Re: [Haskell-cafe] [Ann] group-theory

Absolutely. I do think that the moment you start caring about representations or presentations, as I believe emily has mentioned, some stronger assumptions for certain families of interfaces is worth considering to allow for cool stuff when those assumptions are admissible. On Sun, Dec 13, 2020 at 1:51 AM Zemyla wrote:

A simpler example of groups with no computable equality: the Sum group on computable reals. x + negate x === 0 for all x, but we can never fully prove it, only demonstrate it for each number of digits. And yet this is a perfectly well behaved abelian group.

On Sat, Dec 12, 2020, 23:41 Viktor Dukhovni wrote:

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On Sun, Dec 13, 2020 at 12:19:18AM -0500, Carter Schonwald wrote:

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Having a decidable equality seems important for reasoning about groups. Or computing on representations thereof.

This is of course not always possible. If a group is presented as a quotient of a free group on a set of generators via some set of relators, then deciding whether two words are equal is IIRC known to be generally intractable. One can still perform the group operation, but there is no known terminating algorithm for constructing a canonical form for an element, performing equality tests, ...

Of course one might take the view that groups where equality is not computable are not "useful", but that might rule out some applications.

-- Viktor. _______________________________________________ Haskell-Cafe mailing list To (un)subscribe, modify options or view archives go to: http://mail.haskell.org/cgi-bin/mailman/listinfo/haskell-cafe Only members subscribed via the mailman list are allowed to post.

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