Data Reasoning Intensity (DRI)

• Data Reasoning Intensity (DRI) is a metric that quantifies latent logical reasoning complexity by decomposing expressions, predicates, constants, and nesting depth.
• DRI informs data-centric training strategies by identifying low-value samples for pruning and high-intensity samples as reasoning catalysts.
• Its computation employs detailed formulas that account for operation counts and nesting depth, enhancing data selection, reward shaping, and model evaluation.

Searching arXiv for the primary DRI paper and closely related work. Data Reasoning Intensity (DRI) is a scalar score in $[0,1]$ introduced to measure how much latent logical reasoning complexity a data sample contains, particularly in the training and evaluation of LLMs on logical reasoning tasks (Bi et al., 29 Sep 2025). In its explicit formulation, DRI is not a proxy for length, verbosity, or surface difficulty; it is derived from decomposed logical structures, including predicates, constants, expressions, nesting depth, and multi-step reasoning chains. Closely related work uses “reasoning intensity” in a broader sense for retrieval, where the emphasis shifts from logical decomposition to the marginal benefit of reasoning for distinguishing relevant from irrelevant documents (Chen et al., 9 Oct 2025). Taken together, the emerging literature treats DRI as a data-centric lens on reasoning: it asks not only what a model can do, but what reasoning potential is present in the data and how effectively models exploit it.

1. Conceptual definition and scope

The explicit introduction of Data Reasoning Intensity appears in “Pushing LLMs to Their Logical Reasoning Bound: The Role of Data Reasoning Intensity,” which defines DRI as a scalar score in $[0,1]$ that measures, for each training sample, how much latent logical reasoning complexity it contains (Bi et al., 29 Sep 2025). The paper argues that LLM logical reasoning performance is jointly constrained by the reasoning potential of the data and the cognitive capacity or cost of the model, formalized as

$$\eta(\mathcal{M}, \mathcal{D}) = \frac{E(\mathcal{D})}{C(\mathcal{M})}.$$

Within this formulation, DRI is the mechanism that makes the data-side quantity $E(\mathcal{D})$ measurable at the sample level.

In that formulation, DRI is intended to capture several structural properties at once: how many logical expressions are involved, how deeply those expressions are nested, how many distinct predicates and constants must be tracked, and how long and operator-heavy the reasoning chain is for each option (Bi et al., 29 Sep 2025). The emphasis is therefore on latent logical structure rather than on token count or stylistic complexity. A short but highly nested logical puzzle may have high DRI, whereas a long but narratively simple passage may not.

Related retrieval work does not always use the label “DRI,” but it defines closely aligned notions. ReasonEmbed defines the reasoning intensity of a retrieval sample as “the extent to which reasoning contributes to distinguishing relevant from irrelevant documents,” and operationalizes it through the performance gap between an original query and a reasoning-augmented query (Chen et al., 9 Oct 2025). This suggests that, across subfields, DRI functions as a family of data-centric measures for the incremental value of reasoning structure.

2. Formalization and computation

In the logical-reasoning formulation, DRI is computed by first extracting logical elements from a question or sample $Q$. A distillation function $f$ produces predicates, constants, and logical expressions: $$f(Q) \Rightarrow \{ \mathcal{P}, \mathcal{C}, \mathcal{E}\}.$$ A second function $F$ then combines these elements with answer options $A$ to produce a precondition structure and a reasoning trajectory: $$\text{Trajectory} = F(\text{Element}, A) \Rightarrow \{ \bar{\mathcal{E}}, \mathcal{S} \}.$$ Each reasoning step $[0,1]$0 is represented as

$[0,1]$1

The decomposition therefore treats reasoning as a structured object with countable local features rather than as an uninterpreted text span (Bi et al., 29 Sep 2025).

