From RAM to SAT

Common presentations of the NP-completeness of SAT suffer from two drawbacks which hinder the scope of this flagship result. First, they do not apply to machines equipped with random-access memory, also known as direct-access memory, even though this feature is critical in basic algorithms. Second, they incur a quadratic blow-up in parameters, even though the distinction between, say, linear and quadratic time is often as critical as the one between polynomial and exponential. But the landmark result of a sequence of works overcomes both these drawbacks simultaneously! [HS66, Sch78, PF79, Coo88, GS89, Rob91] The proof of this result is simplified by Van Melkebeek in [vM06, §2.3.1]. Compared to previous proofs, this proof more directly reduces random-access machines to SAT, bypassing sequential Turing machines, and using a simple, well-known sorting algorithm: Odd-Even Merge sort [Bat68]. In this work we give a self-contained rendering of this simpler proof. For context, we note that the impressive works [BSCGT12b, BSCGT12a] give the stronger type of reduction where a candidate satisfying assignment to the SAT instance can be verified probabilistically in polylogarithmic time.