🤖 AI Summary
This paper investigates the computational nature of result sequences generated by fixed-subroutine iteration in infinite-loop algorithms, focusing on their cyclicity, reverse-limit behavior, and non-determinism within the Weihrauch complexity framework. Methodologically, it integrates Weihrauch reducibility, the injective recursion theorem, and the infinite independent choice theorem, while systematically comparing parallelization with the diamond operator composition. The key contribution is the formal introduction and analysis of the *inverse limit operator*, which is proven monotone but not closed under composition; it is shown to be closed under the weak König lemma and a broad class of non-deterministically computable problems. Crucially, the inverse limit operator is strictly stronger than the combination of parallelization and the diamond operator, and exhibits well-behaved properties on important problem classes—including single-valued and Turing-degree-related problems—thereby establishing a novel paradigm for characterizing the structural complexity of iterative computation.
📝 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 KH{o}nig'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.