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Symmetry
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Applications Based on AI in Mathematics and Asymmetry/Symmetry
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Applications Based on AI in Mathematics and Asymmetry/Symmetry

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 30 April 2025 | Viewed by 6570

Special Issue Editor

Prof. Dr. Zhenbing Zeng

E-Mail Website
Guest Editor
Department of Mathematics, Shanghai University, Shanghai 200444, China
Interests: combinatorial geometry; global optimization; symbolic computation and computer algebra; machine proof; algorithms in artificial intelligence
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue invites you to contribute your original research work and review articles on “Applications Based on AI in Mathematics and Asymmetry/Symmetry”, including the recent advances in the theoretical research of AI algorithms and computer-aided research in geometry and their applications, with emphasis on symmetry and asymmetry aspects, as well as new research methods such as symbolic and numerical computation. We hope that this Special Issue will provide the most up-to-date information on important findings and innovative tools in the research of related fields.

Scope: Potential topics including, but not limited to, the following subheadings are deemed suitable for publication:

  • The recent advance of formal methods in automated reasoning of mathematics theorems;
  • Innovative AI-supported computer algebraic algorithms for solving problems in graph theory, combinatorial geometry, and discrete geometry;
  • Symbolic/numerical computations in computer-aided research for finding new theorems and new conjectures;
  • Review of classical results on symmetry/asymmetry/weak symmetry/local symmetry of graphs related to regular or semi-regular polyhedral graphs, Eulerian graphs, and Hamiltonian graphs;
  • Symmetry and asymmetry of graphs or extremal configurations generated in geometric optimization problems and the classification of symmetric graphs;
  • Optimal problems and geometric inequalities related to finite point-configurations in the sphere of three-dimensional space and higher-dimensional spaces;
  • Graph coloring problems, Ramsey numbers, and advances in unsolved problems in graph theory;
  • Semantic graphs/networks, phylogenetic graphs/networks, optimal graph searches, graph neuro-networks, and applications of graph theory.

Prof. Dr. Zhenbing Zeng
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • AI algorithms
  • symmetry
  • asymmetry
  • formal method
  • global optimization
  • extremal configuration

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Further information on MDPI's Special Issue polices can be found here.

Published Papers (5 papers)

