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POHow should I implement a Cayley Table in Haskell?
 

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  1. POHow should I implement a Cayley Table in Haskell?
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    <p>I'm interested in generalizing some computational tools to use a <a href="http://en.wikipedia.org/wiki/Cayley_table">Cayley Table</a>, meaning a lookup table based multiplication operation. </p> <p>I could create a minimal implementation as follows :</p> <pre><code>date CayleyTable = CayleyTable { ct_name :: ByteString, ct_products :: V.Vector (V.Vector Int) } deriving (Read, Show) instance Eq (CayleyTable) where (==) a b = ct_name a == ct_name b data CTElement = CTElement { ct_cayleytable :: CayleyTable, ct_index :: !Int } instance Eq (CTElement) where (==) a b = assert (ct_cayleytable a == ct_cayleytable b) $ ct_index a == ct_index b instance Show (CTElement) where show = ("CTElement" ++) . show . ctp_index a **** b = assert (ct_cayleytable a == ct_cayleytable b) $ ((ct_cayleytable a) ! a) ! b </code></pre> <p>There are however numerous problems with this approach, starting with the run time type checking via <code>ByteString</code> comparisons, but including the fact that <code>read</code> cannot be made to work correctly. Any idea how I should do this correctly?</p> <p>I could imagine creating a family of newtypes <code>CTElement1</code>, <code>CTElement2</code>, etc. for <code>Int</code> with a <code>CTElement</code> typeclass that provides the multiplication and verifies their type consistency, except when doing IO.</p> <p>Ideally, there might be some trick for passing around only one copy of this <code>ct_cayleytable</code> pointer too, perhaps using an implicit parameter like <code>?cayleytable</code>, but this doesn't play nicely with multiple incompatible Cayley tables and gets generally obnoxious. </p> <p>Also, I've gathered that an index into a vector can be viewed as a comonad. Is there any nice comonad instance for vector or whatever that might help smooth out this sort of type checking, even if ultimately doing it at runtime? </p>
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    1. USJeff Burdges
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      1. POHow should I implement a Cayley Table in Haskell?
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      1. POHow should I implement a Cayley Table in Haskell?
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    1. COWhy use a ByteString? Though a Read instance won't be possible unless you can derive the cayley table from just the name and the index.
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      1. POHow should I implement a Cayley Table in Haskell?
      1. USivanm
    2. CONo reason, ct_name exists only to make `Eq CayleyTable` faster because the Cayley table might have millions entries. An `Int` works fine too. Ideally, `Read` should learn the specific Cayley table from the type system, presumably `read "0" :: CTElementFoo` should always return a reasonable value, or perhaps using 1 instead if indices are 1 based.
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