Research

2026

1. A Complete Bounded Theory with Unbounded Types

   Hongyu Zhu

   Journal of Symbolic Logic, 2026

   Published online ahead of print

   Abs DOI arXiv Bib HTML PDF

   One measure of the complexity of a first-order theory, and similarly a type, is the complexity of the formulas required to axiomatize it. We say a theory is bounded if there is an axiomatization involving only \(\forall_n\)-formulas for some finite \(n\), and unbounded otherwise. One might expect bounded theories to have only bounded types. In fact, an analogue holds in infinitary logic, where the complexity of a Scott sentence roughly agrees with the complexity of the most complicated automorphism orbit. Our main result, however, shows this is not the case in the first-order setting: Namely, there can be a bounded theory, in fact \(\forall_1\)-axiomatizable, which has unbounded types.

   @article{Bdd_Unbdd,
     doi = {10.1017/jsl.2026.10208},
     journal = {Journal of Symbolic Logic},
     url = {https://doi.org/10.1017/jsl.2026.10208},
     author = {Zhu, Hongyu},
     keywords = {Logic (math.LO), FOS: Mathematics, FOS: Mathematics, 03C65, 03C10, 03E15},
     title = {A Complete Bounded Theory with Unbounded Types},
     year = {2026},
     copyright = {Creative Commons Attribution 4.0 International},
     note = {Published online ahead of print},
   }

2025

1. The Borel Complexity of the Class of Models of First-Order Theories

   Uri Andrews, David Gonzalez, Steffen Lempp, Dino Rossegger, and Hongyu Zhu

   Proceedings of the American Mathematical Society, 2025

   Abs DOI arXiv Bib HTML PDF

   We investigate the descriptive complexity of the set of models of first-order theories. Using classical results of Knight and Solovay, we give a sharp condition for complete theories to have a \(\boldsymbolΠ_ω^0\)-complete set of models. We also give sharp conditions for theories to have a \(\boldsymbolΠ^0_n\)-complete set of models. Finally, we determine the Turing degrees needed to witness the completeness.

   @article{AGLRZ,
     author = {Andrews, Uri and Gonzalez, David and Lempp, Steffen and Rossegger, Dino and Zhu, Hongyu},
     title = {The {B}orel Complexity of the Class of Models of First-Order
                   Theories},
     journal = {Proceedings of the American Mathematical Society},
     volume = {153},
     year = {2025},
     number = {9},
     pages = {4013--4024},
     issn = {0002-9939,1088-6826},
     mrclass = {03C62 (03C52 03E15)},
     mrnumber = {4936351},
     doi = {10.1090/proc/17308},
     url = {https://doi.org/10.1090/proc/17308},
   }

2021

1. An Analysis of COVID-19 Knowledge Graph Construction and Applications

   Dominic Flocco, Bryce Palmer-Toy, Ruixiao Wang, Hongyu Zhu, Rishi Sonthalia, Junyuan Lin, Andrea L. Bertozzi, and P. Jeffrey Brantingham

   2021

   Abs DOI arXiv Bib HTML PDF

   The construction and application of knowledge graphs have seen a rapid increase across many disciplines in recent years. Additionally, the problem of uncovering relationships between developments in the COVID-19 pandemic and social media behavior is of great interest to researchers hoping to curb the spread of the disease. In this paper we present a knowledge graph constructed from COVID-19 related tweets in the Los Angeles area, supplemented with federal and state policy announcements and disease spread statistics. By incorporating dates, topics, and events as entities, we construct a knowledge graph that describes the connections between these useful information. We use natural language processing and change point analysis to extract tweet-topic, tweet-date, and event-date relations. Further analysis on the constructed knowledge graph provides insight into how tweets reflect public sentiments towards COVID-19 related topics and how changes in these sentiments correlate with real-world events.

   @misc{https://doi.org/10.48550/arxiv.2110.04932,
     doi = {10.48550/ARXIV.2110.04932},
     url = {https://arxiv.org/abs/2110.04932},
     author = {Flocco, Dominic and Palmer-Toy, Bryce and Wang, Ruixiao and Zhu, Hongyu and Sonthalia, Rishi and Lin, Junyuan and Bertozzi, Andrea L. and Brantingham, P. Jeffrey},
     keywords = {Social and Information Networks (cs.SI), Computation and Language (cs.CL), FOS: Computer and information sciences, FOS: Computer and information sciences},
     title = {An Analysis of COVID-19 Knowledge Graph Construction and Applications},
     publisher = {arXiv},
     year = {2021},
     copyright = {arXiv.org perpetual, non-exclusive license},
   }

In preparation

1. Degree Spectra of Structures with Low Scott Rank

   Uri Andrews, David Gonzalez, Joseph S. Miller, and Hongyu Zhu

   In preparation
2. Infinite Unfriendly Jump Inversion

   Uri Andrews, David Gonzalez, and Hongyu Zhu

   In preparation
3. Strongly Non-Computable Degrees

   Matthew Harrison-Trainor and Hongyu Zhu

   In preparation
4. The Type \(\omega\)-Vaught’s Conjecture

   Hongyu Zhu

   In preparation