Abstract
On page 9 of Rogers’ computability book he presents two functions each based on eventual, currently unknown patterns in the decimal expansion of π. One of them is easily (classically) seen to be computable, but the proof is highly non-constructive and, conceptually interestingly, there is no known example algorithm for it. For the other, it is unknown as to whether it is computable. In the future, though, these unknown patterns in the decimal expansion of π may be sufficiently resolved, so that, for the one, we shall know a particular algorithm for it, and/or, for the other whether it’s computable. The present paper provides a “safer” real to replace π so that the associated one function retains its trivial computability but has unprovability of the correctness of any particular program for it. Re the other function, a real r to replace π is given with each bit of this r uniformly linear time computable in the length of its position and so that the Rogers’ other function associated with r is provably uncomputable.
We are grateful for anonymous referees’ helpful corrections and suggestions.
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Case, J., Ralston, M. (2013). Beyond Rogers’ Non-constructively Computable Function. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds) The Nature of Computation. Logic, Algorithms, Applications. CiE 2013. Lecture Notes in Computer Science, vol 7921. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39053-1_6
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