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 ${\displaystyle \exists ^{*}\forall ^{*}}$ quantifier prefix and do not contain any function symbols.

Ramsey proved that, if ${\displaystyle \phi }$ is a formula in the Bernays–Schönfinkel class with one free variable, then either ${\displaystyle \{x\in \mathbb {N} :\phi (x)\}}$ is finite, or ${\displaystyle \{x\in \mathbb {N} :\neg \phi (x)\}}$ is finite.¹

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.²

Efficient algorithms for deciding satisfiability of EPR have been integrated into SMT solvers.³