A Potpourri of Reason Maintenance Methods

We present novel methods to compute changes to materialized views in logic databases like those used by rule-based reasoners. Such reasoners have to address the problem of changing axioms in the presence of materializations of derived atoms. Existing approaches have drawbacks: some require to generate and evaluate large transformed programs that are in Datalog - while the source program is in Datalog and significantly smaller; some recompute the whole extension of a predicate even if only a small part of this extension is affected by the change. The methods presented in this article overcome these drawbacks and derive additional information useful also for explanation, at the price of an adaptation of the semi-naive forward chaining

Constructive Reasoning for Semantic Wikis

One of the main design goals of social software, such as wikis, is to support and facilitate interaction and collaboration. This dissertation explores challenges that arise from extending social software with advanced facilities such as reasoning and semantic annotations and presents tools in form of a conceptual model, structured tags, a rule language, and a set of novel forward chaining and reason maintenance methods for processing such rules that help to overcome the challenges. Wikis and semantic wikis were usually developed in an ad-hoc manner, without much thought about the underlying concepts. A conceptual model suitable for a semantic wiki that takes advanced features such as annotations and reasoning into account is proposed. Moreover, so called structured tags are proposed as a semi-formal knowledge representation step between informal and formal annotations. The focus of rule languages for the Semantic Web has been predominantly on expert users and on the interplay of rule languages and ontologies. KWRL, the KiWi Rule Language, is proposed as a rule language for a semantic wiki that is easily understandable for users as it is aware of the conceptual model of a wiki and as it is inconsistency-tolerant, and that can be efficiently evaluated as it builds upon Datalog concepts. The requirement for fast response times of interactive software translates in our work to bottom-up evaluation (materialization) of rules (views) ahead of time – that is when rules or data change, not when they are queried. Materialized views have to be updated when data or rules change. While incremental view maintenance was intensively studied in the past and literature on the subject is abundant, the existing methods have surprisingly many disadvantages – they do not provide all information desirable for explanation of derived information, they require evaluation of possibly substantially larger Datalog programs with negation, they recompute the whole extension of a predicate even if only a small part of it is affected by a change, they require adaptation for handling general rule changes. A particular contribution of this dissertation consists in a set of forward chaining and reason maintenance methods with a simple declarative description that are efficient and derive and maintain information necessary for reason maintenance and explanation. The reasoning methods and most of the reason maintenance methods are described in terms of a set of extended immediate consequence operators the properties of which are proven in the classical logical programming framework. In contrast to existing methods, the reason maintenance methods in this dissertation work by evaluating the original Datalog program – they do not introduce negation if it is not present in the input program – and only the affected part of a predicate’s extension is recomputed. Moreover, our methods directly handle changes in both data and rules; a rule change does not need to be handled as a special case. A framework of support graphs, a data structure inspired by justification graphs of classical reason maintenance, is proposed. Support graphs enable a unified description and a formal comparison of the various reasoning and reason maintenance methods and define a notion of a derivation such that the number of derivations of an atom is always finite even in the recursive Datalog case. A practical approach to implementing reasoning, reason maintenance, and explanation in the KiWi semantic platform is also investigated. It is shown how an implementation may benefit from using a graph database instead of or along with a relational database

ONTIC: A Knowledge Representation System for Mathematics

Ontic is an interactive system for developing and verifying mathematics. Ontic's verification mechanism is capable of automatically finding and applying information from a library containing hundreds of mathematical facts. Starting with only the axioms of Zermelo-Fraenkel set theory, the Ontic system has been used to build a data base of definitions and lemmas leading to a proof of the Stone representation theorem for Boolean lattices. The Ontic system has been used to explore issues in knowledge representation, automated deduction, and the automatic use of large data bases