Binary simple homogeneous structures are supersimple with finite rank

Suppose that M is an infinite structure with finite relational vocabulary such that every relation symbol has arity at most 2. If M is simple and homogeneous then its complete theory is supersimple with finite SU-rank which cannot exceed the number of complete 2-types over the empty set

A limit law of almost ll-partite graphs

For integers $l \geq 2$, $d \geq 1$ we study (undirected) graphs with vertices $1, ..., n$ such that the vertices can be partitioned into $l$ parts such that every vertex has at most $d$ neighbours in its own part. The set of all such graphs is denoted \mbP_n(l,d). We prove a labelled first-order limit law, i.e., for every first-order sentence $\varphi$, the proportion of graphs in \mbP_n(l,d) that satisfy $\varphi$ converges as $n \to \infty$. By combining this result with a result of Hundack, Pr\"omel and Steger \cite{HPS} we also prove that if $1 \leq s_1 \leq ... \leq s_l$ are integers, then \mb{Forb}(\mcK_{1, s_1, ..., s_l}) has a labelled first-order limit law, where \mb{Forb}(\mcK_{1, s_1, ..., s_l}) denotes the set of all graphs with vertices $1, ..., n$, for some $n$, in which there is no subgraph isomorphic to the complete $(l+1)$-partite graph with parts of sizes $1, s_1, ..., s_l$. In the course of doing this we also prove that there exists a first-order formula $\xi$ (depending only on $l$ and $d$) such that the proportion of \mcG \in \mbP_n(l,d) with the following property approaches 1 as $n \to \infty$: there is a unique partition of $\{1, ..., n\}$ into $l$ parts such that every vertex has at most $d$ neighbours in its own part, and this partition, viewed as an equivalence relation, is defined by $\xi$

Random graphs with bounded maximum degree: asymptotic structure and a logical limit law

For any fixed integer $R \geq 2$ we characterise the typical structure of undirected graphs with vertices $1, ..., n$ and maximum degree $R$, as $n$ tends to infinity. The information is used to prove that such graphs satisfy a labelled limit law for first-order logic. If $R \geq 5$ then also an unlabelled limit law holds

On sets with rank one in simple homogeneous structures

We study definable sets $D$ of SU-rank 1 in $M^{eq}$, where $M$ is a countable homogeneous and simple structure in a language with finite relational vocabulary. Each such $D$ can be seen as a `canonically embedded structure', which inherits all relations on $D$ which are definable in $M^{eq}$, and has no other definable relations. Our results imply that if no relation symbol of the language of $M$ has arity higher than 2, then there is a close relationship between triviality of dependence and $D$ being a reduct of a binary random structure. Somewhat more preciely: (a) if for every $n \geq 2$, every $n$-type $p(x_1, ..., x_n)$ which is realized in $D$ is determined by its sub-2-types $q(x_i, x_j) \subseteq p$, then the algebraic closure restricted to $D$ is trivial; (b) if $M$ has trivial dependence, then $D$ is a reduct of a binary random structure

On the relative asymptotic expressivity of inference frameworks

Let $\sigma$ be a first-order signature and let $\mathbf{W}_n$ be the set of all $\sigma$-structures with domain $\{1, \ldots, n\}$. By an inference framework we mean a class $\mathbf{F}$ of pairs $(\mathbb{P}, L)$, where $\mathbb{P} = (\mathbb{P}_n : n = 1, 2, 3, \ldots)$ and $\mathbb{P}_n$ is a probability distribution on $\mathbf{W}_n$, and $L$ is a logic with truth values in the unit interval $[0, 1]$. An inference framework $\mathbf{F}'$ is asymptotically at least as expressive as another inference framework $\mathbf{F}$ if for every $(\mathbb{P}, L) \in \mathbf{F}$ there is $(\mathbb{P}', L') \in \mathbf{F}'$ such that $\mathbb{P}$ is asymptotically total-variation-equivalent to $\mathbb{P}'$ and for every $\varphi(\bar{x}) \in L$ there is $\varphi'(\bar{x}) \in L'$ such that $\varphi'(\bar{x})$ is asymptotically equivalent to $\varphi(\bar{x})$ with respect to $\mathbb{P}$. This relation is a preorder and we describe a partial order on the equivalence classes of some inference frameworks that seem natural in the context of machine learning and artificial intelligence. Several previous results about asymptotic (or almost sure) equivalence of formulas or convergence in probability can be formulated in terms of relative asymptotic strength of inference frameworks. We incorporate these results in our classification of inference frameworks and prove two new results. Both concern sequences of probability distributions defined by directed graphical models that use ``continuous'' aggregation functions. The first considers queries expressed by a logic with truth values in $[0, 1]$ which employs continuous aggregation functions. The second considers queries expressed by a two-valued conditional logic that can express statements about relative frequencies.Comment: 52 page