The context-level score is defined as

$[0,1]$2

where $[0,1]$3 is the number of logical expressions, $[0,1]$4 is the average nesting depth, and $[0,1]$5 are the counts of distinct predicates and constants. Each answer option $[0,1]$6 is scored separately: $[0,1]$7 The raw intensity is then aggregated as

$[0,1]$8

Finally, the score is normalized to $[0,1]$9 with a log transform and sigmoid: $$\eta(\mathcal{M}, \mathcal{D}) = \frac{E(\mathcal{D})}{C(\mathcal{M})}.$$0 with $$\eta(\mathcal{M}, \mathcal{D}) = \frac{E(\mathcal{D})}{C(\mathcal{M})}.$$1, $$\eta(\mathcal{M}, \mathcal{D}) = \frac{E(\mathcal{D})}{C(\mathcal{M})}.$$2, and $$\eta(\mathcal{M}, \mathcal{D}) = \frac{E(\mathcal{D})}{C(\mathcal{M})}.$$3 (Bi et al., 29 Sep 2025).

This design gives nesting depth a quadratic contribution, both in the context term and in option-level reasoning steps. The paper’s motivation is that deeper nesting imposes disproportionately greater cognitive burden, so depth should not be treated as merely linear. In that sense, DRI is structurally biased toward compositional and multi-step reasoning.

The paper also provides illustrative score ranges. A very simple entailment example, “john knows how to play the piano” with the query “does this entail that someone has the ability to play the piano?”, receives a ReasoningIntensityScore of $$\eta(\mathcal{M}, \mathcal{D}) = \frac{E(\mathcal{D})}{C(\mathcal{M})}.$$4. A medium-complexity Roosevelt story is reported at approximately $$\eta(\mathcal{M}, \mathcal{D}) = \frac{E(\mathcal{D})}{C(\mathcal{M})}.$$5, while a long, abstract paragraph on the service industry and “industrial hollowing” is reported at approximately $$\eta(\mathcal{M}, \mathcal{D}) = \frac{E(\mathcal{D})}{C(\mathcal{M})}.$$6 (Bi et al., 29 Sep 2025). These examples are not a taxonomy, but they show how the scoring function separates shallow entailment from highly structured reasoning.

3. Data potential, model capacity, and empirical behavior

The central empirical claim of the DRI framework is that reasoning performance depends jointly on data potential and model capacity. The paper decomposes model-side cognitive cost as $$\eta(\mathcal{M}, \mathcal{D}) = \frac{E(\mathcal{D})}{C(\mathcal{M})}.$$7, where $$\eta(\mathcal{M}, \mathcal{D}) = \frac{E(\mathcal{D})}{C(\mathcal{M})}.$$8 is model scale, $$\eta(\mathcal{M}, \mathcal{D}) = \frac{E(\mathcal{D})}{C(\mathcal{M})}.$$9 is reasoning computational complexity, and $E(\mathcal{D})$0 denotes architectural constraints; it decomposes data-side potential as $E(\mathcal{D})$1, where $E(\mathcal{D})$2 is the presence and richness of structured reasoning traces, $E(\mathcal{D})$3 is logical component density, and $E(\mathcal{D})$4 is semantic coherence between reasoning steps (Bi et al., 29 Sep 2025). DRI is intended to make the $E(\mathcal{D})$5 and $E(\mathcal{D})$6 side operational at the sample level.

The reported DRI distribution is roughly bell-shaped, with mean $E(\mathcal{D})$7 and standard deviation $E(\mathcal{D})$8. When model error is plotted against DRI bins, error rises from $E(\mathcal{D})$9 at DRI $Q$0 to $Q$1 at DRI $Q$2, then plateaus around $Q$3 beyond $Q$4 (Bi et al., 29 Sep 2025). This establishes DRI as an empirical difficulty axis rather than a purely formal construct. The plateau is especially important: it indicates a regime in which raising reasoning intensity no longer reveals finer gradations of competence because the tasks have already exceeded the model’s current reasoning boundary.

