Paper Citation Record · LEDGER

Learning Compositional Functions with Transformers from Easy-to-Hard Data

As of 14 August 2026, this Paper Citation Record lists 71 of 71 outbound references and 11 inbound Pith citation observations for arXiv:2505.23683.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2505.23683 v1

Coverage vector

measured 71 of 71 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-07T12:46:40.614748Z

measured 82 of 82 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-14T06:32:32.682623+00:00

measured 11 of 11 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-06T21:34:15.603859Z

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: arxiv_reference, observed 2026-06-29T14:33:30.598717Z

Reference resolution

71 of 71 outbound references displayed

verified exact3
verified fuzzy31
unresolved37
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation c791e2bc-b13d-47aa-97a8-afcc6650bf6b · outbound

This paper cites The staircase property: How hierarchical structure can guide deep learning.Advances in Neural Information Processing Systems, 34:26989–27002, 2021.

Learning Compositional Functions with Transformers from Easy-to-Hard Data The staircase property: How hierarchical structure can guide deep learning.Advances in Neural Information Processing Systems, 34:26989–27002, 2021

Reference 1

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:32.494326Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation ddd8ead5-d8c7-406c-80cd-4067f3f7b8da · outbound

This paper cites The merged-staircase property: a necessary and nearly sufficient condition for sgd learning of sparse functions on two-layer neural networks.

Learning Compositional Functions with Transformers from Easy-to-Hard Data The merged-staircase property: a necessary and nearly sufficient condition for sgd learning of sparse functions on two-layer neural networks

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:48.442515Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation a03d8f2f-fec3-480a-9a4c-524dcb10114c · outbound

This paper cites Provable advantage of curriculum learning on parity targets with mixed inputs.Advances in Neural Information Processing Systems, 36:24291–24321, 2023.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Provable advantage of curriculum learning on parity targets with mixed inputs.Advances in Neural Information Processing Systems, 36:24291–24321, 2023

Reference 3

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:32.708882Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 831013f1-6797-4024-ab3a-b34907bde771 · outbound

This paper cites How far can transformers reason? The locality barrier and inductive scratchpad.

Learning Compositional Functions with Transformers from Easy-to-Hard Data How far can transformers reason? The locality barrier and inductive scratchpad

Reference 4

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:48.275011Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 773f1920-9aa9-4517-95ed-c7f90f9bde19 · outbound

This paper cites Transformers learn to implement preconditioned gradient descent for in-context learning.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Transformers learn to implement preconditioned gradient descent for in-context learning

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:48.160610Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation c86c7eff-538c-4a24-8d82-21a2ad229fac · outbound

This paper cites Graph streaming lower bounds for parameter estimation and property testing via a streaming xor lemma.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Graph streaming lower bounds for parameter estimation and property testing via a streaming xor lemma

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:48.033495Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 4ab05c7b-4394-466c-b552-cbf39e6178de · outbound

This paper cites The pitfalls of next-token prediction.

Learning Compositional Functions with Transformers from Easy-to-Hard Data The pitfalls of next-token prediction

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:47.870785Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation b6ed973d-856d-422d-89bd-59af7f89cea5 · outbound

This paper cites Curriculum learning.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Curriculum learning

Reference 8

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:33.334050Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 36d3242c-1566-4cde-b737-5d22627eb4f5 · outbound

This paper cites Separations in the Representational Capabilities of Transformers and Recurrent Architectures.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Separations in the Representational Capabilities of Transformers and Recurrent Architectures

Reference 9

Resolution: verified exact; local_arxiv, observed 2026-08-07T12:46:42.789421Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation ccb8cf36-5b8b-4f33-a4be-6228c304285c · outbound

This paper cites Birth of a transformer: A memory viewpoint.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Birth of a transformer: A memory viewpoint

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:47.760483Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation c834b064-facb-4bf2-9fb7-c7c5926c5ae1 · outbound

This paper cites Data distributional properties drive emergent in-context learning in transformers.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Data distributional properties drive emergent in-context learning in transformers

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:47.608752Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 23b29014-a63e-4589-8fc7-e3563d33c822 · outbound

This paper cites Theoretical limitations of multi-layer Transformer.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Theoretical limitations of multi-layer Transformer

Reference 12

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:33.978754Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a597f3a7-8913-43cb-bfa4-167279020fab · outbound

