We define the well-known puzzle of Battleships as a decision problem and prove it to be NPcomplete, by means of a parsimonious reduction. By applying Valiant and Vazirani’s (1986) result we immediately obtain that the variant of this problem, viz. promising that there is a unique solution, is NP-complete as well, under randomized reductions. As this finding is in sheer contrast with the general...

Abstract The “somewhat vague, intuitive” notion from computability theory of an effective procedure (method) or algorithm can be fairly precisely defined even if it is not sufficiently formal and precise to belong mathematics proper (in a narrow sense)—and (as many have asserted) for that reason the Church–Turing thesis unprovable. It proved logically class procedures decidable, i.e., there no ...