Simplification logic for the management of unknown information

This paper aims to contribute to the extension of classical Formal Concept Analysis (FCA), allowing the management of unknown information. In a preliminary paper, we define a new kind of attribute implications to represent the knowledge from the information currently available. The whole FCA framework has to be appropriately extended to manage unknown information. This paper introduces a new logic for reasoning with this kind of implications, which belongs to the family of logics with an underlying Simplification paradigm. Specifically, we introduce a new algebra, named weak dual Heyting Algebra, that allows us to extend the Simplification logic for these new implications. To provide a solid framework, we also prove its soundness and completeness and show the advantages of the Simplification paradigm. Finally, to allow further use of this extension of FCA in applications, an algorithm for automated reasoning, which is directly built from logic, is defined.Funding for open access charge: Universidad de Málaga / CBUA This article is Supported by Grants TIN2017-89023-P, PRE2018-085199 and PID2021-127870OB-I00 of the Ministry of Science and Innovation of Spain and UMA2018-FEDERJA-001 of the Junta de Andalucia and European Social Fund

Automatic Construction of Implicative Theories for Mathematical Domains

Implication is a logical connective corresponding to the rule of causality "if ... then ...". Implications allow one to organize knowledge of some field of application in an intuitive and convenient manner. This thesis explores possibilities of automatic construction of all valid implications (implicative theory) in a given field. As the main method for constructing implicative theories a robust active learning technique called Attribute Exploration was used. Attribute Exploration extracts knowledge from existing data and offers a possibility of refining this knowledge via providing counter-examples. In frames of the project implicative theories were constructed automatically for two mathematical domains: algebraic identities and parametrically expressible functions. This goal was achieved thanks both pragmatical approach of Attribute Exploration and discoveries in respective fields of application. The two diverse application fields favourably illustrate different possible usage patterns of Attribute Exploration for automatic construction of implicative theories