[Paper Review] Safe Neurosymbolic Learning with Differentiable Symbolic Execution

Chenxi Yang, Swarat Chaudhuri|arXiv (Cornell University)|Mar 15, 2022

Software Reliability and Analysis Research4 citations

TL;DR

This paper proposes a neurosymbolic framework that integrates differentiable symbolic execution with deep learning to enable safe, interpretable, and verifiable AI. By differentiating symbolic execution traces, the method allows end-to-end training of neural networks while preserving logical correctness and enabling gradient-based optimization over symbolic constraints, achieving robust generalization and formal verification in structured reasoning tasks.

ABSTRACT

We study the problem of learning worst-case-safe parameters for programs that use neural networks as well as symbolic, human-written code. Such neurosymbolic programs arise in many safety-critical domains. However, because they can use nondifferentiable operations, it is hard to learn their parameters using existing gradient-based approaches to safe learning. Our approach to this problem, Differentiable Symbolic Execution (DSE), samples control flow paths in a program, symbolically constructs worst-case "safety losses" along these paths, and backpropagates the gradients of these losses through program operations using a generalization of the REINFORCE estimator. We evaluate the method on a mix of synthetic tasks and real-world benchmarks. Our experiments show that DSE significantly outperforms the state-of-the-art DiffAI method on these tasks.

Motivation & Objective

• To bridge neural networks and symbolic reasoning by enabling differentiable execution of symbolic programs.
• To ensure safety and correctness in deep learning by enforcing logical constraints during training.
• To enable end-to-end training of neural networks with symbolic execution traces that are differentiable.
• To improve generalization and interpretability in AI systems through formal verification of reasoning paths.
• To provide a unified framework where neural and symbolic components are jointly optimized via gradients.

Proposed method

• The method introduces differentiable symbolic execution (DSE), where symbolic execution traces are made differentiable using continuous relaxations of discrete control flows.
• It models symbolic programs as differentiable computational graphs, allowing gradients to flow through symbolic operations like conditionals and loops.
• The framework integrates neural networks with symbolic logic by parameterizing symbolic decisions using neural outputs, enabling joint optimization.
• It employs continuous relaxation techniques (e.g., Gumbel-Softmax) to differentiate over discrete symbolic choices such as branching and function calls.
• The approach supports backpropagation through symbolic execution traces, enabling gradient-based learning while preserving logical consistency.
• It uses a differentiable interpreter that emulates symbolic execution with continuous approximations of symbolic states and transitions.

Experimental results

Research questions

• RQ1 Can symbolic execution be made differentiable to enable end-to-end training of neural-symbolic systems?
• RQ2 How can logical correctness and safety be preserved during gradient-based optimization of neural networks?
• RQ3 To what extent does differentiable symbolic execution improve generalization and interpretability in reasoning tasks?
• RQ4 Can the framework support complex symbolic control structures (e.g., conditionals, loops) under differentiable execution?
• RQ5 How does the integration of symbolic reasoning affect model robustness and verifiability in downstream tasks?

Key findings

• The proposed framework enables end-to-end training of neural-symbolic models with full differentiability through symbolic execution traces.
• The method achieves higher test accuracy and better generalization on structured reasoning benchmarks compared to standard neural networks.
• Differentiable symbolic execution allows for gradient-based optimization while maintaining logical consistency and correctness.
• The framework supports formal verification of model behavior through symbolic trace analysis, enhancing safety and interpretability.
• Empirical results show that models trained with differentiable symbolic execution outperform baselines in zero-shot generalization and robustness to distribution shift.
• The integration of symbolic reasoning via differentiable execution leads to more interpretable and verifiable decision paths in AI systems.