Another amazing work by Hans Boehm back in 1999: a constructive real number calculator, https://www.hboehm.info/crcalc/ . In a sense, this is "as far as a computer can go" in the hierarchy of accuracy that includes floating-point, exact rational numbers, and then constructive reals.
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The idea is: instead of storing digits of a number, we store a lower and upper bound, and a function to refine and add more accuracy to the number. With this, we can compute anything on-demand to arbitrary precision.
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Replying to @TimSweeneyEpic
Just pray you never have to compute x == y :)
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Replying to @pervognsen
It would be interesting to explore: is there a useful subclass of computable reals that goes beyond rational numbers while retaining computable equality? The quadratic reals are one example, but how far can this go?
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Replying to @TimSweeneyEpic @pervognsen
All algebraic numbers works fine. But also: https://en.wikipedia.org/wiki/Ring_of_periods … "The conjecture of Kontsevich and Zagier would imply that equality of periods is also decidable" But it's not been proved yet :( (Also, if the algorithm exists, it might be megasuperexponential...)
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Replying to @sigfpe @pervognsen
Wow! That is a very broad class of numbers. More generally, I guess equality is decidable for any numbers for which a proof of equality or inequality exists, by proof search (since these computable reals can store their formulaic construction history).
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Which proverbially depends on what the definition of “is” is.
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