Investigating a general hierarchy of polynomially decidable classes of CNF's based on short tree-like resolution proofs

We investigate a hierarchy Gk (U; S) of classes of conjunctive normal forms, recognizable and SAT-decidable in polynomial time, with special emphasize on the corresponding hardness parameter hU ;S(F) for clause-sets F (the rst level of inclusion). At level 0 an (incomplete, poly-time) oracle U for unsatissability detection and an oracle S for satissability detection is used. The hierarchy from Pretolani 96] is improved in this way with respect to strengthened satissability handling, simpliied recognition and consistent relativization. Also a hierarchy of canonical poly-time reductions with Unit-clause propagation at the rst level is obtained. General methods for upper and lower bounds on hU ;S(F) are developed and applied to a number of well-known examples. hU ;S(F) admits several diierent characterizations, including the space complexity of tree-like resolution and the use of pebble games as in Esteban, Torr an 99]. Using for S the class of linearly satissable clause-sets (based on linear programming) q-Horn clause-sets Boros, Cramer, Hammer 90] are contained at level 2, and for k 1 the \k-times nested Horn clause-sets" from Gallo, Scutelll a 88] are contained at level k. The unsatissable clause-sets in Gk (U; S) are exactly those refutable by relativized k-times nested input resolution, and the SAT decision algorithm 1 searching through the levels from below quasi-automatizes relativized tree-like resolution (using oracle U), while by means of hU (F) nearly precise general bounds on the (relativized) complexity of tree-like resolution (with oracle U) are obtained. In order to cope also with full resolution, a (more comprehensive) hierarchy Wk (U) of unsatissable clause-sets is introduced, based on a new form of width-restricted resolution, and relativized general upper and lower bounds for full resolution are derived, generalizing Ben-Sasson, Wigder-son 99] and also releasing the lower bound from its dependence on the maximal input clause length. Motivated by Bonet, Galesi 99] we give a simpliied example where the lower bound is tight.