Primitive digraphs with large exponents and slowly synchronizing automata

We present several infinite series of synchronizing automata for which the minimum length of reset words is close to the square of the number of states. All these automata are tightly related to primitive digraphs with large exponent.Comment: 23 pages, 11 figures, 3 tables. This is a translation (with a slightly updated bibliography) of the authors' paper published in Russian in: Zapiski Nauchnyh Seminarov POMI [Kombinatorika i Teorija Grafov. IV], Vol. 402, 9-39 (2012), see ftp://ftp.pdmi.ras.ru/pub/publicat/znsl/v402/p009.pdf Version 2: a few typos are correcte

On Synchronizing Colorings and the Eigenvectors of Digraphs

An automaton is synchronizing if there exists a word that sends all states of the automaton to a single state. A coloring of a digraph with a fixed out-degree k is a distribution of k labels over the edges resulting in a deterministic finite automaton. The famous road coloring theorem states that every primitive digraph has a synchronizing coloring. We study recent conjectures claiming that the number of synchronizing colorings is large in the worst and average cases. Our approach is based on the spectral properties of the adjacency matrix A(G) of a digraph G. Namely, we study the relation between the number of synchronizing colorings of G and the structure of the dominant eigenvector v of A(G). We show that a vector v has no partition of coordinates into blocks of equal sum if and only if all colorings of the digraphs associated with v are synchronizing. Furthermore, if for each b there exists at most one partition of the coordinates of v into blocks summing up to b, and the total number of partitions is equal to s, then the fraction of synchronizing colorings among all colorings of G is at least (k-s)/k. We also give a combinatorial interpretation of some known results concerning an upper bound on the minimal length of synchronizing words in terms of v

On the interplay between Babai and Cerny's conjectures

Motivated by the Babai conjecture and the Cerny conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with $n$ states in this class, we prove that the reset thresholds are upper-bounded by $2n^2-6n+5$ and can attain the value $\tfrac{n(n-1)}{2}$. In addition, we study diameters of the pair digraphs of permutation automata and construct $n$-state permutation automata with diameter $\tfrac{n^2}{4} + o(n^2)$.Comment: 21 pages version with full proof

Principal ideal languages and synchronizing automata

We study ideal languages generated by a single word. We provide an algorithm to construct a strongly connected synchronizing automaton for which such a language serves as the language of synchronizing words. Also we present a compact formula to calculate the syntactic complexity of this language.Comment: 15 pages, 9 figure

Faster Exploration of Some Temporal Graphs

A temporal graph G = (G_1, G_2, ..., G_T) is a graph represented by a sequence of T graphs over a common set of vertices, such that at the i-th time step only the edge set E_i is active. The temporal graph exploration problem asks for a shortest temporal walk on some temporal graph visiting every vertex. We show that temporal graphs with n vertices can be explored in O(k n^{1.5} log n) days if the underlying graph has treewidth k and in O(n^{1.75} log n) days if the underlying graph is planar. Furthermore, we show that any temporal graph whose underlying graph is a cycle with k chords can be explored in at most 6kn days. Finally, we demonstrate that there are temporal realisations of sub cubic planar graphs that cannot be explored faster than in ?(n log n) days. All these improve best known results in the literature

The K-Centre Problem for Necklaces

In graph theory, the objective of the k-centre problem is to find a set of $k$ vertices for which the largest distance of any vertex to its closest vertex in the $k$-set is minimised. In this paper, we introduce the $k$-centre problem for sets of necklaces, i.e. the equivalence classes of words under the cyclic shift. This can be seen as the k-centre problem on the complete weighted graph where every necklace is represented by a vertex, and each edge has a weight given by the overlap distance between any pair of necklaces. Similar to the graph case, the goal is to choose $k$ necklaces such that the distance from any word in the language and its nearest centre is minimised. However, in a case of k-centre problem for languages the size of associated graph maybe exponential in relation to the description of the language, i.e., the length of the words l and the size of the alphabet q. We derive several approximation algorithms for the $k$-centre problem on necklaces, with logarithmic approximation factor in the context of l and k, and within a constant factor for a more restricted case

Attainable Values of Reset Thresholds

An automaton is synchronizing if there exists a word that sends all states of the automaton to a single state. The reset threshold is the length of the shortest such word. We study the set RT_n of attainable reset thresholds by automata with n states. Relying on constructions of digraphs with known local exponents we show that the intervals [1, (n^2-3n+4)/2] and [(p-1)(q-1), p(q-2)+n-q+1], where 2 n, gcd(p,q)=1, belong to RT_n, even if restrict our attention to strongly connected automata. Moreover, we prove that in this case the smallest value that does not belong to RT_n is at least n^2 - O(n^{1.7625} log n / log log n). This value is increased further assuming certain conjectures about the gaps between consecutive prime numbers. We also show that any value smaller than n(n-1)/2 is attainable by an automaton with a sink state and any value smaller than n^2-O(n^{1.5}) is attainable in general case. Furthermore, we solve the problem of existence of slowly synchronizing automata over an arbitrarily large alphabet, by presenting for every fixed size of the alphabet an infinite series of irreducibly synchronizing automata with the reset threshold n^2-O(n)

Site-Net: Using global self-attention and real-space supercells to capture long-range interactions in crystal structures

Site-Net is a transformer architecture that models the periodic crystal structures of inorganic materials as a labelled point set of atoms and relies entirely on global self-attention and geometric information to guide learning. Site-Net processes standard crystallographic information files to generate a large real-space supercell, and the importance of interactions between all atomic sites is flexibly learned by the model for the prediction task presented. The attention mechanism is probed to reveal Site-Net can learn long-range interactions in crystal structures, and that specific attention heads become specialized to deal with primarily short- or long-range interactions. We perform a preliminary hyperparameter search and train Site-Net using a single graphics processing unit (GPU), and show Site-Net achieves state-of-the-art performance on a standard band gap regression task.Comment: 23 pages, 13 figure

Computational Complexity of Synchronization under Regular Constraints

Many variations of synchronization of finite automata have been studied in the previous decades. Here, we suggest studying the question if synchronizing words exist that belong to some fixed constraint language, given by some partial finite automaton called constraint automaton. We show that this synchronization problem becomes PSPACE-complete even for some constraint automata with two states and a ternary alphabet. In addition, we characterize constraint automata with arbitrarily many states for which the constrained synchronization problem is polynomial-time solvable. We classify the complexity of the constrained synchronization problem for constraint automata with two states and two or three letters completely and lift those results to larger classes of finite automata