Unified Foundations of Team Semantics via Semirings

Semiring semantics for first-order logic provides a way to trace how facts represented by a model are used to deduce satisfaction of a formula. Team semantics is a framework for studying logics of dependence and independence in diverse contexts such as databases, quantum mechanics, and statistics by extending first-order logic with atoms that describe dependencies between variables. Combining these two, we propose a unifying approach for analysing the concepts of dependence and independence via a novel semiring team semantics, which subsumes all the previously considered variants for first-order team semantics. In particular, we study the preservation of satisfaction of dependencies and formulae between different semirings. In addition we create links to reasoning tasks such as provenance, counting, and repairs

Process modelling for information system description

My previous experiences and some preliminary studies of the relevant technical literature allowed me to identify several reasons for which the current state of the database theory seemed unsatisfactory and required further research. These reasons included: insufficient formalism of data semantics, misinterpretation of NULL values, inconsistencies in the concept of the universal relation, certain ambiguities in domain definition, and inadequate representation of facts and constraints. The commonly accepted ’sequentiality’ principle in most of the current system design methodologies imposes strong restrictions on the processes that a target system is composed of. They must be algorithmic and must not be interrupted during execution; neither may they have any parallel subprocesses as their own components. This principle can no longer be considered acceptable. In very many existing systems multiple processors perform many concurrent actions that can interact with each other. The overconcentration on data models is another disadvantage of the majority of system design methods. Many techniques pay little (or no) attention to process definition. They assume that the model of the Real World consists only of data elements and relationships among them. However, the way the processes are related to each other (in terms of precedence relation) may have considerable impact on the data model. It has been assumed that the Real World is discretisable, i.e. it may be modelled by a structure of objects. The word object is to be interpreted in a wide sense so it can mean anything within the boundaries of this part of the Real World that is to be represented in the target system. An object may then denote a fact or a physical or abstract entity, or relationships between any of these, or relationships between relationships, or even a still more complex structure. The fundamental hypothesis was formulated stating the necessity of considering the three aspects of modelling - syntax, semantics and behaviour, and these to be considered integrally. A syntactic representation of an object within a target system is called a construct A construct which cannot be decomposed further (either syntactically or semantically) is defined to be an atom. Any construct is a result of the following production rules: construct ::= atom I function construct; function ::= atom I construct. This syntax forms a sentential notation. The sentential notation allows for extensive use of denotational semantics. The meaning of a construct may be defined as a function mapping from a set of syntactic constructs to the appropriate semantic domains; these in turn appear to be sets of functions since a construct may have a meaning in more than one class of objects. Because of its functional form the meaning of a construct may be derived from the meaning of its components. The issue of system behaviour needed further investigation and a revision of the conventional model of computing. The sequentiality principle has been rejected, concurrency being regarded as a natural property of processes. A postulate has been formulated that any potential parallelism should be constructively used for data/process design and that the process structure would affect the data model. An important distinction has been made between a process declaration - considered as a form of data or an abstraction of knowledge - and a process application that corresponds to a physical action performed by a processor, according to a specific process declaration. In principle, a process may be applied to any construct - including its own representation - and it is a matter of semantics to state whether or not it is sensible to do so. The process application mechanism has been explained in terms of formal systems theory by introducing an abstract machine with two input and two output types of channels. The system behaviour has been described by defining a process calculus. It is based on logical and functional properties of a discrete time model and provides a means to handle expressions composed of process-variables connected by logical functors. Basic terms of the calculus are: constructs and operations (equivalence, approximation, precedence, incidence, free-parallelism, strict-parallelism). Certain properties of these operations (e.g. associativity or transitivity) allow for handling large expressions. Rules for decomposing/integrating process applications, analogous in some sense to those forming the basis for structured programming, have been derived

Query Rewriting and Optimization for Ontological Databases

Ontological queries are evaluated against a knowledge base consisting of an extensional database and an ontology (i.e., a set of logical assertions and constraints which derive new intensional knowledge from the extensional database), rather than directly on the extensional database. The evaluation and optimization of such queries is an intriguing new problem for database research. In this paper, we discuss two important aspects of this problem: query rewriting and query optimization. Query rewriting consists of the compilation of an ontological query into an equivalent first-order query against the underlying extensional database. We present a novel query rewriting algorithm for rather general types of ontological constraints which is well-suited for practical implementations. In particular, we show how a conjunctive query against a knowledge base, expressed using linear and sticky existential rules, that is, members of the recently introduced Datalog+/- family of ontology languages, can be compiled into a union of conjunctive queries (UCQ) against the underlying database. Ontological query optimization, in this context, attempts to improve this rewriting process so to produce possibly small and cost-effective UCQ rewritings for an input query.Comment: arXiv admin note: text overlap with arXiv:1312.5914 by other author

On the Translatability of View Updates

Abstract We revisit the view update problem and the abstract functional framework by Bancilhon and Spyratos in a setting where views and updates are exactly given by functions that are expressible in first-order logic. We give a characterisation of views and their inverses based on the notion of definability, and we introduce a general method for checking whether a view update can be uniquely translated as an update of the underlying database under the constant complement principle. We study the setting consisting of a single database relation and two views defined by projections and compare our general criterion for translatability with the known results for the case in which the constraints on the database are given by functional dependencies. We extend the setting to any number of projective views, full dependencies (that is, egd’s and full tgd’s) as database constraints, and classes of updates rather than single updates.