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Research

22 pages, 570 KiB  
Article
A Novel Discrete Linear-Exponential Distribution for Modeling Physical and Medical Data
by Khlood Al-Harbi, Aisha Fayomi, Hanan Baaqeel and Amany Alsuraihi
Symmetry 2024, 16(9), 1123; https://doi.org/10.3390/sym16091123 - 29 Aug 2024
Viewed by 796
Abstract
In real-life data, count data are considered more significant in different fields. In this article, a new form of the one-parameter discrete linear-exponential distribution is derived based on the survival function as a discretization technique. An extensive study of this distribution is conducted [...] Read more.
In real-life data, count data are considered more significant in different fields. In this article, a new form of the one-parameter discrete linear-exponential distribution is derived based on the survival function as a discretization technique. An extensive study of this distribution is conducted under its new form, including characteristic functions and statistical properties. It is shown that this distribution is appropriate for modeling over-dispersed count data. Moreover, its probability mass function is right-skewed with different shapes. The unknown model parameter is estimated using the maximum likelihood method, with more attention given to Bayesian estimation methods. The Bayesian estimator is computed based on three different loss functions: a square error loss function, a linear exponential loss function, and a generalized entropy loss function. The simulation study is implemented to examine the distribution’s behavior and compare the classical and Bayesian estimation methods, which indicated that the Bayesian method under the generalized entropy loss function with positive weight is the best for all sample sizes with the minimum mean squared errors. Finally, the discrete linear-exponential distribution proves its efficiency in fitting discrete physical and medical lifetime count data in real-life against other related distributions. Full article
(This article belongs to the Special Issue Applications Based on AI in Mathematics and Asymmetry/Symmetry)
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13 pages, 291 KiB  
Article
Floating-Point Embedding: Enhancing the Mathematical Comprehension of Large Language Models
by Xiaoxiao Jin, Chenyang Mao, Dengfeng Yue and Tuo Leng
Symmetry 2024, 16(4), 478; https://doi.org/10.3390/sym16040478 - 15 Apr 2024
Cited by 1 | Viewed by 1642
Abstract
The processing and comprehension of numerical information in natural language represent pivotal focal points of scholarly inquiry. Across diverse applications spanning text analysis to information retrieval, the adept management and understanding of the numerical content within natural language are indispensable in achieving task [...] Read more.
The processing and comprehension of numerical information in natural language represent pivotal focal points of scholarly inquiry. Across diverse applications spanning text analysis to information retrieval, the adept management and understanding of the numerical content within natural language are indispensable in achieving task success. Specialized encoding and embedding techniques tailored to numerical data offer an avenue toward improved performance in tasks involving masked prediction and numerical reasoning, inherently characterized by numerical values. Consequently, treating numbers in text merely as words is inadequate; their numerical semantics must be underscored. Recent years have witnessed the emergence of a range of specific encoding methodologies designed explicitly for numerical content, demonstrating promising outcomes. We observe similarities between the Transformer architecture and CPU architecture, with symmetry playing a crucial role. In light of this observation and drawing inspiration from computer system theory, we introduce a floating-point representation and devise a corresponding embedding module. The numerical representations correspond one-to-one with their semantic vector values, rendering both symmetric regarding intermediate transformation methods. Our proposed methodology facilitates the more comprehensive encoding and embedding of numerical information within a predefined precision range, thereby ensuring a distinctive encoding representation for each numerical entity. Rigorous testing on multiple encoder-only models and datasets yielded results that stand out in terms of competitiveness. In comparison to the default embedding methods employed by models, our approach achieved an improvement of approximately 3.8% in Top-1 accuracy and a reduction in perplexity of approximately 0.43. These outcomes affirm the efficacy of our proposed method. Furthermore, the enrichment of numerical semantics through a more comprehensive embedding contributes to the augmentation of the model’s capacity for semantic understanding. Full article
(This article belongs to the Special Issue Applications Based on AI in Mathematics and Asymmetry/Symmetry)
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14 pages, 745 KiB  
Article
FGeo-DRL: Deductive Reasoning for Geometric Problems through Deep Reinforcement Learning
by Jia Zou, Xiaokai Zhang, Yiming He, Na Zhu and Tuo Leng
Symmetry 2024, 16(4), 437; https://doi.org/10.3390/sym16040437 - 5 Apr 2024
Cited by 1 | Viewed by 1183
Abstract
Human-like automatic deductive reasoning has always been one of the most challenging open problems in the interdisciplinary field of mathematics and artificial intelligence. This paper is the third in a series of our works. We built a neural-symbolic system, named FGeo-DRL, to automatically [...] Read more.
