to Monika

Temporal active databases are an extension of the traditional database concept. On the temporal side, data is associated to time points or periods and, on the active side, temporal rules provide a means by which some temporal data in the database may force the insertion/deletion of some other temporal data. Thus, the temporal rules establish a \temporal link" between the data, either via a procedural execution mechanism or through a declarative historical interpretation. In this database framework, we are interested in analysing and detecting the e ects that a change in (data associated with) the past may cause in the database. Temporal logic techniques are applied to describe and analyse the problems related to the evolution of temporal databases. For that, we start by presenting the mathematical framework of two-dimensional temporal logics. Several methods of combining two one-dimensional temporal logics are described. Each combination method generates a family of two-dimensional temporal logics. The objective of this study is to establish for each combination method whether some logical properties of the one-dimensional systems, e.g. completeness and decidability, are transferred to their two-dimensional combination. A temporal logical description of data representation and temporal queries in temporal databases is provided, and two-dimensional temporal logic is used to describe temporal data evolution through updates. We use this two-dimensional description to characterise the di erences between the transaction-time and valid-time types of temporal databases. A two-dimensional imperative execution semantics and a one-dimensional declarative valid-time interpretation for temporal rules are then presented. It is shown that, under the execution semantics, updates in the past, and in general any historical update, may cause a violation of the valid-time interpretation, generating a time paradox . A classi cation of time paradoxes is proposed and their occurrences are then considered as the e ects of changing the past . Finally, several algorithms for the detection of occurrences of some types of time paradoxes are presented and their correctness is proved.