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## Similarity preservation in default logic

### Annals of Mathematics and Artificial Intelligence (1999-07-01) 25: 137-160 , July 01, 1999

The paper identifies a problem in default reasoning in Reiter’s Default Logic and related systems: elements which are *similar* given the axioms only, become distinguishable in extensions. We explain why, sometimes, this is considered undesirable. Two approaches are presented for guaranteeing *similarity preservation*: One approach formalizes a way of uniformly applying the defaults to all similar elements by introducing *generic extensions*, which depend only on similarity types of objects. According to the second approach, for a restricted class of default theories, a default theory is viewed as a “shorthand notation” to what is “really meant” by its formulation. In this approach we propose a *rewriting* of defaults in a form that guarantees similarity preservation of the modified theory. It turns out that the above two approaches yield the same result.

## Ontology-based semantic search on the Web and its combination with the power of inductive reasoning

### Annals of Mathematics and Artificial Intelligence (2012-07-01) 65: 83-121 , July 01, 2012

Semantic Web search is currently one of the hottest research topics in both Web search and the Semantic Web. In previous work, we have presented a novel approach to Semantic Web search, which allows for evaluating ontology-based complex queries that involve reasoning over the Web relative to an underlying background ontology. We have developed the formal model behind this approach, and provided a technique for processing Semantic Web search queries, which consists of an offline ontological inference step and an online reduction to standard Web search. In this paper, we continue this line of research. We further enhance the above approach by the use of inductive rather than deductive reasoning in the offline inference step. This increases the robustness of Semantic Web search, as it adds the important ability to handle inconsistencies, noise, and incompleteness, which are all very likely to occur in distributed and heterogeneous environments such as the Web. The inductive variant also allows to infer new (not logically deducible) knowledge (from training individuals). We report on a prototype implementation of (both the deductive and) the inductive variant of our approach in desktop search, and we provide extensive new experimental results, especially on the running time and the precision and the recall of our new approach.

## Introducing the mathematical category of artificial perceptions

### Annals of Mathematics and Artificial Intelligence (1998-11-01) 23: 267-298 , November 01, 1998

Perception is the recognition of elements and events in the environment, usually through integration of sensory impressions. It is considered here as a broad, high-level, concept (different from the sense in which computer vision/audio research takes the concept of perception). We propose and develop premises for a formal approach to a fundamental phenomenon in AI: the diversity of artificial perceptions. A mathematical substratum is proposed as a basis for a rigorous theory of artificial perceptions. A basic mathematical category is defined. Its objects are *perceptions*, consisting of *world elements, connotations*, and a three-valued (*true, false, undefined*) predicative correspondence between them. Morphisms describe paths between perceptions. This structure serves as a basis for a mathematical theory. This theory provides a way of extending and systematizing certain intuitive pre-theoretical conceptions about perception, about improving and/or completing an agent's perceptual grasp, about transition between various perceptions, etc. Some example applications of the theory are analyzed.

## Quasi-classical reasoning in paraconsistent databases

### Annals of Mathematics and Artificial Intelligence (2017-01-17): 1-29 , January 17, 2017

The well-founded model for any general deductive database computed using the paraconsistent relational model, based on four-valued logic, does not support inference rules such as disjunctive syllogism. In order to support disjunctive syllogism, we utilize the generalization of the relational model to quasi-classic (QC) logic, whose inference power is much closer to classical logic. As the paraconsistent relational model is capable of representing incomplete and inconsistent data, we propose an algorithm to find QC model for inconsistent positive extended disjunctive deductive databases. We also provide the proof for the algorithm.

