An Explicit Solution to Post's Problem over the Reals

TL;DR

This paper constructs explicit examples of semi-decidable but undecidable problems over the reals within the BCSS model, demonstrating a rich structure of intermediate degrees unlike the classical discrete case.

Contribution

It provides the first explicit construction of intermediate Turing degrees over the reals in the BCSS model, including an uncountable family of such degrees and extensions to the linear BCSS model.

Findings

Explicit semi-decidable but undecidable language over the reals.

Existence of uncountably many incomparable semi-decidable degrees below the real Halting problem.

Results extend to the linear BCSS model.

Abstract

In the BCSS model of real number computations we prove a concrete and explicit semi-decidable language to be undecidable yet not reducible from (and thus strictly easier than) the real Halting Language. This solution to Post's Problem over the reals significantly differs from its classical, discrete variant where advanced diagonalization techniques are only known to yield the existence of such intermediate Turing degrees. Strengthening the above result, we construct (that is, obtain again explicitly) as well an uncountable number of incomparable semi-decidable Turing degrees below the real Halting problem in the BCSS model. Finally we show the same to hold for the linear BCSS model, that is over (R,+,-,<) rather than (R,+,-,*,/,<).