An Ehrenfeucht-Fraïssé game approach to collapse results in database theory

Abstract

Pursuing an Ehrenfeucht-Fraïssé game approach to collapse results in database theory, we show that, in principle, every natural generic collapse result may be proved via a translation of winning strategies for the duplicator in an Ehrenfeucht-Fraïssé game. Following this approach we can deal with certain infinite databases where previous, highly involved methods fail. We prove the natural generic collapse for Z -embeddable databases over any linearly ordered context structure with arbitrary monadic predicates, and for N -embeddable databases over the context structure 〈 R , < , + , Mon Q , Groups 〉 , where Groups is the collection of all subgroups of 〈 R , + 〉 that contain the set of integers and Mon Q is the collection of all subsets of a particular infinite set Q of natural numbers. This, in particular, implies the collapse for arbitrary databases over 〈 N , < , + , Mon Q 〉 and for N -embeddable databases over 〈 R , < , + , Z , Q 〉 . That is, first-order logic with < can express the same order-generic queries as first-order logic with <, +, etc. Restricting the complexity of the formulas that may be used to formulate queries to Boolean combinations of purely existential first-order formulas, we even obtain the collapse for N -embeddable databases over any linearly ordered context structure with arbitrary predicates. Finally, we develop the notion of N -representable databases, which is a natural generalisation of the notion of finitely representable databases. We show that natural generic collapse results for N -embeddable databases can be lifted to the larger class of N -representable databases. To obtain, in particular, the collapse result for 〈 N , < , + , Mon Q 〉 , we explicitly construct a winning strategy for the duplicator in the presence of the built-in addition relation +. This, as a side product, also leads to an Ehrenfeucht-Fraïssé game proof of the theorem of Ginsburg and Spanier, stating that the spectra of FO (<,+)-sentences are semi-linear.