Softened Symbol Grounding for Neuro-symbolic Systems

Abstract: Neuro-symbolic learning generally consists of two separated worlds, i.e., neural network training and symbolic constraint solving, whose success hinges on symbol grounding, a fundamental problem in AI. This paper presents a novel, softened symbol grounding process, bridging the gap between the two worlds, and resulting in an effective and efficient neuro-symbolic learning framework. Technically, the framework features (1) modeling of symbol solution states as a Boltzmann distribution, which avoids expensive state searching and facilitates mutually beneficial interactions between network training and symbolic reasoning;(2) a new MCMC technique leveraging projection and SMT solvers, which efficiently samples from disconnected symbol solution spaces; (3) an annealing mechanism that can escape from %being trapped into sub-optimal symbol groundings. Experiments with three representative neuro symbolic learning tasks demonstrate that, owining to its superior symbol grounding capability, our framework successfully solves problems well beyond the frontier of the existing proposals.

• The paper presents a novel probabilistic approach employing Boltzmann distribution modeling, MCMC sampling, and annealing to soften symbol grounding, leading to improved performance across diverse tasks.
• It integrates MCMC sampling with SMT solvers to effectively bridge the semantic gap between continuous neural outputs and discrete symbolic reasoning.
• Experimental evaluations demonstrate enhanced accuracy in handwritten formula evaluation, visual Sudoku classification, and shortest path prediction, showcasing significant practical impact.

Overview of "Softened Symbol Grounding for Neuro-symbolic Systems"

The paper "Softened Symbol Grounding for Neuro-symbolic Systems" introduces a novel approach aimed at integrating neural networks with symbolic reasoning more efficiently. The research focuses on addressing the symbol grounding problem by softening the traditional hard boundary methods through probabilistic techniques, thus promoting interactions between the neural and symbolic components in neuro-symbolic systems.

Problem Statement

In neuro-symbolic systems, which combine neural network-based perception with symbolic logic-based reasoning, the symbol grounding problem is pivotal. This problem involves mapping raw inputs to discrete symbols that can be further processed symbolically. Typical methods for symbol grounding are handicapped by a significant semantic gap between continuous neural outputs and discrete symbolic reasoning requirements, often leading to limited performance and generalizability.

Proposed Framework

The paper proposes three major technical innovations to address these shortcomings:

1. Modeling with Boltzmann Distribution: Instead of relying on deterministic state searches, the authors propose modeling symbol solutions as a Boltzmann distribution. This probabilistic approach eases the bridging of neural and symbolic learning by providing a softened mapping from inputs to symbols that are refined over time via an annealing mechanism.
2. MCMC Sampling via Projection and SMT Solvers: They introduce a novel Markov Chain Monte Carlo (MCMC) sampling technique that uses projection and Satisfiability Modulo Theories (SMT) solvers. This method efficiently samples from sparse solution spaces by improving connectivity, thus expediting the search process in complex symbolic spaces.
3. Annealing Mechanism: An annealing strategy is employed to gradually refine the probabilistic mappings to deterministic ones. This process helps in avoiding local optima traps by exploiting mixed-strategy games from game theory, enhancing the interaction between neural perception and logical reasoning.

   Figure 1: An example neural-symbolic system for handwritten formula evaluation. It takes a handwritten arithmetic expression #1{x} as input and evaluates the expression to output #1{y}.

Experimental Evaluation

Experiments were conducted on tasks like handwritten formula evaluation, visual Sudoku classification, and shortest path problems. The proposed method demonstrated significant performance improvements over traditional techniques.

• Handwritten Formula Evaluation: The framework achieved higher accuracy than baseline methods, attributable to the efficient softened symbol grounding facilitating better symbolic reasoning.

  Figure 2: Accuracy (%) of the HWF task. Our methods (i.e., Stage~(1)+(2)) perform much better than comparison methods.

• Visual Sudoku Classification: The approach outperformed state-of-the-art methods in solving visual Sudoku puzzles, verifying effective symbol grounding even with limited initial information.
• Shortest Path Search: The framework showed superior performance in predicting shortest paths in graphs, demonstrating near-perfect accuracy compared to direct supervision methods.

Figure 3: Training curves (the first row) and test curves (the second row) of different approaches. We only plot the curves for some of the methods for brevity.

Theoretical Implications and Future Work

The paper highlights a significant theoretical advancement by integrating a probabilistic approach to bridge the gap between continuous and discrete domains effectively. Future work could focus on extending this framework to more complex systems where the symbolic reasoning logic is not predefined and involves learning the logical constructs alongside perception.

In conclusion, this research advances neuro-symbolic integration by offering a robust probabilistic framework that could significantly impact the scalability and effectiveness of AI systems requiring both neural learning and logical reasoning capabilities.