On Semantic Generalizations of the Bernays-Sch\"onfinkel-Ramsey Class with Finite or Co-finite Spectra
Abstract
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 . A class in this family is denoted EBS(σ), where σ is a subset of . Formulae in EBS(σ) are preserved under substructures modulo a bounded core and modulo re-interpretation of predicates outside σ. We study properties of the family EBS = EBS(σ) | σ ⊂eq , e.g. classes in EBS are spectrally indistinguishable, EBS() is semantically equivalent to BSR over , and EBS() is the set of all FO formulae over with finite or co-finite spectra. Furthermore, (EBS, ⊂eq) forms a lattice isomorphic to the powerset lattice ((), ⊂eq). This gives a natural semantic generalization of BSR as ascending chains in (EBS, ⊂eq). Many well-known FO classes are semantically subsumed by EBS() or EBS(). Our study provides alternative proofs of interesting results like the Lo\'s-Tarski Theorem and the semantic subsumption of the L\"owenheim class with equality by BSR. We also present a syntactic sub-class of EBS(σ) called EDP(σ) and give an expression for the size of the bounded cores of models of EDP(σ) formulae. We show that the EDP(σ) classes also form a lattice structure. Finally, we study some closure properties and applications of the classes presented.