arXiv subjects

Bakhadyr Khoussainov

Publications and source records attributed to Bakhadyr Khoussainov.

13 recordsLinked to original sources

arXiv2024

Large Scale Geometries of Infinite Strings

We introduce geometric consideration into the theory of formal languages. We aim to shed light on our understanding of global patterns that occur on infinite strings. We utilise methods of geometric group theory. Our emphasis is on large scale geometries. Two infinite strings have the same large scale geometry if there are colour preserving bi-Lipschitz maps with distortions between the strings. Call these maps quasi-isometries. Introduction of large scale geometries poses several questions. The first question asks to study the partial order induced by quasi-isometries. This partial order compares large scale geometries; as such it presents an algebraic tool for classification of global patterns. We prove there is a greatest large scale geometry and infinitely many minimal large scale geometries. The second question is related to understanding the quasi-isometric maps on various classes of strings. The third question investigates the sets of large scale geometries of strings accepted by computational models, e.g. Büchi automata. We provide an algorithm that describes large scale geometries of strings accepted by Büchi automata. This links large scale geometries with automata theory. The fourth question studies the complexity of the quasi-isometry problem. We show the problem is $Σ_3^0$-complete thus providing a bridge with computability theory. Finally, the fifth question asks to build algebraic structures that are invariants of large scale geometries. We invoke asymptotic cones, a key concept in geometric group theory, defined via model-theoretic notion of ultra-product. Partly, we study asymptotic cones of algorithmically random strings thus connecting the topic with algorithmic randomness.

cs.LO↗

arXiv2023

String Compression in FA-Presentable Structures

We construct a FA-presentation $ψ: L \rightarrow \mathbb{N}$ of the structure $(\mathbb{N};\mathrm{S})$ for which a numerical characteristic $r(n)$ defined as the maximum number $ψ(w)$ for all strings $w \in L$ of length less than or equal to $n$ grows faster than any tower of exponents of a fixed height. This result leads us to a more general notion of a compressibility rate defined for FA-presentations of any FA-presentable structure. We show the existence of FA-presentations for the configuration space of a Turing machine and Cayley graphs of some groups for which it grows faster than any tower of exponents of a fixed height. For FA-presentations of the Presburger arithmetic $(\mathbb{N};+)$ we show that it is bounded from above by a linear function.

cs.FL↗

arXiv2022

Quasi-Isometric Graph Simplifications

Quasi-isometries are mappings on graphs, with distance-distortions parameterized by a multiplicative factor and an additive constant. The distance-distortions of quasi-isometries are in a general form that captures a wide range of distance-approximating graph simplifications. This paper introduces quasi-isometries into the field of graph simplifications, which is becoming increasingly important as large-scale graphs gain more and more prevalence. We discuss some general goals of graph simplification under the framework of quasi-isometries, and investigate several constructions of quasi-isometric graph simplifications, namely one based on maximal independent sets and one based on grouping vertices. For the latter construction, we prove that it preserves the centers and medians of trees.

cs.DS↗

arXiv2022

Listing Maximal k-Plexes in Large Real-World Graphs

Listing dense subgraphs in large graphs plays a key task in varieties of network analysis applications like community detection. Clique, as the densest model, has been widely investigated. However, in practice, communities rarely form as cliques for various reasons, e.g., data noise. Therefore, $k$-plex, -- graph with each vertex adjacent to all but at most $k$ vertices, is introduced as a relaxed version of clique. Often, to better simulate cohesive communities, an emphasis is placed on connected $k$-plexes with small $k$. In this paper, we continue the research line of listing all maximal $k$-plexes and maximal $k$-plexes of prescribed size. Our first contribution is algorithm ListPlex that lists all maximal $k$-plexes in $O^*(γ^D)$ time for each constant $k$, where $γ$ is a value related to $k$ but strictly smaller than 2, and $D$ is the degeneracy of the graph that is far less than the vertex number $n$ in real-word graphs. Compared to the trivial bound of $2^n$, the improvement is significant, and our bound is better than all previously known results. In practice, we further use several techniques to accelerate listing $k$-plexes of a given size, such as structural-based prune rules, cache-efficient data structures, and parallel techniques. All these together result in a very practical algorithm. Empirical results show that our approach outperforms the state-of-the-art solutions by up to orders of magnitude.

