Download Abstracts Logic and Random Graphs

Abstracts
Logic and Random Graphs
Lorentz Center Leiden, 31 Aug – 4 Sept 2015
Ove Ahlman (Uppsala University)
Almost sure theories approximating simple structures
For each n 2 N let Kn be a set of graphs (or in general, relational structures) with universe
[n] and let µn be the uniform measure on Kn . A sentence ' in the language is called almost
sure for K = (K, µn )n2N if
lim µn ({N 2 Kn : N |= '}) = 1
n!1
The set of almost sure sentences for K, denoted TK is called the almost sure theory. When Kn
consists of all graphs with node set [n], we get the Rado graph as the countable model for TK .
From a model theoretic point of view the Rado graph have many interesting properties. It is
! categorical, supersimple with SU-rank 1 and has trivial pregeometry. From a combinatorial
point of view the Rado graph is interesting due to its extension properties: If A, B are disjoint
finite sets of nodes, then there is a node c such that c is adjacent to all nodes in A but none
in B.
Looking at further examples, such as if Kn consist of all l partite graphs, Kn consist of all
partial orders or Kn consist of all graphs without rigid automorphism group (but with fixed
structure of the support) we get the same model theoretic properties as for the Rado graph.
In this talk I will explain why this is the case and we will see that this is strongly connected
to the notion of extension properties. I will also show that all ! categorical, supersimple
binary structures with SU-rank 1 and trivial pregeometry are axiomatizeable using almost
sure theories. These results lead to many new open questions regarding which structures are
possible to axiomatize using almost sure theories, but also probabilistic questions regarding
how the approximations may be done.
Justin Brody (Goucher College)
Model Theory of Shelah-Spencer Graphs
For ↵ 2 [0, 1], the Erdös-Renyi model G(n, n ↵ ) is a probability space on graphs with n
vertices in which each potential edge occurs with probability n ↵ . For ↵ irrational, Shelah
and Spencer proved the following 0 1 law:
Theorem 1. For
in the language of graphs, we have either
lim P G(n, n
↵
) |=
=0
lim P G(n, n
↵
) |=
=1
n!1
or
n!1
Thus the set of first-order sentences for which limn!1 P (G(n, n
theory, the almost-sure theory T↵ . We will explore three results:
↵)
|= ) = 1 is a complete
1. (Baldwin and Shelah) For irrational ↵, a graph which is a model of T↵ can be obtained
via an amalgamation construction;
2. (Laskowski) For irrational ↵, T↵ can be axiomatized in a particularly simple way (using
an 89 axiomatizaiton);
3. (Brody and Laskowski) For rational ↵, there are two natural model-theoretic analogues
of the almost-sure theory. One is !-stable and the other is essentially undecidable.
Cameron Freer (MIT)
Exchangeable constructions via model theory
Consider a random structure (e.g., graph) with a fixed countably infinite underlying set. The
random structure is exchangeable when its joint distribution is invariant under permutations
of the underlying set. For example, the countably infinite Erdős-Rényi graph is exchangeable;
moreover, it is almost surely isomorphic to a particular graph, known as the Rado graph. It
is natural to ask what other structures admit exchangeable constructions.
In joint work with Ackerman and Patel, we provide a model-theoretic characterization of
those countably infinite structures admitting exchangeable constructions, and demonstrate
the use of techniques from logic to provide a setting in which such constructions can be
explicitly carried out. We also address related questions, such as what structures admit a
unique exchangeable construction. In the case of graphs, our work is closely tied to the theory
of graphons, and we describe this connection.
Joint work with Nathanael Ackerman, Alex Kruckman, Aleksandra Kwiatkowska, Jaroslav
Nešetřil, Rehana Patel, and Jan Reimann.
Simi Haber (Bar-Ilan University)
First order properties of random geometric graphs
A graph property is first order expressible if it can be written as a formal sentence using the
universal and existential quantifiers with variables ranging over the vertices of the graph, the
usual connectives and the relations = and ⇠, where x ⇠ y stands for adjacency.
First order expressible properties have been studied using random models. That is, by looking
on the possible behavior of first order properties given a probability space of models. The
most extensively studied probability space of graphs is the Erdős-Rényi model. A number of
very attractive and surprising results have been obtained, and by now we have a fairly full
description of the behaviour of first order expressible properties on this model.
The Gilbert model of random graphs is obtained as follows. We take n points uniformly at
random from the d-dimensional unit torus, and join two points by an edge if and only their
distance is at most r. In this talk I will discuss a few results telling a nearly complete story on
first order expressible properties of the Gilbert random graph model. In particular we settle
several conjectures of McColm and of Amit Agarwal and Joel Spencer.
