Infinite Trees Research Articles

Ambiguity Hierarchy of Regular Infinite Tree Languages

An automaton is unambiguous if for every input it has at most one accepting computation. An automaton is k-ambiguous (for k > 0) if for every input it has at most k accepting computations. An automaton is boundedly ambiguous if it is k-ambiguous for some $k \in \mathbb{N}$. An automaton is finitely (respectively, countably) ambiguous if for every input it has at most finitely (respectively, countably) many accepting computations. The degree of ambiguity of a regular language is defined in a natural way. A language is k-ambiguous (respectively, boundedly, finitely, countably ambiguous) if it is accepted by a k-ambiguous (respectively, boundedly, finitely, countably ambiguous) automaton. Over finite words every regular language is accepted by a deterministic automaton. Over finite trees every regular language is accepted by an unambiguous automaton. Over $\omega$-words every regular language is accepted by an unambiguous B\"uchi automaton and by a deterministic parity automaton. Over infinite trees Carayol et al. showed that there are ambiguous languages. We show that over infinite trees there is a hierarchy of degrees of ambiguity: For every k > 1 there are k-ambiguous languages that are not k - 1 ambiguous; and there are finitely (respectively countably, uncountably) ambiguous languages that are not boundedly (respectively finitely, countably) ambiguous.

Entire functions arising from trees

Given any infinite tree in the plane satisfying certain topological conditions, we construct an entire function f with only two critical values ±1 and no asymptotic values such that f−1([−1, 1]) is ambiently homeomorphic to the given tree. This can be viewed as a generalization of the result of Grothendieck (see Schneps (1994)) to the case of infinite trees. Moreover, a similar idea leads to a new proof of the result of Nevanlinna (1932) and Elfving (1934).

On spherical unitary representations of groups of spheromorphisms of Bruhat–Tits trees

Consider an infinite homogeneous tree \mathcal{T}_n of valence n+1 , its group Aut (\mathcal{T}_n) of automorphisms, and the group Hier (\mathcal{T}_n) of its spheromorphisms (hierarchomorphisms), i.e., the group of homeomorphisms of the boundary of \mathcal{T}_n that locally coincide with transformations defined by automorphisms. We show that the subgroup Aut (\mathcal{T}_n) is spherical in Hier (\mathcal{T}_n) , i.e., any irreducible unitary representation of Hier (\mathcal{T}_n) contains at most one Aut (\mathcal{T}_n) -fixed vector. We present a combinatorial description of the space of double cosets of Hier (\mathcal{T}_n) with respect to Aut (\mathcal{T}_n) and construct a “new” family of spherical representations of Hier (\mathcal{T}_n) . We also show that the Thompson group Th has PSL (2,\mathbb{Z}) -spherical unitary representations.

Shannon-McMillan theorem and strong law of large numbers for Markov chains indexed by generalized spherically symmetric trees

In this paper, we investigate some strong limit properties for Markov chains indexed by a class of infinite trees, which are the generalizations of the spherically symmetric tree. We first give the definition of the generalized spherically symmetric tree model, then obtain the strong law of large numbers for Markov chain indexed by such tree. Finally, we establish the Shannon-McMillan theorem for Markov chains indexed by generalized spherically symmetric tree by applying the Stolz theorem. The obtained properties are generalizations of some known results in the existing literature.

Continuous phase transitions on Galton–Watson trees

AbstractDistinguishing between continuous and first-order phase transitions is a major challenge in random discrete systems. We study the topic for events with recursive structure on Galton–Watson trees. For example, let $\mathcal{T}_1$ be the event that a Galton–Watson tree is infinite and let $\mathcal{T}_2$ be the event that it contains an infinite binary tree starting from its root. These events satisfy similar recursive properties: $\mathcal{T}_1$ holds if and only if $\mathcal{T}_1$ holds for at least one of the trees initiated by children of the root, and $\mathcal{T}_2$ holds if and only if $\mathcal{T}_2$ holds for at least two of these trees. The probability of $\mathcal{T}_1$ has a continuous phase transition, increasing from 0 when the mean of the child distribution increases above 1. On the other hand, the probability of $\mathcal{T}_2$ has a first-order phase transition, jumping discontinuously to a non-zero value at criticality. Given the recursive property satisfied by the event, we describe the critical child distributions where a continuous phase transition takes place. In many cases, we also characterise the event undergoing the phase transition.

