[Haskell-cafe] Lattice and calculation of Least Upper Bounds

Aaron Gray aaronngray.lists at gmail.com
Fri Jun 29 00:15:29 UTC 2018


Thanks for the reply. Answer inline.

On Tue, 19 Jun 2018 at 20:53, Olaf Klinke <olf at aatal-apotheke.de> wrote:

> Aaron,
> the lattices package provides some modules to extend a given lattice by
> some elements, e.g. new top and bottoms. There are also derived typeclass
> instances for combinations like tuples, endomorphisms and so forth. But the
> way of choice really depends on what you know about your multiple
> inheritance hierarchy

AFAICS all that needs to be known about the inheritance hierarchy in order
to create the lattice is the sub type relations, and also top and bottom
reference types.

Ideally I would like to iterate through the hierarchy starting with top and
then the base classes and add them incrementally with their subtype
dependencies through to bottom.

> In universal algebra one powerful method of constructing (semi-)lattices
> is by generators and relations. That means you define the lattice as a
> quotient of a free lattice. The quotient itself is defined as a set of
> ineqalities on the generators. I don't know how one would implement that
> without dependent types, though, as the type would be another type together
> with a function. To make things worse, the word problem is undecidable in
> general.
> Looking at Algebra.Lattice.Free I'm surprised that the free (semi-)lattice
> types don't have a Monad instance. Does anyone know why they are not
> implemented? Under the hood the free lattice types are identical to the
> continuation monad.

I think the Lattice classes may well be still in flux as I looked about a
month ago ad there seemed to be AFAICR a Least Upper Bound operation taking
a list of elements and returning an element.


> Olaf
> >Hi,
> >
> >I am trying to work out how to use the Algebra.Lattice family of Lattice
> >data structures.
> >
> >Firstly how do I construct a lattice ?
> >
> >What I am wanting to do is to be able to construct a lattice to represent
> a
> >multiple inheritance hierarchy. Then I to be able to find the Least Upper
> >Bound of a set of classes/types. This is in order to find the type of a
> >multiple case expression.
> >
> >I am not sure if the Haskell classes are actually applicable ? but if they
> >are how do I apply them to the following problem please ?

Aaron Gray

Independent Open Source Software Engineer, Computer Language Researcher,
Information Theorist, and amateur computer scientist.
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