Shmuel Lifsches, Saharon Shelah
We continue the works of Gurevich-Shelah and Lifsches-Shelah by showing that it is consistent with ZFC that the first-order theory of random graphs is not interpretable in the monadic theory of all chains. It is provable from ZFC that the theory of random graphs is not interpretable in the monadic second order theory of short chains (hence, in the monadic theory of the real line).
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We study the question of whether, for a given class of finite graphs, one can define, for each graph of the class, a linear ordering in monadic second-order logic, possibly with the help of monadic parameters. We consider two variants of monadic second-order logic: one where we can only quantify over sets of vertices and one where we can also quantify over sets of edges. For several special cases, we present combinatorial characterisations of when such a linear ordering is definable. In some cases, for instance for graph classes that omit a fixed graph as a minor, the presented conditions are …
We consider the domino problem on Schreier graphs of self-similar groups, and more generally their monadic second-order logic. On the one hand, we prove that if the group is bounded then the graph's monadic second-order logic is decidable. This covers, for example, the Sierpiński gasket graphs and the Schreier graphs of the Basilica group. On the other hand, we already prove undecidability of the domino problem for a class of self-similar groups, answering a question by Barbieri and Sablik, and some examples including one of linear growth.
In this paper we survey some surprising connections between group theory, the theory of automata and formal languages, the theory of ends, infinite games of perfect information, and monadic second-order logic.
In 2001, J.-M. Le Bars disproved the zero-one law (that says that every sentence from a certain logic is either true asymptotically almost surely (a.a.s.), or false a.a.s.) for existential monadic second order sentences (EMSO) about undirected graphs. He proved that there exists an EMSO sentence $ϕ$ such that ${\sf P}(G_n \modelsϕ)$ does not converge as $n\to\infty$ (here, the probability distribution is uniform over the set of all graphs on the labeled set of vertices $\{1, \dots, n\}$). In the same paper, he conjectured that, for EMSO sentences with 2 first order variables, the zero-one law …
We study two notions of being well-structured for classes of graphs that are inspired by classic model theory. A class of graphs $C$ is monadically stable if it is impossible to define arbitrarily long linear orders in vertex-colored graphs from $C$ using a fixed first-order formula. Similarly, monadic dependence corresponds to the impossibility of defining all graphs in this way. Examples of monadically stable graph classes are nowhere dense classes, which provide a robust theory of sparsity. Examples of monadically dependent classes are classes of bounded rankwidth (or equivalently, bounded …
Nowhere dense classes of graphs are classes of sparse graphs with rich structural and algorithmic properties, however, they fail to capture even simple classes of dense graphs. Monadically stable classes, originating from model theory, generalize nowhere dense classes and close them under transductions, i.e. transformations defined by colorings and simple first-order interpretations. In this work we aim to extend some combinatorial and algorithmic properties of nowhere dense classes to monadically stable classes of finite graphs. We prove the following results. - In monadically stable classes …
First-Order Model Checking on Monadically Stable Graph Classes
本文证明一阶模型检测在所有单调稳定图类上固定参数可解，并证明边稳定但不单调稳定的遗传类上一阶模型检测是 AW[*]-难和 W[1]-难的，从而刻画了可解性边界。
In this paper we found an upper bound for the minimal quantifier depth of the first part of a monadic second-order sentence without asymptotic probability described by Jerzy Tyszkiewicz, which express the extension grid axiom in the Erdős-Rényi model of random graphs $G(n,n^{-α})$ for some irrational $α$.
