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

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

• Prenex normal form

• 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 

0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/Bernays–Schönfinkel class. Read more