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 [math]\displaystyle{ \exists^*\forall^* }[/math] quantifier prefix and do not contain any function symbols.

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

Applications

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

See also

• Prenex normal form

Notes

References

• Ramsey, F. (1930), "On a problem in formal logic", Proc. London Math. Soc. 30: 264–286, doi:10.1112/plms/s2-30.1.264 
• Piskac, R.; de Moura, L.; Bjorner, N. (December 2008), "Deciding Effectively Propositional Logic with Equality", Microsoft Research Technical Report (2008-181), https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/tr-2008-181.pdf 

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