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Faith and Fate: Limits of Transformers on Compositionality
Nouha Dziri, Ximing Lu, Melanie Sclar, Xiang Lorraine Li, Liwei Jiang, Bill Yuchen Lin, Peter West, Chandra Bhagavatula, Ronan Le Bras, Jena D. Hwang, Soumya Sanyal, Sean Welleck, Xiang Ren, Allyson Ettinger, Zaid Harchaoui, Yejin Choi
TL;DR
The paper asks whether transformer LLMs’ failures on seemingly simple multi-step tasks reveal fundamental compositionality limits. It studies multiplication, logic puzzles, and dynamic programming using computation graphs, finding that models often match linearized subgraphs rather than develop systematic problem-solving skills, while theoretical error accumulation worsens with complexity.
Problem
The paper investigates why transformers excel on seemingly complex tasks yet fail on simple compositional problems, and whether these failures reflect systematic reasoning limits.
Method
The authors formulate three compositional tasks as computation graphs to measure complexity, analyze intermediate procedures, and compare model behavior across graph structures.
Results
Across tasks, transformers’ performance deteriorates as compositional complexity increases, with training yielding near-perfect in-domain results but drastic failures on wider or deeper out-of-distribution graphs.
Takeaways & Limitations
The findings suggest that transformers often solve compositional tasks through linearized subgraph or analogical pattern matching rather than systematic multi-step reasoning.
Takeaways & Limitations
The analysis cannot inspect the exact tokens models attend to, and alternative reasoning linearizations may produce different performance.
Abstract
from arXiv · showhide
Transformer large language models (LLMs) have sparked admiration for their exceptional performance on tasks that demand intricate multi-step reasoning. Yet, these models simultaneously show failures on surprisingly trivial problems. This begs the question: Are these errors incidental, or do they signal more substantial limitations? In an attempt to demystify transformer LLMs, we investigate the limits of these models across three representative compositional tasks -- multi-digit multiplication, logic grid puzzles, and a classic dynamic programming problem. These tasks require breaking problems down into sub-steps and synthesizing these steps into a precise answer. We formulate compositional tasks as computation graphs to systematically quantify the level of complexity, and break down reasoning steps into intermediate sub-procedures. Our empirical findings suggest that transformer LLMs solve compositional tasks by reducing multi-step compositional reasoning into linearized subgraph matching, without necessarily developing systematic problem-solving skills. To round off our empirical study, we provide theoretical arguments on abstract multi-step reasoning problems that highlight how autoregressive generations' performance can rapidly decay with\,increased\,task\,complexity.
1 Introduction
The paper examines whether transformers’ striking successes and failures reflect systematic limits in compositional reasoning. It studies three multi-step tasks and hypothesizes that models rely on linearized path matching and suffer compounding errors as complexity grows.
- Transformers can perform impressively on complex tasks yet achieve only 55% and 59% accuracy on 3-digit multiplication in ChatGPT and GPT4, respectively.The contrast motivates a closer examination of their problem-solving capabilities.
- The study evaluates long-form multiplication, logic grid puzzles, and dynamic programming as representative compositional tasks requiring combined reasoning operations.These tasks involve following computational paths to unique solutions.
- The authors hypothesize that transformers reduce multi-step reasoning to linearized path matching rather than learning systematic rules.This shortcut may work when similar compositional patterns appeared during training but generalize poorly to uncommon or complex examples.
- They also hypothesize that early computational errors propagate and compound, preventing correct solutions on high-complexity tasks.The proposed mechanism concerns novel patterns and increasingly long computational processes.
- Computation graphs decompose tasks into functional steps, enabling structured complexity measurements and verbalized reasoning sequences for language-model evaluation.The authors additionally use information gain to predict learnable patterns without executing full graph computations.
- Training on task-specific data produces near-perfect in-domain performance at low compositional complexity but fails drastically outside that region.The gap suggests that maximum-likelihood training does not necessarily produce systematic problem-solving abilities, even with human-like reasoning steps.
2 Measuring Limitations of Transformers in Compositional Tasks
The paper represents algorithms and compositional tasks as computation graphs, then measures reasoning complexity through graph structure. It applies this framework to multiplication, logic puzzles, and dynamic programming while relating graph complexity to transformer performance.
- 2.1 Computation Graph Definition: A computation graph represents an algorithm’s variables as nodes and primitive-function arguments as directed edges.Each nonsource node stores the result of applying one primitive to its parent values.
