Data.BitVector

Bit-vector arithmetic inspired by SMT-LIB http://smt-lib.org/ and Cryptol http://cryptol.net/.

Bit-vectors are represented as a pair size and value, where sizes are of type Int and values are Integer.

• Bit-vectors are interpreted as unsigned integers (i.e. natural numbers) except for some specific signed operations.
• Most operations are in some way size-polymorphic and, if required, will perform padding to adjust the size of input bit-vectors.

For documentation purposes we will write [n]k to denote a bit-vector of size n representing the natural number k.

Bit-vectors

type BitVector = BV

An alias for BV.

data BV

Big-endian pseudo size-polymorphic bit-vectors.

Instances

Enum BV

Eq BV

Integral BV

Data BV

Num BV

Ord BV

Real BV

Show BV

Monoid BV

Bits BV

Typeable * BV

size :: BV -> Int

The size of a bit-vector.

width :: BV -> Int

An alias for size.

nat :: BV -> Integer

The value of a bit-vector, as a natural number.

uint :: BV -> Integer

An alias for nat.

int :: BV -> Integer

2's complement value of a bit-vector.

>>> int [2]3
-1

>>> int [4]12
-4

Creation

nil :: BV

The empty bit-vector, ie. [0]0.

bitVec :: Integral a => Int -> a -> BV

Create a bit-vector given a size and an integer value.

>>> bitVec 4 3
[4]3

This function also handles negative values.

>>> bitVec 4 (-1)
[4]15

bitVecs :: Integral a => Int -> [a] -> [BV]

List of bit-vector literals of the same size

When a list of integer literals is interpreted as a list of bit-vectors, fromInteger is applied to each element invidually:

>>> [1,3,5] :: [BV]
[ [1]1, [2]3, [3]5 ]

Sometimes we want to specify a list of bit-vectors literals of the same size, and for that we can use bitVects:

>>> bitVecs 3 [1,3,5]
[ [3]1, [3]3, [3]5 ]

ones :: Int -> BV

Create a mask of ones.

zeros :: Int -> BV

Create a mask of zeros.

Test

isNat :: BV -> Bool

Test if the signed value of a bit-vector is a natural number.

isPos :: BV -> Bool

Test if the signed value of a bit-vector is a positive number.

Comparison

(==.) :: BV -> BV -> Bool infix 4

Fixed-size equality.

In contrast with ==, which is size-polymorphic, this equality requires both bit-vectors to be of equal size.

>>> [n]k ==. [m]k
False

>>> [n]k ==. [n]k
True

(/=.) :: BV -> BV -> Bool infix 4

Fixed-size inequality.

The negated version of ==..

(<.) :: BV -> BV -> Bool infix 4

Fixed-size less-than.

(<=.) :: BV -> BV -> Bool infix 4

Fixed-size less-than-or-equals.

(>.) :: BV -> BV -> Bool infix 4

Fixed-size greater-than.

(>=.) :: BV -> BV -> Bool infix 4

Fixed-size greater-than-or-equals.

slt :: BV -> BV -> Bool infix 4

Fixed-size signed less-than.

sle :: BV -> BV -> Bool infix 4

Fixed-size signed less-than-or-equals.

sgt :: BV -> BV -> Bool infix 4

Fixed-size signed greater-than.

sge :: BV -> BV -> Bool infix 4

Fixed-size signed greater-than-or-equals.

Indexing

(@.) :: Integral ix => BV -> ix -> Bool infixl 9

Bit indexing.

u @. i stands for the i-th bit of u.

>>> [4]2 @. 0
False

>>> [4]2 @. 1
True

index :: Integral ix => ix -> BV -> Bool

index i a == a @. i

(@@) :: Integral ix => BV -> (ix, ix) -> BV infixl 9

Bit-string extraction.

u @@ (j,i) == fromBits (map (u @.) [j,j-1..i])

>>> [4]7 @@ (3,1)
[3]3

extract :: Integral ix => ix -> ix -> BV -> BV

extract j i a == a @@ (j,i)

(@:) :: Integral ix => BV -> [ix] -> BV infixl 9

Bit list indexing.

u @: is ==. fromBits $ List.map (u @.) is

(!.) :: Integral ix => BV -> ix -> Bool infixl 9

Reverse bit-indexing.

Index starting from the most significant bit.

u !. i == u @. (size u - i - 1)

>>> [3]3 !. 0
False

least :: Integral ix => ix -> BV -> BV

Take least significant bits.

least m u == u @@ (m-1,0)

most :: Integral ix => ix -> BV -> BV

Take most significant bits.

most m u == u @@ (n-1,n-m)

msb :: BV -> Bool

Most significant bit.

msb u == u !. 0

lsb :: BV -> Bool

Least significant bit.

lsb u == u @. 0

msb1 :: BV -> Int

Most significant 1-bit.

Pre: input must be non-zero.

>>> msb1 [4]2
1

>>> msb1 [4]4
2

lsb1 :: BV -> Int

Least significant 1-bit.

Pre: input must be non-zero.

>>> msb1 [4]3
0

>>> msb1 [4]6
1

Arithmetic

signumI :: Integral a => BV -> a

Bit-vector signum as an Integral.

pow :: Integral exp => BV -> exp -> BV

Bit-vector exponentiation.

pow [n]k e computes k raised to e modulo n.

