SlowGuess/NeurIPS_2023_OCR huggingface dataset:
"output": "Yes, it is possible to reach the goal of having just one black square on an  $8 \times 8$  chessboard. One way to do this is to repaint all of the squares in the first row and first column black, and then repaint the top-left square of the  $2 \times 2$  square white. This results in one black square and all other squares white."
G.5 Symbolic-Integration
Examples from our Symbolic-Integration subdataset based on [13].
"prompt": "What is the integral of  $4x^{7} + \sin(1 + x)$ ", "output": "The integral of  $4x^{7}$  is  $(4/8)x^{8} + C$ , where  $C$  is the constant of integration. The integral of  $\sin(1 + x)$  is  $-\cos(1 + x) + C$ . So the integral of  $4x^{7} + \sin(1 + x)$  is  $(4/8)x^{8} - \cos(1 + x) + C$ ."
"prompt": "What is the integral of  $2048 + 32x^3$ ", "output": "The antiderivative or indefinite integral of  $2048 + 32x^3$  with respect to  $x$  is  $\int (2048 + 32x^3)dx = 2048x + (32/4)x^4 + C$  where  $C$  is the constant of integration."
"prompt": "What is the integral of  $12 + 6\cosh(x)$ ", "output": "The integral of  $12 + 6\cosh(x)$  is  $12x + 6\sinh(x) + C$ , where  $C$  is the constant of integration."
"prompt": "What is the integral of  $(4 + x)[4 + 3x][e^{x*(4 + x)^2}]$ ", "output": "The integral of  $(4 + x)[4 + 3x][e^{x*(4 + x)^2}]$  is not expressible in terms of elementary functions."
"prompt": "What is the integral of  $3x - 4x^{2}[\cos (x + 3)]$ ", "output": "The antiderivative of  $3x - 4x^{2}isx^{2} - (4 / 3)x^{3}$  and the antiderivative of  $[\cos (x + 3)]$  is  $-\sin (x + 3)$ . Therefore, the integral of  $3x - 4x^{2}[\cos (x + 3)]$  is  $x^{2} - (4 / 3)x^{3} - \sin (x + 3) + C$ , where  $C$  is the constant of integration."
"prompt": "What is the integral of  $1 / (2 + \ln(3/2))$ ",  
"output": "The integral of  $1 / (2 + \ln(3/2))$  cannot be expressed in terms of elementary functions and must be approximated numerically or solved using special functions."
H Datasheet for the GHOSTS Dataset
This appendix provides a datasheet for the GHOSTS dataset. The format of this datasheet was introduced in [64] and consolidates the motivation, creation process, composition, and intended uses of our dataset as a series of questions and answers.
H.1 Motivation
Q1. For what purpose was the dataset created? Was there a specific task in mind? Was there a specific gap that needed to be filled? Please provide a description.
The existing datasets of natural-language mathematics are far from covering all the typical tasks professional mathematicians encounter in daily life, making it unclear whether language models can be of any help in this regard. Existing datasets mostly cover elementary mathematics or resemble standard tests like SATs (see Sections 2 and 3). Hence, they do not offer any insight into the usage of ChatGPT as a tool for mathematicians. In this work, we have made the first attempt towards filling this gap, going beyond math problems that are yes-no rated, and proposed a benchmark made and curated by working researchers in the field that tests different dimensions of mathematical reasoning.
Q2. Who created this dataset (e.g., which team, research group) and on behalf of which entity (e.g., company, institution, organization)?
The authors of this work created GHOSTS; see Appendix B.6 for more information.
Q3. Who funded the creation of the dataset? If there is an associated grant, please provide the name of the grantor and the grant name and number.
There is no associated grant or funding which has been used to create the GHOSTS dataset.
Q4. Any other comments?
H.2 Composition
Q5. What do the instances that comprise the dataset represent (e.g., documents, photos, people, countries)? Are there multiple types of instances (e.g., movies, users, and ratings; people and interactions between them; nodes and edges)? Please provide a description.
