On Semantic Generalizations of the Bernays-Schönfinkel-Ramsey Class with Finite or Co-finite Spectra [article]

Abhisekh Sankaran, Supratik Chakraborty
2010 arXiv   pre-print
Motivated by model-theoretic properties of the BSR class, we present a family of semantic classes of FO formulae with finite or co-finite spectra over a relational vocabulary Σ. A class in this family is denoted EBS_Σ(σ), where σ is a subset of Σ. Formulae in EBS_Σ(σ) are preserved under substructures modulo a bounded core and modulo re-interpretation of predicates outside σ. We study properties of the family EBS_Σ = EBS_Σ(σ) | σ⊆Σ, e.g. classes in EBS_Σ are spectrally indistinguishable,
more » ... ) is semantically equivalent to BSR over Σ, and EBS_Σ(∅) is the set of all FO formulae over Σ with finite or co-finite spectra. Furthermore, (EBS_Σ, ⊆) forms a lattice isomorphic to the powerset lattice ((Σ), ⊆). This gives a natural semantic generalization of BSR as ascending chains in (EBS_Σ, ⊆). Many well-known FO classes are semantically subsumed by EBS_Σ(Σ) or EBS_Σ(∅). Our study provides alternative proofs of interesting results like the Loś-Tarski Theorem and the semantic subsumption of the Löwenheim class with equality by BSR. We also present a syntactic sub-class of EBS_Σ(σ) called EDP_Σ(σ) and give an expression for the size of the bounded cores of models of EDP_Σ(σ) formulae. We show that the EDP_Σ(σ) classes also form a lattice structure. Finally, we study some closure properties and applications of the classes presented.
arXiv:1002.4334v2 fatcat:erzc5b3h45cnhnoxwbctygb3ou