The same paper uses DRI to study selective training ranges. Training on $Q$5, which removes the lowest-intensity $Q$6 of samples, outperforms full-data training; $Q$7 remains similar to or slightly better than full-data training; and narrower mid-range bands such as $Q$8 or $Q$9 produce severe performance drops (Bi et al., 29 Sep 2025). The paper summarizes these observations as “low DRI data can be pruned,” “high DRI data are catalysts,” and “too narrow DRI coverage hurts.” This suggests that DRI is valuable not only as a descriptive measure but also as a data-selection variable: it can identify both low-value trivial samples and very high-value reasoning catalysts.

4. DRI-guided optimization and training strategies

The principal optimization method built around DRI is “Re-Cognizing Optimization,” a two-phase strategy designed to reshape and then intensify reasoning-oriented training (Bi et al., 29 Sep 2025). Phase I, “Model Cognition Reshaping,” uses one uniformly shuffled epoch over the whole DRI spectrum. Phase II, “Cognitive Reasoning Enhancement,” prioritizes high-intensity samples by defining

$f$0

so that samples with higher DRI receive higher sampling probability. The method therefore increases effective reasoning exposure without increasing dataset size.

Empirical results show consistent gains over direct fine-tuning, curriculum learning, and bin-based progressive learning. On the unbalanced test set, the average accuracies reported for LLaMA3.1-8B and Qwen2.5-7B are as follows (Bi et al., 29 Sep 2025):

Method	LLaMA3.1-8B	Qwen2.5-7B
Direct	0.806	0.778
Curriculum	0.811	0.823
Bin-based progressive	0.826	0.812
Re-Cognizing Optimization	0.843	0.858
w/o Stage1	0.771	0.827
w/o Stage2	0.809	0.784

On the balanced test set, Qwen2.5-7B reaches $f$1 under the full method, compared with $f$2 for direct fine-tuning, $f$3 for curriculum, and $f$4 for bin-based progressive learning (Bi et al., 29 Sep 2025). The ablations indicate that both stages matter: removing either the initial reshaping stage or the DRI-guided enhancement stage reduces performance substantially.

The same work also integrates DRI into reinforcement learning. Under a GRPO-based setup with Qwen2.5-7B, correct answers on higher-DRI samples receive higher rewards, and the model is additionally rewarded for adhering to a structured output format using > and <answer> tags (Bi et al., 29 Sep 2025). The paper reports that DRI-guided RL improves over both the base model and direct RL with fixed accuracy-based rewards. This extends DRI from data curation and supervised ordering into reward shaping.

5. Retrieval-oriented reasoning intensity and evaluation substrates

Reasoning intensity has also become a central concept in retrieval. ReasonEmbed defines it explicitly as “the extent to which reasoning contributes to distinguishing relevant from irrelevant documents,” and quantifies it using

$f$5

where $f$6 is the original query, $f$7 is a reasoning-augmented query, and $f$8 (Chen et al., 9 Oct 2025). This sample-wise score then drives Redapter, a weighted RI-InfoNCE objective. The accompanying ReMixer pipeline produces 81,659 final queries with average query length 221 tokens, and ReasonEmbed-Qwen3-8B reaches an average nDCG@10 of 38.1 on BRIGHT and 43.18 on R2MED (Chen et al., 9 Oct 2025). This does not use the same formalism as logical DRI, but it establishes a parallel principle: reasoning intensity can be measured by the marginal value of additional reasoning operations.

DIVER frames “reasoning-intensive information retrieval” as retrieval in which relevance is not determined by direct lexical or semantic matching but by “abstract reasoning, analogical thinking, or multi-step inference” (Long et al., 11 Aug 2025). Its four-stage pipeline combines DIVER-DChunk, DIVER-QExpand, a reasoning-enhanced retriever, and hybrid pointwise-listwise reranking. On BRIGHT, DIVER reports nDCG@10 scores of 45.8 overall and 28.9 on original queries, and its results show that reasoning-aware expansions are especially effective when combined with reasoning-aware dense retrieval rather than lexical retrieval alone (Long et al., 11 Aug 2025).

