Ashley Yakeley wrote:

I considered doing something very like this for real (computable) numbers, but because I couldn't properly make the type an instance of Eq, I left it. Actually it was worse than that. Suppose I'm adding two numbers, both of which are actually 1, but I don't know that:

1.000000000.... + 0.999999999....

The trouble is, as far as I know with a finite number of digits, the answer might be 1.9999999999937425 or it might be 2.0000000000013565

...so I can't actually generate any digits at all. So I can't even make the type an instance of my Additive class.

You can, unless you are so ambitious that you want to have an ideal solution. Doing the stuff lazily means that you will have a thunk used in further computations, and the digits will be generated according to your needs. You *MAY* generate these digits physically ('1' or '2' in your case) if you permit yourself to engage in a possibly bottom-less recursive pit, which in most interesting cases actually *has* a bottom, and the process stops. Please look my "Pi" crazy essay. Once the decision concerning the carry is taken, the process becomes "sane", generative, co-recursive, until the next ambiguity. There are of course more serious approaches: intervals, etc. The infinite- precision arithmetic is a mature domain, developed by many people. Actually the Gosper arithmetic of continued fractions is also based on co-recursive expansion, although I have never seen anybody implementing it using a lazy language, and a lazy protocol. Anybody wants to do it with me? (Serious offers only...) Jerzy Karczmarczuk

Dear All, A really, really simple version in Haskell 1.2 has been available from ftp://ftp.cs.man.ac.uk/pub/arithmetic/Haskell/Era1/Era.hs for some considerable time. Of course the only reason for producing it was to show that the language designers didn't get it right. Take it from me, they never do [Is he being ironic here?]. In particular, although (<) and (==) are not computable functions over the computable reals, min and max are! Similarly, signum isn't computable, but abs is. The classes Ord and Num are a theoreticians nightmare. [For the mathematically sophisticated, the problem with these operations revolves around their continuity (in a weird numeric _and_ information theoretic sense) properties given the discrete nature of their range.] Then there's the rounding operation in Haskell 1.2. I must have wasted more hours fixing bugs caused by my naive understanding of _that_ operation than an other single misapprehension about a programming language construct in any langauge! In other files hanging off of http://www.cs.man.ac.uk/arch/dlester/exact.html you'll find a PVS development that shows that (most of) the implementation is correct. A paper on this theorem-proving work has been accepted for TCS, but I don't know whether it'll be published in my life time; and, I'm unsure about the ethics involved in puting up a copy of the paper on my website. However, a summary is: 4 bugs found, 3 only becoming manifest on doing the PVS work. I guess there's something to this formal methods lark after all. I'd planned to do a "Functional Pearl" about this library/package, but the theoretical development is embarassingly inelegant when compared to Richard Bird's classic functional pearls. If you think it'd be worth it, I'll see what I can do. I trust that this will save anyone wasting too much more time over this topic. David Lester. On Thu, 10 Oct 2002, Ashley Yakeley wrote:

At 2002-10-10 01:29, Ketil Z. Malde wrote:

...

I realize it's probably far from trivial, e.g. comparing two equal numbers could easily not terminate, and memory exhaustion would probably arise in many other cases.

I considered doing something very like this for real (computable) numbers, but because I couldn't properly make the type an instance of Eq, I left it. Actually it was worse than that. Suppose I'm adding two numbers, both of which are actually 1, but I don't know that:

1.000000000.... + 0.999999999....

The trouble is, as far as I know with a finite number of digits, the answer might be

1.9999999999937425

or it might be

2.0000000000013565

...so I can't actually generate any digits at all. So I can't even make the type an instance of my Additive class. Not very useful...

On Thu, Oct 10, 2002 at 02:25:59AM -0700, Ashley Yakeley wrote:

At 2002-10-10 01:29, Ketil Z. Malde wrote:

...

I realize it's probably far from trivial, e.g. comparing two equal numbers could easily not terminate, and memory exhaustion would probably arise in many other cases.

I considered doing something very like this for real (computable) numbers, but because I couldn't properly make the type an instance of Eq, I left it. Actually it was worse than that. Suppose I'm adding two numbers, both of which are actually 1, but I don't know that:

1.000000000.... + 0.999999999.... ...

The solution to such quandries is to allow non-unique representation. For instance, you might consider a binary system with allowed digits +1, 0, and -1, so that a number starting 0.xxxxxx can be anything between -1 and 1, and 0.1xxxxx can be anything between 0 and 1, etc. It is then possible to guarantee being able to output digits in a finite amount of time. With a scheme like this, the cases that blow up are ones you expect, like trying to compute 1/0; there are ways around that, too. As Jerzy Karczmarczuk mentioned, there is really extensive literature on this. It's beautiful stuff. Part of my motivation for revising the numeric parts of the Prelude was to make it possible to implement all this elegantly in Haskell. --Dylan Thurston