Natural Language Rewrite Rules (NLR2s)

• NLR2s are a class of domain-agnostic, human-readable transformation rules that guide rewriting and optimization across queries, texts, and programs.
• They are operationalized within formal frameworks and LLM-driven pipelines to enhance query optimization, semantic refinement, and verification tasks.
• Their adaptation via counterexample-guided refinement and equality saturation enables robust, efficient rewriting in diverse domains such as NLP and data management.

Natural Language Rewrite Rules (NLR2s) are a class of domain-agnostic, human-readable transformations for formal or natural language objects—designed to express, guide, and generalize rewriting and optimization logic using textual descriptions rather than rigid, low-level patterns. NLR2s can be leveraged by reasoning systems and learning algorithms, especially LLMs, to enhance the coverage, generalization, and efficiency of rewriting tasks across data management, NLP, and formal verification. Their formalization and integration with machine learning and knowledge-based retrieval pipelines enables adaptive, utility-driven transfer of rewriting expertise between queries, texts, or programs.

1. Formal Definition and Structure of NLR2s

Natural Language Rewrite Rules are concise textual statements describing query, program, or sentence transformation strategies in general terms. Unlike classic syntactic rewritings—often encoded as pattern-replacement pairs (e.g., $p \to r$ in a grammar or algebra)—NLR2s are “meta-rules” or transformation hints expressed in natural language. They may encapsulate optimization logic (“split complex queries into separate CTEs”), correctness criteria, or redundancy elimination (“remove unnecessary UNION ALL operations”), and are typically parameterized to apply across many instances.

NLR2s are not limited to a particular grammar or notation and, by abstraction, allow reasoning systems to apply, combine, or adapt them to unseen cases. For example, in query optimization contexts, an NLR2 may be:

• “Decompose subqueries connected by UNION ALL into distinct CTEs to simplify conditions.”
• “Eliminate filters that entirely subsume the effects of previous joins.”
• “Minimize repeated traversal over the same data structure in linear algebra expressions.”

Their structure is characteristic of guidance or heuristic statements and may reference both semantic and structural transformation principles.

2. Operationalization in Formal Rewriting Frameworks

NLR2s can be instantiated as operational rewrite rules within various formal frameworks. In nominal rewriting systems (NRSs) and combinatory reduction systems (CRSs), transformations involving variable binding and scope (crucial for natural language movement and quantifiers) are formalized and then mapped to higher-order equivalents (Domínguez et al., 2015). A translation function $\mathcal{T}(\Delta, t)$ recursively decomposes nominal terms into CRS meta-terms:

$$\mathcal{T}(\Delta, \pi \cdot X) = X(\overline{xs}), \quad \overline{xs} \triangleq \pi \cdot xs,$$

where $xs$ is an ordered list of potentially captured atoms.

Nominal rewrite rules of form $\nabla \vdash l \to r$ are mapped to CRS rules via:

$$\mathcal{T}^\mathcal{R}(\nabla, l, r) = \mathcal{T}(\nabla, l) \Rightarrow \mathcal{T}(\nabla, r).$$

Preservation of the rewriting relation enables operational correspondence, import of termination and confluence analyses, and transfer of higher-order techniques to NLR2 scenarios involving binding and scope (e.g., quantifier normalization in logical forms). This is critical for rigorous treatment of linguistic transformations such as pronoun binding, movement, and scope resolution.

3. Semantic-Refinement and Rule-Based NL Verbalization

In ontology verbalization and controlled NL generation, NLR2s underpin semantic-refinement pipelines that abstract away redundancy and repetition in logical forms before natural language realization (V et al., 2016). Starting from a label-set $L(x)$ of all logical conditions satisfied by an entity, semantic-refinement applies ordered, rule-based transformations (concept refinement, superclass reduction, existential/universal merging, non-vacuous role construction):

• For $L(x)$ containing both $\exists R.\mathsf{Cat}$ and $\exists R.\mathsf{Animal}$ with $\mathsf{Cat} \sqsubseteq \mathsf{Animal}$, an NLR2 effect is to remove the animal constraint.
• Pairwise combinations of existential and universal role restrictions are collapsed into non-vacuous forms $\mathcal{J}R.C$ for concise NL descriptions.

Algorithmically, rules are applied in a sequence, marking provisionally reduced elements and iteratively pruning label-sets to minimize redundancy. This yields concise, validation-friendly NL output; empirical validation demonstrates superiority over verbatim verbalization—dramatically improving expert ratings and comprehension efficiency.

