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Zhixing You

Publications and source records attributed to Zhixing You.

8 recordsLinked to original sources

More notions of forcing add a square

Foreman and Magidor showed that the continuum hypothesis implies the existence of a countably-closed $\aleph_2$-cc forcing notion $\mathbb P$ for adding $\square_{\aleph_1}$. Here, we show that $\mathbb P$ may consistently be realized as an $\aleph_2$-Souslin tree. More generally, we prove that $\square_\lambda$ may be added by a $\lambda^+$-Souslin tree, providing the first analog of the Foreman--Magidor forcing at the level of successors of singular cardinals. Our construction is uniform and extends to inaccessible cardinals as well.

math.LO

D-Mem: A Dual-Process Memory System for LLM Agents

Driven by the development of persistent, self-adapting autonomous agents, equipping these systems with high-fidelity memory access for long-horizon reasoning has emerged as a critical requirement. However, prevalent retrieval-based memory frameworks often follow an incremental processing paradigm that continuously extracts and updates conversational memories into vector databases, relying on semantic retrieval when queried. While this approach is fast, it inherently relies on lossy abstraction, frequently missing contextually critical information and struggling to resolve queries that rely on fine-grained contextual understanding. To address this, we introduce D-Mem, a dual-process memory system. It retains lightweight vector retrieval for routine queries while establishing an exhaustive Full Deliberation module as a high-fidelity fallback. To achieve cognitive economy without sacrificing accuracy, D-Mem employs a Multi-dimensional Quality Gating policy to dynamically bridge these two processes. Experiments on the LoCoMo and RealTalk benchmarks using GPT-4o-mini and Qwen3-235B-Instruct demonstrate the efficacy of our approach. Notably, our Multi-dimensional Quality Gating policy achieves an F1 score of 53.5 on LoCoMo with GPT-4o-mini. This outperforms our static retrieval baseline, Mem0$^\ast$ (51.2), and recovers 96.7\% of the Full Deliberation's performance (55.3), while incurring significantly lower computational costs.

cs.AI

A new model for all $C$-sequences are trivial

We construct a model in which all $C$-sequences are trivial, yet there exists a $κ$-Souslin tree with full vanishing levels. This answers a question of Lambie-Hanson and Rinot, and provides an optimal combination of compactness and incompactness. It is obtained by incorporating a so-called mutually exclusive ascent path to Kunen's original forcing construction.

math.LO

Ketonen's question and other cardinal sins

Answering a question of Ketonen from the late 1970's, it is proved that a weakly compact cardinal carrying an indecomposable ultrafilter need not be measurable. The result is obtained by analyzing the limit of a decreasing sequence of models of ZFC. The utility of this proof technique is demonstrated further in this paper, where a problem by Bagaria and Magidor concerning strong compactness, and a problem by Lambie-Hanson and Rinot concerning the $C$-sequence number are solved as well.

math.LO

Full Souslin trees at small cardinals

A $κ$-tree is said to be full if each of its limit levels omits no more than one potential branch. Kunen asked whether a full $κ$-Souslin tree may consistently exist. Shelah gave an affirmative answer of height a strong limit Mahlo cardinal. Here, it is shown that these trees may consistently exist at small cardinals. Indeed, there can be $\aleph_3$ many full $\aleph_2$-trees such that the product of any countably many of them is an $\aleph_2$-Souslin tree.

math.LO

The vanishing levels of a tree

We initiate the study of the spectrum $Vspec(κ)$ of sets that can be realized as the vanishing levels $V(T)$ of a normal $κ$-tree $T$. The latter is an invariant in the sense that if $T$ and $T'$ are club-isomorphic, then the symmetric difference of $V(T)$ and $V(T')$ is nonstationary. Additional features of this invariant imply that $Vspec(κ)$ is closed under finite unions and intersections. The set $V(T)$ must be stationary for an homogeneous normal $κ$-Aronszajn tree $T$, and if there exists a special $κ$-Aronszajn tree, then there exists one $T$ that is homogeneous and satisfies $V(T)=κ$ (modulo clubs). It is consistent (from large cardinals) that there is an $\aleph_2$-Souslin tree, and yet $V(T)$ is co-stationary for every $\aleph_2$-tree $\mathbf T$. Both $V(T)=\emptyset$ and $V(T)=κ$ (modulo clubs) are shown to be feasible using $κ$-Souslin trees even at some large cardinal close to a weakly compact. It is also possible to have a family of $2^κ$ many $κ$-Souslin trees for which the corresponding family of vanishing levels forms an antichain modulo clubs.

math.LO

How far is almost strong compactness from strong compactness

Bagaria and Magidor introduced the notion of almost strong compactness, which is very close to the notion of strong compactness. Boney and Brooke-Taylor asked whether the least almost strongly compact cardinal is strongly compact. Goldberg gives a positive answer in the case $\mathrm{SCH}$ holds from below and the least almost strongly compact cardinal has uncountable cofinality. In this paper, we give a negative answer for the general case. Our result also gives an affirmative answer to a question of Bagaria and Magidor.

math.LO

On the cofinality of the least $λ$-strongly compact cardinal

In this paper, we characterize the possible cofinalities of the least $λ$-strongly compact cardinal. We show that, on the one hand, for any regular cardinal, $δ$, that carries a $λ$-complete uniform ultrafilter, it is consistent, relative to the existence of a supercompact cardinal above $δ$, that the least $λ$-strongly compact cardinal has cofinality $δ$. On the other hand, provably the cofinality of the least $λ$-strongly compact cardinal always carries a $λ$-complete uniform ultrafilter.

math.LO