Towards a better understanding of SAT translations

Of fundamental importance for SAT solving is the translation to CNF. One of the basic tasks is to find metrics for determining what are “good” translations, i.e., what makes the resulting SAT problem easy to solve. We introduce a new measure hd(F ), the “hardness”, for formulas F in conjunctive normal form (i.e., clause-sets). hd(F ) for unsatisfiable clause-sets has been studied in [12,14]. However, now we treat satisfiable clause-sets differently. We consider the Advanced Encryption Standard (AES) and Data Encryption Standard (DES) as examples. The key discovery for these ciphers present examples of hard problems, and we investigate translations of these problems geared towards minimising hd(F ). We present the SAT representation hypothesis: The task of solving a SAT problem efficiently is captured by constructing a representation of the underlying boolean function which is of low hardness.