This paper cites Training Dynamics of Multi-Head Softmax Attention for In-Context Learning: Emergence, Convergence, and Optimality.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Training Dynamics of Multi-Head Softmax Attention for In-Context Learning: Emergence, Convergence, and Optimality

Reference 13

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:34.110434Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b134bc0c-7e37-4fb7-824d-df2c5462488e · outbound

This paper cites Unveiling Induction Heads: Provable Training Dynamics and Feature Learning in Transformers.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Unveiling Induction Heads: Provable Training Dynamics and Feature Learning in Transformers

Reference 14

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:34.250512Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3ed040cd-8986-4385-b60e-f88d2bf4c500 · outbound

This paper cites Understanding the Interplay between Parametric and Contextual Knowledge for Large Language Models.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Understanding the Interplay between Parametric and Contextual Knowledge for Large Language Models

Reference 15

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:34.394535Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 8bde2b42-80a5-4895-a919-a11e66fefed9 · outbound

This paper cites Neural networks can learn represen- tations with gradient descent.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Neural networks can learn represen- tations with gradient descent

Reference 16

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:34.489754Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b5054e96-626f-42fc-be1a-26f48263814f · outbound

This paper cites From Explicit CoT to Implicit CoT: Learning to Internalize CoT Step by Step.

Learning Compositional Functions with Transformers from Easy-to-Hard Data From Explicit CoT to Implicit CoT: Learning to Internalize CoT Step by Step

Reference 17

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:34.662611Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9bfa3d55-b086-4846-a561-d6cee1fbf8ce · outbound

This paper cites The Llama 3 Herd of Models.

Learning Compositional Functions with Transformers from Easy-to-Hard Data The Llama 3 Herd of Models

Reference 18

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:34.805557Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation c841ccb3-8b76-48cc-a404-53bbb7898b27 · outbound

This paper cites Faith and Fate: Limits of Transformers on Compositionality.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Faith and Fate: Limits of Transformers on Compositionality

Reference 19

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:34.917688Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation e4bc1d60-d879-4948-895a-d503ea825cc7 · outbound

This paper cites Learning and development in neural networks: The importance of starting small.Cognition, 48(1):71–99, 1993.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Learning and development in neural networks: The importance of starting small.Cognition, 48(1):71–99, 1993

Reference 20

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:35.061289Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b543bd7a-7bda-4750-b189-27547524ec73 · outbound

This paper cites Towards revealing the mystery behind chain of thought: a theoretical perspective.Advances in Neural Information Processing Systems, 36, 2024.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Towards revealing the mystery behind chain of thought: a theoretical perspective.Advances in Neural Information Processing Systems, 36, 2024

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:47.382613Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 2ef8ca9c-4295-4c58-b682-80c33d7bab6e · outbound

This paper cites Global Convergence in Training Large-Scale Transformers.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Global Convergence in Training Large-Scale Transformers

Reference 22

Resolution: verified exact; local_arxiv, observed 2026-08-07T12:46:42.265485Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation d40ad934-5101-49d1-ae2f-38f0ad56451f · outbound

This paper cites Better & Faster Large Language Models via Multi-token Prediction.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Better & Faster Large Language Models via Multi-token Prediction

Reference 23

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:35.454047Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 732e7025-32e6-4375-a0f9-6c2d23667c6c · outbound

This paper cites Interleaved group products.SIAM Journal on Computing, 48(2):554–580, 2019.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Interleaved group products.SIAM Journal on Computing, 48(2):554–580, 2019

Reference 24

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:47.203328Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 9fb45c7f-1a84-423b-a32d-f6bf87f6f129 · outbound

This paper cites Transformers Learn to Implement Multi-step Gradient Descent with Chain of Thought.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Transformers Learn to Implement Multi-step Gradient Descent with Chain of Thought

Reference 25

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:35.648813Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 7a1ba3e9-0933-4e24-b9c2-4b1e1948cc63 · outbound

This paper cites In-Context Convergence of Transformers.

