[Paper Review] Foundations of Reasoning with Uncertainty via Real-valued Logics

Ronald Fagin, Ryan Riegel+1 more|arXiv (Cornell University)|Aug 6, 2020

Advanced Algebra and Logic4 citations

TL;DR

This paper presents a sound and strongly complete axiomatization for reasoning with uncertainty using real-valued logics, enabling precise, mathematically grounded logical inference in neuro-symbolic AI systems. It introduces a parametrized framework that supports weighted subformulas and provides a linear programming-based decision procedure for key real-valued logics, establishing the first provably sound and complete inference system for logical neural networks (LNNs).

ABSTRACT

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.

Motivation & Objective

• To establish a formal, mathematically rigorous foundation for logical inference in real-valued logics, which underlie many modern neuro-symbolic AI systems.
• To address the lack of strong completeness and sound axiomatizations in real-valued logics, despite their widespread use in probabilistic, fuzzy, and neural-symbolic reasoning.
• To extend logical inference to handle weighted subformulas, which model the relative influence of components in systems like Logical Neural Networks (LNNs).
• To provide a decision procedure based on linear programming for determining logical implication in a range of real-valued logics under natural assumptions.
• To close a longstanding gap in the formalization of reasoning with uncertainty, particularly for continuous truth values in [0,1].

Proposed method

• Introduces a two-layer logical framework: an inner layer of real-valued formulas with connectives (e.g., & for 'and', ⊕ for 'or'), and an outer layer of classical (0-1) sentences about the truth values of these formulas.
• Develops a parametrized, sound, and strongly complete axiomatization that applies to essentially all standard real-valued logics, including Łukasiewicz, product, and Gödel logics.
• Extends the axiomatization to include weighted formulas, where subformulas are assigned influence coefficients, enabling modeling of neural network-style activation functions.
• Proposes a decision procedure based on linear programming to determine whether a set of outer-layer sentences logically implies another, valid for logics with continuous connectives.
• Uses the framework to formally validate inference in systems like LNNs, proving their logical correctness and completeness for the first time.
• Leverages known results from fuzzy logic and rational Pavelka logic (RPL) to contextualize and justify the generality and expressiveness of the proposed system.

Experimental results

Research questions

• RQ1 Can a sound and strongly complete axiomatization be developed for real-valued logics that supports both standard connectives and weighted subformulas?
• RQ2 How can logical inference over real-valued formulas be formalized such that all valid inferences are derivable from a finite set of axioms?
• RQ3 What decision procedure can be used to determine logical implication in real-valued logics under natural assumptions, particularly for Łukasiewicz, product, and Gödel logics?
• RQ4 Can the proposed framework be applied to prove the correctness and completeness of existing neuro-symbolic systems like Logical Neural Networks (LNNs)?
• RQ5 To what extent can the framework express complex truth-value constraints, such as intervals or inequalities, and what are its limitations?

Key findings

• The paper presents a sound and strongly complete axiomatization for a broad class of real-valued logics, ensuring that all logically valid inferences are derivable from the axioms.
• The framework is parametrized to cover all standard real-valued logics, including Łukasiewicz, product, and Gödel logics, making it widely applicable.
• The extension to weighted subformulas enables formal reasoning in systems like LNNs, where inputs have varying influence, and provides the first proof of logical completeness for such systems.
• A decision procedure based on linear programming is developed for determining logical implication in a range of real-valued logics, particularly those with continuous connectives.
• The framework can express complex constraints such as intervals of truth values (e.g., 'φ ≥ 0.5'), though it cannot express strict inequalities like 'φ > 0.5' due to continuity constraints.
• The results establish the first provably sound and complete logical inference system for LNNs, filling a critical gap in neuro-symbolic AI formalization.