Loop Termination and Generalized Collatz Sequences
For program verification researchers, it provides a theoretical link between loop termination and number theory, but the result is conditional on an unproven conjecture.
The paper establishes a connection between termination of one-variable linear-constraint loops over integers and generalized Collatz sequences, proving that termination is decidable in polynomial time conditional on a conjecture about Collatz sequences, and that any decision procedure would resolve open instances of that conjecture.
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.