Bernays–Schönfinkel class

The Bernays–Schönfinkel class (also known as Bernays–Schönfinkel–Ramsey class) of formulas, named after Paul Bernays, Moses Schönfinkel and Frank P. Ramsey, is a fragment of first-order logic formulas where satisfiability is decidable.

It is the set of sentences that, when written in prenex normal form, have an $$\exists^*\forall^*$$ quantifier prefix and do not contain any function symbols.

Ramsey proved that, if $$\phi$$ is a formula in the Bernays–Schönfinkel class with one free variable, then either $$\{x \in \N : \phi(x)\} $$ is finite, or $$\{x \in \N : \neg \phi(x)\} $$ is finite.

This class of logic formulas is also sometimes referred as effectively propositional (EPR) since it can be effectively translated into propositional logic formulas by a process of grounding or instantiation.

The satisfiability problem for this class is NEXPTIME-complete.

Applications
Efficient algorithms for deciding satisfiability of EPR have been integrated into SMT solvers.