Bridging Equational Properties and Patterns on Graphs: an AI-Based Approach

Oguzhan Keskin, Alisia Maria Lupidi, Stefano Fioravanti, Lucie Charlotte Magister, Pietro Barbiero, Pietro Lio, Francesco Giannini
Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML), PMLR 221:156-168, 2023.

Abstract

AI-assisted solutions have recently proven successful when applied to Mathematics and have opened new possibilities for exploring unsolved problems that have eluded traditional approaches for years or even centuries. Following this direction, this paper presents an innovative approach aiming at establishing correlations between equational properties of algebraic structures that can be represented through graphs and specific sub-portions of their topological representation. The methodology incorporates the utilization of graph neural architectures to validate theorems or conjectures, complemented by Explainability (XAI) metrics that lend support to these statements. In particular, we examine the distributive and modular properties of algebraic lattices, whose characterization is well-known in universal algebra, hence using these properties as an experimental test bench. The findings of this study demonstrate the effectiveness of the proposed approach in identifying and retrieving established subpatterns that characterize the equational properties under investigation. Moreover, the approach exhibits the capability to generate novel and noteworthy candidates as theorem suggesters, thereby offering valuable prospects for further exploration by mathematicians.

Cite this Paper

BibTeX
@InProceedings{pmlr-v221-keskin23a, title = {Bridging Equational Properties and Patterns on Graphs: an AI-Based Approach}, author = {Keskin, Oguzhan and Lupidi, Alisia Maria and Fioravanti, Stefano and Magister, Lucie Charlotte and Barbiero, Pietro and Lio, Pietro and Giannini, Francesco}, booktitle = {Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML)}, pages = {156--168}, year = {2023}, editor = {Doster, Timothy and Emerson, Tegan and Kvinge, Henry and Miolane, Nina and Papillon, Mathilde and Rieck, Bastian and Sanborn, Sophia}, volume = {221}, series = {Proceedings of Machine Learning Research}, month = {28 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v221/keskin23a/keskin23a.pdf}, url = {https://proceedings.mlr.press/v221/keskin23a.html}, abstract = {AI-assisted solutions have recently proven successful when applied to Mathematics and have opened new possibilities for exploring unsolved problems that have eluded traditional approaches for years or even centuries. Following this direction, this paper presents an innovative approach aiming at establishing correlations between equational properties of algebraic structures that can be represented through graphs and specific sub-portions of their topological representation. The methodology incorporates the utilization of graph neural architectures to validate theorems or conjectures, complemented by Explainability (XAI) metrics that lend support to these statements. In particular, we examine the distributive and modular properties of algebraic lattices, whose characterization is well-known in universal algebra, hence using these properties as an experimental test bench. The findings of this study demonstrate the effectiveness of the proposed approach in identifying and retrieving established subpatterns that characterize the equational properties under investigation. Moreover, the approach exhibits the capability to generate novel and noteworthy candidates as theorem suggesters, thereby offering valuable prospects for further exploration by mathematicians.} }
Endnote
%0 Conference Paper %T Bridging Equational Properties and Patterns on Graphs: an AI-Based Approach %A Oguzhan Keskin %A Alisia Maria Lupidi %A Stefano Fioravanti %A Lucie Charlotte Magister %A Pietro Barbiero %A Pietro Lio %A Francesco Giannini %B Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML) %C Proceedings of Machine Learning Research %D 2023 %E Timothy Doster %E Tegan Emerson %E Henry Kvinge %E Nina Miolane %E Mathilde Papillon %E Bastian Rieck %E Sophia Sanborn %F pmlr-v221-keskin23a %I PMLR %P 156--168 %U https://proceedings.mlr.press/v221/keskin23a.html %V 221 %X AI-assisted solutions have recently proven successful when applied to Mathematics and have opened new possibilities for exploring unsolved problems that have eluded traditional approaches for years or even centuries. Following this direction, this paper presents an innovative approach aiming at establishing correlations between equational properties of algebraic structures that can be represented through graphs and specific sub-portions of their topological representation. The methodology incorporates the utilization of graph neural architectures to validate theorems or conjectures, complemented by Explainability (XAI) metrics that lend support to these statements. In particular, we examine the distributive and modular properties of algebraic lattices, whose characterization is well-known in universal algebra, hence using these properties as an experimental test bench. The findings of this study demonstrate the effectiveness of the proposed approach in identifying and retrieving established subpatterns that characterize the equational properties under investigation. Moreover, the approach exhibits the capability to generate novel and noteworthy candidates as theorem suggesters, thereby offering valuable prospects for further exploration by mathematicians.
APA
Keskin, O., Lupidi, A.M., Fioravanti, S., Magister, L.C., Barbiero, P., Lio, P. & Giannini, F.. (2023). Bridging Equational Properties and Patterns on Graphs: an AI-Based Approach. Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML), in Proceedings of Machine Learning Research 221:156-168 Available from https://proceedings.mlr.press/v221/keskin23a.html.

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