# User:WillNess

(Difference between revisions)
 Revision as of 08:07, 7 August 2011 (edit)← Previous diff Current revision (16:54, 19 November 2011) (edit) (undo) (14 intermediate revisions not shown.) Line 1: Line 1: - I am a newbie, interested in Haskell. + A perpetual Haskell newbie. I like ''[http://ideone.com/qpnqe this semi-one-liner]'': - + - I like ''[http://ideone.com/qpnqe this]'': + - primes = 2 : g (fix g) -- double staged production idea due to M. O'Neill + -- inifinte folding idea due to Richard Bird - where + -- double staged production idea due to Melissa O'Neill - g xs = 3 : (gaps 5 \$ foldi (\x:xs -> (x:) . union xs) + -- tree folding idea Dave Bayer / simplified formulation Will Ness - [[p*p, p*p+2*p..] | p <- xs]) + primes = 2 : g (fix g) - gaps k s@(x:xs) -- | k<=x = minus [k,k+2..] xs + where - -- inlined to avoid a space leak + g xs = 3 : gaps 5 (foldi (\(c:cs) -> (c:) . union cs) [] - = if k < x + [[x*x, x*x+2*x..] | x <- xs]) - then k : gaps (k+2) s + - else gaps (k+2) xs + - fix g = xs where xs = g xs + fix g = xs where xs = g xs -- global defn to avoid space leak + + gaps k s@(c:t) -- == minus [k,k+2..] (c:t), k<=c, + | k < c = k : gaps (k+2) s -- fused to avoid a space leak + | True = gaps (k+2) t - foldi is on [[Fold#Tree-like_folds|Tree-like folds]]. More at [[Prime numbers#Sieve_of_Eratosthenes|Prime numbers]]. + foldi is on [[Fold#Tree-like_folds|Tree-like folds]] page. union and more at [[Prime numbers#Sieve_of_Eratosthenes|Prime numbers]]. + + The constructive definition of primes is the Sieve of Eratosthenes: + + ::::$\textstyle\mathbb{S} = \mathbb{N}_{2} \setminus \bigcup_{p\in \mathbb{S}} \{p\,q:q \in \mathbb{N}_{p}\}$ + using standard definition + ::::$\textstyle\mathbb{N}_{k} = \{ n \in \mathbb{N} : n \geq k \}$   . . . or,  $\textstyle\mathbb{N}_{k} = \{k\} \bigcup \mathbb{N}_{k+1}$   :) :) . + + Trial division sieve is: + + ::::$\textstyle\mathbb{T} = \{n \in \mathbb{N}_{2}: (\forall p \in \mathbb{T})(2\leq p\leq \sqrt{n}\, \Rightarrow \neg{(p \mid n)})\}$ + + If you're put off by self-referentiality, just replace $\mathbb{S}$ or $\mathbb{T}$ on the right-hand side of equations with $\mathbb{N}_{2}$, but even ancient Greeks knew better.

## Current revision

A perpetual Haskell newbie. I like this semi-one-liner:

--   inifinte folding idea due to Richard Bird
--   double staged production idea due to Melissa O'Neill
--   tree folding idea Dave Bayer / simplified formulation Will Ness
primes = 2 : g (fix g)
where
g xs = 3 : gaps 5 (foldi (\(c:cs) -> (c:) . union cs) []
[[x*x, x*x+2*x..] | x <- xs])

fix g = xs where xs = g xs        -- global defn to avoid space leak

gaps k s@(c:t)                    -- == minus [k,k+2..] (c:t), k<=c,
| k < c = k : gaps (k+2) s     --     fused to avoid a space leak
| True  =     gaps (k+2) t

foldi is on Tree-like folds page. union and more at Prime numbers.

The constructive definition of primes is the Sieve of Eratosthenes:

$\textstyle\mathbb{S} = \mathbb{N}_{2} \setminus \bigcup_{p\in \mathbb{S}} \{p\,q:q \in \mathbb{N}_{p}\}$

using standard definition

$\textstyle\mathbb{N}_{k} = \{ n \in \mathbb{N} : n \geq k \}$   . . . or,  $\textstyle\mathbb{N}_{k} = \{k\} \bigcup \mathbb{N}_{k+1}$   :) :) .

Trial division sieve is:

$\textstyle\mathbb{T} = \{n \in \mathbb{N}_{2}: (\forall p \in \mathbb{T})(2\leq p\leq \sqrt{n}\, \Rightarrow \neg{(p \mid n)})\}$

If you're put off by self-referentiality, just replace $\mathbb{S}$ or $\mathbb{T}$ on the right-hand side of equations with $\mathbb{N}_{2}$, but even ancient Greeks knew better.