GHOSTS consists of textual prompts, in natural language, representing mathematical questions. For each prompt, GHOSTS contains one or more instances of outputs of (Chat)GPT and corresponding fine-grained evaluation by the authors.
Q6. How many instances are there in total (of each type, if appropriate)?
There are 709 prompts in GHOSTS; a selection of 170 of these makes up miniGHOSTS, and a selection of 14 of those makes up microGHOSTS. For 24 of the questions from the GHOSTS dataset, light prompt engineering variations have been carried out. Each of the  $709 + 24$  questions from GHOSTS has been evaluated on ChatGPT, version 9-January-2023 and 30-January-2023, and 170 questions from miniGHOSTS have been evaluated on GPT-4 (and thus on microGHOSTS too). Thus, in total,  $(709 + 24) \times 2 + 170 = 1636$  outputs and evaluations have been carried out. See also Appendix B.6 for more information.
Q7. Does the dataset contain all possible instances or is it a sample (not necessarily random) of instances from a larger set? If the dataset is a sample, then what is the larger set? Is the sample representative of the larger set (e.g., geographic coverage)? If so, please describe how this representativeness was validated/verified. If it is not representative of the larger set, please describe why not (e.g., to cover a more diverse range of instances because instances were withheld or unavailable).
GHOSTS tries to cover a wide range of mathematical questions from 78 different MSC codes; see Appendix B.1 and B.2. However, due to the prohibitive cost of human evaluation, which cannot be fully automated away (see Section 3.3), it is not feasible to
represent all mathematical fields across all dimensions of "mathematical behavior" and all types of mathematical questions (overview questions, fact-stating questions, etc.).
Q8. What data does each instance consist of? "Raw" data (e.g., unprocessed text or images) or features? In either case, please provide a description.
GHOSTS, miniGHOSTS, and microGHOSTS consist of a collection of JSON objects (one for each data point), and each JSON object consists of 10 key-values pairs as detailed in Appendix B.2.
Q9. Is there a label or target associated with each instance? If so, please provide a description.
No, we do not explicitly define a label or target for the instances. However, the rating of the output can potentially be used to select good and bad mathematical conversations of (Chat)GPT in order to fine-tune models and the errorcodes and warningcodes can be used to make a more fine-grained classification possible.
Q10. Is any information missing from individual instances? If so, please provide a description, explaining why this information is missing (e.g., because it was unavailable). This does not include intentionally removed information but might include, e.g., redacted text.
Q11. Are relationships between individual instances made explicit (e.g., users' movie ratings, social network links)? If so, please describe how these relationships are made explicit.
Relations between instances are explicitly given by the same values on (subsets) of the fields, e.g., the same prompt, the same model version, or the same MSC code. Prompt-engineered variations of the same question are represented as an array of JSON objects, one object for each variation.
Q12. Are there recommended data splits (e.g., training, development/validation, testing)? If so, please provide a description of these splits, explaining the rationale behind them.
Q13. Are there any errors, sources of noise, or redundancies in the dataset? If so, please provide a description.
The evaluation of the prompts included in GHOSTS underlies human errors. However, we tried to mitigate these errors; see Appendix B.5.
Q14. Is the dataset self-contained, or does it link to or otherwise rely on external resources (e.g., websites, tweets, other datasets)? If it links to or relies on external resources,
(a) Are there guarantees that they will exist, and remain constant, over time?
(b) Are there official archival versions of the complete dataset (i.e., including the external resources as they existed at the time the dataset was created)?
(c) Are there any restrictions (e.g., licenses, fees) associated with any of the external resources that might apply to a future user? Please provide descriptions of all external resources and any restrictions associated with them, as well as links or other access points, as appropriate.
