Understanding Inconsistency -- A Contribution to the Field of Non-monotonic Reasoning

Conflicting information in an agent's knowledge base may lead to a semantical defect, that is, a situation where it is impossible to draw any plausible conclusion. Finding out the reasons for the observed inconsistency and restoring consistency in a certain minimal way are frequently occurring issues in the research area of knowledge representation and reasoning. In a seminal paper Raymond Reiter proves a duality between maximal consistent subsets of a propositional knowledge base and minimal hitting sets of each minimal conflict -- the famous hitting set duality. We extend Reiter's result to arbitrary non-monotonic logics. To this end, we develop a refined notion of inconsistency, called strong inconsistency. We show that minimal strongly inconsistent subsets play a similar role as minimal inconsistent subsets in propositional logic. In particular, the duality between hitting sets of minimal inconsistent subsets and maximal consistent subsets generalizes to arbitrary logics if the stronger notion of inconsistency is used. We cover various notions of repairs and characterize them using analogous hitting set dualities. Our analysis also includes an investigation of structural properties of knowledge bases with respect to our notions. Minimal inconsistent subsets of knowledge bases in monotonic logics play an important role when investigating the reasons for conflicts and trying to handle them, but also for inconsistency measurement. Our notion of strong inconsistency thus allows us to extend existing results to non-monotonic logics. While measuring inconsistency in propositional logic has been investigated for some time now, taking the non-monotony into account poses new challenges. In order to tackle them, we focus on the structure of minimal strongly inconsistent subsets of a knowledge base. We propose measures based on this notion and investigate their behavior in a non-monotonic setting by revisiting existing rationality postulates, and analyzing the compliance of the proposed measures with these postulates. We provide a series of first results in the context of inconsistency in abstract argumentation theory regarding the two most important reasoning modes, namely credulous as well as skeptical acceptance. Our analysis includes the following problems regarding minimal repairs: existence, verification, computation of one and characterization of all solutions. The latter will be tackled with our previously obtained duality results. Finally, we investigate the complexity of various related reasoning problems and compare our results to existing ones for monotonic logics

Modal logics are coalgebraic

Applications of modal logics are abundant in computer science, and a large number of structurally different modal logics have been successfully employed in a diverse spectrum of application contexts. Coalgebraic semantics, on the other hand, provides a uniform and encompassing view on the large variety of specific logics used in particular domains. The coalgebraic approach is generic and compositional: tools and techniques simultaneously apply to a large class of application areas and can moreover be combined in a modular way. In particular, this facilitates a pick-and-choose approach to domain specific formalisms, applicable across the entire scope of application areas, leading to generic software tools that are easier to design, to implement, and to maintain. This paper substantiates the authors' firm belief that the systematic exploitation of the coalgebraic nature of modal logic will not only have impact on the field of modal logic itself but also lead to significant progress in a number of areas within computer science, such as knowledge representation and concurrency/mobility

FO(FD): Extending classical logic with rule-based fixpoint definitions

We introduce fixpoint definitions, a rule-based reformulation of fixpoint constructs. The logic FO(FD), an extension of classical logic with fixpoint definitions, is defined. We illustrate the relation between FO(FD) and FO(ID), which is developed as an integration of two knowledge representation paradigms. The satisfiability problem for FO(FD) is investigated by first reducing FO(FD) to difference logic and then using solvers for difference logic. These reductions are evaluated in the computation of models for FO(FD) theories representing fairness conditions and we provide potential applications of FO(FD).Comment: Presented at ICLP 2010. 16 pages, 1 figur

Knowledge Representation Concepts for Automated SLA Management

Outsourcing of complex IT infrastructure to IT service providers has increased substantially during the past years. IT service providers must be able to fulfil their service-quality commitments based upon predefined Service Level Agreements (SLAs) with the service customer. They need to manage, execute and maintain thousands of SLAs for different customers and different types of services, which needs new levels of flexibility and automation not available with the current technology. The complexity of contractual logic in SLAs requires new forms of knowledge representation to automatically draw inferences and execute contractual agreements. A logic-based approach provides several advantages including automated rule chaining allowing for compact knowledge representation as well as flexibility to adapt to rapidly changing business requirements. We suggest adequate logical formalisms for representation and enforcement of SLA rules and describe a proof-of-concept implementation. The article describes selected formalisms of the ContractLog KR and their adequacy for automated SLA management and presents results of experiments to demonstrate flexibility and scalability of the approach.Comment: Paschke, A. and Bichler, M.: Knowledge Representation Concepts for Automated SLA Management, Int. Journal of Decision Support Systems (DSS), submitted 19th March 200