Re: type class Boolean

Hi all, let's start at lattices, a set with ∨ and ∧, where a∧(a∨b) ≈ a, a∨(a∧b) ≈ a, and ∨ and ∧ are commutative, associative, and idempotent. To get to Boolean algebras, we need to add a. bounds, b. distributivity, and c. complements. Given any lattice, you can always add a +∞ and a -∞ to get a bounded lattice (there is a functor ...). So I think that from the point of view of class structure Lattice and Bounded can exist independently. At least for lattices, upper and lower bounds are independent, so ??maybe?? a OneSidedBounded class. Not all lattices are distributive [i.e., they don't autmoatically satisfy a∧(c∨b) ≈ (a∧c)∨(a∧b), a∨(c∧b) ≈ (a∨c)∧(a∨b) ]. A class between lattice and DistributiveLattice that might be useful is ModularLattice (lattices with a notion of "height"). There are also meet semi-distributive ... (I'm not an expert on lattice theory, but I know several). Most people are most familiar with distributive lattices. Similarly, not all bounded distributive lattices have a complement operation. A bounded distributive lattice with a complement operation that satisfies De Morgan's laws and a′′≈a (alternatively a′∨a≈⊤) is a Boolean algebra. There is a strictly weaker notion of a Heyting algebra, which might be useful as a class, because it uses implication (→) as a fundamental operation in place of not (′). Usually in a Heyting algebra a′ is defined to mean a→⊥. There are Heyting algebras where a′∨a≈⊤ doesn't hold. As a programmer, I haven't seen a non-Boolean Heyting algebra in "the wild", or at least not recognized them, but it doesn't mean they wouldn't be useful. At any rate, these are some of the things that I would think about while simultaneously trying to avoid to much bikeshedding on the one hand, and suddenly forcefully retro-fitting Applicative (or Semigroup) on the other. hth, stay safe! Happy Solstice! David -- David Casperson Computer Science David Feuer, on 2020-12-21, you wrote:

From: David Feuer To: Tikhon Jelvis Date: Mon, 21 Dec 2020 15:49:18 Cc: Ben Franksen , Haskell Libraries Subject: Re: type class Boolean Message-ID:

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I imagine you'd want (&&) and (||) to live in a Lattice class, `bottom` and `top` in `BoundedBelow` and `BoundedAbove`, respectively, and `not` in a `Boolean` one. The relationship between Lattice, JoinSemilattice, and MeetSemilattice is unfortunately awkward (much like that between Ring, AbelianGroup, and Monoid), but mumble mumble something.

On Mon, Dec 21, 2020 at 6:32 PM Tikhon Jelvis wrote:

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For me, the most useful kind of instance would be symbolic booleans like SBV's SBool type.

A real bonus would be making if-then-else polymorphic as well—although that doesn't really fit with the lattice abstraction. I know we can do that with RebindableSyntax, but that is a *heavyweight* extension to enable!

Breaking boolean behavior up into a few different classes would work for this application. Boolean algebras for and/not/etc, Conditional for ifThenElse and maybe even Boolish or something for True and False pattern synonyms (or just true and false constants).

I don't think all of this belongs in base, but making booleans and boolean operators more polymorphic would definitely be useful.

On Mon, Dec 21, 2020 at 3:21 PM Ben Franksen wrote:

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Am 21.12.20 um 19:54 schrieb Tom Ellis:

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I think it's worth seeing more instances. As it is I don't understand in what situations one would use these polymorphically and therefore why `liftA2 (&&)`, `fmap not`, `pure True` and friends wouldn't suffice.

As with all overloading it's partly a matter of convenience. For instance you can't use 'liftA2 (&&)' as an operator. And your list of alternatives above demonstrates that one has to remember which lifting operator (pure, fmap, liftA2) to use, depending on arity.

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