Mickel González Sánchez, Maksim Evgenievich Zhukovskii
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 $α$.
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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 …
In this paper, we study existential monadic second order (EMSO) properties of undirected graphs. In 2001, J.-M. Le Bars proved that there exists an EMSO sentence about undirected graphs such that the probability that it is true does not converge (here, the probability distribution is uniform over the set of all graphs on the fixed set of vertices). In the same paper, he conjectured that, for EMSO sentences with 2 first order variables, the 0-1 law holds (every sentence is either true asymptotically almost surely (a.a.s.), or false a.a.s.). We give an example of EMSO sentence with 1 monadic var…
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 conjecture that the class of frame matroids can be characterised by a sentence in the monadic second-order logic of matroids, and we prove that there is such a characterisation for the class of bicircular matroids. The proof does not depend on an excluded-minor characterisation.
Alternating quantifier depth is a natural measure of difficulty required to express first order logical sentences. We define a sequence of first order properties on rooted, locally finite trees in a recursive manner, and provide rigorous arguments for finding the alternating quantifier depth of each property in the sequence, using Ehrenfeucht-Fraïssé games.
For any fixed positive integer $k$, let $α_{k}$ denote the smallest $α\in (0,1)$ such that the random graph sequence $\left\{G\left(n, n^{-α}\right)\right\}$ does not satisfy the zero-one law for the set $\mathcal{E}_{k}$ of all existential first order sentences that are of quantifier depth at most $k$. This paper finds upper and lower bounds on $α_{k}$, showing that as $k \rightarrow \infty$, we have $α_{k} = \left(k - 2 - t(k)\right)^{-1}$ for some function $t(k) = Θ(k^{-2})$. We also establish the precise value of $α_{k}$ when $k = 4$.
Spectrum of a first order sentence is the set of all $α$ such that $G(n, n^{-α})$ does not obey zero-one law w.r.t. this sentence. We have proved that the minimal number of quantifier alternations of a first order sentence with an infinite spectrum equals 3. We have also proved that the spectrum of a first order sentence with a quantifier depth 4 has no limit points except possibly the points 1/2 and 3/5.
Monadic transductions and definable classes of matroids
本文系统建立拟阵单子二阶转导理论，证明后向翻译定理，并据此证明格路拟阵类与尖刺子式类可由单子二阶逻辑定义。
We study the computational complexity of short sentences in Presburger arithmetic (Short-PA). Here by "short" we mean sentences with a bounded number of variables, quantifiers, inequalities and Boolean operations; the input consists only of the integer coefficients involved in the linear inequalities. We prove that satisfiability of Short-PA sentences with $m+2$ alternating quantifiers is $Σ_{P}^m$-complete or $Π_{P}^m$-complete, when the first quantifier is $\exists$ or $\forall$, respectively. Counting versions and restricted systems are also analyzed. Further application are given to hardne…
The classical zero-one law for first-order logic on random graphs says that for any first-order sentence $ϕ$ in the theory of graphs, as n approaches infinity, the probability that the random graph G(n, p) satisfies $ϕ$ approaches either 0 or 1. It is well known that this law fails to hold for any formalism that can express the parity quantifier: for certain properties, the probability that G(n, p) satisfies the property need not converge, and for others the limit may be strictly between 0 and 1. In this paper, we capture the limiting behavior of properties definable in first order logic augme…
We prove that, for every $\ell\geq 4$, there exists an $\ell$-vertex graph and a first order sentence having a quantifier depth at most $\ell-1$ defining the property of having an induced subgraph isomorphic to the given one. We prove that a first order sentence defining the property of containing an induced subgraph on $\ell$ vertices isomorphic to a given disjoint union of isomorphic complete multipartite graphs has a quantifier depth at least $\ell$. Finally, we prove that, for every graph on $\ell\leq 5$ vertices a sentence defining the property of containing an induced subgraph isomorphic…
Spectrum of FO logic with quantifier depth 4 is finite
本文研究随机图 $G(n,n^{-α})$ 上一阶逻辑的0-1律，证明使 $k$-谱无限的最小量词深度为5，从而说明深度4的谱是有限的。
Monadic Second-Order Logic of Permutations
本文研究置换的 monadic second-order logic 的表达能力与模型检测复杂度，证明了其表达力强于一阶逻辑但无法表达不动点性质，并给出了基于 tree-width 的线性时间算法及在固定遗传类下的困难性证明。
We discuss the definability of finite graphs in first-order logic with two relation symbols for adjacency and equality of vertices. The logical depth $D(G)$ of a graph $G$ is equal to the minimum quantifier depth of a sentence defining $G$ up to isomorphism. The logical width $W(G)$ is the minimum number of variables occurring in such a sentence. The logical length $L(G)$ is the length of a shortest defining sentence. We survey known estimates for these graph parameters and discuss their relations to other topics (such as the efficiency of the Weisfeiler-Lehman algorithm in isomorphism testing…
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 disprove EMSO(FO$^2$) convergence law for the binomial random graph $G(n,p)$ for any constant probability $p$. More specifically, we prove that there exists an existential monadic second order sentence with 2 first order variables such that, for every $p\in(0,1)$, the probability that it is true on $G(n,p)$ does not converge.
We say that a first order sentence A defines a graph G if A is true on G but false on any graph non-isomorphic to G. Let L(G) (resp. D(G)) denote the minimum length (resp. quantifier rank) of a such sentence. We define the succinctness function s(n) (resp. its variant q(n)) to be the minimum L(G) (resp. D(G)) over all graphs on n vertices. We prove that s(n) and q(n) may be so small that for no general recursive function f we can have f(s(n))\ge n for all n. However, for the function q^*(n)=\max_{i\le n}q(i), which is the least monotone nondecreasing function bounding q(n) from above, we have …
The logical depth of a graph $G$ is the minimum quantifier depth of a first order sentence defining $G$ up to isomorphism in the language of the adjacency and the equality relations. We consider the case that $G$ is a dissection of a convex polygon or, equivalently, a biconnected outerplanar graph. We bound the logical depth of a such $G$ from above by a function of combinatorial parameters of the dual tree of $G$.
The {\em spectrum} of a first-order logic sentence is the set of natural numbers that are cardinalities of its finite models. In this paper we show that when restricted to using only two variables, but allowing counting quantifiers, the spectra of first-order logic sentences are semilinear and hence, closed under complement. At the heart of our proof are semilinear characterisations for the existence of regular and biregular graphs, the class of graphs in which there are a priori bounds on the degrees of the vertices. Our proof also provides a simple characterisation of models of two-variable …
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