Sorting, linear time and the satisfiability problem

Let DLIN denote the class of problems that are computable withinO(n/logn) uniform time on a RAM which can read its input (of lengthn) in blocks and which only uses integersO(n/logn). We prove that the sorting problem belongs to DLIN and formulate the Linear Time Thesis: “Each problem computable by a linear time bounded algorithm (in an intuitive sense) belongs to DLIN”. We also study the subclass DTIMESORT(n) of problems computable within linear time on a Turing machine using in addition a fixed number of sortings, and we show how the reductions of this class, so-called sort-lin reductions, are useful to classify NP problems: e.g. there is a sort-lin reduction from SAT to 3-SAT (using no sorting) and a sort-lin reduction from the NP-complete graph problem KERNEL to SAT (but we do not know of any similar reduction in DTIME(n)). Similarly, problem 2-SAT is linearly equivalent to the problem of Horn renaming (via sort-lin reductions). Finally, SAT is compared with many other combinatorial problems. A problemA is SAT-easy (resp. SAT-hard) if there is a sort-lin reduction fromA to SAT (resp. SAT toA). A SAT-equivalent problem is a problem both SAT-easy and SAT-hard. It is shown that the class of SAT-easy (resp. SAT-equivalent) problems is very large and that its generalization to so-called special clauses or, more generally, regular clauses, does not enlarge it. Moreover, we justify our opinion that problem SAT is, in some sense, a minimal NP-complete problem.