Shota Rustaveli National Science Foundation of GeorgiaStart Date: 2026-03-10 End Date: 2029-03-10
Classical logic operates with strict notions of truth and rigid deduction rules, but many real-world domains, such as decision-making, knowledge representation, and natural language understanding, demand richer, more flexible reasoning frameworks. To meet these needs, various logical systems have been developed that extend classical logic by incorporating degrees of truth, uncertainty, similarity, or cost, including approaches based on fuzzy logic, probabilistic logic, quantitative algebras, etc. In these contexts, quantitative inference plays a central role: deductions involve computing degrees, weights, or costs, offering a model of human reasoning, where conclusions are often graded rather than strictly true or false.
Many existing approaches attach quantitative information directly to formulas or inference rules. However, in applications like declarative programming or knowledge representation by linguistic concepts, this tight coupling can be unnatural. An alternative methodology, known as the Logic + Vague Knowledge + Control approach, separates logical structure from vague knowledge and control mechanisms, handling imprecision through modifications of computational processes. This idea has driven advances in quantitative unification, constraint solving, and their integration into logic programming, expert systems, and knowledge-based reasoning, with applications ranging from neuro-symbolic inference to natural language understanding.
This project aims to advance the state of the art by developing resolution- and tableau-based inference methods within a framework where approximate information is modeled by quantale-valued relations in both first- and higher-order languages. Moving beyond fuzzy relations to a more general quantale-based setting opens powerful new avenues for modeling complex phenomena, such as resources, distances, accessibility, and contextual relevance. An interesting application of this approach would involve adapting Natural Logic (a higher-order framework that reasons directly on linguistic form) to handle vague and imprecise information. This research will contribute new techniques for quantitative, flexible reasoning across a variety of knowledge-intensive domains.