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 quantifier prefix and do not contain any function symbols.
Ramsey proved that, if is a formula in the Bernays–Schönfinkel class with one free variable, then either is finite, or 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
Notes
- ^ Pratt-Hartmann, Ian (2023-03-30). Fragments of First-Order Logic. Oxford University Press. p. 186. ISBN 978-0-19-196006-2.
- ^ Lewis, Harry R. (1980), "Complexity results for classes of quantificational formulas", Journal of Computer and System Sciences, 21 (3): 317–353, doi:10.1016/0022-0000(80)90027-6, MR 0603587
- ^ de Moura, Leonardo; Bjørner, Nikolaj (2008). Armando, Alessandro; Baumgartner, Peter; Dowek, Gilles (eds.). "Deciding Effectively Propositional Logic Using DPLL and Substitution Sets". Automated Reasoning. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer: 410–425. doi:10.1007/978-3-540-71070-7_35. ISBN 978-3-540-71070-7.
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" (PDF), Microsoft Research Technical Report (2008–181)
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