4. Integration with LLMs and Query Optimization

In scalable query rewriting systems, NLR2s function as the communication and reasoning layer between user intent, formal system, and LLM agent (Liu et al., 14 Mar 2024, Sun et al., 2 Dec 2024). Systems such as GenRewrite and R-Bot both extract, regularize, and apply NLR2s to guide LLMs through multi-step rewriting tasks:

• NLR2s are generated as LLM explanations of beneficial rewrites, stored in a repository, and used as hints via contextual prompting.
• Relevance and utility of NLR2s are scored via embedding-based similarity and benefit attribution formulas:

$$\text{score}(rg) = \sum_{i=1}^k \text{weight}(n_i) \cdot \text{indicator}(n_i, rg) \cdot \text{benefit}(rg),$$

with $\text{weight}(n_i)$ derived from query embedding distances.

Hybrid structure-semantics retrieval (combining structural matching functions, masked query templates, and semantic embedding fusion) ensures contextually optimal hint selection. The rewriting process is iterative and reflective: each step applies a selection and ordering of NLR2s/rules informed by retrieved evidence and observed query costs, mitigating LLM hallucination and compositional error while accommodating rule interdependencies.

5. Mechanisms for Correctness, Generality, and Efficiency

Correctness (semantic equivalence) in NLR2-driven rewriting is enforced via counterexample-guided refinement: LLMs analyze semantic and syntactic discrepancies between original and rewritten artifacts, generate counterexamples, and iteratively modify rewrites to converge on correctness (Liu et al., 14 Mar 2024). This process is generally more effective than zero-shot LLM prompting or pure heuristic rewriting.

Generality and compactness of NLR2 sets are achieved via equality saturation (Nandi et al., 2021), where e-graphs represent large equivalence classes of terms, allowing inference and selection of compact, orthogonal rule sets. Candidate rules $\ell \to r$ are filtered via characteristic vectors and saturated across term space:

$$C = \{ \ell \to r \mid \forall \sigma: \llbracket \sigma(\ell) \rrbracket_D = \llbracket \sigma(r) \rrbracket_D \}.$$

This compaction facilitates efficient reasoning and application of NLR2s in high-dimensional, redundancy-rich domains, such as paraphrasing normalization or program optimization.

6. Application Domains and Transfer Mechanisms

NLR2s are applied in:

• Query optimization: Human-readable NLR2s guide LLMs to transform poorly written or suboptimal SQL queries into efficient forms—demonstrably increasing coverage and speedup on benchmarks such as TPC-H and TPC-DS (Liu et al., 14 Mar 2024, Sun et al., 2 Dec 2024).
• Reasoning about equivalence: Transformer-based models output rewrite sequences that provably imply program equivalence, usable in automated grading, compiler optimization validation, and robust paraphrasing (Kommrusch et al., 2021).
• Ontology validation and NL generation: Semantic-refinement NLR2s yield fully specified, redundancy-free descriptions for knowledge graph entities, driving ontological correctness and user comprehension (V et al., 2016).
• Formal logic and NLP phenomena involving binding: Higher-order rewriting systems backed by NLR2s enable analysis and manipulation of variable binding, quantifier scope, and pronoun resolution under strict semantic preservation (Domínguez et al., 2015).

7. Theoretical Properties and Future Directions

NLR2s, when formalized and integrated with transport mechanisms (see heterogeneous equality in $\lambda\Pi$-calculus modulo) (Blot et al., 14 Feb 2024), allow theories to maintain conservative extension properties: translation from rewrite rules to axioms preserves provability and semantic content. This ensures that type or category changes effected by NLR2s do not disrupt semantic consistency—analogous to ensuring paraphrase or optimizer rewrites do not change true meaning.

Future advances may include:

• Automated NLR2 synthesis using equality saturation, transfer learning, or neural architecture search to expand the coverage of rewrite knowledge;
• Deployment of NLR2s across multi-modal reasoning contexts;
• Precise benchmarking of NLR2-driven systems against classic rule-based architectures with regard to compositional generalization, robustness to distribution shift, and interpretability.

A plausible implication is that NLR2s, as an abstraction, can support systematic transfer of expert rewriting intuition from one formal or natural language domain to another, thus accelerating the convergence of adaptive AI-driven reasoning and optimization systems.