(Joint work with T. Muller)
Peter Heinig (Universität Hamburg)
Hamilton-based versus Hamilton-generated flow-lattices of random graphs
This talk o↵ers three things related to random graphs, integer flows and the complexity of
sublanguages of the language of finite undirected graphs: (1) an unproven conjecture likely
to be approachable by someone well-versed in NP-completeness proofs, (2) an unproven challenging conjecture about G(n, p) whose plausibility is in stark contrast to how inconceivable
a proof seems to be with current methods (a new method seems to be necessary), and (3) the
presentation of related results that I proved in my dissertation.
Lefteris Kirousis (National & Kapodistrian University of Athens and
Computer Technology Institute & Press “Diophantus”)
On the Algorithmic Lovász Local Lemma
with applications on Acyclic Edge Coloring
and (possibly) Satisfiability
The algorithm for Lovász Local Lemma by Moser and Tardos gives a constructive way to
prove the existence of combinatorial objects that satisfy a system of constraints. We present
an alternative probabilistic analysis of the algorithm that does not involve reconstructing the
history of the algorithm. We apply our technique to improve the best known upper bound
to acyclic chromatic index. We will also hint on possible applications of this approach to the
Satisfiability Problem
Teresa Klimosova (University of Warwick)
Infinite dimensional finitely forcible graphon
Graphons are analytic objects associated with convergent sequences of graphs. Problems
from extremal combinatorics and theoretical computer science led to a study of graphons
determined by finitely many subgraph densities, which are referred to as finitely forcible. We
show that there exists a finitely forcible graphon such that the topological space of its typical
vertices has infinite Lebesgue covering dimension, disproving the conjecture by Lovász and
Szegedy. The talk is based on joint work with Roman Glebov and Dan Král’.
Phokion G. Kolaitis (University of California, Santa Cruz and IBM Research-Almaden)
Random Graphs and the Parity Quantifier
The classical zero-one law for first-order logic asserts that the asymptotic probability of every
first-order definable property of finite graphs always exists and is either zero or one. Over the
years, zero-one laws have been established for numerous extensions of first-order logic, including least-fixed point logic, finite-variable infinitary logics, and certain fragments of existential
second-order logic. The zero-one law, however, fails to hold for any logical formalism that is
powerful enough to express parity, that is, the property there is an odd number of elements;
in fact, for such logical formalisms, even the convergence law fails to hold.
In this work, we turn the parity barrier into a feature and systematically investigate the
asymptotic probabilities of properties of finite graphs expressible in first-order logic augmented
with the parity quantifier. Our main result is a modular convergence law that captures the
limiting behavior of properties expressible in this extension of first-order logic on finite graphs.
This is joint work with Swastik Kopparty, Rutgers University.
Vera Koponen (Uppsala University)
On logical limit laws that are not zero-one laws
In some natural cases of finite graphs/structures a logical limit law, but not zero-one law,
arises. This happens, for example, if one considers regular graphs, graphs with a fixed bound
on the maximum degree, or graphs in which a certain fixed graph may not appear as a
subgraph. In proving these results the so-called configuration model plays a crucial role.
From a model theoretic point of view one may note that, in the mentioned cases, it makes
sense to speak about “limit theories” (as analogies of the “the almost sure theory” in the cases
of a zero-one law). In each case there are 2! limit theories, each one of which is supersimple
with SU-rank 1 but not !-categorical.
Another case in which limit, but not zero-one, laws arise is if one considers classes of finite
structures which contain only nonrigid structures. Here the methods are more “traditional”,
evolving around extension axioms, but of course the nonrigidness of the structures considered
causes some complications. In this context the well known zero-one law of finite relational
structures due to Glebskii et al and Fagin appears as a particular instance of an infinite “hierarchy” of limit laws (but not necessarily zero-one laws) for structures with some “minimum
complexity” of their automorphism group. In each instance one gets a finite number of “limit
theories” each one of which is supersimple with SU-rank 1 and !-categorical.
I will mention some results and problems concerning these themes.
Alex Kruckman (UC Berkeley)
Properly ergodic random structures
One natural notion of ”random L-structure” is a probability measure on the space of Lstructures with domain ! which is invariant and ergodic for the natural action of S1 =
Sym(!) on this space. Such measures arise naturally as limits of sequences of finite structures
which are convergent in the appropriate sense, generalizing the graph limits of Lovsz and
Szegedy. Ergodicity tells us that the set of models of any sentence of the infinitary logic L!1 ,!
has measure 0 or 1, so it makes sense to talk about the complete theory of a random structure.
Some random structures are almost surely isomorphic to a given countable structure, and the
countable structures which admit an invariant measure in this sense have been completely
characterized by Ackerman, Freer, and Patel. In this talk, we will consider properly ergodic
random structures, those which do not concentrate on any single isomorphism type. The
main result is a counting types characterization of those sentences of L!1 ,! which admit
properly ergodic random models. As a corollary, we show that the complete L!1 ,! -theory of
any properly ergodic random structure has no models (of any cardinality), but its restriction
to any countable fragment has continuum-many countable models. This is joint work with
Ackerman, Freer, and Patel.
