Personal tools

User:WillNess

From HaskellWiki

(Difference between revisions)
Jump to: navigation, search
(16 intermediate revisions by one user 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]'':
+
  
 
<haskell>
 
<haskell>
primes = 2 : g (fix g) -- double staged production idea due to M. O'Neill
+
--   inifinte folding idea due to Richard Bird
fix g = xs where xs = g xs
+
--  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        -- 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   
 
</haskell>
 
</haskell>
  
<code>foldi</code> is on [[Fold#Tree-like_folds|Tree-like folds]]. More at [[Prime numbers]].
+
<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:
 +
 
 +
::::<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> &emsp; . . . or, &ensp;<math>\textstyle\mathbb{N}_{k} = \{k\} \bigcup \mathbb{N}_{k+1}</math> &emsp; :)&emsp;:) .
 +
 
 +
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>, but even ancient Greeks knew better.

Revision as of 16:54, 19 November 2011

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.