The work reported here lays the foundations of data exchange in the presence of probabilistic data. This requires rethinking the very basic concepts of traditional data exchange, such as solution, universal solution, and the certain answers of target queries. We develop a framework for data exchange over probabilistic databases, and make a case for its coherence and robustness. This framework applies to arbitrary schema mappings, and finite or countably infinite probability spaces on the source and target instances. After establishing this framework and formulating the key concepts, we study the application of the framework to a concrete and practical setting where probabilistic databases are compactly encoded by means of annotations formulated over random Boolean variables. In this setting, we study the problems of testing for the existence of solutions and universal solutions, materializing such solutions, and evaluating target queries (for unions of conjunctive queries) in both the exact sense and the approximate sense. For each of the problems, we carry out a complexity analysis based on properties of the annotation, in various classes of dependencies. Finally, we show that the framework and results easily and completely generalize to allow not only the data, but also the schema mapping itself to be probabilistic.

The class of typed template dependencies is a class of data dependencies that includes embedded multivalued and join dependencies. It is shown that the implication and the_/Inite implication problems for this class are unsolvable. An immediate corollary is that this class has no formal system for finite implication. It is also shown how to construct a finite set of typed template dependencies whose implication and finite implication problems are unsolvable. The class of projected join dependencies is a proper subclass of the above class, and it generalizes slightly embedded join dependencies. It is shown that the implication and the finite implication problems for this class are also unsolvable. An immediate corollary is that this class has no universe-bounded formal system for either implication or finite implication. 1. INTR~OUCTI~N In the relational model one views the database as a collection of relations, where each relation is a set of tuples over some domain of values [ 141. One notable feature of this model is its being almost devoid of semantics. A tuple in a relation represents a relationship between certain values, but from the mere syntactic definition of the

Various restrictions on the terms allowed for substitution give rise to different cases of semi-unification. Semi-unification on finite and regular terms has already been considered in the literature. We introduce a general case of semi-unification where substitutions are allowed on non-regular terms, and we prove the equivalence of this general case to a well-known undecidable data base dependency problem , thus establishing the undecidability of general semi-unification. We present a unified way of looking at the various problems of semi-unification. We give some properties that are common to all the cases of semi-unification. We also the principality property and the solution set for those problems. We prove that semi-unification on general terms has the principality property. Finally, we present a recursive inseparability result between semi-unification on regular terms and semi-unification on general terms. Partly supported by NSF grant CCR-9113196. Address: Department of Compu...

Abstract. We give an overview of decidability results for modal logics having a binary modality. We put an emphasis on the demonstration of proof-techniques, and hope that this will also help in finding the borderlines between decidable and undecidable fragments of usual first-order logic. Key words: poly-modal and multi-modal logics, decision problems, Arrow logics, algebraic logic, relation algebra, associativity, dynamic logics, action algebras, Boolean algebras with operators We investigate here decidability problems concerning logics having an extra binary connective "o " beside the Boolean ones. These logics are strongly related to ordinary first-order logic, see Henkin et al. (1985, ch. 5.3) on this connection in an algebraic setting. Our most important aims are to give a transparent overview of the results and to stress the crucial points and ideas of the proofs, especially when the extra binary connective is associative 9 The emphasis is on those parts of the proof methods which have been well known for the specialists. (The reason for this is the didactic character of the present paper 9 For the details of those ideas which are our own contributions we give references to more technical papers. Since associativity of"o " corresponds to commutativity of first-order existential quantifiers in some sense (cf. op cit), our results and techniques can help to find decidable fragments of first-order logic as well (for such results see N6meti (1985, 1987, 1992)) 9