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My research is in the field of logic, a broad scientific discipline located in a fascinating intersection of several distinct areas, which has a lot to do with one of the defining human traits: our ability to reason correctly. Therefore, logic is very relevant for artificial intelligence and my research can be indeed framed in the study of logical foundations of AI.
In particular, I am interested in many-valued logics, that is, logical systems with a semantics that (contrary to the classical logic restricted only to true or false statements) uses more than two values with a wide variety of interpretations. Their mathematical study has spanned through all levels of logical formalism (propositional, modal, first- and higher-order predicate logics) and their syntactical and semantical aspects. You can consult the Stanford Encyclopedia entry on fuzzy logic (cowritten with Petr Cintula and Chris Fermüller) or the three volumes of the Handbook of Mathematical Fuzzy Logic (coedited with Petr Hájek, Petr Cintula, and Chris Fermüller). From the point of view of philosophy and linguistics, many-valued logics are relevant tools for formalizing reasoning with graded properties, which are ubiquitious in rational action and discourse.
In computer science and foundations of AI, I am contributing to the many-valued approach by obtaining results in many-valued finite model theory (in particular, zero-one laws), weighted logics and corresponding models of computation and descriptive complexity, a generalization of Codd's Theorem and results on the complexity of queries in databases annotated on semirings, and a multidimensional paradigm for many-valued reasoning.
More generally, I am interested in the study of non-classical logics (and their applications): reasoning systems that, in a variety of ways, go beyond the traditional paradigm of binary analysis in terms of true and false. Together with Petr Cintula, we have proposed a general approach to logics with implication, in the style of abstract algebraic logic. We have collected the results of a series of joint articles in a book, Logic and Implication. As we wrote on the back cover, "This monograph presents a general theory of weakly implicative logics, a family covering a vast number of non-classical logics studied in the literature, concentrating mainly on the abstract study of the relationship between logics and their algebraic semantics. It can also serve as an introduction to (abstract) algebraic logic, both propositional and first-order, with special attention paid to the role of implication, lattice and residuated connectives, and generalized disjunctions."
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