The dataset is self-contained. However, some of the prompts cannot be publicly released since they are taken or adapted from sources that are protected by copyright, for which no license was given, see Appendix B.3; though we do release the output of the models on these prompts, together with our evaluation of the output. Further, we provide a reference to the original, copyrighted materials, so that a user can easily retrieve the original prompts.
Q15. Does the dataset contain data that might be considered confidential (e.g., data that is protected by legal privilege or by doctor-patient confidentiality, data that includes the content of individuals non-public communications)?
If so, please provide a description.
Q16. Does the dataset contain data that, if viewed directly, might be offensive, insulting, threatening, or might otherwise cause anxiety? If so, please describe why.
Q17. Does the dataset relate to people? If not, you may skip remaining questions in this section.
Q18. Does the dataset identify any subpopulations (e.g., by age, gender)? If so, please describe how these subpopulations are identified and provide a description of their respective distributions within the dataset.
Q19. Is it possible to identify one or more natural persons, either directly or indirectly (i.e., in combination with other data) from the dataset? If so, please describe how.
Q20. Does the dataset contain data that might be considered sensitive in any way (e.g., data that reveals racial or ethnic origins, sexual orientations, religious beliefs, political opinions or union memberships, or locations; financial or health data; biometric or genetic data; forms of government identification, such as social security numbers; criminal history)? If so, please provide a description.
Q21. Any other comments?
H.3 Collection Process
Q22. How was the data associated with each instance acquired? Was the data directly observable (e.g., raw text, movie ratings), reported by subjects (e.g., survey responses), or indirectly inferred/derived from other data (e.g., part-of-speech tags, model-based guesses for age or language)? If data was reported by subjects or indirectly inferred/derived from other data, was the data validated/verified? If so, please describe how.
We collected and constructed prompts from various sources, see Table 1 and Section 3. For the evaluation, we captured the corresponding outputs of (Chat)GPT and rated them according to the instructions in Appendix B.2 and B.4.
Q23. What mechanisms or procedures were used to collect the data (e.g., hardware apparatus or sensor, manual human curation, software program, software API)? How were these mechanisms or procedures validated?
To query (Chat)GPT, we used the GUI web interface at the URL chat.openai.com/ chat; see Appendix B.6 for detailed reasons for using the GUI interface.
Q24. If the dataset is a sample from a larger set, what was the sampling strategy?
The prompts of the MATH and Symbolic-Integration subdatasets have been randomly sampled from [12] and [13], across different files from those datasets.
The prompts taken from all the other sources (books and math.stackexchange.com, similarly have been randomly selected.
For our miniGHOSTS dataset, we sampled 10 prompts from each of the 17 files in GHOSTS in the following way: Our results in Section 4 indicate that the 9-January-2023 and the 30-January-2023 ChatGPT versions have similar overall performance; however, the behavior differs on a more fine-grained level and was marginally better for the 30-January-2023 version. Hence, we assembled miniGHOSTS by computing all subsets of 10 prompts having approximately the same mean rating and standard deviation as the original file from GHOSTS, rated on the 30-January-2023 version
of ChatGPT. A manual inspection of these subsets, in order to pick a subset with appropriate mathematical content (we want to have a mathematically diverse dataset), then led to the final selection of the miniGHOSTS dataset. The microGHOSTS dataset was created from manual inspection of the miniGHOSTS dataset, by isolating in total 14 questions which were deemed difficult for language models.
Q25. Who was involved in data collection process (e.g., students, crowd-workers, contractors) and how were they compensated (e.g., how much were crowd-workers paid)?
Only we have been involved in the data collection process. No payment (other than one made through regular employment) in relation to creating this dataset and writing this article was made.
Q26. Over what timeframe was the data collected? Does this timeframe match the creation timeframe of the data associated with the instances (e.g., recent crawl of old news articles)? If not, please provide a description of the timeframe.
