In the supplementary material, we describe the domain specific languages used in our experiments (Section 1), demonstrate how the proposed CKY-E2 method works by a concrete example (Section 2.1), show formal properties of CKY-E2 (Section 2.2), present dataset setups and analyze model behaviors (Section 3), and list environmental details for experiments (Section??). In this section, we will present and discuss the domain-specific languages (DSLs) we use for two domains: visual reasoning and language-guided navigation. We will further introduce the neurosymbolic module we have designed for executing programs in these two domains. Overall, each DSL contains a set of types and a set of deterministic modules that have been manually designed for realizing necessary operations in these domains. However, in contrast to realizing them as we do in standard programming languages (with for-loops and if-conditions), we will be using tensor operations (e.g., tensor additions and multiplications) to realize them so that the output of each program is differentiable with respect to all of its inputs. We refer readers to the original papers for a detailed introduction to the DSL and neuro-symbolic program execution. Here we only highlight the key aspects of our language and its neuro-symbolic realization, and discuss the difference between our implementation and the ones in original papers. Our visual reasoning DSL is a subset of CLEVR, containing 6 types and 8 primitive operations. Table 1 illustrates all 6 types and how they are internally represented in neuro-symbolic execution. Table 2 further shows all operations in the DSL. There are two main differences between the DSL used by G2L2 and the original CLEVRDSL.

Moreover, our 1B model achieves exceptional performance, surpassing GPT -3.5-Turbo and Claude-3 Haiku.

Recent divide-and-conquer reasoning approaches, particularly those based on Chain-of-Thought (CoT), have substantially improved the Text-to-SQL capabilities of Large Language Models (LLMs). However, when applied to complex enterprise databases, such methods struggle to maintain coherent reasoning due to limited context capacity, unreliable schema linking, and weak grounding in database semantics. To overcome these issues, we introduce DSR-SQL, a \textbf{D}ual-\textbf{S}tate \textbf{R}easoning framework that models Text-to-SQL as an interaction between an adaptive context state and a progressive generation state. The first constructs a compact, semantically faithful environment by refining large schemas and selecting relevant structures, while the second formalizes SQL synthesis as feedback-guided state transitions, enabling the model to self-correct and align with user intent. Without any post-training or in-context examples, DSR-SQL achieves competitive performance, reaching 35.28\% execution accuracy on Spider 2.0-Snow and 68.32\% on BIRD development set. Our implementation will be open-sourced at: https://github.com/DMIRLAB-Group/DSR-SQL.

This paper introduces an Error Correction through Prompt Tuning for NL-to-SQL, leveraging the latest advancements in generative pre-training-based LLMs and RAG. Our work addresses the crucial need for efficient and accurate translation of natural language queries into SQL expressions in various settings with the growing use of natural language interfaces. We explore the evolution of NLIDBs from early rule-based systems to advanced neural network-driven approaches. Drawing inspiration from the medical diagnostic process, we propose a novel framework integrating an error correction mechanism that diagnoses error types, identifies their causes, provides fixing instructions, and applies these corrections to SQL queries. This approach is further enriched by embedding fine-tuning and RAG, which harnesses external knowledge bases for improved accuracy and transparency. Through comprehensive experiments, we demonstrate that our framework achieves a significant 12 percent accuracy improvement over existing baselines, highlighting its potential to revolutionize data access and handling in contemporary data-driven environments.

Text-to-SQL is a pivotal task that bridges natural language understanding and structured data access, yet it remains fundamentally challenging due to semantic ambiguity and complex compositional reasoning. While large language models (LLMs) have greatly advanced SQL generation though prompting, supervised finetuning and reinforced tuning, the shift toward test-time scaling exposes a new bottleneck: selecting the correct query from a diverse candidate pool. Existing selection approaches, such as self-consistency or best-of-$N$ decoding, provide only shallow signals, making them prone to inconsistent scoring, fragile reasoning chains, and a failure to capture fine-grained semantic distinctions between closely related SQL candidates. To this end, we introduce JudgeSQL, a principled framework that redefines SQL candidate selection through structured reasoning and weighted consensus tournament mechanism. JudgeSQL develops a reasoning-based SQL judge model that distills reasoning traces with reinforcement learning guided by verifiable rewards, enabling accurate and interpretable judgments. Building on this, a weighted consensus tournament integrates explicit reasoning preferences with implicit generator confidence, yielding selections that are both more reliable and more efficient. Extensive experiments on the BIRD benchmark demonstrate that JudgeSQL exhibits superior SQL judgment capabilities and good cross-scale generalization and robustness to generator capacity.