Beyond Geometry: How Neural Networks Learn to Reason with Symbols

Author: Denis Avetisyan

New research reveals that transformer models can develop abstract reasoning skills by learning symbolic strategies, even without explicit training on pre-defined concepts.

Probes trained on the closure subspace accurately predict variable presence and exhibit a weak alignment with the model’s unembedding direction for each token, demonstrating a correlation between learned representations and identifiable features within the model's internal state. — Probes trained on the closure subspace accurately predict variable presence and exhibit a weak alignment with the model’s unembedding direction for each token, demonstrating a correlation between learned representations and identifiable features within the model’s internal state.

This study demonstrates that transformer models trained on tasks involving variable meanings exhibit emergent symbolic reasoning, offering insights into their interpretability and ability to generalize beyond geometric patterns.

While large language models excel at pattern recognition, their capacity for abstract reasoning-particularly without relying on pre-encoded knowledge-remains an open question. This paper, ‘In-Context Algebra’, investigates how transformer models learn to solve arithmetic problems where variable symbols lack fixed meanings, a setting distinct from prior work demonstrating geometric embeddings with fixed values. We find that, despite this challenge, models achieve near-perfect accuracy by developing symbolic reasoning mechanisms-specifically, copying answers, recognizing identity elements, and utilizing closure-based cancellation-rather than relying on geometric representations. Could these emergent symbolic strategies represent a fundamental step towards more robust and generalizable in-context learning capabilities?

The Illusion of Arithmetic: Exposing the Limits of Statistical Mimicry

Despite demonstrating remarkable proficiency in generating human-like text, current large language models often falter when confronted with even basic arithmetic. This isn’t a matter of lacking knowledge of numerical facts-the models can readily recall that $2 + 2 = 4$-but rather an inability to consistently apply arithmetic principles to solve novel problems. Errors arise not from computational mistakes in the traditional sense, but from a reliance on statistical patterns within the training data rather than a genuine understanding of mathematical operations. The models excel at mimicking the form of correct reasoning, but frequently struggle with the underlying logic, leading to inconsistent and unreliable results even in seemingly straightforward calculations. This highlights a fundamental limitation: fluency in language does not equate to competency in formal reasoning.

A novel approach to enhancing arithmetic reasoning in large language models centers on the principles of abstract algebra, specifically leveraging the structure of algebraic groups. This framework moves beyond the limitations of current models, which often rely on statistical pattern matching, and instead grounds calculations in a formal, symbolic system. By representing arithmetic operations within the context of group theory – where operations combine elements to produce other elements within the same set, adhering to properties like associativity and identity – the system can enforce mathematical consistency. This isn’t simply about achieving correct answers; it’s about building a computational foundation where each step is logically derived from established axioms, much like a mathematical proof. The application of algebraic groups provides a rigorous structure for representing numbers and operations, allowing the model to manipulate symbols according to defined rules, and ultimately fostering a more reliable and transparent approach to arithmetic tasks, extending beyond superficial fluency to genuine understanding of numerical relationships, for example, solving equations like $x + 2 = 5$ with guaranteed consistency.

Unlike current large language models that often rely on identifying patterns within training data, this framework emphasizes symbolic manipulation – treating numerical problems as abstract algebraic expressions. This mirrors the process of mathematical proof, where problems are solved not through memorization, but by applying established rules to transform equations and isolate variables. Instead of recognizing that $2 + 2 = 4$ as a frequently observed pairing, the system would manipulate symbols according to the axioms of arithmetic, demonstrating why the equation holds true. This shift from pattern recognition to formal deduction aims to unlock a level of reliability and generalizability currently absent in purely statistical approaches, enabling the model to tackle novel arithmetic challenges with the same rigor as a human mathematician.

The model generalizes well to unseen algebraic groups, achieving comparable performance to in-distribution groups, but struggles with non-group structures like quasigroups and magmas, particularly in hold-out sequence prediction.

Constructing Logical Sequences: Encoding Arithmetic Within Group Theory

Input sequences are constructed by concatenating tokens representing established facts about algebraic groups and specific variable assignments. These facts, formalized within the system, define relationships and properties inherent to the group structure, such as group axioms and element interactions. Variable assignments establish concrete values for variables used within the arithmetic problems, effectively grounding the abstract group properties to specific numerical instances. The order of concatenation is critical; facts and assignments are sequenced to provide the model with the necessary information for subsequent operations, ensuring a deterministic and predictable outcome based on the defined group theory and variable values. This method allows for the programmatic construction of inputs tailored to leverage the model’s capacity for arithmetic reasoning within the framework of abstract algebra.

