Disjunctively incomplete information in relational databases: modeling and related issues

In this dissertation, the issues related to the information incompleteness in relational databases are explored. In general, this dissertation can be divided into two parts. The first part extends the relational natural join operator and the update operations of insertion and deletion to I-tables, an extended relational model representing inclusively indefinite and maybe information, in a semantically correct manner. Rudimentary or naive algorithms for computing natural joins on I-tables require an exponential number of pair-up operations and block accesses proportional to the size of I-tables due to the combinatorial nature of natural joins on I-tables. Thus, the problem becomes intractable for large I-tables. An algorithm for computing natural joins under the extended model which reduces the number of pair-up operations to a linear order of complexity in general and in the worst case to a polynomial order of complexity with respect to the size of I-tables is proposed in this dissertation. In addition, this algorithm also reduces the number of block accesses to a linear order of complexity with respect to the size of I-tables;The second part is related to the modeling aspect of incomplete databases. An extended relational model, called E-table, is proposed. E-table is capable of representing exclusively disjunctive information. That is, disjunctions of the form P[subscript]1\mid P[subscript]2\mid·s\mid P[subscript]n, where ǁ denotes a generalized logical exclusive or indicating that exactly one of the P[subscript]i\u27s can be true. The information content of an E-table is precisely defined and relational operators of selection, projection, difference, union, intersection, and cartisian product are extended to E-tables in a semantically correct manner. Conditions under which redundancies could arise due to the presence of exclusively disjunctive information are characterized and the procedure for resolving redundancies is presented;Finally, this dissertation is concluded with discussions on the directions for further research in the area of incomplete information modeling. In particular, a sketch of a relational model, IE-table (Inclusive and Exclusive table), for representing both inclusively and exclusively disjunctive information is provided

Default reasoning and neural networks

In this dissertation a formalisation of nonmonotonic reasoning, namely Default logic, is discussed. A proof theory for default logic and a variant of Default logic - Prioritised Default logic - is presented. We also pursue an investigation into the relationship between default reasoning and making inferences in a neural network. The inference problem shifts from the logical problem in Default logic to the optimisation problem in neural networks, in which maximum consistency is aimed at The inference is realised as an adaptation process that identifies and resolves conflicts between existing knowledge about the relevant world and external information. Knowledge and data are transformed into constraint equations and the nodes in the network represent propositions and constraint equations. The violation of constraints is formulated in terms of an energy function. The Hopfield network is shown to be suitable for modelling optimisation problems and default reasoning.Computer ScienceM.Sc. (Computer Science

Efficient Maximum A-Posteriori Inference in Markov Logic and Application in Description Logics

Maximum a-posteriori (MAP) query in statistical relational models computes the most probable world given evidence and further knowledge about the domain. It is arguably one of the most important types of computational problems, since it is also used as a subroutine in weight learning algorithms. In this thesis, we discuss an improved inference algorithm and an application for MAP queries. We focus on Markov logic (ML) as statistical relational formalism. Markov logic combines Markov networks with first-order logic by attaching weights to first-order formulas. For inference, we improve existing work which translates MAP queries to integer linear programs (ILP). The motivation is that existing ILP solvers are very stable and fast and are able to precisely estimate the quality of an intermediate solution. In our work, we focus on improving the translation process such that we result in ILPs having fewer variables and fewer constraints. Our main contribution is the Cutting Plane Aggregation (CPA) approach which leverages symmetries in ML networks and parallelizes MAP inference. Additionally, we integrate the cutting plane inference (Riedel 2008) algorithm which significantly reduces the number of groundings by solving multiple smaller ILPs instead of one large ILP. We present the new Markov logic engine RockIt which outperforms state-of-the-art engines in standard Markov logic benchmarks. Afterwards, we apply the MAP query to description logics. Description logics (DL) are knowledge representation formalisms whose expressivity is higher than propositional logic but lower than first-order logic. The most popular DLs have been standardized in the ontology language OWL and are an elementary component in the Semantic Web. We combine Markov logic, which essentially follows the semantic of a log-linear model, with description logics to log-linear description logics. In log-linear description logic weights can be attached to any description logic axiom. Furthermore, we introduce a new query type which computes the most-probable 'coherent' world. Possible applications of log-linear description logics are mainly located in the area of ontology learning and data integration. With our novel log-linear description logic reasoner ELog, we experimentally show that more expressivity increases quality and that the solutions of optimal solving strategies have higher quality than the solutions of approximate solving strategies