Positive First-Order Logic Is NP-Complete

The decision problem for positive first-order logic with equality is NP-complete. More generally, if Σ is a finite set of atomic sentences (i.e., atomic formulas of the form t
1
= t
2
or Rt
1
… t
n
containing no variables) and negations of atomic sentences and if ɸ is a positive first-order sentence, then the problem of determining whether ɸ is true in all models of Σ is NP-complete.