Martin Grohe
We prove that graphs G, G' satisfy the same sentences of first-order logic with counting of quantifier rank at most k if and only if they are homomorphism-indistinguishable over the class of all graphs of tree depth at most k. Here G, G' are homomorphism-indistinguishable over a class C of graphs if for each graph F in C, the number of homomorphisms from F to G equals the number of homomorphisms from F to G'.
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Distinguishing Graphs by Counting Homomorphisms from Sparse Graphs
该研究证明了排除某些minor的图类中，同构不可区分性不同于同构，并展示了相关图类的同构区分闭合性及严格层次结构。
Complexity classification of counting graph homomorphisms modulo a prime number
本研究证实了Faben与Jerrum提出的素数模下图同态计数复杂性猜想，并开发出可拓展至一般计数CSP的模计数归约技术，推进了模计数问题的复杂性理论研究。
A classical result due to Lovasz (1967) shows that the isomorphism type of a graph is determined by homomorphism counts. That is, graphs G and H are isomorphic whenever the number of homomorphisms from K to G is the same as the number of homomorphisms from K to H for all graphs K. Variants of this result, for various classes of finite structures, have been exploited in a wide range of research fields, including graph theory and finite model theory. We provide a categorical approach to homomorphism counting based on the concept of polyadic (finite) set. The latter is a special case of the notio…
Counting Homomorphisms from Hypergraphs of Bounded Generalised Hypertree Width: A Logical Characterisation
本研究提出双类型计数逻辑$GC^k$，证明其等价于广义超树宽度至多k的超图类上的同态不可区分性，将图领域的相关经典结论推广至超图场景。
Going deep and going wide: Counting logic and homomorphism indistinguishability over graphs of bounded treedepth and treewidth
研究计数逻辑的表达力，特别是k变量和量词秩q片段，通过同构不可区分性分析。分离图类并证明同构不可区分性关系的语义分离。
Going Deep and Going Wide: Counting Logic and Homomorphism Indistinguishability over Graphs of Bounded Treedepth and Treewidth
本文用初等技巧重新证明计数一阶逻辑片段 $\mathsf{C}^k_q$ 与图类 $\mathcal{T}^k_q$ 上同态不可区分性的对应关系，并通过图论分析将其与有界树宽和树深图类的交集分离，同时证明 $\mathcal{TD}_q$ 同态区分封闭。
Modular Counting over 3-Element and Conservative Domains
本文研究一般关系结构同态的模计数问题，提出将其归约到更小域的新方法，并刻画了3元素域与保守域上该问题的计算复杂度。
Limitations of Game Comonads via Homomorphism Indistinguishability
本文通过研究线性代数逻辑中的可逆映射等价（IM），利用同态不可区分性证明了在图范畴上不存在有限秩comonad能刻画IM等价，揭示了博弈comonad方法的局限性。
Game comonads have brought forth a new approach to studying finite model theory categorically. By representing model comparison games semantically as comonads, they allow important logical and combinatorial properties to be exressed in terms of their Eilenberg-Moore coalgebras. As a result, a number of results from finite model theory, such as preservation theorems and homomorphism counting theorems, have been formalised and parameterised by comonads, giving rise to new results simply by varying the comonad. In this paper we study the limits of the comonadic approach in the combinatorial and h…
Homomorphism Indistinguishability and Game Comonads for Restricted Conjunction and Requantification
本文扩展了同态不可区分性框架以处理受限重量化的逻辑，通过引入结合可重用性的路径/树分解和追逃博弈，推广了Lovász型定理，并构建新comonads统一了逻辑、组合与范畴视角。
A classical result by Lovász asserts that two graphs $G$ and $H$ are isomorphic if and only if they have the same left profile, that is, for every graph $F$, the number of homomorphisms from $F$ to $G$ coincides with the number of homomorphisms from $F$ to $H$. Dvor{á}k and later on Dell, Grohe, and Rattan showed that restrictions of the left profile to a class of graphs can capture several different relaxations of isomorphism, including equivalence in counting logics with a fixed number of variables (which contains fractional isomorphism as a special case) and co-spectrality (i.e., two graphs…
On Homomorphism Indistinguishability and Hypertree Depth
本文证明超图上守卫深度至多 $k$ 的 $GC^k$ 语句等价性，恰对应严格超树深度至多 $k$ 类上的同态不可区分性，并将图上的相关结果推广到超图。
Adaptive Query Algorithms for Relational Structures Based on Homomorphism Counts
本文系统比较了基于同态计数的左、右查询算法在非自适应、有界自适应和无界自适应设置下的表达能力，并分析了布尔半环下仅记录同态存在性的情形。
We show that constant-depth Frege systems with counting axioms modulo $m$ polynomially simulate Nullstellensatz refutations modulo $m$. Central to this is a new definition of reducibility from formulas to systems of polynomials with the property that, for most previously studied translations of formulas to systems of polynomials, a formula reduces to its translation. When combined with a previous result of the authors, this establishes the first size separation between Nullstellensatz and polynomial calculus refutations. We also obtain new, small refutations for certain CNFs by constant-depth …
