An Ehrenfeucht-Fraı̈ssé Game Approach to Collapse Results in Database Theory

We present a new Ehrenfeucht-Fraı̈ssé 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-Fraı̈ssé 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, <,+,MonQ,Groups〉, where Groups is the collection of all subgroups of 〈R,+〉 that contain the set of integers and MonQ 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, <,+,MonQ〉 and for N-embeddable databases over 〈R, <,+,Z,Q〉. I.e., 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 Nembeddable databases over any linearly ordered context structure with arbitrary predicates. Finally, we develop the notion of N-representable databases, which is a natural generalization 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, <,+,MonQ〉, 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-Fraı̈ssé game proof of the theorem of Ginsburg and Spanier, stating that the spectra of FO(<,+)-sentences are semi-linear. ACM-classification: F.4.1 [Mathematical Logic and Formal Languages]: Computational Logic; H.2.3 [Database Management]: Query Languages