Existential Second Order Logic Expression With Horn First Order for Maximum Clique (Decision Version) (1004.1814v5)

Abstract: We show that the maximum clique problem (decision version) can be expressed in existential second order (ESO) logic, where the first order part is a Horn formula in second-order quantified predicates. Without ordering, the first order part is $\Pi_2$ Horn; if ordering is used, then it is universal Horn (in which case, the second order variables can be determined in polynomial time). UPDATE: Manuscript withdrawn, because results are incorrect. If phi = phi_1 AND phi_2, and phi is a Horn formula, it does NOT mean that both phi_1 and phi_2 are Horn formulae. Furthermore, the cardinality constraint CANNOT be expressed as a universal Horn sentence in ESO (NOT even when the structure is ordered). Graedel's theorem is valid at a lower (machine) level, but probably NOT at a higher level.