Expressibility at the machine level versus structure level: ESO   universal Horn Logic and the class P
  We show that ESO universal Horn logic (existential second logic where the first order part is a universal Horn formula) is insufficient to capture P, the class of problems decidable in polynomial time. This statement is true in the presence of a successor relation in the input vocabulary. We provide two proofs --- one based on reduced products of two structures, and another based on approximability theory (the second proof is under the assumption that P is not the same as NP). We show that the difference between the results here and those in Gr\"{a}del (1991), is due to the fact that the expressions this paper deals with are at the "structure level", whereas the expressions in Gr\"{a}del (1991) are at the "machine level" --- a case of Easier done than said. 
Prabhu Manyem
cs.CC cs.LO