Flipper games for monadically stable graph classes
本文通过 Flipper 博弈刻画单调稳定图类，并给出两种证明：一种利用模型论并揭示类型可定义性相关性质，另一种基于 flip-wideness 提供计算 Flipper 获胜走法的高效算法。
Monadic dependence from reducts, and applications to twin-width of oriented graphs
研究二元关系结构的单论依赖性，通过约化给出充分条件，并应用于有向图的双宽度，证明FO模型检测的固定参数可处理性。
In this paper, we study zero-one laws for the Erdős--Rényi random graph model $G(n,p)$ in the case when $p = n^{-α}$ for $α>0$. For a given class $\mathcal{K}$ of logical sentences about graphs and a given function $p=p(n)$, we say that $G(n,p)$ obeys the zero-one law (w.r.t. the class $\mathcal{K}$) if each sentence $φ\in\mathcal{K}$ either a.a.s. true or a.a.s. false for $G(n,p)$. In this paper, we consider first order properties and monadic second order properties of bounded \textit{quantifier depth} $k$, that is, the length of the longest chain of nested quantifiers in the formula expressi…
We prove that for any monotone class of finite relational structures, the first-order theory of the class is NIP in the sense of stability theory if, and only if, the collection of Gaifman graphs of structures in this class is nowhere dense. This generalises to relational structures a result previously known for graphs and answers an open question posed by Adler and Adler (2014). The result is established by the application of Ramsey-theoretic techniques and shows that the property of being NIP is highly robust for monotone classes. We also show that the model-checking problem for first-order …
Existential Positive Transductions of Sparse Graphs
研究提出稀疏化猜想，通过存在性正 FO 公式特征化 co-matching-free 单子稳定类，并发现其与 MSO 的表达能力相同。
A class of graphs is structurally nowhere dense if it can be constructed from a nowhere dense class by a first-order transduction. Structurally nowhere dense classes vastly generalize nowhere dense classes and constitute important examples of monadically stable classes. We show that the first-order model checking problem is fixed-parameter tractable on every structurally nowhere dense class of graphs. Our result builds on a recently developed game-theoretic characterization of monadically stable graph classes. As a second key ingredient of independent interest, we provide a polynomial-time alg…
Positive First-order Logic on Words and Graphs
本文研究有限词上一阶逻辑片段 FO+，证明存在单调但无法用 FO+ 定义的 FO 可定义语言，借此说明 Lyndon 保持定理在有限结构上失效，并将该结论推广至有限图，同时证明判断正则语言是否 FO+ 可定义的问题不可判定。
Constant time testability of first-order logic with modulo counting on finitary graphs
本文研究了在有界度模型下，常数运行时间的属性测试算法元定理。针对有界组件大小的图类，证明了一阶逻辑带模计数（FOMOD）的常数时间可测试性，并引入了数论中的'补充分条件'来推断全局信息。
Logical transductions provide a very useful tool to encode classes of structures inside other classes of structures. In this paper we study first-order (FO) transductions and the quasiorder they induce on infinite classes of finite graphs. Surprisingly, this quasiorder is very complex, though shaped by the locality properties of first-order logic. This contrasts with the conjectured simplicity of the monadic second order (MSO) transduction quasiorder. We first establish a local normal form for FO transductions, which is of independent interest. Then we prove that the quotient partial order is …
Logical properties of random graphs from small addable classes
本文证明了单调二阶逻辑在若干连通图类上满足零一律，而去掉连通性后仅满足收敛律。
We address questions of logic and expressibility in the context of random rooted trees. Infiniteness of a rooted tree is not expressible as a first order sentence, but is expressible as an existential monadic second order sentence (EMSO). On the other hand, finiteness is not expressible as an EMSO. For a broad class of random tree models, including Galton-Watson trees with offspring distributions that have full support, we prove the stronger statement that finiteness does not agree up to a null set with any EMSO. We construct a finite tree and a non-null set of infinite trees that cannot be di…
Monadic Second-Order Logic in HOL: Deep and Shallow with Automated Faithfulness (Extended Preprint)
在Isabelle/HOL中为MSO开发三种嵌入，提出两类替换机制实现自动化忠实性，并机械化证明向下Loewenheim-Skolem定理，区分一般与标准解释，通过经典例子展示差异。
Paul Erdős and Alfred Renyi considered the evolution of the random graph G(n,p) as p ``evolved'' from 0 to 1. At p=1/n a sudden and dramatic change takes place in G. When p=c/n with c<1 the random G consists of small components, the largest of size Theta(log n). But by p=c/n with c>1 many of the components have ``congealed'' into a ``giant component'' of size Theta (n). Erdős and Renyi called this the double jump, the terms phase transition (from the analogy to percolation) and Big Bang have also been proferred. Now imagine an observer who can only see G through a logical fog. He may refer to …
Let $\mathcal G$ be an addable, minor-closed class of graphs. We prove that the zero-one law holds in monadic second-order logic (MSO) for the random graph drawn uniformly at random from all {\em connected} graphs in $\mathcal G$ on $n$ vertices, and the convergence law in MSO holds if we draw uniformly at random from all graphs in $\mathcal G$ on $n$ vertices. We also prove analogues of these results for the class of graphs embeddable on a fixed surface, provided we restrict attention to first order logic (FO). Moreover, the limiting probability that a given FO sentence is satisfied is indepe…
On first-order transductions of classes of graphs
本文研究图类的一阶转导拟序，证明转导的局部范式，并探讨其复杂性、极值类、路径宽度层次及弱稀疏类的作用。
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