[Haskell-cafe] Re: Set Operations In Haskell's Type System

On May 9, 4:46 am, wren ng thornton wrote:

John Creighton wrote:

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On May 6, 4:30 am, Bartek Ćwikłowski wrote:

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2010/5/6 John Creighton :

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"a" isa "d" if their exists a "b" and "c" such that the following conditions hold: "a" isa subset of "b", "b" isa "c" "c" is a subset of "d" This definition doesn't make sense - it's recursive, but there's no base case, unless this is some kind of co-recursion.

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Are you sure that "subset" isn't what you really want? With subset you can already ask questions such as "is tabby cat an animal?". If so, my code (from hpaste) already has this (iirc isDescendentOf ).

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When I succeed in implementing it I'll show you the result. Anyway, some perspective (perhaps), I once asked, "what is the difference between a subset and an element of a set:

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http://www.n-n-a.com/science/about33342-0-asc-0.html

And it's truly an interesting question. Too bad it didn't get a better discussion going (from what I read of it). Though the link Peter_Smith posted looks interesting.

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note 1) Okay I'm aware some will argue my definitions here and if it helps I could choose new words, the only question really is, is the relationship isa which I described a useful abstraction.

I think the key issue comes down to what you want to do with it. I'm not entirely sure what the intended reading is for "isa subset of", but I'll assume you mean the same as "is a subset of"[1]. One apparent side effect of the definition above is that it collapses the hierarchy.

That is, with traditional predicates for testing element and subset membership, we really do construct a hierarchy. If A `elem` B and B `elem` C, it does not follow that A `elem` C (and similar examples). But with your definition it seems like there isn't that sort of stratification going on. If the requirements are A `subset` B, B `elem` C, and C `subset` D--- well we can set C=D, and now: A `elem` D = A `subset` B && B `elem` D.

Depending on the ontology you're trying to construct, that may be perfectly fine, but it's certainly a nonstandard definition for elements and subsets. I don't know if this mathematical object has been worked on before, but it's not a hierarchy of sets.

[1] My other, equivalent, guess would be you mean "A isa (powerset B)" but avoided that notation because it looks strange.

-- Live well, ~wren _______________________________________________ Haskell-Cafe mailing list Haskell-C...@haskell.orghttp://www.haskell.org/mailman/listinfo/haskell-cafe

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Keep in mind that my recent definition of "is a"

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"a" isa "d" if their exists a "b" and "c" such that the following conditions hold: "a" isa subset of "b", "b" isa "c" "c" is a subset of "d"

is distinct from the question I asked a long time ago of the difference between a set and an element. The question I asked a long time ago is largely philosophical but can have axiomatic consequences in set theory. Ignoring the philosophical meanings behind a set, both the operations of subset and "element of", define a partial order. The subset relationship seems to define things that are more similar then the "element of" relationship.