A closed view of a database schema is one which is totally encapsulated. Insofar as the user is concerned, the view is the database schema. The rest of the database system is not visible through the view, and is is not required for complete use of the view. Similarly, the updates which may be effected through the view have their scope limited entirely to that view. In this paper, we lay the mathematical foundations for the systematic support of such views. The proper context is shown to be that of update translation under constant meet complement, a refinement of the constant complement strategy of Bancilhon and Spyratos. The central complexity result for relational schemata is that checking the legality of updates is "infinitely" simpler than blindly checking that the new state is legal for the view schema, and in the particular case that the base schema is constrained by functional dependencies, may always be performed in constant time, even if the view schema is not finitely axiomatizable. We further establish that, under very natural assumptions, update strategies for closed views are unique.

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Abstract. It is well known that the complexity of testing the correctness of an arbitrary update to a database view can be far greater than the complexity of testing a corresponding update to the main schema. However, views are generally managed according to some protocol which limits the admissible updates to a subset of all possible changes. The question thus arises as to whether there is a more tractable relationship between these two complexities in the presence of such a protocol. In this paper, this question is answered in the affirmative for closed update strategies, which are based upon the constant-complement approach of Bancilhon and Spyratos. Working within a very general framework which is independent of any particular data model, but which recaptures relational schemata constrained by so-called equality-generating dependencies (EGDs), (which include functional dependencies (FDs)), it is shown that the complexity of testing the correctness of a view update which follows a closed update strategy is no greater than that of testing a corresponding update to the main schema. In particular, if the main schema is relational and constrained by FDs, then there exists a set of FDs on the view, against which any candidate update may be tested for correctness. This holds even though the entire view may not be finitely axiomatizable, much less constrained by FDs alone.

The problem of updating incomplete infor mation databases is vieweda sap rogramming problem. From this point of view, for mal denotational semantics are developed for twoa pplicativep rogramming languages, BLU and HLU. BLU is a ver y simple language with only fivep rimitives,a nd is designed primar ily as a tool for the implementation of higher-levell anguages.T he semantics of BLU are for mally developed at twol ev els,p ossible wor lds and clausal, and the latter is shown to be a correct implementation of the for mer. HLU is a "user level" update language.I ti s defined entirely in terms of BLU,a nd so immediately inherits its semantic definition from that language.T his demonstrates a levelo fc ompleteness for BLU as a levelo f pr imitives for update language implementation. The necessity of a particular BLU pr imitive, masking, suggests that there is a high degree of inherent complexity in updating logical databases. IntroductionDatabase systems maybeviewedasconsisting of twocomponents.Adatabase schema specifies the general structure of admissible data, and remains constant. Database instances,onthe other hand, record the actual state of the wor ld at a given point in time,and changes upon update.I nthe case of complete infor mation, there is exactly one instance associated with the system at anyg iven point, whereas in the incomplete infor mation case,t here is a collection of alternativei nstances,o rpossible worlds.In the complete infor mation case,t he representation of the system state is a usually a direct one (such as a set of relations in the relational case), although indirect representation is also possible,a si nt he negation as failure, or closed wor ld clausal representation [3].In the case of incomplete infor mation, on the other hand, direct representation is -1-

In earlier work, Bancilhon and Spyratos introduced the concept of a complement to a database schema, and showed how this notion could be used in theories of decomposition and update semantics. However, they also showed that, except in trivial cases, even minimal complements are never unique, so that many desirable results, such as canonical decompositions, cannot be realized. Their work dealt with database schemata which are sets and database mappings which are functions, without further structure. In this work, we show that by adding a modest amount of additional structure, many important uniqueness results may be obtained. Specifically, we work with database schemata whose legal states form partially ordered sets (posets) with least elements, and with database mappings which are isotonic and which preserve this least element. This is a natural algebraic structure which is inherent in many important examples, including relational schemata constrained by data dependencies, with views constructed by composition of projection, restriction, and selection. Other examples include deductive database schemata in which views are defined by rules, and general first-order logic databases. Within this context of posets, we show that direct (i.e., independent) complements must be unique, and that in fact the directly complementable views have the structure, in a very natural sense, of a Boolean algebra. Decompositions of the schema then become identifiable with finite subalgebras of this Boolean algebra. To demonstrate the utility of our approach, we examine in some detail its applicability to the relational model. Particularly, we establish that under the condition that the schema is constrained by universal Horn sentences, there is a unique ultimate decomposition into a finite set of type restrictions. The latter are a special class of views which includes classical projections which occur in direct decompositions. In particular, classical join-based decomposition is completely recovered within a framework which explicitly axiomatizes independence via null values.