Copyright | (c) 2017 Andrew Lelechenko |
---|---|

License | MIT |

Maintainer | Andrew Lelechenko <andrew.lelechenko@gmail.com> |

Safe Haskell | None |

Language | Haskell2010 |

Helpers for continuous mappings, useful to memoize
functions on `Int`

(instead of `Word`

only) and
functions over two and three arguments.

**Example 1**

Imagine writing a program to simulate
Rule 90.
This is a cellular automaton,
which consists of an infinite one-dimensional line of cells,
each being either dead (`False`

) or alive (`True`

).
If two neighbours of a cell are equal,
it becomes dead on the next step, otherwise alive.

Usually cellular automata are modelled by a finite vector. This is a bit suboptimal, because cellular automata may grow in different directions over time, but with a finite vector one has to define a bounding segment well beforehand. Moreover, what if we are interested to explore an evolution of an essentially infinite initial configuration?

It would be natural to encode an initial configuration
as a function `Int`

`->`

`Bool`

, which takes a coordinate
and returns the status of the corresponding cell. Define
a function, which translates the automaton to the next step:

step :: (Int -> Bool) -> (Int -> Bool) step current = \n -> current (n - 1) /= current (n + 1)

Unfortunately, iterating `step`

would be extremely slow
because of branching recursion. One
could suggest to introduce a caching layer:

step :: (Int -> Bool) -> (Int -> Bool) step current = \n -> current' (n - 1) /= current' (n + 1) where current' = memoize (current . fromIntegral) . fromIntegral

Unfortunately, it would not work well,
because `fromIntegral`

`::`

`Int`

`->`

`Word`

maps `-1`

to `maxBound`

and it would take ages to memoize
everything up to `maxBound`

.
But continuous mappings `intToWord`

and `wordToInt`

avoid this issue:

step :: (Int -> Bool) -> (Int -> Bool) step current = \n -> current' (n - 1) /= current' (n + 1) where current' = memoize (current . wordToInt) . intToWord

**Example 2**

What about another famous cellular automaton:
Conway's Game of Life?
It is two-dimensional, so its state can be represented as
a function `Int`

`->`

`Int`

`->`

`Bool`

. Following the approach above,
we would like to memoize such functions.
Namely, cast the state to `Word`

`->`

`Bool`

, ready for memoization:

cast :: (Int -> Int -> Bool) -> (Word -> Bool) cast f = \n -> let (x, y) = fromZCurve n in f (word2int x) (word2int y)

and then back:

uncast :: (Word -> Bool) -> (Int -> Int -> Bool) uncast g = \x y -> g (toZCurve (int2word x) (int2word y))

# Documentation

intToWord :: Int -> Word Source #

Total map, which satisfies

abs (intToWord x - intToWord y) <= 2 * abs (x - y)

Note that usual `fromIntegral`

`::`

`Int`

`->`

`Word`

does not
satisfy this inequality,
because it has a discontinuity between −1 and 0.

`>>>`

[9,7,5,3,1,0,2,4,6,8,10]`map intToWord [-5..5]`

wordToInt :: Word -> Int Source #

Inverse for `intToWord`

.

`>>>`

[0,-1,1,-2,2,-3,3,-4,4,-5,5]`map wordToInt [0..10]`

toZCurve :: Word -> Word -> Word Source #

Total map from plain to line, continuous almost everywhere. See Z-order curve.

Only lower halfs of bits of arguments are used (32 bits on 64-bit architecture).

`>>>`

[0,2,8,10,1,3,9,11,4,6,12,14,5,7,13,15]`[ toZCurve x y | x <- [0..3], y <- [0..3] ]`

fromZCurve :: Word -> (Word, Word) Source #

Inverse for `toZCurve`

.
See Z-order curve.

`>>>`

[(0,0),(1,0),(0,1),(1,1),(2,0),(3,0),(2,1),(3,1),(0,2),(1,2),(0,3),(1,3),(2,2),(3,2),(2,3),(3,3)]`map fromZCurve [0..15]`

toZCurve3 :: Word -> Word -> Word -> Word Source #

Total map from space to line, continuous almost everywhere. See Z-order curve.

Only lower thirds of bits of arguments are used (21 bits on 64-bit architecture).

`>>>`

[0,4,32,36,2,6,34,38,16,20,48,52,18,22,50,54,1,5,33,37,3,7,35,39,17,21,49,53,19,23,51,55, 8,12,40,44,10,14,42,46,24,28,56,60,26,30,58,62,9,13,41,45,11,15,43,47,25,29,57,61,27,31,59,63]`[ toZCurve3 x y z | x <- [0..3], y <- [0..3], z <- [0..3] ]`

fromZCurve3 :: Word -> (Word, Word, Word) Source #

Inverse for `toZCurve3`

.
See Z-order curve.

`>>>`

[(0,0,0),(1,0,0),(0,1,0),(1,1,0),(0,0,1),(1,0,1),(0,1,1),(1,1,1),(2,0,0),(3,0,0),(2,1,0),(3,1,0),(2,0,1),(3,0,1),(2,1,1),(3,1,1), (0,2,0),(1,2,0),(0,3,0),(1,3,0),(0,2,1),(1,2,1),(0,3,1),(1,3,1),(2,2,0),(3,2,0),(2,3,0),(3,3,0),(2,2,1),(3,2,1),(2,3,1),(3,3,1), (0,0,2),(1,0,2),(0,1,2),(1,1,2),(0,0,3),(1,0,3),(0,1,3),(1,1,3),(2,0,2),(3,0,2),(2,1,2),(3,1,2),(2,0,3),(3,0,3),(2,1,3),(3,1,3), (0,2,2),(1,2,2),(0,3,2),(1,3,2),(0,2,3),(1,2,3),(0,3,3),(1,3,3),(2,2,2),(3,2,2),(2,3,2),(3,3,2),(2,2,3),(3,2,3),(2,3,3),(3,3,3)]`map fromZCurve3 [0..63]`