Nans Lefevbre (Université Paris Diderot)
Convergence laws for random hyper-graphs with prescribed degrees
The most famous result in the logical study of random structures is the zero-one law on
random graphs, which states that every first-order definable property is asymptotically true
on almost all graphs or on almost none. In more recent years, Lynch proved that every firstorder property has a limit probability on classes of graphs with prescribed degree sequences.
We show how to extend this to a general theorem on hyper-graphs, that allows to recover
classical results on random graphs as special cases. It relies on using the incidence graph as
logical structure, and using the bipartite configuration model to generate hyper-graphs with
chosen degree sequences.
Tomasz Luczak (Adam Mickiewicz University, Poznań)
The logic of the phase transition
The well known result of Erdős-Rényi states that if p = c/n and c < 1 then a.a.s. the random
graph G(n, p) consists of small components of size O(log n), while for c > 1 a.a.s. its structure
is dominated by a giant component which covers a positive fraction of all vertices of the graph.
In 1994 Shelah and Spencer proved that the point of the phase transition is also pivotal for
the monadic second order properties of G(n, p). If pn = c + o(1) for some 0 < c < 1, then
the probability of each such property of G(n, p) converges. On the other hand, if pn = c > 1,
then there exists a monadic second order property such that the probability that G(n, p)
has
approaches both 0 and 1 infinitely often. In the talk we study the monadic second
order properties of G(n, p) near the point of the phase transition, when np = 1 + o(1).
James F. Lynch (Clarkson University)
Logic, Random Graphs, and Systems Biology
The talk will be focused on a class of random structures that originated in the study of
biochemical reaction systems. These systems are often formalized as Markov chains whose
states are finite structures called site graphs. We will describe the biology that motivated
this formalization and argue that it leads naturally to the consideration of certain probability
distributions on site graphs. We state a convergence law for a first-order logic of site graphs:
the probability of every sentence in the logic converges to a limit as the size of the graphs
increases. An outline of the proof will be sketched.
Marc Noy (UPC Barcelona)
Logical limit laws for classes of graphs
Zero-one laws have been studied not only for the G(n, p) model, but also for constrained
classes of graphs, such as regular graphs, H-free graphs or trees. After reviewing results on
these classes, we focus on minor-closed classes. Extending a result by McColm (2002) on
labelled trees, we prove the zero-one law in monadic second order (MSO) logic for connected
graphs in a minor-closed class G which in addition is addable, that is, closed under taking
components and blocks. For arbitrary graphs in G we prove the convergence law.
The class GS of graphs embeddable in a fixed surface S is minor-closed but not addable.
For connected graphs in GS we show the zero-one law in first order logic. In contrast, we
construct formulas in MSO whose limiting probability is any rational number between 0 and
1.
Moumanti Podder (New York University)
Probability of any given neighbourhood of the root, conditioned on the
tree being infinite
Infinite Poisson Galton-Watson trees almost surely contain universal subtrees at sufficiently
large distance away from the root. The Ehrenfeucht value of such trees is determined by
the neighbourhood of the root of radius approximately 4k , where k is the fixed number of
moves in the Ehrenfeucht game. This is why we need to be able to compute the probability of
having a given neighbourhood of the root, conditioned on the tree being infinite. In this talk,
I describe a recursive procedure to compute these conditional probabilities in a systematic
way, where the recursion is on the number of generations away from the root up to which we
consider the neighbourhood. This talk links up closely with Joel Spencer’s talk “The First
Order World of Galton-Watson Trees”.
Co-author(s): Joel Spencer, Courant Institute of Mathematical Sciences, New York University
Andrei M. Raigorodskii (Moscow State University)
Zero-one laws for random subgraphs of some distance graphs
In 1969 Glebskii, Kogan, Liogonkii, and Talanov initiated the study of the so-called zero-one
laws for random graphs in the Erdős–Rényi binomial model. More precisely, let G(n, p) be
a random graph on n vertices whose edges are drawn independently, each with probability
p = p(n). Glebskii et al. proved that any graph property that can be expressed in the firstorder language has probability that tends either to 1 or to 0 as n ! 1, provided p = 12 . The
same is true for many other functions p = p(n), and this is what we mean by a zero-one law.
In our work, we study a model of random subgraphs of some distance graphs in Rn . For this
model, we prove analogs of zero-one laws. At the same time, we show that, in some regimes,
even for a constant edge probability, no zero-one law hold.
Joel Spencer (New York University)
The First Order World of Galton-Watson Trees
In the classic Galton-Watson tree each node has, independently, Poisson mean (a parameter)
children. We concentrate on first order properties A of rooted trees. We let f ( ) denote the
probability of A holding. We show that f ( ) is determined by a finite set of equations
involving only polynomials and exponentials. We further show that f ( ) is a real analytic
function of . Central to the argument is the locality of the corresponding Ehrenfeucht game
on rooted trees.
