We propose a framework for modeling uncertainty where both belief and doubt can be given independent, first-class status. We adopt probability theory as the mathematical formalism for manipulating uncertainty. An agent can express the uncertainty in her knowledge about a piece of information in the form of a confidence level, consisting of a pair of intervals of probability, one for each of her belief and doubt. The space of confidence levels naturally leads to the notion of a trilattice, similar in spirit to Fitting's bilattices. Intuitively, the points in such a trilattice can be ordered according to truth, information, or precision. We develop a framework for probabilistic deductive databases by associating confidence levels with the facts and rules of a classical deductive database. While the trilattice structure offers a variety of choices for defining the semantics of probabilistic deductive databases, our choice of semantics is based on the truth-ordering, which we find to be closest to the classical framework for deductive databases. In addition to proposing a declarative semantics based on valuations and an equivalent semantics based on fixpoint theory, we also propose a proof procedure and prove it sound and complete. We show that while classical Datalog query programs have a polynomial time data complexity, certain query programs in the probabilistic deductive database framework do not even terminate on some input databases. We identify a large natural class of query programs of practical interest in our framework, and show that programs in this class possess polynomial time data complexity, i.e. not only do they terminate on every input database, they are guaranteed to do so in a number of steps polynomial in the input database size.

We provide a principled extension of SQL, called SchemaSQL, that offers the capability of uniform manipulation of data and schema in relational multidatabase systems. We develop a precise syntax and semantics of SchemaSQL in a manner that extends traditional SQL syntax and semantics, and demonstrate the following. (1) SchemaSQL retains the flavor of SQL while supporting querying of both data and schema. (2) It can be used to transform data in a database in a structure substantially different from original database, in which data and schema may be interchanged.(3) It also permits the creation of views whose schema is dynamically dependent on the contents of the input instance. (4) While aggregation in SQL is restricted to values occurring in one column at a time, SchemaSQL permits "horizontal" aggregation and even aggregation over more general "blocks" of information. (5) SchemaSQL provides a useful facility for interoperability and data/schema manipulation in relational multidatabase systems. We provide many examples to illustrate our claims. We clearly spell out the formal semantics of SchemaSQL that accounts for all these features. We describe an architecture for the implementation of SchemaSQL and develop implementation algorithms based on available database technology that allows for powerful integration of SQL based relational DBMS. We also discuss the applicability of SchemaSQL for handling semantic heterogeneity arising in a multidatabase system.

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