On the time complexity of computer viruses

Computer viruses can disable computer systems not only by destroying data or modifying a system's configuration, but also by consuming most of the computing resources such as CPU time and storage. The latter effects are related to the computational complexity of computer viruses. In this correspondence, we investigate some issues concerning the time complexity of computer viruses, and prove some known experimental results mathematically. We prove that there exist computer viruses with arbitrarily long running time, not only in the infecting procedure but in the executing procedure. Moreover, we prove that there are computer viruses with arbitrarily large time complexity in the detecting procedure, and there are undecidable computer viruses that have no "minimal" detecting procedure.