"Non-deterministic multi-valued matrices (Nmatrices) are a new field of research rapidly developing towards a foundational logical theory. The idea of Nmatrices is in extending the usual algebraic deterministic semantics of logical systems by importing the idea of non-deterministic computations from Computer Science, and allowing the truth-value of a formula to be chosen non-deterministically out of a given set of options. The goal of the proposed project is to extensively develop the semantic framework which is provided by the idea of Nmatrices. This includes developing a foundational methodology for providing modular non-deterministic semantics for non-classical logics, and exploring its applications in the following directions: - Proof Theory: using Nmatrices to characterize cut-elimination and other important syntactic phenomena in various analytic calculi, including substructural calculi and hypersequents. - Automated Deduction: automatizing deduction in non-deterministic logics by extending the multi-valued expert system MULtlog, developed by the host group. - Approximating Logics: investigating effective approximations of logics using finite multi-valued Nmatrices. - Non-deterministic Fuzzy Logic: developing new non-deterministic foundations for fuzzy logics based on Nmatrices and exploring their applications."