The model interprets input tokens not as literal values, but as representations of elements within a defined algebraic group. This mapping establishes a context-dependent semantic meaning for each token, determining its role in subsequent operations. Specifically, the group structure dictates how tokens interact; operations are not performed on numerical values directly, but on the group elements to which the tokens are mapped. Consequently, the meaning of a token, and therefore the operation it initiates, is dynamically determined by its position within the generated sequence and the established group context. This allows for complex arithmetic to be performed by controlling the relationships between these mapped elements, rather than relying on pre-defined arithmetic rules.

The model’s arithmetic capabilities are directly enabled by the structured input sequence; specific token arrangements function as instructions. By carefully ordering facts related to algebraic groups and variable assignments within this sequence, we provide the model with the data necessary to execute calculations. This control over the input sequence bypasses the need for explicit training on arithmetic operations; instead, the model leverages its existing language modeling capabilities to interpret the sequence and produce the correct numerical result. Essentially, the sequence acts as a program, dictating the steps the model takes to solve the given arithmetic problem, with each token contributing to the overall computational logic.

The data generation process creates diverse sequences by sampling algebraic groups, assigning vocabulary symbols, and mapping facts through a latent space, ensuring that symbol meanings vary across each generated sequence.

Dissecting the Mechanism: Verbatim Recall and Commutative Strategies

The model demonstrates a high degree of efficiency in arithmetic tasks through the mechanism of ‘Verbatim Copying’. This process involves retrieving previously encountered factual sequences directly from memory, bypassing the need for recomputation. Quantitative analysis indicates an accuracy of 99.5% when performing sequence copying tasks, suggesting a robust capability for factual recall. This functionality significantly contributes to the model’s overall performance by minimizing computational load and enabling rapid responses for known arithmetic problems.

The model utilizes a ‘Commutative Copying’ strategy by recognizing and exploiting the commutative property of group operations – specifically, that the order of operands does not affect the result when applied to certain arithmetic problems. This is achieved by copying and reordering input sequences where applicable, effectively reducing the computational complexity. For instance, in addition ($a + b = b + a$), the model can identify equivalent problem formulations through sequence matching and retrieve previously computed results or apply learned operations in a different, yet equivalent, order. This strategy is particularly effective when dealing with operations where commutativity holds, allowing for efficient problem solving by leveraging existing knowledge and minimizing the need for novel computation.

The model demonstrates problem simplification through two mechanisms: Identity Element Recognition and Closure-Based Cancellation. Following training on copying sequences, the model achieves 50% accuracy in identifying identity elements within arithmetic operations. More significantly, Closure-Based Cancellation consistently achieves 100% accuracy. This process relies on learned subspaces and a top-K matching strategy to identify and eliminate terms that, when combined, result in a closed, solvable component. This effectively reduces the complexity of initial problems by isolating and removing redundant or canceling elements, allowing for efficient computation of the remaining components.

Analysis of the copying head reveals it prioritizes verbatim facts when available, shifting to commutative facts if the exact fact is absent, self-attending when both are missing, and attending to all answer slots when presented with corrupted facts, demonstrating a clear strategy for retrieving and applying prior knowledge.

Probing Internal Logic: Attention as a Window into Reasoning

Within the Transformer architecture, the attention mechanism functions as a dynamic weighting system, selectively emphasizing pertinent elements of an input sequence while diminishing the influence of irrelevant ones. This process allows the model to focus computational resources on the most critical information for a given task, effectively mimicking cognitive prioritization. Rather than treating all input tokens equally, attention assigns a scalar value – an ‘attention weight’ – to each, signifying its importance. These weights are then used to create a weighted sum of the input embeddings, producing a context-aware representation that captures the relationships between different parts of the sequence. Consequently, the model doesn’t merely process a string of tokens; it discerns a hierarchy of relevance, allowing it to perform complex reasoning and achieve state-of-the-art results in various natural language processing applications. The strength of this mechanism lies in its ability to learn these relationships directly from data, without requiring explicit programming of linguistic rules or structural dependencies.

Researchers employed a technique called causal intervention, utilizing Householder transformations to directly manipulate the attention weights within a neural network. This allowed for a targeted examination of how specific attention patterns contribute to the model’s problem-solving process. By altering these weights, the study could effectively ‘probe’ the network’s internal reasoning, determining which elements the model deemed crucial for arriving at a correct solution. The Householder transformation, a method from linear algebra, provided a precise way to modify attention without completely disrupting the network’s structure, enabling a nuanced understanding of the causal relationship between attention and performance. This approach moves beyond simply observing attention patterns to actively testing their functional role, revealing the model’s reliance on key inputs for accurate computation.

Investigations into the attention mechanisms of the Transformer architecture reveal a surprising capacity for prioritizing information critical to accurate computation. Through targeted manipulation of attention weights – a process known as causal intervention – researchers found the model consistently focused on elements fundamentally important for solving algebraic problems. This isn’t simply pattern recognition; the model demonstrates an understanding of underlying mathematical structure, consistently selecting the precise components needed for correct results. Remarkably, intervention accuracy reached 100% not only on the data used for training, but also on entirely new, unseen validation sets, suggesting a robust and generalizable comprehension of the algebraic relationships within the presented tasks.