Learning Compositional Functions with Transformers from Easy-to-Hard Data In-Context Convergence of Transformers

Reference 26

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:35.756325Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation dcb56841-e383-407a-af90-d80388e83cbe · outbound

This paper cites Mas- sively parallel computation: Algorithms and applications.Foundations and Trends® in Optimization, 5(4):340–417, 2023.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Mas- sively parallel computation: Algorithms and applications.Foundations and Trends® in Optimization, 5(4):340–417, 2023

Reference 27

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:47.058801Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation ccca6d86-49a0-4459-8eea-37c03d64ca10 · outbound

This paper cites Vision transformers provably learn spatial structure.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Vision transformers provably learn spatial structure

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:46.916436Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 7d3ab2f4-6822-4c76-afe7-47e5270748c0 · outbound

This paper cites Repeat After Me: Transformers are Better than State Space Models at Copying.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Repeat After Me: Transformers are Better than State Space Models at Copying

Reference 29

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:36.145807Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 99bacc77-02fd-418d-9507-034ef2093aaf · outbound

This paper cites Efficient noise-tolerant learning from statistical queries.J.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Efficient noise-tolerant learning from statistical queries.J

Reference 30

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:46.776875Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 2322fda7-1f28-4a84-bb51-4dec2ba0ffc8 · outbound

This paper cites Transformers Provably Solve Parity Efficiently with Chain of Thought.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Transformers Provably Solve Parity Efficiently with Chain of Thought

Reference 31

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:36.416475Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation dd18b5f5-62d5-4173-bc4b-3b016e719ba7 · outbound

This paper cites Adam: A Method for Stochastic Optimization.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Adam: A Method for Stochastic Optimization

Reference 32

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:36.536688Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation be79f8cf-1577-4a53-8f21-a77e162349bc · outbound

This paper cites Learning to reason and memorize with self-notes.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Learning to reason and memorize with self-notes

Reference 33

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:46.652195Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 070b5a5f-cc81-49cf-bbce-bb5df2caf729 · outbound

This paper cites Solving quantitative reasoning problems with language models.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Solving quantitative reasoning problems with language models

Reference 34

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:46.509475Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation a1257c2b-f6eb-4450-91f6-91492b32782b · outbound

This paper cites How do transformers learn topic structure: Towards a mechanistic understanding.

Learning Compositional Functions with Transformers from Easy-to-Hard Data How do transformers learn topic structure: Towards a mechanistic understanding

Reference 35

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:46.366060Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation a5719318-da93-4fef-a358-989d748c5d6b · outbound

This paper cites Chain of Thought Empowers Transformers to Solve Inherently Serial Problems.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Chain of Thought Empowers Transformers to Solve Inherently Serial Problems

Reference 36

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:37.044242Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 4d5d60aa-d7d8-4822-99a7-c91d5121ced5 · outbound

This paper cites Let's Verify Step by Step.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Let's Verify Step by Step

Reference 37

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:37.158871Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation e23e46eb-224b-41a4-8744-80dcfddca948 · outbound

This paper cites DeepSeek-V3 Technical Report.

Learning Compositional Functions with Transformers from Easy-to-Hard Data DeepSeek-V3 Technical Report

Reference 38

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:37.256187Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation d5258745-2be9-4dfb-b9d2-4182af1f2f97 · outbound

This paper cites Ash, Surbhi Goel, Akshay Krishnamurthy, and Cyril Zhang.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Ash, Surbhi Goel, Akshay Krishnamurthy, and Cyril Zhang

Reference 39

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:46.210499Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation dedc056e-1367-4bac-a297-f4259dc3a96f · outbound

This paper cites RegMix: Data Mixture as Regression for Language Model Pre-training.

Learning Compositional Functions with Transformers from Easy-to-Hard Data RegMix: Data Mixture as Regression for Language Model Pre-training

Reference 40

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:37.549144Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a805c674-e3e4-496c-b700-826fa40ccb80 · outbound

This paper cites The Expressive Power of Transformers with Chain of Thought.

Learning Compositional Functions with Transformers from Easy-to-Hard Data The Expressive Power of Transformers with Chain of Thought

Reference 41

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:37.657775Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation ba40d699-5080-47eb-b246-71d1a32e7455 · outbound

This paper cites The parallelism tradeoff: Limitations of log-precision transformers.Transactions of the Association for Computational Linguistics, 11:531–545, 2023.