- 2.1 Computation Graph Definition: The graph’s source nodes are algorithm inputs, while its sole leaf node represents the algorithm’s output.For A(x), the source values equal x and the leaf value equals A(x).
- 2.1 Computation Graph Definition: Autoregressive language models require a topological ordering to linearize computation graphs into reasoning sequences.This ordering preserves graph dependencies while converting computation steps into input sequences.
- 2.2 Quantifying Compositional Complexity using Graph Metrics: Reasoning depth is the graph’s maximum layer number and serves as a proxy for the maximum multi-hop reasoning required.Layer number is the longest path length from any source to a node.
- 2.2 Quantifying Compositional Complexity using Graph Metrics: Reasoning width measures variables maintained in parallel, while average parallelism is |V| divided by reasoning depth.Width uses the mode of shortest source distances, whereas average parallelism captures average graph width.
- 2.3 Relative Information Gain: Relative Information Gain estimates how observing subsets of input variables informs output elements and can analyze ancestor influence in computation graphs.This supports analysis of partial successes and recognizable surface patterns.
- 2.4 Exploring Three Representative Compositional Tasks: Definitions: Long-form multiplication uses an O(k1k2) algorithm with one-digit multiplication, summation, carry, modulo, and concatenation primitives.Inputs contain at most five digits per number, and intermediate nodes encode partial results.
- 2.4 Exploring Three Representative Compositional Tasks: Definitions: Einstein’s puzzle maps clues to partially filled matrices, with increasing houses and attributes making constraint synthesis more compositionally complex.The computation graph uses an elimination function and outputs a fully filled solution matrix.
3 Testing the Limits of Transformers: Empirical Evidence
Across zero-shot, few-shot, and fine-tuned settings, transformer performance is near-perfect for familiar or low-complexity cases but deteriorates sharply as compositional complexity and distribution shift increase. The experiments indicate that scratchpads and extended training do not produce robust systematic generalization, while correct outputs often correlate with memorized computation subgraphs or surface patterns.
- Zero-shot and few-shot settings: Near-perfect performance deteriorates to zero as problem size or average parallelism increases in zero-shot and few-shot settings.These results indicate that pre-training does not sufficiently teach models to combine basic operations on increasingly complex compositional problems.
- Question-answer training: High in-domain accuracy sharply declines on unseen depth- and width-based splits across the multiplication, puzzle, and dynamic programming tasks.Exhaustive task-specific question-answer training therefore does not yield systematic problem-solving capabilities.
- Explicit scratchpad training: Near-perfect in-distribution performance still fails entirely on wider or deeper OOD graphs after explicit scratchpad training.Scratchpad supervision does not make component operations generalizable, and prompting gains diminish to zero as complexity increases.
- Grokking: Extended training beyond overfitting produces no improvement in OOD generalization on the multiplication task.The authors hypothesize that task difficulty may impede learning the structured representations associated with grokking, while noting that prolonged training would be inefficient and unscalable.
- Linearized subgraph matching: Correct predictions correlate with more frequent appearances of their full computation subgraphs in training data, whereas these frequencies tend toward zero at large depths.This pattern supports pattern matching as an explanation for low-complexity and in-domain success, with efficiency declining as tasks become more complex.
- Linearized subgraph matching: Fully correct nodes sharply decrease toward zero with graph depth, while propagation errors are usually more common than local errors.The results suggest that models can perform single-step operations but fail to plan and compose several steps into an overall correct solution.
4 Error Propagations: The Theoretical Limits
Theoretical analysis models compositional reasoning as repeated function estimation and shows that small per-step errors can make final errors grow rapidly with problem size. Under stated assumptions, incorrect-prediction probability converges exponentially toward one for independent or iterated applications.
- Independent and iterated applications: The paper analyzes transformers as estimators executing algorithms composed of independent or repeated applications of a function.This framework examines how the probability of reaching the correct answer changes as problem size increases.
- Independent and iterated applications: For n independent function applications, independent estimator errors make the incorrect-prediction probability increase exponentially with n.Proposition 4.1 assumes a perfect estimator for the combining function and a bounded collision rate.
- Independent and iterated applications: When the collision rate satisfies c_n ≤ βα^n, independent-composition error probability tends exponentially to 1 as n increases.The result formalizes rapid failure as the number of required independent applications grows.
