Foundations of Reasoning with Uncertainty via Real-valued Logics
This work addresses the foundational problem of ensuring correctness and inferential power in neuro-symbolic AI systems, which is incremental as it builds on existing real-valued logic frameworks.
The paper tackles the lack of quantitative characterization of logical inference in real-valued logics used in neuro-symbolic approaches by providing a sound and strongly complete axiomatization that covers essentially every real-valued logic, including fuzzy logics, and offers a decision procedure based on linear programming for logical implication under certain conditions.
Real-valued logics underlie an increasing number of neuro-symbolic approaches, though typically their logical inference capabilities are characterized only qualitatively. We provide foundations for establishing the correctness and power of such systems. We give a sound and strongly complete axiomatization that can be parametrized to cover essentially every real-valued logic, including all the common fuzzy logics. Our class of sentences are very rich, and each describes a set of possible real values for a collection of formulas of the real-valued logic, including which combinations of real values are possible. Strong completeness allows us to derive exactly what information can be inferred about the combinations of real values of a collection of formulas given information about the combinations of real values of several other collections of formulas. We then extend the axiomatization to deal with weighted subformulas. Finally, 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.