On completeness of historical relational query languages

Numerous proposals for extending the relational data model to incorporate the temporal dimension of data have appeared in the past several years. These proposals have differed considerably in the way that the temporal dimension has been incorporated both into the structure of the extended relations of these temporal models and into the extended relational algebra or calculus that they define. Because of these differences, it has been difficult to compare the proposed models and to make judgments as to which of them might in some sense be equivalent or even better. In this paper we define temporally grouped and temporally ungrouped historical data models and propose two notions of historical relational completeness, analogous to Codd's notion of relational completeness, one for each type of model. We show that the temporally ungrouped models are less expressive than the grouped models, but demonstrate a technique for extending the ungrouped models with a grouping mechanism to capture the additional semantic power of temporal grouping. For the ungrouped models, we define three different languages, a logic with explicit reference to time, a temporal logic, and a temporal algebra, and motivate our choice for the first of these as the basis for completeness for these models. For the grouped models, we define a many-sorted logic with variables over ordinary values, historical values, and times. Finally, we demonstrate the equivalence of this grouped calculus and the ungrouped calculus extended with a grouping mechanism. We believe the classification of historical data models into grouped and ungrouped models provides a useful framework for the comparison of models in the literature, and furthermore, the exposition of equivalent languages for each type provides reasonable standards for common, and minimal, notions of historical relational completeness.