- Independent and iterated applications: For repeated function applications, the incorrect-prediction probability also increases exponentially, approaching at least 1 − c/(c + ϵ).Here ϵ is the estimator’s error probability and c bounds recovery by chance from an incorrect input.
- Assumptions and scope: The repeated-application analysis uses a single-valid-reasoning assumption and notes a possible state-transition generalization with error probability ϵ and recovery probability at most c.The propositions are framed for high-performing estimators, while experiments focus on out-of-the-box transformers.
- Application to evaluated tasks: The propositions apply directly to multiplication, dynamic programming, and the puzzle because their computations decompose into the analyzed function applications.The tasks are also characterized as low-collision settings.
5 Discussion
The discussion argues that transformers can appear compositional while relying on analogical pattern matching rather than robust multi-step reasoning. It connects this empirical account with theoretical limits and recommends shorter compositions, lenient evaluation, and planning augmentation.
- Collapsed compositionality: Transformers can collapse multi-step compositional operations through analogical pattern matching when training data contains suitable input-output patterns.This shortcut can produce acceptable solutions but poor generalization beyond observed complexity.
- Theoretical implications: Theoretical proofs show that, under reasonable assumptions, incorrect-prediction probability converges exponentially to approximately 1 for abstract compositional tasks.The discussion attributes this limitation to greedy next-word prediction and constrained error recovery.
- Practical implications: The paper suggests using transformers for tasks requiring only a few compositional steps or allowing approximate solutions that avoid executing the entire computation graph.It also proposes augmenting transformers with planning modules.
- Limitations and future work: The authors could not push empirical limits further because of compute constraints and limited access to the largest language models, including GPT4.They invite broader investigation with more training data, epochs, and resources.
6 Related Work
Related work examines transformer reasoning, compositional challenges, generalization, grokking, and theoretical expressiveness. The paper distinguishes its focus on fundamental limits and full mastery from work primarily targeting improved performance or expressiveness bounds.
- Reasoning abilities in transformer LLMs: Transformers show strong reasoning performance across many tasks, but related research also documents difficulties across domains and in algorithmic reasoning.Scaling larger models and datasets is associated with improved performance.
- Challenges in compositional tasks: Prior compositionality work uses implicit-answer fine-tuning, explicit reasoning generation, and training-example selection to improve performance and generalization.These approaches generally seek stronger performance without requiring complete mastery.
- Challenges in generalization: Transformer generalization research covers easy-to-hard transfer, length generalization, and symbolic mathematical integration.The cited studies investigate multiple facets of generalization.
- Grokking: Grokking research studies delayed generalization and structured representations in arithmetic and sparse parity tasks.Subsequent work characterizes when and why grokking arises.
- Theoretical expressiveness: Expressiveness research studies what transformers can represent under perfect parameters, whereas all tasks here belong to an expressible class.This suggests known expressiveness upper bounds may not be tight for the studied setting.
7 Conclusions
The paper argues that understanding transformer failures is increasingly important as these models are widely deployed. Its compositionality analysis supports pursuing robust generalization and systematic problem-solving for more reliable AI systems.
- Conclusions: The study emphasizes developing models that generalize robustly and solve compositional tasks systematically, beyond settings with abundant training examples.The stated goal is more reliable AI systems capable of precise compositional reasoning.
8 Limitations
The computation-graph analysis cannot establish which tokens a model attends to when generating scratchpad predictions. Alternative reasoning linearizations may therefore produce different performance.
- Scratchpad analysis establishes correlation with preceding context, not the exact tokens attended to during prediction.The limitation follows from lacking access to the studied models’ activations.
- Alternative approaches to linearizing reasoning processes may yield different performances and warrant further exploration.
Appendices
The appendices include a subsection on empirical surface-pattern analysis for multiplication with GPT4, ChatGPT, and GPT3.
- The appendices contain an empirical surface-pattern analysis of multiplication for GPT4, ChatGPT, and GPT3.
A Compositional Tasks
The paper constructs three compositional tasks and represents their solution procedures as computation graphs, with multiplication, logic puzzles, and dynamic programming each given explicit generation and solving procedures.
- Multiplication: Multiplication data contain two numbers with up to five digits, and prompts use either question-answer pairs or scratchpads.The dataset exhaustively generates 9 × 10^(k−1) combinations per number with k set to 5; the depicted scratchpad performed best among tested variants.
- Logic grid puzzles: Logic-grid puzzles represent solutions as a K × M matrix mapping houses to attributes and are generated from randomly sampled and permuted property values.The construction ensures one sampled property is Name to improve readability and coherence.