Learning Compositional Functions with Transformers from Easy-to-Hard Data The parallelism tradeoff: Limitations of log-precision transformers.Transactions of the Association for Computational Linguistics, 11:531–545, 2023

Reference 42

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:37.789402Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 5255c7ec-42a6-43ae-b389-7051503a870d · outbound

This paper cites How transformers learn causal structure with gradient descent.

Learning Compositional Functions with Transformers from Easy-to-Hard Data How transformers learn causal structure with gradient descent

Reference 43

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:45.972428Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation d36466a5-8576-412c-b84a-5474d5d7895d · outbound

This paper cites Understanding Factual Recall in Transformers via Associative Memories.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Understanding Factual Recall in Transformers via Associative Memories

Reference 44

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:38.010378Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 367d7a22-b668-4a37-b725-678b2bc061ab · outbound

This paper cites Rounds in communication complexity revisited.SIAM Journal on Computing, 22(1):211–219, 1993.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Rounds in communication complexity revisited.SIAM Journal on Computing, 22(1):211–219, 1993

Reference 45

Resolution: verified exact; doi, observed 2026-08-07T12:46:40.801870Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 45205473-97b1-49c4-b5fb-f8495ed7503f · outbound

This paper cites Show Your Work: Scratchpads for Intermediate Computation with Language Models.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Show Your Work: Scratchpads for Intermediate Computation with Language Models

Reference 46

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:38.260594Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 53ea21a2-0bc5-4f93-9ba7-922355824d0e · outbound

This paper cites In-context learning and induction heads.Transformer Cir- cuits Thread, 2022.

Learning Compositional Functions with Transformers from Easy-to-Hard Data In-context learning and induction heads.Transformer Cir- cuits Thread, 2022

Reference 47

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:45.790112Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation ffc1f16e-3425-486b-bab0-270f6c72b8ff · outbound

This paper cites Progressive distillation induces an implicit curriculum.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Progressive distillation induces an implicit curriculum

Reference 48

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:38.497645Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation de5a35ec-1470-4b72-9c7b-c858563207da · outbound

This paper cites Papadimitriou and Michael Sipser.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Papadimitriou and Michael Sipser

Reference 49

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:38.611842Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation e56752ed-55ec-4177-885f-087b1798635a · outbound

This paper cites On limitations of the trans- former architecture.

Learning Compositional Functions with Transformers from Easy-to-Hard Data On limitations of the trans- former architecture

Reference 50

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:45.565431Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 8f8dcf86-ec50-4dfd-8d3c-c5265be5cb83 · outbound

This paper cites Learning and transferring sparse contextual bi- grams with linear transformers.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Learning and transferring sparse contextual bi- grams with linear transformers

Reference 51

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:45.433429Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation a0aaa9a3-4f95-4cec-b64e-24a101960671 · outbound

This paper cites Representational strengths and limitations of transformers.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Representational strengths and limitations of transformers

Reference 52

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:45.235854Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation f8097cbb-e197-4a3b-b5ea-cb4eb9dceede · outbound

This paper cites Understanding Transformer Reasoning Capabilities via Graph Algorithms.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Understanding Transformer Reasoning Capabilities via Graph Algorithms

Reference 53

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:39.144554Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation e1f0a159-eac9-4ca1-8d46-d41662a12b6f · outbound

This paper cites Transformers, parallel computation, and logarithmic depth.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Transformers, parallel computation, and logarithmic depth

Reference 54

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:45.078056Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 9792657b-2058-465f-9fe8-172cdcf3327b · outbound

This paper cites End-to-end memory networks.

Learning Compositional Functions with Transformers from Easy-to-Hard Data End-to-end memory networks

Reference 55

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:44.894174Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 19db7bda-130f-4046-83f8-0764f09a3f47 · outbound

This paper cites Characterizing statistical query learning: simplified notions and proofs.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Characterizing statistical query learning: simplified notions and proofs

Reference 56

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:44.758978Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 36aa5752-892f-40e3-ba6c-081c9d010c5c · outbound

This paper cites Scan and snap: Understanding training dynamics and token composition in 1-layer transformer.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Scan and snap: Understanding training dynamics and token composition in 1-layer transformer

Reference 57

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:44.593968Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 22e35c1f-1df0-4579-a963-34b49678ef39 · outbound

This paper cites Solving math word problems with process- and outcome-based feedback.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Solving math word problems with process- and outcome-based feedback

