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

Conference Program

Document provides a list of the sessions, speakers, workshops, and committees of the 32nd Summer Conference on Topology and Its Applications

Fuzzy relational Galois connections between fuzzy transitive digraphs

We present a fuzzy version of the notion of relational Galois connection between fuzzy transitive directed graphs (fuzzy T-digraphs) on the specific setting in which the underlying algebra of truth values is a complete Heyting algebra. The components of such fuzzy Galois connection are fuzzy relations satisfying certain reasonable properties expressed in terms of the so-called full powering. Moreover, we provide a necessary and sufficient condition under which it is possible to construct a right adjoint for a given fuzzy relation between a fuzzy T-digraph and an unstructured set.This research is partially supported by the State Agency of Research (AEI), the Spanish Ministry of Science, Innovation and Universities (MCIU), the European Social Fund (FEDER), the Junta de Andalucía (JA), and the Universidad de Málaga (UMA) through the research projects with reference PGC2018-095869-B-I00, PID2021-127870OB-I00, (MCIU/AEI/FEDER, UE) and UMA18-FEDERJA-001 (JA/ UMA/ FEDER, UE). B. De Baets was supported by the Flemish Government under the “Onderzoeksprogramma Artificiële Intelligentie (AI) Vlaanderen” programme. Funding for open access charge: Universidad de Málaga / CBU

Algebraic Uncertainty Theory : A Unifying Perspective on Reasoning under Uncertainty

The question of how to represent and process uncertainty is of fundamental importance to the scientific process, but also in everyday life. Currently there exist a lot of different calculi for managing uncertainty, each having its own advantages and disadvantages. Especially, almost all are defining the domain and structure of uncertainty values a priori, e.g., one real number, two real numbers, a finite domain, and so on, but maybe uncertainty is best measured by complex numbers, matrices or still another mathematical structure. This thesis investigates the notion of uncertainty from a foundational point of view, provides an ontology and axiomatic core system for uncertainty and derives and not defines the structure of uncertainty. The main result, the ring theorem, stating that uncertainty values are elements of the [0,1]-interval of a partially ordered ring, is used to derive a general decomposition theorem for uncertainty values, splitting them into a numerical interval and an ``interaction term''. In order to illustrate the unifying power of these results, the relationship to Dempster-Shafer theory is discussed and it is shown that all Dempster-Shafer measures over finite domains can be represented by ring-valued uncertainty measures. Finally, the historical development of approaches to modeling uncertainty which have led to the results of this thesis are reviewed