The collection date matches the creation time. It is specified in the timestamp key in each data point from GHOSTS and spans a timeframe from January 9, 2023, to now. Using the timestamp, the version of ChatGPT that was used can be inferred, see Appendix C.
Q27. Were any ethical review processes conducted (e.g., by an institutional review board)? If so, please provide a description of these review processes, including the outcomes, as well as a link or other access point to any supporting documentation. Not applicable.
Q28. Does the dataset relate to people? If not, you may skip remaining questions in this section. No.
Q29. Did you collect the data from the individuals in question directly, or obtain it via third parties or other sources (e.g., websites)? Not applicable.
Q30. Were the individuals in question notified about the data collection? If so, please describe (or show with screenshots or other information) how notice was provided, and provide a link or other access point to, or otherwise reproduce, the exact language of the notification itself. Not applicable.
Q31. Did the individuals in question consent to the collection and use of their data? If so, please describe (or show with screenshots or other information) how consent was requested and provided, and provide a link or other access point to, or otherwise reproduce, the exact language to which the individuals consented. Not applicable.
Q32. If consent was obtained, were the consenting individuals provided with a mechanism to revoke their consent in the future or for certain uses? If so, please provide a description, as well as a link or other access point to the mechanism (if appropriate).
Q33. Has an analysis of the potential impact of the dataset and its use on data subjects (e.g., a data protection impact analysis) been conducted? If so, please provide a description of this analysis, including the outcomes, as well as a link or other access point to any supporting documentation.
Q34. Any other comments? No.
H.4 Preprocessing, Cleaning, and/or Labeling
Q35. Was any preprocessing/cleaning/labeling of the data done (e.g., discretization or bucketing, tokenization, part-of-speech tagging, SIFT feature extraction, removal of instances, processing of missing values)? If so, please provide a description. If not, you may skip the remainder of the questions in this section.
We corrected various minor issues and inconsistencies that could arise in the process of manual evaluation, see Appendix B.5.
Q36. Was the "raw" data saved in addition to the preprocessed/cleaned/ labeled data (e.g., to support unanticipated future uses)? If so, please provide a link or other access point to the "raw" data.
The output key in each JSON object contains the raw output from (Chat)GPT—unless ChatGPT used rendered  $\mathrm{IATEX}$  in which case our policy was to transcribe it. In very few cases, potential copy-paste errors were noticed, which were fixed.
Q37. Is the software used to preprocess/clean/label the instances available? If so, please provide a link or other access point.
The raw output of (Chat)GPT in the output key has not been cleaned, see Q36. Cleaning of the other values has been done first using Python scripts, in an automated way, and subsequently by hand, to correct any further, unforeseen mistakes, see Appendix B.5. The Python scripts are available upon request.
Q38. Any other comments?
H.5 Uses
Q39. Has the dataset been used for any tasks already? If so, please provide a description.
We have used the GHOSTS dataset to evaluate and compare the mathematical capabilities of different LLMs, in particular, different (Chat)GPT versions; see Section 4.
Q40. Is there a repository that links to any or all papers or systems that use the dataset? If so, please provide a link or other access point.
Future work citing the GHOSTS dataset will be listed by citation trackers such as Google Scholar and Semantic Scholar.
Q41. What (other) tasks could the dataset be used for?
If the dataset is growing further, we anticipate that GHOSTS can be used as training data for fine-tuning LLMs.
Q42. Is there anything about the composition of the dataset or the way it was collected and preprocessed/cleaned/labeled that might impact future uses?
For example, is there anything that a future user might need to know to avoid uses that could result in unfair treatment of individuals or groups (e.g., stereotyping, quality of service issues) or other undesirable harms (e.g., financial harms, legal risks) If so, please provide a description. Is there anything a future user could do to mitigate these undesirable harms?
Q43. Are there any tasks for which the dataset should not be used? If so, please provide a description.
Q44. Any other comments?