- Logic grid puzzles: A greedy puzzle solver repeatedly applies the minimum number of rules needed to solve cells, iterating through clues and clue combinations.
- Dynamic programming: The dynamic-programming task computes maximum sums of nonadjacent subsequences using base cases and a recurrence over neighboring dynamic-programming values.The recurrence compares dpi+1 with ai + dpi+2, while allowing the empty sequence through a nonnegative lower bound.
- Dynamic programming: The DP solution reconstructs a lexicographically smallest maximizing subsequence from computed dp values while preventing adjacent selections.
- Experimental setup: Experiments evaluate six LLMs with zero-shot, few-shot, and GPT3 fine-tuning, comparing question-answer and question-scratchpad formats.The study reports faster GPT3 convergence with question-scratchpad pairs and measures performance as task complexity increases.
B.3 Limits of Transformers with question-answer Training
Question-answer fine-tuning produces near-perfect performance on in-domain examples but poor out-of-domain generalization, while scratchpad prompting improves many few-shot results before performance declines with complexity.
- Question-answer training: GPT3 fine-tuned on question-answer pairs performs near-perfectly in-domain but lacks generalization across problem-size, depth, and width splits.
- Question-answer training: Question-answer fine-tuning on multiplication spans 1-digit by 1-digit through 4-digit by 2-digit examples, totaling 1.8M pairs.
- Question-answer training: Question-answer fine-tuning fails on out-of-domain puzzle sizes despite training on puzzles up to 4×4.
- Question-scratchpad training: Question-scratchpad fine-tuning also achieves perfect in-domain accuracy but fails to generalize to wider or deeper computation graphs.
- Few-shot prompting: GPT4 question-scratchpad few-shot accuracy improves for most problem sizes across the three tasks but declines toward zero as complexity increases.
- Surface-pattern analysis: Partial-response analyses report surface-pattern accuracy exceeding full-answer accuracy, including relatively easy prediction of multiplication’s trailing zeros.
C.3 Relative Information Gain Predictions for Dynamic Programming Task
Relative Information Gain predicts that DP outputs are easiest to infer at boundary positions and from nearby input elements, patterns largely matched by the analyzed models.
- Single-element predictions: a1 and an−1 are especially informative predictors of DP output elements, consistent with boundary-sensitive task structure.The analysis considers DP problems with sizes 2 ≤ n ≤ 10; an−1 denotes the last sequence element regardless of its index.
- Single-element predictions: The first input element weakly helps predict its neighbor, with RelativeIG>0.1 only for a1 predicting the second output-related value.A high a1 indicates that o2 is unlikely to be selected in the final subsequence.
- Model correspondence: All analyzed models predict o1 most accurately, while GPT4 also places on−1 among its easiest predictions.GPT3 analyses were limited for on−1 because the model often generated output sequences shorter than required.
D.1 Error accumulates with larger parallel applications of an estimated function (width)
Theoretical analysis shows that repeated applications of an estimated function can make small local errors accumulate, driving final-task errors toward one as compositional width grows. The framework applies to multiplication, dynamic programming, and puzzle elimination tasks under stated assumptions.
- Formal framework: The analysis models multi-step reasoning through repeated applications of a base function and tracks how estimation errors accumulate.It considers independent repeated applications and uses an error-estimation and accumulation perspective.
- Formal framework: P(fn ≠ bfn) approaches 1 when each component has error rate ϵ > 0, the combining function is nearly injective, and component errors are independent.The result follows from the exponential decay of P(X = Y) = (1 − ϵ)^n under the stated assumptions.
- Application to multiplication: Long-form multiplication therefore becomes unreliable as the number of multiplied digits grows, even when shifted addition is solved perfectly.The corollary states that solving fixed m-digit by 1-digit multiplication imperfectly causes m-digit by n-digit multiplication accuracy to tend toward zero as n increases.
- Assumptions and implications: The bounds depend on assumptions about recovery from incorrect intermediate states, and empirical estimates require approximating parameters such as ϵ and α.The multiplication proof identifies selection and arithmetic errors as separate possible sources of exponential accumulation.
- Application to representative tasks: The same proposition applies directly to dynamic-programming recursions, m-by-1 digit multiplication, and puzzle elimination because each repeatedly applies a fixed reasoning step.These task structures satisfy the repeated-function formulation used in the theoretical analysis.