An Algorithmic Meta Theorem for Homomorphism Indistinguishability
本文证明了对于有界树宽且可在CMSO2逻辑中定义的图类，同态不可区分性问题存在随机多项式时间算法，并给出了判定该问题的首个通用算法及其复杂性分析。
Expressive Power of Deep Homomorphism Networks over Relational Databases
本文通过将深度同态网络与一阶逻辑的片段建立联系，研究了其在关系数据库上的表达能力，并分析了空问题和包含问题的可判定性。
Bounding the Weisfeiler-Leman Dimension via a Depth Analysis of I/R-Trees
本研究引入 WL 深度概念，通过分析 I/R 树分支深度，证明计数逻辑下 n/4+o(n) 个变量足以定义任意 n 顶点图，并给出基于顶点覆盖数的新界限。
Kuske and Schweikardt introduced the very expressive first-order counting logic FOC(P) to model database queries with counting operations. They showed that there is an efficient model-checking algorithm on graphs with bounded degree, while Grohe and Schweikardt showed that probably no such algorithm exists for trees of bounded depth. We analyze the fragment FO({>0}) of this logic. While we remove for example subtraction and comparison between two non-atomic counting terms, this logic remains quite expressive: We allow nested counting and comparison between counting terms and arbitrarily large …
Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing
本文研究了同态不可区分性与多重自动机等价性问题，证明了其复杂度为C_L完全，并展示了算法在特定图类上的最优性。
When do homomorphism counts help in query algorithms?
本文研究同态计数在查询算法中的作用，证明对同态等价封闭的性质而言，布尔半环 B 与非负整数半环 N 上的左查询算法能力等价，并刻画了同时允许左右查询算法的性质。
Lovász (1967) showed that two finite relational structures A and B are isomorphic if, and only if, the number of homomorphisms from C to A is the same as the number of homomorphisms from C to B for any finite structure C. Soon after, Pultr (1973) proved a categorical generalisation of this fact. We propose a new categorical formulation, which applies to any locally finite category with pushouts and a proper factorisation system. As special cases of this general theorem, we obtain two variants of Lovász' theorem: the result by Dvořák (2010) that characterises equivalence of graphs in the k-dime…
Given a connected graph $G$ and its vertex $x$, let $U_x(G)$ denote the universal cover of $G$ obtained by unfolding $G$ into a tree starting from $x$. Let $T=T(n)$ be the minimum number such that, for graphs $G$ and $H$ with at most $n$ vertices each, the isomorphism of $U_x(G)$ and $U_y(H)$ surely follows from the isomorphism of these rooted trees truncated at depth $T$. Motivated by applications in theory of distributed computing, Norris [Discrete Appl. Math. 1995] asks if $T(n)\le n$. We answer this question in the negative by establishing that $T(n)=(2-o(1))n$. Our solution uses basic too…
Simulating Logspace-Recursion with Logarithmic Quantifier Depth
本文证明 LREC= 可定义性质可由带计数量词的一阶逻辑 k 变量片段以对数量词深度表达，并推出常数维 Weisfeiler-Leman 算法可在对数次迭代内识别区间图与弦状无爪图。
We prove that, on bounded expansion classes, every first-order formula with modulo counting is equivalent, in a linear-time computable monadic expansion, to an existential first-order formula. As a consequence, we derive, on bounded expansion classes, that first-order transductions with modulo counting have the same encoding power as existential first-order transductions. Also, modulo-counting first-order model checking and computation of the size of sets definable in modulo-counting first-order logic can be achieved in linear time on bounded expansion classes. As an application, we prove that…
The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, and elsewhere. Its structural and algorithmic properties have demonstrated to play a crucial role in many of those applications. For instance, in the decision CSPs, structural properties of the relational structures involved---like, for example, dismantlability---and their logical characterizations have been instrumental for determining the complexity and other properties of the problem. Topological properties of the solution set such as connecte…
Evaluating Restricted First-Order Counting Properties on Nowhere Dense Classes and Beyond
本文研究带计数扩展的一阶逻辑在无处稠密图类上的高效评估问题，证明无量词公式具有几乎线性FPT算法，并指出在几乎无处稠密类上精确求解不可能但可加性近似，从而为部分覆盖问题给出固定参数算法。
On Numbers of Simplicial Walks and Equivalent Canonizations for Graph Recognition
通过简单折线行走数量定义的SW细化过程，用于识别有限路径宽图的属性，其计算复杂度为$O(kn^{3k})$。
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