H.6 Distribution
Q45. Will the dataset be distributed to third parties outside of the entity (e.g., company, institution, organization) on behalf of which the dataset was created? If so, please provide a description.
Yes, the GHOSTS, miniGHOSTS, and microGHOSTS datasets will be made publicly available. It consists of three JSON files, one per ChatGPT version (9-January-2023, 30-January-2023, and GPT-4, where the latter was evaluated on miniGHOSTS, while the former two on the full GHOSTS dataset; the microGHOSTS prompts are tagged within the miniGHOSTS dataset, see the comment key). Some prompts will not be available due to copyright issues (see Appendix B.3), but a precise reference where the original prompt can be found will be included instead.
Q46. How will the dataset be distributed (e.g., tarball on website, API, GitHub) Does the dataset have a digital object identifier (DOI)?
The dataset will be made available on GitHub in the public repository github.com/friederrr/GHOSTS as a collection of JSON files.
Q47. When will the dataset be distributed?
The dataset is already available.
Q48. Will the dataset be distributed under a copyright or other intellectual property (IP) license, and/or under applicable terms of use (ToU)? If so, please describe this license and/or ToU, and provide a link or other access point to, or otherwise reproduce, any relevant licensing terms or ToU, as well as any fees associated with these restrictions.
We release the GHOSTS, miniGHOSTS, and microGHOSTS datasets under the following Creative Commons license: Attribution-NonCommercial 4.0 International (CC BY-NC 4.0), unless we are bound by licenses of individual prompts or files from various subdatasets to release those prompts or files under more restrictive licenses; see Appendix B.3 for more information.
Q49. Have any third parties imposed IP-based or other restrictions on the data associated with the instances? If so, please describe these restrictions, and provide a link or other access point to, or otherwise reproduce, any relevant licensing terms, as well as any fees associated with these restrictions.
IP-restrictions apply only to those prompts that were not solely created by the authors (which are under the CC BY-NC 4.0, as explained above), see Appendix B.3 for these cases.
Q50. Do any export controls or other regulatory restrictions apply to the dataset or to individual instances? If so, please describe these restrictions, and provide a link or other access point to, or otherwise reproduce, any supporting documentation. No.
Q51. Any other comments?
H.7 Maintenance
Q52. Who will be supporting/hosting/maintaining the dataset?
The dataset will be hosted on a GitHub repository; see Q46. All the information about the dataset, including links to the paper and future announcements, will be written in the README file of the GitHub repository.
Q53. How can the owner/curator/manager of the dataset be contacted (e.g., email address)?
The email addresses of the authors are publicly available. Moreover, it is possible to raise an issue on GitHub.
Q54. Is there an erratum? If so, please provide a link or other access point.
Future changes will be documented in the README file of the GitHub repository. Differences in single files can be tracked in the Git history.
Q55. Will the dataset be updated (e.g., to correct labeling errors, add new instances, delete instances)? If so, please describe how often, by whom, and how updates will be communicated to users (e.g., mailing list, GitHub)?
We will continue to maintain the dataset to fix any errors that may have occurred. We will allow either direct email contact or GitHub pull requests in this sense. Beyond fixing such errors, we consider the dataset to be frozen.
Q56. If the dataset relates to people, are there applicable limits on the retention of the data associated with the instances (e.g., were individuals in question told that their data would be retained for a fixed period of time and then deleted)? If so, please describe these limits and explain how they will be enforced.
Q57. Will older versions of the dataset continue to be supported/hosted/maintained? If so, please describe how. If not, please describe how its obsolescence will be communicated to users.
Q58. If others want to extend/augment/build on/contribute to the dataset, is there a mechanism for them to do so? If so, please provide a description. Will these contributions be verified? If so, please describe how. If not, why not? Is there a process for communicating/distributing these contributions to other users? If so, please provide a description.
Any external contribution to our dataset is strongly encouraged. Every addition to the dataset will be carefully reviewed by the authors. For other details, please see Q55.