Ordinals (empty) → 0.0.0.1
raw patch · 4 files changed
+169/−0 lines, 4 filesdep +basesetup-changed
Dependencies added: base
Files
- LICENSE +12/−0
- Math/Ordinals/MultiSet.hs +137/−0
- Ordinals.cabal +17/−0
- Setup.lhs +3/−0
+ LICENSE view
@@ -0,0 +1,12 @@+Copyright (c) 2010, Ki Yung Ahn+All rights reserved.++Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:++ * Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.+ * Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.+ * Neither the name of the Portland State University nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.++THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.++
+ Math/Ordinals/MultiSet.hs view
@@ -0,0 +1,137 @@+-- Encoding of ordinals up to epsilon_0 as an iterated multiset:+-- definition in Basic Proof Theory by Troelstra and Schwichenberg.+-- Note, this is not the most efficient way to calculate ordinals.+-- This library is better than having none.+-- I think CNF representation would be more efficent,+-- planning to add in the next version of this library.++-- For further analysis on efficiency of implementations on ordinals see+-- http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.91.8089++-- FYI, An ordinal calculator that covers wider range beyond epsion_0+-- can be found at http://www.mtnmath.com/ord/ which is written C++.+-- However, I found a serious errors in this calculator:+-- the property a^b * a^c == a^(b+c) doesn't hold.+-- (e.g., +-- ordCalc> 3 ^(w^w + w + 1) * 3^(w^w) == 3 ^ (w^w + w + 1 + w^w)+-- FALSE )+-- This calculator didn't seem to run QuickCeck style tests+-- although it does have large tests cases some of them seg faults sometimes+-- depending on the machine I built this ord calcualtor.++++module Math.Ordinals.MultiSet where++import Data.List (groupBy, intersperse)++newtype Ordinal = O [Ordinal] deriving Eq++w o = O [o]++instance Show Ordinal where+ show o@(O os)+ | o < O[1] = show (length os)+ | otherwise = "("+ ++ (concat . intersperse " + ")+ ["w "++ show o' ++ showFactor(length os') | os'@(o':_)<-oss ]+ ++ ")"+ where showFactor k = if k==1 then "" else (" *"++show k)+ oss = groupBy (==) os++instance Ord Ordinal where+ compare 0 0 = EQ+ compare 0 _ = LT+ compare _ 0 = GT+ compare (O(a:as)) (O(b:bs)) = case compare a b of+ EQ -> compare as bs+ r -> r++-- well formedness of ordinals+wf (O []) = True+wf (O [o]) = wf o+wf (O (o:os@(o':_))) = o >= o' && wf (O os)+++instance Enum Ordinal where -- TODO+ toEnum = fromInt+ fromEnum = toInt++instance Real Ordinal where -- TODO+ +instance Integral Ordinal where+ -- works only within Int limit+ toInteger o@(O as) | all (0 ==) as = fromIntegral (toInt o)+ toInteger _ = error "ordinal not less than omega"++ div a b = fst (divMod a b)++ mod a b = snd (divMod a b)++ divMod = quotRem -- TODO -- somebody help figure this out++ quotRem _ _ = error "Ordinal quotRem not yet defined"+++instance Num Ordinal where+ -- to parse in haskell integer literals within Int limit+ fromInteger k = fromInt (fromIntegral k)++ abs = id++ signum 0 = 0+ signum (O(o:os)) = O[o]++ o + 0 = o+ O as + O bs@(b:_) = O (takeWhile (>=b) as ++ bs)++ -- for a <= b exists r = b - a such that a + r = b+ -- i.e., a + (b - a) = b for a <= b+ o - 0 = o+ 0 - _ = 0+ o1@(O(a:as)) - o2@(O(b:bs)) = case compare a b of + LT -> 0+ EQ -> O as - O bs+ GT -> o1++ -- something similar to+ -- http://www.volny.cz/behounek/logic/papers/ordcalc/index.html+ 0 * _ = 0+ o1@(O(a:as)) * O bs+ | null bs' = sum [o1 | _ <- zs] -- finite+ | null zs = O [a+b | b<-bs'] -- limit ordinal+ | otherwise = o1 * O bs' + o1 * O zs -- non-limit ordinal+ where (bs',zs) = span (>0) bs+ (as1,as2) = span (a==) (a:as)++-- exponentiation : define new op since neither ^ nor ^^ will work+infixr 8 ^:+_ ^: 0 = 1+0 ^: _ = 0+1 ^: _ = 1+o1@(O osA@(a:_)) ^: o2@(O osB@(b:_))+ | finiteB = o1^: (o2 - 1) * o1 -- finite power+ | mB > 0 = o1^: O osB' * o1^: mB -- non-limit ordinals+ | finiteA = O [o2]+ | otherwise = O [a * o2]+ where+ (osA',zsA) = span (>0) osA+ (osB',zsB) = span (>0) osB+ mA = O zsA+ mB = O zsB+ finiteB = null osB'+ finiteA = null osA'+++toInt (O as) | all (0 ==) as = length as+toInt _ = error "ordinal not less than omega"++fromInt k = O (replicate k (O []))++infixr 5 .:+o .: O os = O (o:os)+infixr 5 ++.+as ++. O bs = O (as++bs)+infixr 5 .++.+O as .++. O bs = O (as++bs)+
+ Ordinals.cabal view
@@ -0,0 +1,17 @@+name: Ordinals+version: 0.0.0.1+synopsis: Ordinal arithmetic+description: Ordinal arithmetic implementation up to epsilon_0.+ Currently based on interated multiset representation,+ planning to add CNF based implenetation.+category: Math+license: BSD3+license-file: LICENSE+author: Ki Yung Ahn+maintainer: kya@pdx.edu+homepage: https://patch-tag.com/r/kyagrd/Ordinals/+build-depends: base < 5+build-type: Simple+ghc-options: -Wall+exposed-modules: Math.Ordinals.MultiSet+ghc-prof-options: -auto-all
+ Setup.lhs view
@@ -0,0 +1,3 @@+#!/usr/bin/env runhaskell+> import Distribution.Simple+> main = defaultMain