SAT Has No Wizards

An (encoded) decision problem over Σ is a pair (E,F ) where E=words that encode instances of the problem, F=words to be accepted. We use strings in a technical sense, borrowed from Computability. With any NP problem (E,F ) we associate a set of strings |LogE(F )| called the reduced logogram of F relative to E, which conveys structural information on E, F , and how F is embedded in E. We define notions of internal independence of decision problems in terms of |LogE(F )|. The kernel Ker(P ) of a program P that solves (E,F ) is the set of those strings in |LogE(F )| that are actually used by P in making decisions. There are strict relations between Ker(P ) and the complexity of P . We develop an application to SAT that relies upon a property of strong internal independence of SAT . We show that SAT cannot have in its reduced logogram certain strings that, when present, serve as collective certificates. As a consequence, all the programs that solve SAT have the same kernel Ker(P ) = |LogCNF (SAT )|.