here is the output of t
suc (suc (suc (suc (suc (suc (suc (suc zero))))))).

February 10, 2013