Fuzzy inequational logic

We present a logic for reasoning about graded inequalities which generalizes the ordinary inequational logic used in universal algebra. The logic deals with atomic predicate formulas of the form of inequalities between terms and formalizes their semantic entailment and provability in graded setting which allows to draw partially true conclusions from partially true assumptions. We follow the Pavelka approach and define general degrees of semantic entailment and provability using complete residuated lattices as structures of truth degrees. We prove the logic is Pavelka-style complete. Furthermore, we present a logic for reasoning about graded if-then rules which is obtained as particular case of the general result

Towards Contingent World Descriptions in Description Logics

The philosophical, logical, and terminological junctions between Description Logics (DLs) and Modal Logic (ML) are important because they can support the formal analysis of modal notions of ‘possibility’ and ‘necessity’ through the lens of DLs. This paper introduces functional contingents in order to (i) structurally and terminologically analyse ‘functional possibility’ and ‘functional necessity’ in DL world descriptions and (ii) logically and terminologically annotate DL world descriptions based on functional contingents. The most significant contributions of this research are the logical characterisation and terminological analysis of functional contingents in DL world descriptions. The ultimate goal is to investigate how modal operators can – logically and terminologically – be expressed within DL world descriptions

A map of dependencies among three-valued logics

International audienceThree-valued logics arise in several fields of computer science, both inspired by concrete problems (such as in the management of the null value in databases) and theoretical considerations. Several three-valued logics have been defined. They differ by their choice of basic connectives, hence also from a syntactic and proof-theoretic point of view. Different interpretations of the third truth value have also been suggested. They often carry an epistemic flavor. In this work, relationships between logical connectives on three-valued functions are explored. Existing theorems of functional completeness have laid bare some of these links, based on specific connectives. However we try to draw a map of such relationships between conjunctions, negations and implications that extend Boolean ones. It turns out that all reasonable connectives can be defined from a few of them and so all known three-valued logics appear as a fragment of only one logic. These results can be instrumental when choosing, for each application context, the appropriate fragment where the basic connectives make full sense, based on the appropriate meaning of the third truth-value

Interval Neutrosophic Sets and Logic: Theory and Applications in Computing

A neutrosophic set is a part of neutrosophy that studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. The neutrosophic set is a powerful general formal framework that has been recently proposed. However, the neutrosophic set needs to be specified from a technical point of view. Here, we define the set-theoretic operators on an instance of a neutrosophic set, and call it an Interval Neutrosophic Set (INS). We prove various properties of INS, which are connected to operations and relations over INS. We also introduce a new logic system based on interval neutrosophic sets. We study the interval neutrosophic propositional calculus and interval neutrosophic predicate calculus. We also create a neutrosophic logic inference system based on interval neutrosophic logic. Under the framework of the interval neutrosophic set, we propose a data model based on the special case of the interval neutrosophic sets called Neutrosophic Data Model. This data model is the extension of fuzzy data model and paraconsistent data model. We generalize the set-theoretic operators and relation-theoretic operators of fuzzy relations and paraconsistent relations to neutrosophic relations. We propose the generalized SQL query constructs and tuple-relational calculus for Neutrosophic Data Model. We also design an architecture of Semantic Web Services agent based on the interval neutrosophic logic and do the simulation study

Proceedings of the 5th International Workshop "What can FCA do for Artificial Intelligence?", FCA4AI 2016(co-located with ECAI 2016, The Hague, Netherlands, August 30th 2016)

International audienceThese are the proceedings of the fifth edition of the FCA4AI workshop (http://www.fca4ai.hse.ru/). Formal Concept Analysis (FCA) is a mathematically well-founded theory aimed at data analysis and classification that can be used for many purposes, especially for Artificial Intelligence (AI) needs. The objective of the FCA4AI workshop is to investigate two main main issues: how can FCA support various AI activities (knowledge discovery, knowledge representation and reasoning, learning, data mining, NLP, information retrieval), and how can FCA be extended in order to help AI researchers to solve new and complex problems in their domain. Accordingly, topics of interest are related to the following: (i) Extensions of FCA for AI: pattern structures, projections, abstractions. (ii) Knowledge discovery based on FCA: classification, data mining, pattern mining, functional dependencies, biclustering, stability, visualization. (iii) Knowledge processing based on concept lattices: modeling, representation, reasoning. (iv) Application domains: natural language processing, information retrieval, recommendation, mining of web of data and of social networks, etc

On nearness measures in fuzzy relational data models

AbstractIt has been widely recognized that the imprecision and incompleteness inherent in real-world data suggest a fuzzy extension for information management systems. Various attempts to enhance these systems by fuzzy extensions can be found in the literature. Varying approaches concerning the fuzzification of the concept of a relation are possible, two of which are referred to in this article as the generalized fuzzy approach and the fuzzy-set relation approach. In these enhanced models, items can no longer be retrieved by merely using equality-check operations between constants; instead, operations based on some kind of nearness measures have to be developed. In fact, these models require such a nearness measure to be established for each domain for the evaluation of queries made upon them. An investigation of proposed nearness measures, often fuzzy equivalences, is conducted. The unnaturalness and impracticality of these measures leads to the development of a new measure: the resemblance relation, which is defined to be a fuzzified version of a tolerance relation. Various aspects of this relation are analyzed and discussed. It is also shown how the resemblance relation can be used to reduce redundancy in fuzzy relational database systems