This is faster than Haskell's (^) operator because it performs modulo division just once. Besides, a^0 == [1]0 !!!

sdiv :: BV -> BV -> BV

2's complement signed division.

srem :: BV -> BV -> BV

2's complement signed remainder (sign follows dividend).

smod :: BV -> BV -> BV

2's complement signed remainder (sign follows divisor).

lg2 :: BV -> BV

Ceiling logarithm base 2.

Pre: input bit-vector must be non-zero.

List-like operations

(#) :: BV -> BV -> BV infixr 5

Concatenation of two bit-vectors.

cat :: BV -> BV -> BV

Deprecated: Use (#) or append instead

Concatenation of two bit-vectors.

append :: BV -> BV -> BV

Concatenation of two bit-vectors.

concat :: [BV] -> BV

An alias for join.

zeroExtend :: Integral size => size -> BV -> BV

Logical extension.

>>> zeroExtend 3 [1]1
[4]1

signExtend :: Integral size => size -> BV -> BV

Arithmetic extension.

>>> signExtend 2 [2]1
[4]1

>>> signExtend 2 [2]3
[4]15

foldl :: (a -> Bool -> a) -> a -> BV -> a

foldl f z (fromBits [un, ..., u1, u0]) == ((((z `f` un) `f` ...) `f` u1) `f` u0)

foldl f e = fromBits . foldl f e . toBits

foldl_ :: (a -> Bool -> a) -> a -> BV -> a

Deprecated: Use corresponding versions without underscore

foldl f z (fromBits [un, ..., u1, u0]) == ((((z `f` un) `f` ...) `f` u1) `f` u0)

foldl f e = fromBits . foldl f e . toBits

foldr :: (Bool -> a -> a) -> a -> BV -> a

foldr f z (fromBits [un, ..., u1, u0]) == un `f` (... `f` (u1 `f` (u0 `f` z)))

foldr f e = fromBits . foldr f e . toBits

foldr_ :: (Bool -> a -> a) -> a -> BV -> a

Deprecated: Use corresponding versions without underscore

foldr f z (fromBits [un, ..., u1, u0]) == un `f` (... `f` (u1 `f` (u0 `f` z)))

foldr f e = fromBits . foldr f e . toBits

reverse :: BV -> BV

reverse == fromBits . reverse . toBits

reverse_ :: BV -> BV

Deprecated: Use corresponding versions without underscore

reverse == fromBits . reverse . toBits

replicate :: Integral size => size -> BV -> BV

Pre: if replicate_ n u then n > 0 must hold.

replicate_ n == fromBits . concat . replicate n . toBits

replicate_ :: Integral size => size -> BV -> BV

Deprecated: Use corresponding versions without underscore

Pre: if replicate_ n u then n > 0 must hold.

replicate_ n == fromBits . concat . replicate n . toBits

and :: [BV] -> BV

Conjunction.

Essentially, and == foldr1 (.&.).

Returns [1]1 if the input list is empty.

and_ :: [BV] -> BV

Deprecated: Use corresponding versions without underscore

Conjunction.

Essentially, and == foldr1 (.&.).

Returns [1]1 if the input list is empty.

or :: [BV] -> BV

Disjunction.

Essentially, or == foldr1 (.|.).

Returns [1]0 if the input list is empty.

or_ :: [BV] -> BV

Deprecated: Use corresponding versions without underscore

Disjunction.

Essentially, or == foldr1 (.|.).

Returns [1]0 if the input list is empty.

split :: Integral times => times -> BV -> [BV]

Split a bit-vector k times.

>>> split 3 [4]15
[[2]0,[2]3,[2]3]

group :: Integral size => size -> BV -> [BV]

Split a bit-vector into n-wide pieces.

>>> group 3 [4]15
[[3]1,[3]7]

group_ :: Integral size => size -> BV -> [BV]

Deprecated: Use corresponding versions without underscore

Split a bit-vector into n-wide pieces.

>>> group 3 [4]15
[[3]1,[3]7]

join :: [BV] -> BV

Concatenate a (possibly empty) list of bit-vectors.

>>> join [[2]3,[2]2]
[4]14

Bitwise operations

module Data.Bits

not :: BV -> BV

An alias for complement.

not_ :: BV -> BV

Deprecated: Use corresponding versions without underscore

An alias for complement.

nand :: BV -> BV -> BV

Negated .&..

nor :: BV -> BV -> BV

Negated .|..

xnor :: BV -> BV -> BV

Negated xor.

(<<.) :: BV -> BV -> BV infixl 8

Left shift.

shl :: BV -> BV -> BV infixl 8

Left shift.

(>>.) :: BV -> BV -> BV infixl 8

Logical right shift.

shr :: BV -> BV -> BV infixl 8

Logical right shift.

ashr :: BV -> BV -> BV infixl 8

Arithmetic right shift

(<<<.) :: BV -> BV -> BV infixl 8

Rotate left.

rol :: BV -> BV -> BV infixl 8

Rotate left.

(>>>.) :: BV -> BV -> BV infixl 8

Rotate right.

ror :: BV -> BV -> BV infixl 8

Rotate right.

Conversion

fromBool :: Bool -> BV

Create a bit-vector from a single bit.

fromBits :: [Bool] -> BV

Create a bit-vector from a big-endian list of bits.

>>> fromBits [False, False, True]
[3]1

toBits :: BV -> [Bool]

Create a big-endian list of bits from a bit-vector.

>>> toBits [4]11
[True, False, True, True]

Pretty-printing

showBin :: BV -> String

Show a bit-vector in binary form.

showOct :: BV -> String

Show a bit-vector in octal form.

showHex :: BV -> String

Show a bit-vector in hexadecimal form.