hinze-streams-1.0: Streams and Unique Fixed Points

Data.Stream.Hinze.Stream

Description

Functional Pearl: Streams and Unique Fixed Points Ralf Hinze The 13th ACM SIGPLAN International Conference on Functional Programming (ICFP 2008) Victoria, British Columbia, Canada, September 22-24, 2008

Streams, infinite sequences of elements, live in a coworld: they are given by a coinductive data type, operations on streams are implemented by corecursive programs, and proofs are conducted using coinduction. But there is more to it: suitably restricted, stream equations possess unique solutions, a fact that is not very widely appreciated. We show that this property gives rise to a simple and attractive proof technique essentially bringing equational reasoning to the coworld. In fact, we redevelop the theory of recurrences, finite calculus and generating functions using streams and stream operators building on the cornerstone of unique solutions. The development is constructive: streams and stream operators are implemented in Haskell, usually by one-liners. The resulting calculus or library, if you wish, is elegant and fun to use. Finally, we rephrase the proof of uniqueness using generalised algebraic data types.

Particularly elegant examples are obtained using n+k patterns!

Memo, Idiom, Num (!), Enum, Integral, Fractional, NumExt

The great contribution of this pearl are coherent numeric instances for infinite streams, given by:

```    (+)              =  zip (+)
(-)              =  zip (-)
(*)              =  zip (*)
negate           =  map negate
abs              =  map abs
signum           =  map signum
toEnum i         =  repeat (toEnum i)
div              =  zip div
mod              =  zip mod
quotRem s t      =  unzip (zip quotRem s t)
fromInteger      =  repeat . fromInteger
s / t            =  zip (Prelude./) s t
recip s          =  map recip s
fromRational r   =  repeat (fromRational r)
(^)              =  zip (^)
(/)              =  zip (/)
fact             =  map fact
fall             =  zip fall
choose           =  zip choose
```

Synopsis

# Functions on streams

(<:) :: a -> Stream a -> Stream aSource

Cons for streams

unzip :: Stream (a, b) -> (Stream a, Stream b)Source

unzip two streams

iterate :: (a -> a) -> a -> Stream aSource

(<<) :: [a] -> Stream a -> Stream aSource

# Recurrences

turn :: Integral a => a -> [a]Source

tree :: Integral a => a -> Stream aSource

# Finite calculus

diff :: Num a => Stream a -> Stream aSource

sum :: Num a => Stream a -> Stream aSource

sumv :: Num a => Stream a -> Stream aSource

# Generating functions

const :: Num a => a -> Stream aSource

z :: Num a => Stream aSource

(**) :: Num a => Stream a -> Stream a -> Stream aSource

power :: (Fractional a, Integral b) => Stream a -> b -> Stream aSource