# DiLA: Enhancing LLM Tool Learning with Differential Logic Layer

arXiv: https://arxiv.org/abs/2402.11903 · HTML: https://mlsanity.com/paper/2402.11903/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 20 of its 44 references that are in the map.
- First posted on arXiv 19 Feb 2024.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)
- LLM agents and tool use / [General LLM agent frameworks and benchmarks](https://mlsanity.com/agents/general-llm-agent-frameworks-and-benchmarks/)

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Beyond the Imitation Game: Quantifying and extrapolating the capabilities of language models](https://mlsanity.com/paper/2206.04615/) (2022)
- [Least-to-Most Prompting Enables Complex Reasoning in Large Language Models](https://mlsanity.com/paper/2205.10625/) (2022)
- [PAL: Program-aided Language Models](https://mlsanity.com/paper/2211.10435/) (2022)
- [Logic-LM: Empowering Large Language Models with Symbolic Solvers for Faithful Logical Reasoning](https://mlsanity.com/paper/2305.12295/) (2023)
- [Alphazero-like Tree-Search can Guide Large Language Model Decoding and Training](https://mlsanity.com/paper/2309.17179/) (2023)
- [SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver](https://mlsanity.com/paper/1905.12149/) (2019)
- [LINC: A Neurosymbolic Approach for Logical Reasoning by Combining Language Models with First-Order Logic Provers](https://mlsanity.com/paper/2310.15164/) (2023)
- [On the Self-Verification Limitations of Large Language Models on Reasoning and Planning Tasks](https://mlsanity.com/paper/2402.08115/) (2024)
- [Solving Math Word Problems by Combining Language Models With Symbolic Solvers](https://mlsanity.com/paper/2304.09102/) (2023)
- [Large Language Models Can Learn Temporal Reasoning](https://mlsanity.com/paper/2401.06853/) (2024)
- [SatLM: Satisfiability-Aided Language Models Using Declarative Prompting](https://mlsanity.com/paper/2305.09656/) (2023)
- [Language Models can be Logical Solvers](https://mlsanity.com/paper/2311.06158/) (2023)
- [SATformer: Transformer-Based UNSAT Core Learning](https://mlsanity.com/paper/2209.00953/) (2022)
- [Low-rank semidefinite programming for the MAX2SAT problem](https://mlsanity.com/paper/1812.06362/) (2018)
- [Enhancing Logical Reasoning in Large Language Models to Facilitate Legal Applications](https://mlsanity.com/paper/2311.13095/) (2023)

## Built on by (most cited first, 6 of 6)

- [Empowering LLMs with Logical Reasoning: A Comprehensive Survey](https://mlsanity.com/paper/2502.15652/) (2025)
- [Large Language Models Meet Symbolic Provers for Logical Reasoning Evaluation](https://mlsanity.com/paper/2502.06563/) (2025)
- [Deeper Insights Without Updates: The Power of In-Context Learning Over Fine-Tuning](https://mlsanity.com/paper/2410.04691/) (2024)
- [Neuro-Symbolic Artificial Intelligence: Towards Improving the Reasoning Abilities of Large Language Models](https://mlsanity.com/paper/2508.13678/) (2025)
- [Can Language Models Pretend Solvers? Logic Code Simulation with LLMs](https://mlsanity.com/paper/2403.16097/) (2024)
- [Infeasibility Aware Large Language Models for Combinatorial Optimization](https://mlsanity.com/paper/2604.01455/) (2026)

## Abstract (arXiv)

Considering the challenges faced by large language models (LLMs) in logical reasoning and planning, prior efforts have sought to augment LLMs with access to external solvers. While progress has been made on simple reasoning problems, solving classical constraint satisfaction problems, such as the Boolean Satisfiability Problem (SAT) and Graph Coloring Problem (GCP), remains difficult for off-the-shelf solvers due to their intricate expressions and exponential search spaces. In this paper, we propose a novel differential logic layer-aided language modeling (DiLA) approach, where logical constraints are integrated into the forward and backward passes of a network layer, to provide another option for LLM tool learning. In DiLA, LLM aims to transform the language description to logic constraints and identify initial solutions of the highest quality, while the differential logic layer focuses on iteratively refining the LLM-prompted solution. Leveraging the logic layer as a bridge, DiLA enhances the logical reasoning ability of LLMs on a range of reasoning problems encoded by Boolean variables, guaranteeing the efficiency and correctness of the solution process. We evaluate the performance of DiLA on two classic reasoning problems and empirically demonstrate its consistent outperformance against existing prompt-based and solver-aided approaches.