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

TL;DR

This paper introduces a novel Ehrenfeucht-Fraisse game approach to prove collapse results in database theory, enabling analysis of infinite databases and generalizing classical notions of database representability.

Contribution

It develops a new game-based method for proving collapse results, extends these results to N-representable databases, and provides a game-theoretic proof of a classical theorem on spectra of FO(<,+)-sentences.

Findings

Proves natural generic collapse for Z-embeddable databases over ordered structures.

Establishes collapse for N-embeddable databases over specific structures.

Develops the concept of N-representable databases and lifts collapse results to this class.

Abstract

We present a new Ehrenfeucht-Fraisse game approach to collapse results in database theory and we show that, in principle, this approach suffices to prove every natural generic collapse result. 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). Here, N, Z, R, denote the sets of natural numbers, integers, and real numbers, respectively. Groups is the collection of all subgroups of (R,+) that contain Z, and Mon_Q is the collection of all subsets of a particular infinite subset Q of N. 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 generalization of the classical 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-Fraisse game proof of the theorem of Ginsburg and Spanier, stating that the spectra of FO(<,+)-sentences are semi-linear.