Difference between revisions of "User:WillNess"
Jump to navigation
Jump to search
(13 intermediate revisions by the same user not shown) | |||
Line 1: | Line 1: | ||
+ | [https://wiki.haskell.org/index.php?title=Monad&oldid=63472 Monad is composable computation descriptions]. |
||
− | I am a newbie, interested in Haskell. |
||
+ | ---- |
||
⚫ | |||
+ | |||
⚫ | |||
<haskell> |
<haskell> |
||
− | -- |
+ | -- infinite folding due to Richard Bird |
− | -- double staged |
+ | -- double staged primes production due to Melissa O'Neill |
− | -- tree folding idea |
+ | -- tree folding idea Heinrich Apfelmus / Dave Bayer |
− | primes = 2 : |
+ | primes = 2 : _Y ((3:) . gaps 5 |
− | + | . foldi (\(x:xs) -> (x:) . union xs) [] |
|
− | + | . map (\p-> [p*p, p*p+2*p..])) |
|
⚫ | |||
− | gaps k s@(c:t) |
||
⚫ | |||
⚫ | |||
+ | _Y g = g (_Y g) -- multistage production via Y combinator |
||
− | fix g = xs where xs = g xs -- global defn to avoid space leak |
||
+ | |||
⚫ | |||
⚫ | |||
⚫ | |||
</haskell> |
</haskell> |
||
− | <code>foldi</code> is on [[Fold#Tree-like_folds|Tree-like folds]] page. |
+ | <code>foldi</code> is on [[Fold#Tree-like_folds|Tree-like folds]] page. <code>union</code> and more at [[Prime numbers#Sieve_of_Eratosthenes|Prime numbers]]. |
+ | |||
+ | The constructive definition of primes is the Sieve of Eratosthenes, '''P''' = '''N'''<sub><sub>2</sub></sub>\'''N'''<sub><sub>2</sub></sub><sub>*</sub>'''N'''<sub><sub>2</sub></sub> = '''N'''<sub><sub>2</sub></sub>\'''P'''<sub>*</sub>'''N'''<sub><sub>2</sub></sub> : |
||
+ | |||
+ | ::::<math>\textstyle\mathbb{S} = \mathbb{N}_{2} \setminus \bigcup_{p\in \mathbb{S}} \{p\,q:q \in \mathbb{N}_{p}\}</math> |
||
+ | using standard definition |
||
+ | ::::<math>\textstyle\mathbb{N}_{k} = \{ n \in \mathbb{N} : n \geq k \}</math>   . . . or,  <math>\textstyle\mathbb{N}_{k} = \{k\} \bigcup \mathbb{N}_{k+1}</math> . |
||
+ | |||
+ | Trial division sieve is: |
||
+ | |||
+ | ::::<math>\textstyle\mathbb{T} = \{n \in \mathbb{N}_{2}: (\forall p \in \mathbb{T})(2\leq p\leq \sqrt{n}\, \Rightarrow \neg{(p \mid n)})\}</math> |
||
+ | |||
+ | If you're put off by self-referentiality, just replace <math>\mathbb{S}</math> or <math>\mathbb{T}</math> on the right-hand side of equations with <math>\mathbb{N}_{2}</math>, as the ancient Greeks might or mightn't have done, as well. |
Latest revision as of 13:50, 21 February 2023
Monad is composable computation descriptions.
I like this one-liner:
-- infinite folding due to Richard Bird
-- double staged primes production due to Melissa O'Neill
-- tree folding idea Heinrich Apfelmus / Dave Bayer
primes = 2 : _Y ((3:) . gaps 5
. foldi (\(x:xs) -> (x:) . union xs) []
. map (\p-> [p*p, p*p+2*p..]))
_Y g = g (_Y g) -- multistage production via Y combinator
gaps k s@(c:t) -- == minus [k,k+2..] (c:t), k<=c,
| k < c = k : gaps (k+2) s -- fused for better performance
| otherwise = gaps (k+2) t -- k==c
foldi
is on Tree-like folds page. union
and more at Prime numbers.
The constructive definition of primes is the Sieve of Eratosthenes, P = N2\N2*N2 = N2\P*N2 :
using standard definition
- . . . or, .
Trial division sieve is:
If you're put off by self-referentiality, just replace or on the right-hand side of equations with , as the ancient Greeks might or mightn't have done, as well.