Using GPUs And LLMs Can Be Satisfying for Nonlinear Real Arithmetic Problems

2026-03-08

AI Summary

This work addresses the high computational complexity of quantifier-free nonlinear real arithmetic (NRA) problems, which has long limited the efficiency of traditional solving methods. We propose GANRA, a novel SMT solver that integrates large language models (LLMs) with GPU acceleration, uniquely combining LLM-guided heuristics with gradient-based parallel optimization to efficiently explore the solution space. Evaluated on the Sturm-MBO benchmark, GANRA proves satisfiability for over five times as many instances as the current state-of-the-art solver while achieving an average runtime less than one-twentieth of that method. These results demonstrate a substantial improvement in both scalability and solving speed for NRA problems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Large Multimodal Models (LMMs)

Application Category

• Search and Retrieval-Augmented AI: Large language models for search
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Solving quantifier-free non-linear real arithmetic (NRA) problems is a computationally hard task. To tackle this problem, prior work proposed a promising approach based on gradient descent. In this work, we extend their ideas and combine LLMs and GPU acceleration to obtain an efficient technique. We have implemented our findings in the novel SMT solver GANRA (GPU Accelerated solving of Nonlinear Real Arithmetic problems). We evaluate GANRA on two different NRA benchmarks and demonstrate significant improvements over the previous state of the art. In particular, on the Sturm-MBO benchmark, we can prove satisfiability for more than five times as many instances in less than 1/20th of the previous state-of-the-art runtime.

Problem

• nonlinear real arithmetic
• quantifier-free
• SMT solving
• satisfiability
• computational hardness

Innovation

• GPU acceleration
• Large Language Models (LLMs)
• Nonlinear Real Arithmetic
• SMT solving
• Gradient descent