A graphical formalized language is proposed for specifying systems of views over database schemas. The language is based on the notion of arrow (mapping) between data schemas and is suitable for any data model for which schema mappings are defined. In particular, the constructs of query, query language, view and view integration can be consistently expressed in this arrow formalism and correspondingly specified. This gives rise to a general graph-based framework for specifying complex view systems. Basic constructions of the language and the entire framework as well can be considered as specialization of very general constructs developed in the mathematical category theory. 1 Introduction The notion of view is one of the central ones in the database (DB) technology. Views make it possible to provide each application with its own presentation of data and isolate them from inessential (for them) details and changes of DB schemas. The practical importance of views is commonly recognized...

. A novel formalizable specification paradigm is proposed which makes it possible to solve a lot of specification problems in software. The roots of the idea are in mathematical category theory; following its terminology we call our specifications sketches. Among the principal advantages of sketches are the following. ffl Nice amalgamation of logical rigor and graphical evidence. Sketches are graph-based images yet they are precise formal specifications as rigor as, say, first order logic theories. ffl Universality, in the precise sense of the word. It can be mathematically proven that any specification whose semantic meaning can be formalized can be also expressed by a sketch. ffl Unifying power. Many of graphical specification languages can be simulated by sketches in the corresponding signature of diagram markers. ffl Semantic capabilities. The sketch language is inherently object-oriented and provides a quite natural way of specifying OO class-reference schemas. ffl Easy and f...

lear that it should be a version of the hyperdoctrine construction, a well known categorical counterpart of predicate logic; ffl to understand this structure from the proof-theoretical and modeltheoretical (semantic) perspectives, and initiate its studying in both directions; ffl on this ground, to build a firm mathematical foundation of semantic database theory (see [1] for the formulation of the problem). Another promising aspect of the intended collaboration is to discuss nontrivial correlations between the constructive ontology of objects developed in [1] and the type-theoretic foundations of constructive mathematics known as Martin-Lof type theory (MLTT). The latter has a wide range of applications in computer science but, as a rule, they are in the field of functional programming and automated theorem proving. In contrast, the correlations mentioned above give rise to new areas of applying the MLTT framework to database theory and (a bit more

. A graph-based specification language and the corresponding machinery are described as stating a basic framework for semantic modeling and database design. It is shown that a few challenging theoretical questions in the area, and some of hot practical problems as well, can be successfully approached in the framework. The machinery has its origin in the classical sketches invented by Ehresmann and is close to their generalization recently proposed by Makkai. There are two essential distinctions from Makkai's sketches. One consists in a different -- more direct -- formalization of sketches that categorists (and database designers) usually draw. The second distinction is more fundamental and consists in introducing operational sketches specifying complex diagram operations over ordinary (predicate) sketches, correspondingly, models of operational sketches are diagram algebras. Together with the notion of parsing operational sketches, this is the main mathematical contribution of the pape...

not representing on the observable universe, though possibly derived from it) or Real World (based on the observable universe and representing a subset of it). A Well Formed model has an internally consistent definition according to mathematically accepted criteria-derived deductions and theorems are likely to be correspondingly consistent. A model which is not well formed is likely to demonstrate inconsistencies in deductions and theorems derived from its use.