[gambit-list] (Slightly) Off topic: Sample Scheme code for the computable reals

Bradley Lucier lucier at math.purdue.edu
Tue Feb 1 18:22:32 EST 2022


Thank you for the kind words.

The issue for your example is that equality (=) and inequality (<) are 
themselves not computable, this is discussed in Section 4.8.

So the signum function, like the example in the first paragraph of 
Section 5, is not in fact computable.

Brad

On 2/1/22 5:41 PM, John Cowan wrote:
> This is a truly great paper, and I am glad to have read it.  But there 
> is something wrong with the theorem at the beginning of 4.14 ("all 
> computable functions that are defined for all computable real arguments 
> and return computable real results are continuous"), probably that the 
> claim is over-generalized (or maybe just that I am a high-school 
> mathematician).  The domain of the signum function obviously includes 
> the computable reals.  The codomain is {0, 1, -1}, which are all 
> computable.  Yet signum is not continuous at 0.  This can be generalized 
> to other stepwise functions with any countable number of discontinuous 
> points.  What gives?
> 
> 
> On Tue, Feb 1, 2022 at 12:15 PM Bradley Lucier <lucier at math.purdue.edu 
> <mailto:lucier at math.purdue.edu>> wrote:
> 
>     I wrote a paper on the computable reals with a page of (Gambit) Scheme
>     code at the end that implements some of the examples.  The code was
>     organized for clarity of presentation (base 10, easiest algorithms,
>     etc.), so it's not production quality, but there are references for
>     people who are interested:
> 
>     https://scholarship.claremont.edu/jhm/vol12/iss1/25/
>     <https://scholarship.claremont.edu/jhm/vol12/iss1/25/>
> 
>     I made sure that the typesetting didn't screw up the code, that you can
>     just copy it to a Gambit REPL and everything works.
> 
>     And Section 3 tells a funny story about me learning how to compute
>     square roots by hand.  (Perhaps only a mathematician could write that
>     sentence!)
> 
>     Brad
> 
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