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

On the property of T-distributivity

WOS: 000320947500001In this paper, we introduce the notion of T-distributivity for any t-norm on a bounded lattice. We determine a relation between the t-norms T and T', where T' is a T-distributive t-norm. Also, for an arbitrary t-norm T, we give a necessary and sufficient condition for T-D to be T-distributive and for T to be T-boolean AND-distributive. Moreover, we investigate the relation between the T-distributivity and the concepts of the T-partial order, the divisibility of t-norms. We also determine that the T-distributivity is preserved under the isomorphism. Finally, we construct a family of t-norms which are not distributive over each other with the help of incomparable elements in a bounded lattice

An extension of the ordering based on nullnorms

summary:In this paper, we generally study an order induced by nullnorms on bounded lattices. We investigate monotonicity property of nullnorms on bounded lattices with respect to the $F$-partial order. Also, we introduce the set of incomparable elements with respect to the F-partial order for any nullnorm on a bounded lattice. Finally, we investigate the relationship between the order induced by a nullnorm and the distributivity property for nullnorms

A T-partial order obtained from T-norms

summary:A partial order on a bounded lattice $L$ is called t-order if it is defined by means of the t-norm on $L$. It is obtained that for a t-norm on a bounded lattice $L$ the relation $a\preceq_{T}b$ iff $a=T(x,b)$ for some $x\in L$ is a partial order. The goal of the paper is to determine some conditions such that the new partial order induces a bounded lattice on the subset of all idempotent elements of $L$ and a complete lattice on the subset $A$ of all elements of $L$ which are the supremum of a subset of atoms

Multiple instance fuzzy inference.

A novel fuzzy learning framework that employs fuzzy inference to solve the problem of multiple instance learning (MIL) is presented. The framework introduces a new class of fuzzy inference systems called Multiple Instance Fuzzy Inference Systems (MI-FIS). Fuzzy inference is a powerful modeling framework that can handle computing with knowledge uncertainty and measurement imprecision effectively. Fuzzy Inference performs a non-linear mapping from an input space to an output space by deriving conclusions from a set of fuzzy if-then rules and known facts. Rules can be identified from expert knowledge, or learned from data. In multiple instance problems, the training data is ambiguously labeled. Instances are grouped into bags, labels of bags are known but not those of individual instances. MIL deals with learning a classifier at the bag level. Over the years, many solutions to this problem have been proposed. However, no MIL formulation employing fuzzy inference exists in the literature. In this dissertation, we introduce multiple instance fuzzy logic that enables fuzzy reasoning with bags of instances. Accordingly, different multiple instance fuzzy inference styles are proposed. The Multiple Instance Mamdani style fuzzy inference (MI-Mamdani) extends the standard Mamdani style inference to compute with multiple instances. The Multiple Instance Sugeno style fuzzy inference (MI-Sugeno) is an extension of the standard Sugeno style inference to handle reasoning with multiple instances. In addition to the MI-FIS inference styles, one of the main contributions of this work is an adaptive neuro-fuzzy architecture designed to handle bags of instances as input and capable of learning from ambiguously labeled data. The proposed architecture, called Multiple Instance-ANFIS (MI-ANFIS), extends the standard Adaptive Neuro Fuzzy Inference System (ANFIS). We also propose different methods to identify and learn fuzzy if-then rules in the context of MIL. In particular, a novel learning algorithm for MI-ANFIS is derived. The learning is achieved by using the backpropagation algorithm to identify the premise parameters and consequent parameters of the network. The proposed framework is tested and validated using synthetic and benchmark datasets suitable for MIL problems. Additionally, we apply the proposed Multiple Instance Inference to the problem of region-based image categorization as well as to fuse the output of multiple discrimination algorithms for the purpose of landmine detection using Ground Penetrating Radar

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