Loops, Inverse Limits and Non-Determinism
Vasco Brattka
TL;DR
The paper defines and analyzes the inverse limit operator $f^ abla$ in Weihrauch complexity to model infinite-loop computations, situating it relative to existing loop and parallelization operators. It proves monotonicity under strong Weihrauch reducibility while showing $f^ abla$ is not a closure operator, and demonstrates that weak König's lemma is closed under inverse limits, implying non-deterministically computable problems form a closed class under this operation. Key technical tools include an injective version of the recursion theorem and an infinitary independent choice theorem, and the work compares $f^ abla$ to the diamond operator followed by parallelization, with detailed results for single-valued problems and problems on Turing degrees. Altogether, the results illuminate how infinite-loop constructs interact with the Weihrauch lattice and provide groundwork for programming with loops in non-deterministic and degree-theoretic settings.
Abstract
We introduce an operator on problems in Weihrauch complexity, which we call the inverse limit, and which corresponds to an infinite compositional product. This operation arises naturally whenever one implements algorithms that produce a sequence of results in an infinite loop, using some fixed subroutine. We prove that the corresponding operator is monotone with respect to (strong) Weihrauch reducibility but that it is not a closure operator. One of our findings is that weak Kőnig's lemma is closed under inverse limits, which implies that the class of non-deterministically computable problems is also closed under this operation. Consequently, this class allows for a high degree of flexibility in programming. As our main technical tools, we present an injective version of the recursion theorem and an infinitary version of the so-called independent choice theorem. We also show that, in general, the inverse limit operator is more powerful than the composition of the diamond operator followed by the parallelization operator. However, in many practical scenarios, these compositions yield a result, which coincides with the application of the inverse limit operator. Finally, we discuss the special situation of loops for single-valued problems and for problems on Turing degrees.