Reference 58

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:39.598022Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 2259d854-e282-4a79-aa47-a6c1ad55e9a9 · outbound

This paper cites an unresolved cited work.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Unresolved cited work

Reference 59

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:46:44.399090Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 0da4c9a8-52a1-46f5-aef6-94ee1512ebdf · outbound

This paper cites Chain-of-thought prompting elicits reasoning in large language models.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Chain-of-thought prompting elicits reasoning in large language models

Reference 60

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:44.220213Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 4c59a0ee-c65c-4fe3-8215-7a17f3099a43 · outbound

This paper cites From Sparse Dependence to Sparse Attention: Unveiling How Chain-of-Thought Enhances Transformer Sample Efficiency.

Learning Compositional Functions with Transformers from Easy-to-Hard Data From Sparse Dependence to Sparse Attention: Unveiling How Chain-of-Thought Enhances Transformer Sample Efficiency

Reference 61

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:39.840714Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b7d6e0cb-cbc7-4f78-99c0-39d9167d5ef4 · outbound

This paper cites Memory Networks.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Memory Networks

Reference 62

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:39.905563Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 02cb2c74-8a26-46ff-a8aa-d8384dd7a2a4 · outbound

This paper cites Towards AI-Complete Question Answering: A Set of Prerequisite Toy Tasks.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Towards AI-Complete Question Answering: A Set of Prerequisite Toy Tasks

Reference 63

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:39.984911Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation d0bcd8b8-c830-42e1-9a0c-ce3c25732862 · outbound

This paper cites Doremi: Optimizing data mixtures speeds up language model pretraining.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Doremi: Optimizing data mixtures speeds up language model pretraining

Reference 64

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:44.044810Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 2f6f0c7b-8d67-432a-b51b-fb187d7f4894 · outbound

This paper cites Do Large Language Models Latently Perform Multi-Hop Reasoning?.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Do Large Language Models Latently Perform Multi-Hop Reasoning?

Reference 65

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:40.093818Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 6a3e5602-bead-4bd9-99ca-23dab1739c35 · outbound

This paper cites Tree of thoughts: deliberate problem solving with large language models.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Tree of thoughts: deliberate problem solving with large language models

Reference 66

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:43.858525Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 3c0f99ec-21a5-482c-b5d4-71d77c1f7c13 · outbound

This paper cites Pointer chasing via triangular discrimination.Combinatorics, Probability and Computing, 29(4):485–494, 2020.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Pointer chasing via triangular discrimination.Combinatorics, Probability and Computing, 29(4):485–494, 2020

Reference 67

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:43.665821Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation b2fc7f14-4432-4a2a-a85f-c240aa9b09e9 · outbound

This paper cites Trained Transformers Learn Linear Models In-Context.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Trained Transformers Learn Linear Models In-Context

Reference 68

Resolution: unresolved; no resolver link, observed 2026-08-07T12:46:40.385953Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 48c3ed10-53dd-4f8b-98cf-3c616de2aa69 · outbound

This paper cites an unresolved cited work.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Unresolved cited work

Reference 69

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:46:43.471531Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 6efa7547-edbe-489e-869f-cbc4f2d7184a · outbound

This paper cites Since the softmax in layer ℓ is nearly saturated and thus close to one-hot, the update norm is very small and can be bounded as noise terms.

Learning Compositional Functions with Transformers from Easy-to-Hard Data Since the softmax in layer ℓ is nearly saturated and thus close to one-hot, the update norm is very small and can be bounded as noise terms

Reference 70

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:43.368846Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 192e0eaa-499b-4984-bd8d-e32f3cb90dc2 · outbound

This paper cites That means in the tth gradient step, there will be some small gradient updates for later layersℓ′ ≥t introduced by the perturbation.

Learning Compositional Functions with Transformers from Easy-to-Hard Data That means in the tth gradient step, there will be some small gradient updates for later layersℓ′ ≥t introduced by the perturbation

Reference 71

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:46:43.194198Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Pith citing papers

Observation 1e43c969-1d4f-439d-8343-a97345183408 · inbound

Beyond Statistical Learning: Exact Learning Is Essential for General Intelligence cites this paper.

