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

TL;DR

This paper demonstrates that ESO universal Horn logic cannot fully characterize the class P, highlighting a fundamental difference between structure-level and machine-level expressibility in computational complexity.

Contribution

It establishes the limitations of ESO universal Horn logic in capturing P, contrasting structure-level and machine-level expressibility with two distinct proofs.

Findings

ESO universal Horn logic does not capture P with successor relations

Two proofs: reduced products and approximability theory

Difference attributed to structure-level versus machine-level expressions

Abstract

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.