Abstract geometrical computation 11: Slanted firing squad synchronisation on signal machines

Firing Squad Synchronisation on Cellular Automata is the dynamical synchronisation of finitely many cells without any prior knowledge of their range. This can be conceived as a signal with an infinite speed. Most of the proposed constructions naturally translate to the continuous setting of signal machines and generate fractal figures with an accumulation on a horizontal line, i.e. synchronously, in the space-time diagram. Signal machines are studied in a series of articles named Abstract Geometrical Computation.In the present article, we design a signal machine that is able to synchronise/accumulate on any non-infinite slope. The slope is encoded in the initial configuration. This is done by constructing an infinite tree such that each node computes the way the tree expands.The interest of Abstract Geometrical computation is to do away with the constraint of discrete space, while tackling new difficulties from continuous space. The interest of this paper in particular is to provide basic tools for further study of computable accumulation lines in the signal machine model.

Collapsible Pushdown Parity Games

This article studies a large class of two-player perfect-information turn-based parity games on infinite graphs, namely, those generated by collapsible pushdown automata. The main motivation for studying these games comes from the connections from collapsible pushdown automata and higher-order recursion schemes, both models being equi-expressive for generating infinite trees. Our main result is to establish the decidability of such games and to provide an effective representation of the winning region as well as of a winning strategy. Thus, the results obtained here provide all necessary tools for an in-depth study of logical properties of trees generated by collapsible pushdown automata/recursion schemes.

On the intersection conjecture for infinite trees of matroids

Using a new technique, we prove a rich family of special cases of the matroid intersection conjecture. Roughly, we prove the conjecture for pairs of tame matroids which have a common decomposition by 2-separations into finite parts.

On degrees of ambiguity for Büchi tree automata

An automaton is unambiguous if for every input it has at most one accepting computation. An automaton is finitely (respectively, countably) ambiguous if for every input it has at most finitely (respectively, countably) many accepting computations. An automaton is boundedly ambiguous if there is k∈N, such that for every input it has at most k accepting computations. We consider nondeterministic Büchi automata (NBA) over infinite trees and prove that it is decidable in polynomial time, whether an automaton is unambiguous, boundedly ambiguous, finitely ambiguous, or countably ambiguous.

Graphoidally independent infinite graphs

A graphoidal cover of a graph G (not necessarily finite) is a collection ψ of paths in G, called ψ-edges, (not necessarily finite, not necessarily open) satisfying the following axioms: (GC-1) Every vertex of G is an internal vertex of at most one path in ψ, and (GC-2) every edge of G is in exactly one path in ψ. The pair is called a graphoidally covered graph. In a graphoidally covered graph two distinct vertices u and v of G are ψ-adjacent if they are the ends of a finite open path in ψ. A graphoidally covered graph (or G) in which no two distinct vertices are ψ-adjacent is called ψ-independent and a graph G possessing a graphoidal cover ψ such that G is ψ-independent is called a graphoidally independent graph. This paper is an attempt to characterize graphoidally independent infinite graphs. In this paper we establish complete characterization of graphoidally independent infinite trees, infinite unicyclic graphs and infinite 2-edge connected graphs.

Higher-order Recursion Schemes and Collapsible Pushdown Automata: Logical Properties

This article studies the logical properties of a very general class of infinite ranked trees, namely, those generated by higher-order recursion schemes. We consider, for both monadic second-order logic and modal -calculus, three main problems: model-checking, logical reflection (a.k.a. global model-checking, that asks for a finite description of the set of elements for which a formula holds), and selection (that asks, if exists, for some finite description of a set of elements for which an MSO formula with a second-order free variable holds). For each of these problems, we provide an effective solution. This is obtained, thanks to a known connection between higher-order recursion schemes and collapsible pushdown automata and on previous work regarding parity games played on transition graphs of collapsible pushdown automata.