## On stratified disjunctive programs

### Annals of Mathematics and Artificial Intelligence (1990-09-01) 1: 339-357 , September 01, 1990

We address the problem of a consistent fixpoint semantics for general disjunctive programs restricted to stratifiable programs which do not recurse through negative literals. We apply the nonmonotonic fixpoint theory developed by Apt, Blair and Walker to a closure operator*T*^{c} and develop a fixpoint semantics for stratified disjunctive programs. We also provide an iterative definition for negation, called the Generalized Closed World Assumption for Stratified programs (GCWAS), and show that our semantics captures this definition. We develop a model-theoretic semantics for stratified disjunctive programs and show that the least state characterized by the fixpoint semantics corresponds to a stable-state defined in a manner similar to the stable-models of Gelfond and Lifschitz. We also discuss a weaker stratification semantics for general disjunctive programs based on the Weak Generalized Closed World Assumption.

## Programming with non-determinism in deductive databases

### Annals of Mathematics and Artificial Intelligence (1997-03-01) 19: 97-125 , March 01, 1997

While non-determinism has long been established as a key concept in logic pro-gramming, its importance in the context of deductive databases was recognized only recently. This paper provides an overview of recent results on this topic with the aim of providing an introduction to the theory and practice of non-determinism in deductive databases. In particular we (i) recall the main results linking non-deterministic constructs in database languages to the theory of data complexity and the expressibility hierarchy of query languages; (ii) provide a reasoned introduction to effective programming with non-deterministic constructs; (iii) compare the usage of non-deterministic constructs in languages such as LDL++ to that of traditional logic programming languages; (iv) discuss the link between the semantics of logic programs with non-deterministic constructs and the stable-model semantics of logic programs with negation.

## Geometry of relative plausibility and relative belief of singletons

### Annals of Mathematics and Artificial Intelligence (2010-05-01) 59: 47-79 , May 01, 2010

The study of the interplay between belief and probability can be posed in a geometric framework, in which belief and plausibility functions are represented as points of simplices in a Cartesian space. Probability approximations of belief functions form two homogeneous groups, which we call “affine” and “epistemic” families. In this paper we focus on relative plausibility, belief, and uncertainty of probabilities of singletons, the “epistemic” family. They form a coherent collection of probability transformations in terms of their behavior with respect to Dempster’s rule of combination. We investigate here their geometry in both the space of all pseudo belief functions and the probability simplex, and compare it with that of the affine family. We provide sufficient conditions under which probabilities of both families coincide.

## Conditional Independence in A Coherent Finite Setting

### Annals of Mathematics and Artificial Intelligence (2001-08-01) 32: 287-313 , August 01, 2001

A definition of stochastic independence which avoids the inconsistencies (related to events of probability 0 or 1) of the classic one has been proposed by Coletti and Scozzafava for two events. We extend it to *conditional* independence among finite sets of events. In particular, the case of (finite) discrete random variables is studied. We check which of the relevant properties connected with graphical structures hold. Hence, an axiomatic characterization of these independence models is given and it is compared to the classic ones.

## Modeling Connectionist Network Structures: Some Geometric and Categorical Aspects

### Annals of Mathematics and Artificial Intelligence (2002-11-01) 36: 279-301 , November 01, 2002

This contribution deals with an approach for mathematical modeling of the network structures of a certain connectionist network paradigm. Analysis of the structure of an artificial neural network (ANN) in that class of networks shows a possibility to introduce geometric and categorical modeling methods. This can be described briefly as follows. A (noncommutative) geometric space can be interpreted as a so-called geometric net. To a given ANN a corresponding geometric net can be associated. Geometric spaces form a category. Consequently, one obtains a category of geometric nets with a suitable notion of morphism. Then it is natural to interpret a learning step of an ANN as a morphism, thus learning corresponds to a finite sequence of morphisms (the associated networks are the objects). An associated (“local”) geometric net is less complex than the original ANN, but it contains all necessary information about the network structure. The association process together with learning (expressed by morphisms) leads to a commutative diagram corresponding to a suitable natural transformation, in terms of category theory. Commutativity of the diagram can be exploited to make learning “cheaper”. The simplified mathematical network model was used in ANN simulation applied in an industrial project on quality control. The “economy” of the model could be observed in a considerable increase of performance and decrease of production costs. Some prospects on the role of group operations that are induced by the regular structure of the underlying networks conclude the article.