Loop Termination and Generalized Collatz Sequences
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917496190140416 |
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| author | Carelli, Mishel |
| author_facet | Carelli, Mishel |
| contents | Linear-constraint loops are programs whose transition relation is specified by a system of linear inequalities. The termination problem asks, given a loop, whether it admits an infinite computation. Decidability of termination remains open for linear-constraint loops over integers, rationals, and reals. We focus on loops over integers and show that they are tightly connected to generalized Collatz sequences - integer sequences generated by maps that are linear on each residue class modulo a fixed natural number. We prove that termination of one-variable linear-constraint loops is decidable in polynomial time, provided a long-standing conjecture about generalized Collatz sequences holds. Conversely, we show that any decision procedure for one-variable loops would prove or refute specific instances of this conjecture, which remain open. Moreover, we show that if a one-variable loop has a cyclic trace, then it also has a cyclic trace of length at most two. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_15094 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Loop Termination and Generalized Collatz Sequences Carelli, Mishel Logic in Computer Science F.3.1 Linear-constraint loops are programs whose transition relation is specified by a system of linear inequalities. The termination problem asks, given a loop, whether it admits an infinite computation. Decidability of termination remains open for linear-constraint loops over integers, rationals, and reals. We focus on loops over integers and show that they are tightly connected to generalized Collatz sequences - integer sequences generated by maps that are linear on each residue class modulo a fixed natural number. We prove that termination of one-variable linear-constraint loops is decidable in polynomial time, provided a long-standing conjecture about generalized Collatz sequences holds. Conversely, we show that any decision procedure for one-variable loops would prove or refute specific instances of this conjecture, which remain open. Moreover, we show that if a one-variable loop has a cyclic trace, then it also has a cyclic trace of length at most two. |
| title | Loop Termination and Generalized Collatz Sequences |
| topic | Logic in Computer Science F.3.1 |
| url | https://arxiv.org/abs/2605.15094 |