Beyond Statistical Learning: Exact Learning Is Essential for General Intelligence Learning Compositional Functions with Transformers from Easy-to-Hard Data

Reference 26

Resolution: unresolved; no resolver link, observed 2026-08-06T21:34:15.603859Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 67828437-2bbd-46e5-b46c-cd501bce0c1b · inbound

Scaling Latent Reasoning via Looped Language Models cites this paper.

Scaling Latent Reasoning via Looped Language Models Learning Compositional Functions with Transformers from Easy-to-Hard Data

Reference 81

Resolution: metadata mismatch; arxiv_id, observed 2026-05-15T07:43:11.763654Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation feed43f0-78ec-4421-a44f-ce39f10e4457 · inbound

Scaling Latent Reasoning via Looped Language Models cites this paper.

Scaling Latent Reasoning via Looped Language Models Learning Compositional Functions with Transformers from Easy-to-Hard Data

Reference 81

Resolution: unresolved; no resolver link, observed 2026-08-04T07:31:49.270195Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 7ff28ce0-6fcb-4a8c-accc-578779f06471 · inbound

Deep sequence models tend to memorize geometrically; it is unclear why cites this paper.

Deep sequence models tend to memorize geometrically; it is unclear why Learning Compositional Functions with Transformers from Easy-to-Hard Data

Reference 186

Resolution: verified exact; arxiv_id, observed 2026-05-21T20:40:36.310219Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 0b1e143f-b234-4151-8cd5-0c79778c1c00 · inbound

Transformers with RL or SFT Provably Learn Sparse Boolean Functions, But Differently cites this paper.

Transformers with RL or SFT Provably Learn Sparse Boolean Functions, But Differently Learning Compositional Functions with Transformers from Easy-to-Hard Data

Reference 29

Resolution: unresolved; no resolver link, observed 2026-08-03T20:57:12.841539Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3340179c-4d68-4d1f-8b84-d172d3085ec2 · inbound

Breaking the Reversal Curse in Autoregressive Language Models via Identity Bridge cites this paper.

Breaking the Reversal Curse in Autoregressive Language Models via Identity Bridge Learning Compositional Functions with Transformers from Easy-to-Hard Data

Reference 18

Resolution: unresolved; no resolver link, observed 2026-08-03T05:27:44.481599Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation c1fafadd-a6b4-4052-9480-75040c60f282 · inbound

Sharp Capacity Scaling of Spectral Optimizers in Learning Associative Memory cites this paper.

Sharp Capacity Scaling of Spectral Optimizers in Learning Associative Memory Learning Compositional Functions with Transformers from Easy-to-Hard Data

Reference 58

Resolution: metadata mismatch; arxiv_id, observed 2026-05-14T23:38:16.390943Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 2e0e0277-07c5-4180-b57c-d9ec1f0787ec · inbound

The Power of Power Law: Asymmetry Enables Compositional Reasoning cites this paper.

The Power of Power Law: Asymmetry Enables Compositional Reasoning Learning Compositional Functions with Transformers from Easy-to-Hard Data

Reference 53

Resolution: verified exact; arxiv_id, observed 2026-05-11T19:31:08.436718Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 26962014-0ff3-463c-a610-90aef65b491b · inbound

The Power of Power Law: Asymmetry Enables Compositional Reasoning cites this paper.

The Power of Power Law: Asymmetry Enables Compositional Reasoning Learning Compositional Functions with Transformers from Easy-to-Hard Data

Reference 53

Resolution: unresolved; no resolver link, observed 2026-07-12T18:26:05.728364Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 7a4a587c-aaa8-4cc0-8415-c2fd22c76ffe · inbound

Transformers Efficiently Perform In-Context Logistic Regression via Normalized Gradient Descent cites this paper.

Transformers Efficiently Perform In-Context Logistic Regression via Normalized Gradient Descent Learning Compositional Functions with Transformers from Easy-to-Hard Data

Reference 50

Resolution: verified exact; arxiv_id, observed 2026-05-11T19:16:09.597626Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 3836b10a-2e80-4a5b-9cc5-52054adc3b64 · inbound

Transformers Provably Learn to Internalize Chain-of-Thought cites this paper.

Transformers Provably Learn to Internalize Chain-of-Thought Learning Compositional Functions with Transformers from Easy-to-Hard Data

Reference 47

Resolution: metadata mismatch; arxiv_id, observed 2026-06-29T14:33:30.600195Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.