Trees in Positive Entropy Subshifts

I give a simple proof for the fact that positive entropy subshifts contain infinite binary trees where branching happens synchronously in each branch, and that the branching times form a set with positive lower asymptotic density.

Exponential independence in subcubic graphs

A set S of vertices of a graph G is exponentially independent if, for every vertex u in S,∑v∈S∖{u}(12)dist(G,S)(u,v)−1<1, where dist(G,S)(u,v) is the distance between u and v in the graph G−(S∖{u,v}). The exponential independence numberαe(G) of G is the maximum order of an exponentially independent set in G. In the present paper we present several bounds on this parameter and highlight some of the many related open problems. In particular, we prove that subcubic graphs of order n have exponentially independent sets of order Ω(n/log2⁡(n)), that the infinite cubic tree has no exponentially independent set of positive density, and that subcubic trees of order n have exponentially independent sets of order (n+3)/4.

Branching Densities of Cube-Free and Square-Free Words

Binary cube-free language and ternary square-free language are two “canonical” representatives of a wide class of languages defined by avoidance properties. Each of these two languages can be viewed as an infinite binary tree reflecting the prefix order of its elements. We study how “homogenious” these trees are, analysing the following parameter: the density of branching nodes along infinite paths. We present combinatorial results and an efficient search algorithm, which together allowed us to get the following numerical results for the cube-free language: the minimal density of branching points is between 3509/9120≈0.38476 and 13/29≈0.44828, and the maximal density is between 0.72 and 67/93≈0.72043. We also prove the lower bound 223/868≈0.25691 on the density of branching points in the tree of the ternary square-free language.

Алгоритмы преобразования конечных автоматов, соответствующих бесконечным итерационным деревьям

In this paper, we work with some different variants of finite automata, each of which corresponds to an infinite iterative tree constructed for some given morphism. At the same time, each of the automata constructed for a given morphism describes the main properties of this morphism. Besides, in each case (i.e., for each variant of the automaton), the following “inverse problem” also arises: to describe the morphism (or simply specify a pair of languages) for which such a given automaton is obtained. We present a computer program for constructing one such automaton, so-called PRI automaton. After that, we consider a detailed example of a PRI automaton for a pair of different languages. Continuing to consider this example, we use the last automaton to perform usual transformations described and repeatedly applied in our previous publications, i.e., the determination and canonization of the mirror automaton for possible application of the results obtained in the algorithm for minimizing nondeterministic automata. In the considered situation, such a minimal automaton is another automaton constructed on the basis of a given morphism tree, a nondeterministic automaton, the so-called NSPRI# automaton, and we also show the equality of these automata (which implies the equivalence of PRI and NSPRI#) in the paper by an example. Based on the NSPRI# automaton, a non-deterministic NSPRI automaton is constructed using a trivial (but non-equivalent) transformation; a detailed study of this automaton is expected in future publications. Examples of PRI and NSPRI# automata for pairs of matching languages are also of interest, we also give one such example in this paper.

