Algorithmic Primitives and Compositional Geometry of Reasoning in Language Models

TL;DR

The paper investigates how LLMs perform multi-step reasoning by proposing a reusable, compositional library of algorithmic primitives. It models primitives as vectors in activation space that can be injected into the residual stream, formalized as $h_\ell$ and $v^{(p)}_\ell$ with $\tilde{h}_\ell(x,t) = h_\ell(x,t) + \alpha\, v^{(p)}_\ell$, and traces these across benchmarks including $\text{TSP}$, $\text{3-SAT}$, $\text{AIME}$, and GraphNav in models such as Phi-4-Base, Phi-4-Reasoning, and Llama-3-8B. The authors identify cross-domain primitives, propose a geometric, vector-arithmetic framework for composing primitives, and provide mechanistic evidence that reasoning finetuning enhances cross-task generalization and transfer. Their results reveal both universal and task-specific primitives, with reasoning-finetuned models showing more systematic use of verification and path-generation primitives, and demonstrate that cross-task primitive transfer is possible through vector injections. Overall, the work suggests a geometry of reasoning primitives that underpins compositional generalization and informs future algorithmic finetuning and model-human alignment strategies.

Abstract

How do latent and inference time computations enable large language models (LLMs) to solve multi-step reasoning? We introduce a framework for tracing and steering algorithmic primitives that underlie model reasoning. Our approach links reasoning traces to internal activation patterns and evaluates algorithmic primitives by injecting them into residual streams and measuring their effect on reasoning steps and task performance. We consider four benchmarks: Traveling Salesperson Problem (TSP), 3SAT, AIME, and graph navigation. We operationalize primitives by clustering neural activations and labeling their matched reasoning traces. We then apply function vector methods to derive primitive vectors as reusable compositional building blocks of reasoning. Primitive vectors can be combined through addition, subtraction, and scalar operations, revealing a geometric logic in activation space. Cross-task and cross-model evaluations (Phi-4, Phi-4-Reasoning, Llama-3-8B) show both shared and task-specific primitives. Notably, comparing Phi-4 with its reasoning-finetuned variant highlights compositional generalization after finetuning: Phi-4-Reasoning exhibits more systematic use of verification and path-generation primitives. Injecting the associated primitive vectors in Phi-4-Base induces behavioral hallmarks associated with Phi-4-Reasoning. Together, these findings demonstrate that reasoning in LLMs may be supported by a compositional geometry of algorithmic primitives, that primitives transfer cross-task and cross-model, and that reasoning finetuning strengthens algorithmic generalization across domains.

Figures (10)

• Figure 1: Algorithmic Tracing & Steering: Primitive Extraction & Evaluation. We trace algorithmic primitives by a clustering latent representations, b identifying corresponding reasoning traces, c meta-clustering primitives, identifying sequential transition trends among clusters, and comparing cluster similarity across models and tasks. Once we identify primitives, d we extract associated primitive vectors from top heads, and e use causal patching to validate, explore the compositional geometry and cross-task transfer of primitives.
• Figure 2: Primitive vector injection induces associated algorithmic behavior. Injecting the nearest_neighbor or generate_path primitive vectors directly after the prompt a increases expression in output (see examples). b Varying the injection layer and the magnitude modifies the number of unique paths generated (top row) and the proportion of nearest-neighbor paths (bottom row). c Injecting primitive vectors for compute_distance in the middle of the reasoning trace increases relevant behavioral hallmarks in the output. d Primitive vectors from AIME were injected to different layers while solving Traveling Salesperson (TSP), showing cross-task primitive transfer and algorithmic induction.
• Figure 3: Clusters are organized in a hierarchical manner and meta-clusters are repeatedly traversed throughout the response. a Cluster-cluster transition matrix on TSP, organized by meta-cluster (MC). Most transitions occur within each meta-cluster. b Average t-SNE trajectories across different highlighted primitives. c Transition between different clusters for an example output. Different meta-clusters are highlighted by colors. The latter half of the response undergoes a cyclical transition. d Most common transitions between different meta-clusters reveals frequently occurring cycles. e-g Cluster-to-cluster transition matrices structured in terms of the inferred meta-clusters on e 3-SAT, f AIME, and g GraphNav.
• Figure 4: Primitive Cluster Patterns in Phi-4 and Phi-4-Reasoning. a-dtop Normalized dissimilarity of primitive cluster frequencies between Phi-4 and Phi-4-Reasoning for a) TSP, b) 3-SAT, c) AIME, and d) GraphNav. bottom Clusters sorted by whether they appear more frequently in Phi-4-Reasoning responses (positive difference) or in Phi-4 responses (negative difference). The lineplot specifies the differences whereas the rasterplot underneath specifies the relative frequencies of the different clusters per response. eleft Dissimilarity of primitive cluster frequencies between Phi-4-Reasoning responses to AIME and TSP. right Clusters sorted by whether they occur more frequently in AIME or TSP. The rasterplot underneath again specifies the relative frequencies of the different clusters per response.
• Figure 5: Algorithmic primitives operating over graphs, compositional induction, and cross-task transfer. a Injecting the sum of a terminal node recognition primitive vector (PV_TNR) and a reward comparison primitive vector (PV_RC) in a zero-shot setting (ZS) improves model performance on a task requiring a composition of both primitives and recovers few-shot performance (FS). See Fig. \ref{['fig:comp_supp']} for additional models. b Primitive vectors for extracting the first or last node of a presented graph reliably improves performance across different injection layers and across a shuffled few-shot and a zero shot-setting. c After injecting these primitive vectors into complex reasoning traces solving TSP, the model becomes more likely to mention the corresponding node next.