Personal tools

User:WillNess

From HaskellWiki

(Difference between revisions)
Jump to: navigation, search
Current revision (16:54, 19 November 2011) (edit) (undo)
 
(7 intermediate revisions not shown.)
Line 1: Line 1:
-
I'm 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>
Line 9: Line 7:
primes = 2 : g (fix g)
primes = 2 : g (fix g)
where
where
-
g ps = 3 : gaps 5 (foldi (\(q:qs) -> (q:) . union qs)
+
g xs = 3 : gaps 5 (foldi (\(c:cs) -> (c:) . union cs) []
-
[[p*p, p*p+2*p..] | p <- ps])
+
[[x*x, x*x+2*x..] | x <- xs])
-
gaps k s@(c:t)
+
-
| k < c = k : gaps (k+2) s -- | k<=c = minus [k,k+2..] s
+
-
| True = gaps (k+2) t -- fused to avoid a space leak
+
-
fix g = xs where xs = g xs -- global defn to avoid space leak
+
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]] page. <code>union</code> and more at [[Prime numbers#Sieve_of_Eratosthenes|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]].
-
Also, the math formula for Sieve of Eratosthenes,
+
The constructive definition of primes is the Sieve of Eratosthenes:
-
::::<math>\textstyle\mathbb{S} = \mathbb{N}_{2} \setminus \bigcup_{p\in \mathbb{S}} \{n p:n \in \mathbb{N}_{p}\}</math>
+
::::<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;:) .
-
where
+
Trial division sieve is:
-
 
+
-
::::<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:
+
::::<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>
-
::::<math>\textstyle\mathbb{T} = \{n \in \mathbb{N}_{2}: (\not\exists p \in \mathbb{T}) (p\leq \sqrt{n} \and 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.

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.

Retrieved from "http://www.haskell.org/haskellwiki/User:WillNess"