Fudan Logic Student Seminar 2025
July 31
Speaker: 李想 LI, Xiang
Time: 9:30 - 10:30. Location: HGX208.
Interpreting First-Order Structures in Nice Graphs
In this talk, we revisit the well-known result that structures in a finite language are bi-interpretable with graphs satisfying a “niceness” property introduced by Alan Mekler. We identify an oversight in Wilfrid Hodges’ Model Theory and propose a new construction of the interpreting graph that ensures the bi-interpretation holds while guaranteeing that the interpreted graph satisfies the niceness property in Mekler’s Construction.
July 23
Speaker: 侯乃中 HOU, Nai-Chung
Time: 15:30 - 16:30. Location: HGX208.
Higher Structural Reflection and Very Large Cardinals
Gödel’s Program in set theory seeks well-justified extensions of ZFC that that can resolve important mathematical questions. Large Cardinal Axioms (LCAs) are the most significant additional axioms. One approach to justifying LCAs is to formulate them in terms of reflection principles that are intrinsically plausible, an approach favored by Gödel himself. The most systematic attempt along this line is the research program of Joan Bagaria, which aims to reformulate LCAs as instances of principles of Structural Reflection (SR). The original formulation of SR is already very strong. However, it remained unclear whether there are direct and equally plausible generalizations of SR that correspond to even stronger LCAs. This is a key question for the prospects of the SR program. In this talk we present a series of results that bear on the above question. We formulate natural generalizations of SR that relate to stronger LCAs in the same way SR relates to LCAs, thereby supporting the SR program. On the other hand, we also show that further generalizations of the principle lead to contradictions, which suggests the problem of extendibility to inconsistency for the SR program. We also apply our results to answer several open questions concerning LCAs and SR in the literature. The philosophical significance of these results for the evaluation of the SR program is discussed. Slides
Speaker: 李宇轩 LI, Yuxuan
Time: 16:30 - 17:30. Location: HGX208.
A Variant of Chaitin's Omega Function
This is a joint work with Shuheng Zhang, Xiaoyan Zhang and Xuanheng Zhao. We study the continuous function $f$ defined by $$x\mapsto \sum_{\sigma\le_L x }2^{-K(\sigma)}$$ as an variant of Chaitin's $\Omega$ from the perspective of analysis, computability, and algorithmic randomness: (i) $f$ is differentiable precisely at density random points; (ii) $f(x)$ is $x$-random if and only if $x$ is weakly low for $K$ (low for $\Omega$); (iii) the range of $f$ is a null, nowhere dense, perfect $\Pi^0_1(\emptyset')$ class with Hausdorff dimension $1$; (iv) No real of the form $\sum_{x \le _L \sigma \le _L y}2^{-K(\sigma)}$ is $Kurtz$-$random$ relative to $\emptyset'$; (v) $f(x)\oplus x\ge_T\emptyset'$ for all $x$; (vi) there are $2^{\aleph_0}$ many $x$ such that $f(x)$ is not 1-random; (vii) $f$ is not Turing invariant but is Turing invariant in the ideal of $K$-trivial reals. We also introduce a notion of left c.e. interval randomness and prove that this notion is equivalent to $Kurtz$-$random$ relative to $\emptyset'$.
May 29
Speaker: 鞠大恒 JU, Daheng and 宋书浩 SONG, Shuhao
Time: 15:25 - 17:05. Location: HGW2403.
Frucht's Theorem and Its Formalization
Frucht's theorem states that every group is the automorphism group of a graph. This theorem is provable in $\textrm{ZFC-}$ and $\textrm{ZF}$, but not in $\textrm{ZF-}$. We show that it is provable in $\textrm{ZF-}+\textrm{FAFA}_2$, and formalize these results in Lean.
March 25
Speaker: 朱子璇 ZHU, Zixuan. Institut für Mathematische Logik und Grundlagenforschung, Universität Münster
Time: 15:25 - 17:00. Location: HGW2403.
A rank on the imaginaries in the proper pairs of ACF
This talk explores rank notions in beautiful pairs, particularly in the theory of proper pairs of algebraically closed fields (ACF). We first introduce fundamental results on beautiful pairs and mention that, in the case of ACF, the Morley Rank and SU-Rank of real elements coincide, along with a description of their ranks in (ACF)_P. Building on Pillay's results on imaginaries in (ACF)_P, we extend this rank notion to all imaginaries in (ACF)_P^{eq}, ensuring a coherent and well-defined ranking framework.
March 21
Speaker: 张明 ZHANG, Ming. 复旦大学自然语言处理实验室
Time: 13:00 - 14:30. Location: HGW2403.
Comprehensive Analysis of LLM Evaluation Methodologies
Recently, the evaluation of Large Language Models has emerged as a popular area of research. The three crucial questions for LLM evaluation are ``what, where, and how to evaluate''. However, the existing research mainly focuses on the first two questions, which are basically what tasks to give the LLM during testing and what kind of knowledge it should deal with. As for the third question, which is about what standards to use, the types of evaluators, how to score, and how to rank, there hasn't been much discussion. In this paper, we analyze evaluation methods by comparing various criteria with both manual and automatic evaluation, utilizing onsite, crowd-sourcing, public annotators and GPT-4, with different scoring methods and ranking systems. We propose a new dataset, LLMEval and conduct evaluations on 20 LLMs. A total of 2,186 individuals participated, leading to the generation of 243,337 manual annotations and 57,511 automatic evaluation results. We perform comparisons and analyses of different settings and conduct 10 conclusions that can provide some insights for evaluating LLM in the future.