Human-like automatic deductive reasoning has always been one of the most challenging open problems in the interdisciplinary field of mathematics and artificial intelligence. This paper is the third in a series of our works. We built a neural-symbolic system, named FGeo-DRL, to automatically perform human-like geometric deductive reasoning. The neural part is an AI agent based on deep reinforcement learning, capable of autonomously learning problem-solving methods from the feedback of a formalized environment, without the need for human supervision. It leverages a pre-trained natural language model to establish a policy network for theorem selection and employ Monte Carlo Tree Search for heuristic exploration. The symbolic part is a reinforcement learning environment based on geometry formalization theory and FormalGeo, which models geometric problem solving (GPS) as a Markov Decision Process (MDP). In the formal symbolic system, the symmetry of plane geometric transformations ensures the uniqueness of geometric problems when converted into states. Finally, the known conditions and objectives of the problem form the state space, while the set of theorems forms the action space. Leveraging FGeo-DRL, we have achieved readable and verifiable automated solutions to geometric problems. Experiments conducted on the formalgeo7k dataset have achieved a problem-solving success rate of 86.40%. Full article
(This article belongs to the Special Issue Applications Based on AI in Mathematics and Asymmetry/Symmetry)
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15 pages, 711 KiB  
Article
FGeo-TP: A Language Model-Enhanced Solver for Euclidean Geometry Problems
by Yiming He, Jia Zou, Xiaokai Zhang, Na Zhu and Tuo Leng
Symmetry 2024, 16(4), 421; https://doi.org/10.3390/sym16040421 - 3 Apr 2024
Cited by 2 | Viewed by 1037
Abstract
The application of contemporary artificial intelligence techniques to address geometric problems and automated deductive proofs has always been a grand challenge to the interdisciplinary field of mathematics and artificial intelligence. This is the fourth article in a series of our works, in our [...] Read more.
The application of contemporary artificial intelligence techniques to address geometric problems and automated deductive proofs has always been a grand challenge to the interdisciplinary field of mathematics and artificial intelligence. This is the fourth article in a series of our works, in our previous work, we established a geometric formalized system known as FormalGeo. Moreover, we annotated approximately 7000 geometric problems, forming the FormalGeo7k dataset. Despite the fact that FGPS (Formal Geometry Problem Solver) can achieve interpretable algebraic equation solving and human-like deductive reasoning, it often experiences timeouts due to the complexity of the search strategy. In this paper, we introduced FGeo-TP (theorem predictor), which utilizes the language model to predict the theorem sequences for solving geometry problems. The encoder and decoder components in the transformer architecture naturally establish a mapping between the sequences and embedding vectors, exhibiting inherent symmetry. We compare the effectiveness of various transformer architectures, such as BART or T5, in theorem prediction, and implement pruning in the search process of FGPS, thereby improving its performance when solving geometry problems. Our results demonstrate a significant increase in the problem-solving rate of the language model-enhanced FGeo-TP on the FormalGeo7k dataset, rising from 39.7% to 80.86%. Furthermore, FGeo-TP exhibits notable reductions in solution times and search steps across problems of varying difficulty levels. Full article
(This article belongs to the Special Issue Applications Based on AI in Mathematics and Asymmetry/Symmetry)
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18 pages, 407 KiB  
Article
FGeo-SSS: A Search-Based Symbolic Solver for Human-like Automated Geometric Reasoning
by Xiaokai Zhang, Na Zhu, Yiming He, Jia Zou, Cheng Qin, Yang Li and Tuo Leng
Symmetry 2024, 16(4), 404; https://doi.org/10.3390/sym16040404 - 30 Mar 2024
Viewed by 1411
Abstract
Geometric problem solving (GPS) has always been a long-standing challenge in the fields of automated reasoning. Its problem representation and solution process embody rich symmetry. This paper is the second in a series of our works. Based on the Geometry Formalization Theory and [...] Read more.
Geometric problem solving (GPS) has always been a long-standing challenge in the fields of automated reasoning. Its problem representation and solution process embody rich symmetry. This paper is the second in a series of our works. Based on the Geometry Formalization Theory and the FormalGeo geometric formal system, we have developed the Formal Geometric Problem Solver (FGPS) in Python 3.10, which can serve as an interactive assistant or as an automated problem solver. FGPS is capable of executing geometric predicate logic and performing relational reasoning and algebraic computation, ultimately achieving readable, traceable, and verifiable automated solutions for geometric problems. We observed that symmetry phenomena exist at various levels within FGPS and utilized these symmetries to further refine the system’s design. FGPS employs symbols to represent geometric shapes and transforms various geometric patterns into a set of symbolic operation rules. This maintains symmetry in basic transformations, shape constructions, and the application of theorems. Moreover, we also have annotated the formalgeo7k dataset, which contains 6981 geometry problems with detailed formal language descriptions and solutions. Experiments on formalgeo7k validate the correctness and utility of the FGPS. The forward search method with random strategy achieved a 39.71% problem-solving success rate. Full article
(This article belongs to the Special Issue Applications Based on AI in Mathematics and Asymmetry/Symmetry)
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