12,732 research outputs found
On implicational bases of closure systems with unique critical sets
We show that every optimum basis of a finite closure system, in D.Maier's
sense, is also right-side optimum, which is a parameter of a minimum CNF
representation of a Horn Boolean function. New parameters for the size of the
binary part are also established. We introduce a K-basis of a general closure
system, which is a refinement of the canonical basis of Duquenne and Guigues,
and discuss a polynomial algorithm to obtain it. We study closure systems with
the unique criticals and some of its subclasses, where the K-basis is unique. A
further refinement in the form of the E-basis is possible for closure systems
without D-cycles. There is a polynomial algorithm to recognize the D-relation
from a K-basis. Thus, closure systems without D-cycles can be effectively
recognized. While E-basis achieves an optimum in one of its parts, the
optimization of the others is an NP-complete problem.Comment: Presented on International Symposium of Artificial Intelligence and
Mathematics (ISAIM-2012), Ft. Lauderdale, FL, USA Results are included into
plenary talk on conference Universal Algebra and Lattice Theory, June 2012,
Szeged, Hungary 29 pages and 2 figure
Ultimate approximations in nonmonotonic knowledge representation systems
We study fixpoints of operators on lattices. To this end we introduce the
notion of an approximation of an operator. We order approximations by means of
a precision ordering. We show that each lattice operator O has a unique most
precise or ultimate approximation. We demonstrate that fixpoints of this
ultimate approximation provide useful insights into fixpoints of the operator
O.
We apply our theory to logic programming and introduce the ultimate
Kripke-Kleene, well-founded and stable semantics. We show that the ultimate
Kripke-Kleene and well-founded semantics are more precise then their standard
counterparts We argue that ultimate semantics for logic programming have
attractive epistemological properties and that, while in general they are
computationally more complex than the standard semantics, for many classes of
theories, their complexity is no worse.Comment: This paper was published in Principles of Knowledge Representation
and Reasoning, Proceedings of the Eighth International Conference (KR2002
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Reasoning with Data Flows and Policy Propagation Rules
Data-oriented systems and applications are at the centre of current developments of the World Wide Web. In these scenarios, assessing what policies propagate from the licenses of data sources to the output of a given data-intensive system is an important problem. Both policies and data flows can be described with Semantic Web languages. Although it is possible to define Policy Propagation Rules (PPR) by associating policies to data flow steps, this activity results in a huge number of rules to be stored and managed. In a recent paper, we introduced strategies for reducing the size of a PPR knowledge base by using an ontology of the possible relations between data objects, the Datanode ontology, and applying the (A)AAAA methodology, a knowledge engineering approach that exploits Formal Concept Analysis (FCA). In this article, we investigate whether this reasoning is feasible and how it can be performed. For this purpose, we study the impact of compressing a rule base associated with an inference mechanism on the performance of the reasoning process. Moreover, we report on an extension of the (A)AAAA methodology that includes a coherency check algorithm, that makes this reasoning possible. We show how this compression, in addition to being beneficial to the management of the knowledge base, also has a positive impact on the performance and resource requirements of the reasoning process for policy propagation
On the Usability of Probably Approximately Correct Implication Bases
We revisit the notion of probably approximately correct implication bases
from the literature and present a first formulation in the language of formal
concept analysis, with the goal to investigate whether such bases represent a
suitable substitute for exact implication bases in practical use-cases. To this
end, we quantitatively examine the behavior of probably approximately correct
implication bases on artificial and real-world data sets and compare their
precision and recall with respect to their corresponding exact implication
bases. Using a small example, we also provide qualitative insight that
implications from probably approximately correct bases can still represent
meaningful knowledge from a given data set.Comment: 17 pages, 8 figures; typos added, corrected x-label on graph
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