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/
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
Afficher les réponses par date
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@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/
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
Gambit-list mailing list Gambit-list@iro.umontreal.ca https://mailman.iro.umontreal.ca/cgi-bin/mailman/listinfo/gambit-list
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@math.purdue.edu mailto:lucier@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 _______________________________________________ Gambit-list mailing list Gambit-list@iro.umontreal.ca <mailto:Gambit-list@iro.umontreal.ca> https://mailman.iro.umontreal.ca/cgi-bin/mailman/listinfo/gambit-list <https://mailman.iro.umontreal.ca/cgi-bin/mailman/listinfo/gambit-list>
-
Bradley Lucier -
John Cowan