Wave Propagation for Reaction-Diffusion Equations on Infinite Random Trees

The asymptotic wave speed for FKPP type reaction-diffusion equations on a class of infinite random metric trees are considered. We show that a travelling wavefront emerges, provided that the reaction rate is large enough. The wavefront travels at a speed that can be quantified via a variational formula involving the random branching degrees $\vec{d}$ and the random branch lengths $\vec{\ell}$ of the tree. This speed is slower than that of the same equation on the real line $\mathbb{R}$, and we estimate this slow down in terms of $\vec{d}$ and $\vec{\ell}$. Our key idea is to project the Brownian motion on the tree onto a one-dimensional axis along the direction of the wave propagation. The projected process is a multi-skewed Brownian motion, introduced by Ramirez [Multi-skewed Brownian motion and diffusion in layered media, Proc. Am. Math. Soc., Vol. 139, No. 10, pp.3739-3752, 2011], with skewness and interface sets that encode the metric structure $(\vec{d}, \vec{\ell})$ of the tree. Combined with analytic arguments based on the Feynman-Kac formula, this idea connects our analysis of the wavefront propagation to the large deviations principle (LDP) of the multi-skewed Brownian motion with random skewness and random interface set. Our LDP analysis involves delicate estimates for an infinite product of $2\times 2$ random matrices parametrized by $\vec{d}$ and $\vec{\ell}$ and for hitting times of a random walk in random environment. By exhausting all possible shapes of the LDP rate function (action functional), the analytic arguments that bridge the LDP and the wave propagation overcome the random drift effect due to multi-skewness.

Quantum ergodicity for expanding quantum graphs in the regime of spectral delocalization

We consider a sequence of finite quantum graphs with few loops, so that they converge, in the sense of Benjamini-Schramm, to a random infinite quantum tree. We assume these quantum trees are spectrally delocalized in some interval I, in the sense that their spectrum in I is purely absolutely continuous and their Green's functions are well controlled near the real axis. We furthermore suppose that the underlying sequence of discrete graphs is expanding. We deduce a quantum ergodicity result, showing that the eigenfunctions with eigenvalues lying in I are spatially delocalized.

A nonamenable “factor” of a Euclidean space

Answering a question of Benjamini, we present an isometry-invariant random partition of the Euclidean space Rd, d≥3, into infinite connected indistinguishable pieces, such that the adjacency graph defined on the pieces is the 3-regular infinite tree. Along the way, it is proved that any finitely generated one-ended amenable Cayley graph can be represented in Rd as an isometry-invariant random partition of Rd to bounded polyhedra, and also as an isometry-invariant random partition of Rd to indistinguishable pieces. A new technique is developed to prove indistinguishability for certain constructions, connecting this notion to factor of IID’s.

Research on a novel construction of probabilistic visual cryptography scheme (k,n,0,1,1)−PVCS for threshold access structures

In this paper, we propose new algorithms to construct a probabilistic visual cryptography scheme (k,n,0,1,1)−PVCS for threshold access structures based on the deterministic visual cryptography schemes (DVCS) and classify the probabilistic schemes into online and offline ones according to the reference mode of distribution column vectors.First in consideration of the relationship between DVCS and PVCS we propose a new method for expanded DVCS based on a matrix with binary vectors of Vnk as column vectors so as to construct PVCS without pixel expansion.Then, using the extended DVCS we suggest PVCS construction method which does not require storing of the distribution column vectors and delineate the relationship between our online PVCS and visual cryptograms of random grids (VCRG).We also propose a new infinite tree in order to generate binary vectors of Vnk.

The majority vote process and other consensus processes on trees

The majority vote process was one of the first interacting particle systems to be investigated. It can be described briefly as follows. There are two possible opinions at each site of a graph G. At rate 1−e, the opinion at a site aligns with the majority opinion at its neighboring sites and, at rate e, the opinion at a site is randomized due to noise, where e∈[0,1] is a parameter. Despite the simple dynamics of the majority vote process, its equilibrium behavior is difficult to analyze when the noise rate is small but positive. In particular, when the underlying graph is G=Zn with n≥2, it is not known whether the process possesses more than one equilibrium. This is surprising, especially in light of the close analogy between this model and the stochastic Ising model, where much more is known. Here, we study the majority vote process on the infinite tree Td with vertex degree d. For d≥5 and small noise, we show that there are uncountably many mutually singular equilibria, with convergence to such an equilibrium occurring exponentially quickly from nearby initial states. Our methods are quite flexible and extend to a broader class of models, consensus processes. This class includes the stochastic Ising model and other processes in which the dynamics at a site depend on the number of neighbors holding a given opinion. All of our proofs are carried out in this broader context.