Expressibility at the machine level versus structure level: ESO universal Horn Logic and the class P

Consider decision problems derived from optimization problems. For problems in this category, we show that they cannot be expressed in existential second order (ESO) logic with a first order part that is universal Horn (or simply, ESO Π1 Horn) at the structure level, as opposed to machine level. This is true even when we consider ordered structures. We show that an NP-complete problem (Maximum Independent Set) and a polynomially solvable problem (Maximum Matching) have identical ESO Π1 (non-Horn) expressions when the only unknowns are the second order predicates, hence this logic is too weak to distinguish between the two classes. At the structure level, Fagin’s theorem is false in one sense, in that ESO universal sentences under ordered structures cannot capture the class NP. Also at the structure level, under ordered structures, least fixed point logic with counting (LFP+C) is insufficient to capture P, the class of problems decidable in polynomial time.