Foundations of reasoning with uncertainty via real-valued logics

• 成果类型: Article
• 署名作者: Fagin, Ronald; Riegel, Ryan; Gray, Alexander
• 署名单位: International Business Machines (IBM); IBM USA; International Business Machines (IBM); IBM USA
• 刊物名称: PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
• ISSN/ISSBN: 0027-14427
• DOI: 10.1073/pnas.2309905121
• 发表日期: 2024-05-21
• 关键词: fuzzy-logic; THEOREM

摘要

Interest in logics with some notion of real -valued truths has existed since at least Boole and has been increasing in AI due to the emergence of neuro-symbolic approaches, though often their logical inference capabilities are characterized only qualitatively. We provide foundations for establishing the correctness and power of such systems. We introduce a rich class of multidimensional sentences, with a sound and complete axiomatization that can be parameterized to cover many real -valued logics, including all the common fuzzy logics, and extend these to weighted versions, and to the case where the truth values are probabilities. Our multidimensional sentences form a very rich class. Each of our multidimensional sentences describes a set of possible truth values for a collection of formulas of the real -valued logic, including which combinations of truth values are possible. Our completeness result is strong, in the sense that it allows us to derive exactly what information can be inferred about the combinations of truth values of a collection of formulas given information about the combinations of truth values of a finite number of other collections of formulas. We give a decision procedure based on linear programming for deciding, for certain real -valued logics and under certain natural assumptions, whether a set of our sentences logically implies another of our sentences. The generality of this work, compared to many previous works on special cases, may provide insights for both existing and new real -valued logics whose inference properties have never been characterized. This work may also provide insights into the reasoning capabilities of deep learning models. Significance This work introduces a rich class of multidimensional sentences that yield a sound and complete axiomatization for a larger class of real -valued logics than previously considered, including all of the most common fuzzy logics, weighted versions, and probabilistic logics, many of which have garnered renewed interest as a result of the developing field of neuro-symbolic AI. Here, complete axiomatization holds in a strong sense: whenever a finite set F of our sentences logically implies one of our sentences y, that is, whenever every model of F is a model of y, there is a proof of r from F using our axiomatization. A decision procedure for two of the popular such logics, under certain natural assumptions, is presented. This work may also provide insights into the formal reasoning capabilities of deep learning models.