Space-Time Assumptions Behind NP-Hardness of Propositional Satisfiability

For some problems, we know feasible algorithms for solving them. Other computational problems (such as propositional satisfiability) are known to be NP-hard, which means that, unless P=NP (which most computer scientists believe to be impossible), no feasible algorithm is possible for solving all possible instances of the corresponding problem. Most usual proofs of NP-hardness, however, use Turing machine -- a very simplified version of a computer -- as a computation model. While Turing machine has been convincingly shown to be adequate to describe what can be computed in principle, it is much less intuitive that these oversimplified machine are adequate for describing what can be computed effectively; while the corresponding adequacy results are known, they are not easy to prove and are, thus, not usually included in the textbooks. To make the NP-hardness result more intuitive and more convincing, we provide a new proof in which, instead of a Turing machine, we use a generic computational device. This proof explicitly shows the assumptions about space-time physics that underlie NP-hardness: that all velocities are bounded by the speed of light, and that the volume of a sphere grows no more than polynomially with radius. If one of these assumptions is violated, the proof no longer applies; moreover, in such space-times we can potentially solve the satisfiability problem in polynomial time