#### Publication Date

6-2013

#### Abstract

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.

## Comments

Technical Report: UTEP-CS-12-44a