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arXiv · 2601.12772

Logical Undefinability of the Generalized Collatz Transition Relation in Büchi Arithmetic

Madhav Dhiman·Rohan Pandey
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Abstract

Let $q$ be an odd prime and let $d$ be an odd integer. We show that the arbitrary-step transition relation of the generalized Collatz map $T_{q,d}$ is not first-order definable in Base-2 Büchi Arithmetic ($BA_2$). We do this by demonstrating that if the transition relation were definable, the exponential set $P_q = \{q^y : y \in \mathbb{N}\}$ would also be definable in $BA_2$. Since $P_q$ is strictly non-semilinear, this yields a direct contradiction with the Cobham--Semënov theorem. Consequently, we demonstrate that no finite automaton reading base-2 representations can recognize this transition relation.

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Madhav Dhiman, Rohan Pandey. 2026-06-08. Logical Undefinability of the Generalized Collatz Transition Relation in Büchi Arithmetic. https://arxiv.org/abs/2601.12772

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