RaDeR develops another retrieval-side realization of high-intensity data. It trains dense retrievers from retrieval-augmented reasoning trajectories and self-reflective relevance judgments derived from mathematical problem solving, and it reports the first dense retriever that outperforms BM25 when queries are chain-of-thought reasoning steps (Das et al., 23 May 2025). In that setting, positives are documents that help complete successful reasoning trajectories, while hard negatives are retrieved but self-reflectively rejected theorems. This suggests a particularly strong form of DRI: relevance is conditioned on a reasoning state, not only on the initial query.

DRE-Bench does not define DRI directly, but it offers a dynamic evaluation substrate for DRI-like concepts. It organizes 36 abstract reasoning tasks into four cognitive levels—Attribute, Spatial, Sequential, and Conceptual—and reports that although most LLMs achieve competent and robust performance in low-level cognition, they struggle with high-level cognition and exhibit limited generalization as task complexity grows (Yang et al., 3 Jun 2025). A plausible implication is that DRE-Bench provides a natural external axis for validating DRI: if a data metric really captures reasoning intensity, it should correlate with robustness across these increasing cognitive levels.

A plausible precursor to DRI is Data Readiness Level, which defines data quality relative to a query in terms of relevance, coherence, believability, and noisiness, and operationalizes part of this through cosine similarity and Document Disparity (Guan et al., 2017). This suggests a broader family resemblance between readiness-based and intensity-based measures: both are question-conditioned attempts to quantify how strongly data support reasoning.

6. Terminological ambiguity, limitations, and open directions

The acronym “DRI” is not unique across arXiv literatures. In addition to Data Reasoning Intensity, it appears with unrelated meanings in deliberative theory and combinatorial optimization.

Acronym usage	Meaning	Source
DRI	Data Reasoning Intensity	(Bi et al., 29 Sep 2025)
DRI	Deliberative Reason Index	(Veri, 18 Apr 2026)
DRI	Decompose–Route–Improve	(Kerscher et al., 2024)

The Deliberative Reason Index measures how coherently a group’s preferences over policy options are grounded in stated considerations across waves or subgroups, and a 2026 paper proposes a modified DRI with a low-signal penalty to correct inflation under near-zero correlation structure (Veri, 18 Apr 2026). The VRPTW paper uses DRI for a Decompose–Route–Improve framework in which clustering, routing, and pruned local search are driven by a spatial–temporal–demand similarity metric (Kerscher et al., 2024). These works are conceptually unrelated to Data Reasoning Intensity and should not be conflated with it.

The explicit Data Reasoning Intensity framework also has stated limits. The 2025 paper notes that experiments cover only two model families, that logical decomposition relies on an LLM and may introduce noise, that the scoring formula may not capture all aspects of reasoning difficulty, and that the evaluation is performed in fine-tuning rather than full pretraining (Bi et al., 29 Sep 2025). Related retrieval work identifies additional limitations: reasoning-intensity estimates depend on the quality of the LLM that generates reasoning rewrites, the current scope is restricted to reasoning-intensive retrieval, and model dependence remains an open issue when trying to define a more general DRI (Chen et al., 9 Oct 2025).

Future directions are correspondingly broad. The DRI paper proposes broader model and task coverage, refined RL integration, adaptive reasoning routing, hybridization with other data-selection methods, and pretraining-stage integration (Bi et al., 29 Sep 2025). Retrieval-side work points toward finer-grained decomposition of reasoning types, broader domain coverage, and more stable model-independent intensity measures (Chen et al., 9 Oct 2025). Taken together, these directions indicate that DRI is best understood not as a settled universal metric, but as an active research program aimed at quantifying the reasoning potential embedded in data and aligning model training with that potential.