Multiple attention heads collaboratively construct a cancellation set by demoting logits of tokens in fact answer slots that share symbols with the query’s slots, effectively refining the model’s focus.

Beyond Superficial Competence: Phase Transitions and the Pursuit of True Understanding

During the training of these advanced models, a distinct ‘phase transition’ emerges, analogous to shifts observed in physical systems like water changing to ice. This isn’t merely a gradual improvement in performance; instead, the model fundamentally alters how it learns. Initially, the model might rely on memorization or superficial patterns, but as training progresses, it undergoes a qualitative change, transitioning to a state where it effectively exploits the underlying algebraic structure of the mathematical problems. This shift is characterized by a change in the model’s representational capacity – it begins to build more abstract and generalizable internal representations, allowing it to solve previously intractable problems with greater efficiency and accuracy. The observation of this phase transition suggests that the training process isn’t simply optimizing parameters, but is actively reshaping the model’s very approach to reasoning, unlocking a higher level of mathematical competence.

The training process isn’t simply a descent into minimizing error; it exhibits a distinct phase transition guided by the carefully constructed loss function. This function doesn’t merely quantify mistakes, but actively shapes how the model learns to represent mathematical expressions. By penalizing certain errors more than others, the loss function encourages the neural network to discover and leverage the underlying algebraic structure within the data. Essentially, the model is optimized not just for accuracy, but for a particular form of representation – one that mirrors the inherent symmetries and relationships within mathematical equations. This strategic optimization allows the system to generalize effectively, moving beyond rote memorization to achieve a deeper, more robust understanding of mathematical principles, and potentially extending these capabilities to other structured domains.

The development of this phase-transition-guided training paradigm suggests a pathway towards reasoning systems exhibiting enhanced resilience and clarity. By optimizing models to leverage the inherent algebraic structure within problems, researchers anticipate a shift away from brittle, black-box approaches towards solutions that generalize more effectively across diverse mathematical challenges. This isn’t limited to arithmetic; the underlying principles promise applicability to fields requiring complex logical deduction, such as symbolic reasoning, automated theorem proving, and even areas of scientific discovery where identifying and exploiting structural relationships is paramount. Ultimately, this work proposes a future where artificial intelligence doesn’t simply solve problems, but demonstrates a degree of understanding, allowing for more reliable and transparent reasoning processes, potentially unlocking new capabilities in artificial general intelligence and beyond.

Transformer model learning progresses through distinct phases, beginning with structural token prediction and group closure, then advancing to verbatim copying and cancellation, before finally mastering associative sequences, with each phase marked by a drop in training loss and the acquisition of specific skills.

The pursuit of symbolic reasoning within transformer models, as demonstrated in this study of in-context learning, echoes a fundamental tenet of computational elegance. The research highlights how these models develop strategies based on variable meanings, moving beyond mere pattern recognition. This aligns perfectly with John McCarthy’s assertion: “The best way to program is to begin with a formal specification.” Without rigorous definitions – understanding the variables and their relationships – the model’s reasoning remains superficial. The paper’s focus on abstract algebraic groups and the model’s ability to learn symbolic strategies underscores the necessity of a formal, provable foundation, rather than relying on empirically derived approximations. The emphasis on defining the ‘rules of the game’ is central to achieving genuine intelligence in these systems.

Beyond Demonstration

The observation that transformer models adopt symbolic strategies – not as explicitly programmed, but as emergent behavior – shifts the burden of explanation. The question is no longer simply whether these models can perform abstract reasoning, but why this particular form of reasoning arises. The current work establishes a foundation, but the leap from variable manipulation to genuine understanding of algebraic groups remains unbridged. A critical next step involves formalizing the constraints under which these emergent symbolic systems operate, and determining whether they are, in principle, capable of representing the full structure of relevant mathematical objects.

Current evaluation relies heavily on task performance. This is, predictably, insufficient. The true test lies in provability. Can the reasoning processes within these models be extracted and verified through formal methods? Establishing a link between internal representations and demonstrable mathematical truths would elevate this field beyond empirical observation, toward a more rigorous science. The limitations of in-context learning – its sensitivity to prompt design, its lack of explicit generalization – suggest that a more stable, axiomatic foundation is required.

Ultimately, the pursuit of artificial intelligence should not aim to replicate intelligence, but to illuminate the underlying principles of computation itself. This work hints that the language of mathematics is not merely a tool for expressing thought, but may be a fundamental constraint on the very structure of intelligent systems. Further investigation may reveal whether this is an accidental artifact of the training regime, or a deeper truth about the nature of reason.

Original article: https://arxiv.org/pdf/2512.16902.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/