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In this paper the authors analyze the complexity of checking safety and termination properties, for a very simple, yet non-trivial, class of programs with singly-linked list data structures. Since, in general, programs with lists are known to have the power of Turing machines, the authors restrict the control structure, by forbidding nested loops and destructive updates. Surprisingly, even with these simplifying conditions, verifying safety and termination for programs working on heaps with more than one cycle are undecidable, whereas decidability can be established when the input heap may have at most one loop. The proofs for both the undecidability and the decidability results rely on non-trivial number-theoretic results.
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