Caroline Terry (UI Chicago)
Discrete metric spaces: global structure, enumeration, and 0-1 laws
Fix an integer r 3. Given an integer n, we define Mr (n) to be the set of metric spaces with
underlying set {1, . . . , n} such that the distance between any two points lies in {1, . . . , r}.
We present results describing the approximate structure of these metric spaces when n is
large. We then present consequences of these structural results, including an asymptotic
enumeration for Mr (n), and in the case when r is even, a first-order labeled 0-1 law. This is
joint work with Dhruv Mubayi.
Oleg Verbitsky (HU Berlin)
Canonizing almost all graphs
Babai, Erdős, and Selkow (1980) have shown that the color refinement procedure with high
probability splits the vertex set of a random graph into singletons. This gives a very efficient
graph canonization algorithm that succeeds for almost all input graphs. Another consequence
of this result is that almost all graphs are definable in two-variable first-order logic with
counting quantifiers. We give an efficient algorithm recognizing all graphs definable in this
logic and also prove that all such graphs are canonizable in polynomial time.
Moreover, we prove that all graphs definable in two-variable counting logic have integral
polytopes of fractional automorphisms. This implies that Tinhofer’s linear-programming
approach to isomorphism testing has at least as large applicability range as the combinatorial
color refinement method.
This is joint work with V. Arvind, Johannes Köbler, and Gaurav Rattan [FCT’15,MFCS’15].
Lutz Warnke (University of Cambridge)
Counting extensions revisited
Rooted extensions (such as the number of triangles containing a given vertex) play an important rule in the study of zero-one laws for random graphs. In 1990 Spencer gave sufficient
conditions for the event that whp all roots of G(n, p) have the ‘same’ asymptotic number of
extensions, i.e., (1 ± ✏) times their expected number (for constant ✏). Perhaps surprisingly,
the question whether these conditions are necessary has remained open. In this talk we qualitatively resolve this problem for the ‘strictly balanced’ case (also allowing for ✏ n ↵ ). We
also mention some intriguing questions that remain open.
Based on joint work with Matas Šileikis.
M.E. Zhukovskii (Moscow Institute of Physics and Technology)
On infinite spectra of first-order properties
The random graph G(n, p) obeys Zero-One Law if for each first-order property its probability
tends to 0 or tends to 1. In [1], S. Shelah and J. Spencer showed that if ↵ is an irrational
positive number and p(n) = n ↵+o(1) , then G(n, n ↵ ) obeys Zero-One Law.
The random graph G(n, p) obeys Zero-One k-Law if for each first-order property which is
expressed by a first-order formulae with quantifier depth at most k its probability tends to 0
or tends to 1.
We let Lk and L denote the set of properties which are expressed by first-order formulae
with quantifier depthes at most k and the set of all first-order properties respectively. For
any L 2 L, we define two notions of its spectra, S 1 (L) and S 2 (L). The first considers
p = n ↵ . S 1 (L) is the set of ↵ 2 (0, 1) which does not satisfy the following property:
limn!1 P(G(n, n ↵ ) |= L) exists and is either zero or one. The second considers p = n ↵+o(1) .
S 2 (L) is the set of ↵ 2 (0, 1) which does not satisfy the following property: there exists
2 {0, 1} and ✏ > 0 so that when n ↵ ✏ < p(n) < n ↵+✏ , limn!1 P(G(n, p(n)) |= L) = .
It can be shown that for any rational ↵ 2 (0, 1) there is a first-order
S property L such that
↵ 2 S 1 (L). So, the Zero-One Law of Shelah and Spencer implies L2L S 1 (L) = Q \ (0, 1).
It is easy to show that letting L be the property of every two vertices to have a common
neighbor, S 2 (L) = { 12 } while S 1 (L) = ?. Thus, for any L 2 L, we have S 1 (L) ⇢ S 2 (L) but
there may not be the equality. However, in [1] Shelah and Spencer proved that every S 2 (L)
consists only of rational numbers as well. Let Sk1 be the union of S 1 (L) over all L 2 Lk , Sk2
be the union of S 2 (L) over all L 2 Lk .
In the talk, we present the new results on Zero-One k-Law for G(n, n ↵ ) and the distributions
of the elements of Sk1 and Sk2 . In particular, we find bounds on the minimal and the maximal
limit points in Sk1 and Sk2 . Moreover, we prove that the minimal k such that the set Sk1 (Sk2 )
is infinite is either 4 or 5.
References
[1] S. Shelah, J.H. Spencer, Zero-one laws for sparse random graphs, J. Amer. Math. Soc.
1: 97–115, 1988.