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Restarting a piece of code
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Date: -- (:)
From: Alex Baretta <alex@b...>
Subject: Re: [Caml-list] Restarting a piece of code
Andrej Bauer wrote:

> 
> Can we avoid having to pass treshold around directly? You can imagine
> that users won't be too thrilled about this sort of thing.

I think you can. See arithmetic functions as building blocks for
computations---rather than computations themselves---which are
parametric with respect to the threshold value.

let unit_expression _ = assert false
let imaginary_unit_expression _ = assert false
let addition_expression _ = assert false
let negative_expression _ = assert false
let multiplication_expression _ = assert false
let inverse_expression _ = assert false
let exponential_expression _ = assert false
let logarithm_expression _ = assert false

exception Overflow

let rec compute grow_threshold threshold_transformation = fun
  computation
  threshold
->
  try
    computation
      (compute grow_threshold)
      (threshold_transformation threshold)
  with Overflow ->
    let new_threshold = grow_threshold threshold in
      computation
        (compute grow_threshold)
        (threshold_transformation new_threshold)

(* Here we define the primitive operations of the algebraic structure *)
(* We don't mind if the primitive operations are a little heavy, so   *)
(* long as it is easy to compose them to form complex computations.   *)

let one = fun compute threshold -> unit_expression threshold
let i = fun compute threshold -> imaginary_unit_expression threshold

let add x y = fun compute threshold ->
  let x' = compute x threshold in
  let y' = compute y threshold in
    addition_expression x y threshold

let neg x = fun compute threshold ->
  let x' = compute x threshold in
    negative_expression x threshold

let mul x y = fun compute threshold ->
  let x' = compute x threshold in
  let y' = compute y threshold in
    multiplication_expression x y threshold

let inv x = fun compute threshold ->
  let x' = compute x threshold in
    inverse_expression x threshold

let exp x = fun compute threshold ->
  let x' = compute x threshold in
    exponential_expression x threshold

let log x = fun compute threshold ->
  let x' = compute x threshold in
    logarithm_expression x threshold

(* Let's say these are all the basic computations we need. Now we can *)
(* start building more computations on top of these.                  *)

let sub x y = add x (neg y)
let div x y = mul x (inv y)
let pow x y = exp (mul (log x) y)
let root x y = exp (mul (log x) (inv y))
let two = add one one
let twoi = add i i
let cos x = div (add (exp (mul i x)) (exp (neg (mul i x)))) two
let sin x = div (sub (exp (mul i x)) (exp (neg (mul i x)))) twoi

> I sense monads. Or am I looking for dynamic binding?

You are looking for partial evaluation/multistage programming, but you
don't necessarily have to delve into MetaOcaml to solve your problem. As
you can see you can generate a homomorphism from the calculus of
imaginary numbers to the calculus of computations of imaginary numbers
which can be directly represented in Ocaml.

> Best regards,
> 
> Andrej
> 


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