Towards a theory of good SAT representations

We aim at providing a foundation of a theory of "good" SAT representations F of boolean functions f. We argue that the hierarchy UCk of unit-refutation complete clause-sets of level k, introduced by the authors, provides the most basic target classes, that is, F in UCk is to be achieved for k as small as feasible. If F does not contain new variables, i.e., F is equivalent (as a CNF) to f, then F in UC1 is similar to "achieving (generalised) arc consistency" known from the literature (it is somewhat weaker, but theoretically much nicer to handle). We show that for polysize representations of boolean functions in this sense, the hierarchy UCk is strict. The boolean functions for these separations are "doped" minimally unsatisfiable clause-sets of deficiency 1; these functions have been introduced in [Sloan, Soerenyi, Turan, 2007], and we generalise their construction and show a correspondence to a strengthened notion of irredundant sub-clause-sets. Turning from lower bounds to upper bounds, we believe that many common CNF representations fit into the UCk scheme, and we give some basic tools to construct representations in UC1 with new variables, based on the Tseitin translation. Note that regarding new variables the UC_1-representations are stronger than mere "arc consistency", since the new variables are not excluded from consideration.