Abstract
As we work toward artificial general intelligence, it is clear that we must try to imbue agents with faculties which ensure they are trustworthy. We firmly believe that an AGI agent must be able to explain its decision-making in order for it to be considered trustworthy. More specifically, agents must be able to explain themselves in a way that is both logically correct and understandable to humans. We take a first step toward a system that can generate explanations which satisfy this pair of conditions. We created the first model that can produce summaries of modal-logic proofs using a transformer language model. We qualitatively evaluated the model’s outputs on a held-out test set and found that the logical content of the model’s explanations precisely matched the input proofs in 60% of cases.
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Notes
- 1.
As we discuss in Sect. 5, there are systems that can create explanations of modal logic proofs [4], but not summaries. That is, they can produce explanations which have a one-to-one correspondence with the input proof, but cannot synthesize a summary that highlights only the major components of the proof.
- 2.
For a full exposition of exactly what a cognitive calculus is and isn’t, we point the interested reader to Appendix A of [3].
- 3.
Most cognitive calculi subsume first-order logic; some others include also second-, third-, and higher-order logics. For reasons that will be explained later in the paper, the cognitive calculus we utilize herein includes, of extensional logics, only zero-order logic.
- 4.
Such as the Deontic Cognitive Event Calculus (\({\mathcal {{DCEC}}}\)) and its inductive counterpart (\({\mathcal {{IDCEC}}}\)). The interested reader is referred to [3], which utilizes both of these calculi.
- 5.
For brevity, the pre-existing core of the function signatures is excluded as we will not need them for the problems presented herein.
- 6.
Other calculi (e.g. the situation calculus) for modeling commonsense and physical reasoning can be easily switched out in-place of the event calculus.
- 7.
The interested reader is referred to [2] for a thorough analysis of the environmental, financial, and societal concerns surrounding transformers.
- 8.
We certainly do not believe transformers are the only method by which this generation of proof explanations can be achieved. In fact, we expect that methods of natural-language generation which incorporate symbolic reasoning would almost certainly provide better assurances that the resulting explanations match the logical content of the proof. We discuss this more in Sect. 6.
- 9.
See Sect. 2.1.
- 10.
Newlines and indentation have been added for readability.
- 11.
They state that their goal was to produce summaries of proofs similar to what would be seen in “mathematical textbooks”. While they didn’t specify any sub-fields of mathematics, nor the level of rigor (e.g. high-school, undergraduate, or graduate textbooks) they intended their method for, the examples in the paper all involve mathematical-induction proofs of arithmetic properties, e.g. commutativity of addition. It is now known that some classical mathematics beyond merely the textbooks involves third-order logic, and proofs couched therein.
- 12.
We note that, while this is a significant drawback, which we shortly address as pressing future work, AI agents which utilize the type of technology presented herein (i.e. a cognitive calculus for reasoning and a transformer for NLG) would still enact logically correct decision-making, even if its explanation wasn’t correct. That is, the agent’s behavior would still be bound by what it could prove via the calculus’ inference schemata.
References
Alexoudi, M., Zinn, C., Bundy, A.: English summaries of mathematical proofs. In: Second International Joint Conference on Automated Reasoning-Workshop on Computer-Supported Mathematical Theory Development, pp. 49–60. Citeseer (2004)
Bender, E.M., Gebru, T., McMillan-Major, A., Shmitchell, S.: On the dangers of stochastic parrots: can language models be too big? In: Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency, pp. 610–623 (2021)
Bringsjord, S., Govindarajulu, N.S., Giancola, M.: Automated argument adjudication to solve ethical problems in multi-agent environments. Paladyn J. Behav. Robot. 12, 310–335 (2021). https://doi.org/10.1515/pjbr-2021-0009
Felty, A., Hager, G.: Explaining modal logic proofs. In: Proceedings of the 1988 IEEE International Conference on Systems, Man, and Cybernetics, vol. 1, pp. 177–180. IEEE (1988)
Francez, N.: Proof-Theoretic Semantics. College Publications, London (2015)
Giancola, M., Bringsjord, S., Govindarajulu, N.S.: Novel intensional defeasible reasoning for AI: is it cognitively adequate? In: Proceedings of the IJCAI Workshop on “Cognitive Aspects of Knowledge Representation” (CAKR 2022). CEUR-WS (2022). http://ceur-ws.org/Vol-3251/paper9.pdf
Giancola, M., Bringsjord, S., Govindarajulu, N.S., Varela, C.: Making maximally ethical decisions via cognitive likelihood & formal planning. In: Ferreira, M.I.A., Tokhi, O. (eds.) Towards Trustworthy Artificial Intelligent Systems, Intelligent Systems, Control and Automation: Science and Engineering, vol. 102. Springer (2022). https://link.springer.com/chapter/10.1007/978-3-031-09823-9_10
Govindarajulu, N.S., Bringsjord , S., Peveler, M.: On quantified modal theorem proving for modeling ethics. Electron. Proc. Theor. Comput. Sci. 311, 43–49 (2019)
Kowalski, R., Sergot, M.: A logic-based calculus of events. N. Gener. Comput. 4(1), 67–95 (1986)
Leon, I.E.: OntoGen: a knowledge-based approach to natural language generation. Master’s thesis, Rensselaer Polytechnic Institute (2020)
McShane, M., Leon, I.: Language generation for broad-coverage, explainable cognitive systems. arXiv preprint arXiv:2201.10422 (2022)
Zhang, J., Zhao, Y., Saleh, M., Liu, P.: PEGASUS: pre-training with extracted gap-sentences for abstractive summarization. In: International Conference on Machine Learning, pp. 11328–11339. PMLR (2020)
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This research is partially enabled by support from ONR and AFOSR (Award # FA9550-17-1-0191).
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A Fine-Tuning and Evaluation Implementation
A Fine-Tuning and Evaluation Implementation
We used the Hugging Face Transformers library to access Pegasus, fine-tune the model, and evaluate it. Our fork of the model (https://github.com/mjgiancola/transformers) includes scripts for fine-tuning with the parameters we set to enable reproduction of our results (https://github.com/mjgiancola/transformers/tree/main/examples/research_projects/proof-nlg).
The script \(\texttt {fine\_tune\_pegasus.sh}\) runs the fine-tuning process. It is configured to generate the fine-tuned model’s predictions on the test set after fine-tuning is completed. Additionally, the script \(\texttt {get\_predictions\_from\_fine\_}{} \texttt {tuned.py}\) loads the fine-tuned model and outputs pretty-printed results, including the input (a proof), the ground-truth output (a human-generated explanation), and the model’s output.
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Giancola, M., Bringsjord, S., Govindarajulu, N.S. (2023). Toward Generating Natural-Language Explanations of Modal-Logic Proofs. In: Goertzel, B., Iklé, M., Potapov, A., Ponomaryov, D. (eds) Artificial General Intelligence. AGI 2022. Lecture Notes in Computer Science(), vol 13539. Springer, Cham. https://doi.org/10.1007/978-3-031-19907-3_21
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