cs.DS↗

arXiv2019

The isomorphism problem for tree-automatic ordinals with addition

This paper studies tree-automatic ordinals (or equivalently, well-founded linearly ordered sets) together with the ordinal addition operation +. Informally, these are ordinals such that their elements are coded by finite trees for which the linear order relation of the ordinal and the ordinal addition operation can be determined by tree automata. We describe an algorithm that, given two tree-automatic ordinals with the ordinal addition operation, decides if the ordinals are isomorphic.

cs.FL↗

arXiv2019

Random Subgroups of Rationals

This paper introduces and studies a notion of \emph{algorithmic randomness} for subgroups of rationals. Given a randomly generated additive subgroup $(G,+)$ of rationals, two main questions are addressed: first, what are the model-theoretic and recursion-theoretic properties of $(G,+)$; second, what learnability properties can one extract from $G$ and its subclass of finitely generated subgroups? For the first question, it is shown that the theory of $(G,+)$ coincides with that of the additive group of integers and is therefore decidable; furthermore, while the word problem for $G$ with respect to any generating sequence for $G$ is not even semi-decidable, one can build a generating sequence $β$ such that the word problem for $G$ with respect to $β$ is co-recursively enumerable (assuming that the set of generators of $G$ is limit-recursive). In regard to the second question, it is proven that there is a generating sequence $β$ for $G$ such that every non-trivial finitely generated subgroup of $G$ is recursively enumerable and the class of all such subgroups of $G$ is behaviourally correctly learnable, that is, every non-trivial finitely generated subgroup can be semantically identified in the limit (again assuming that the set of generators of $G$ is limit-recursive). On the other hand, the class of non-trivial finitely generated subgroups of $G$ cannot be syntactically identified in the limit with respect to any generating sequence for $G$. The present work thus contributes to a recent line of research studying algorithmically random infinite structures and uncovers an interesting connection between the arithmetical complexity of the set of generators of a randomly generated subgroup of rationals and the learnability of its finitely generated subgroups.

cs.LO↗

arXiv2016

Cayley automatic representations of wreath products

We construct the representations of Cayley graphs of wreath products using finite automata, pushdown automata and nested stack automata. These representations are in accordance with the notion of Cayley automatic groups introduced by Kharlampovich, Khoussainov and Miasnikov and its extensions introduced by Elder and Taback. We obtain the upper and lower bounds for a length of an element of a wreath product in terms of the representations constructed.

math.GR↗

arXiv2014

Dynamic Algorithms for Interval Scheduling on a Single Machine

We investigate dynamic algorithms for the interval scheduling problem. Our algorithm runs in amortised time $O(\log n)$ for query operation and $O(d\log^2 n)$ for insertion and removal operations, where $n$ and $d$ are the maximal numbers of intervals and pairwise overlapping intervals respectively. We also show that for a monotonic set, that is when no interval properly contains another interval, the amortised complexity is $O(\log n)$ for both query and update operations. We compare the two algorithms for the monotonic interval sets using experiments.

cs.DS↗

arXiv2011

From automatic structures to automatic groups

In this paper we introduce the concept of a Cayley graph automatic group (CGA group or graph automatic group, for short) which generalizes the standard notion of an automatic group. Like the usual automatic groups graph automatic ones enjoy many nice properties: these group are invariant under the change of generators, they are closed under direct and free products, certain types of amalgamated products, and finite extensions. Furthermore, the Word Problem in graph automatic groups is decidable in quadratic time. However, the class of graph automatic groups is much wider then the class of automatic groups. For example, we prove that all finitely generated 2-nilpotent groups and Baumslag-Solitar groups B(1,n) are graph automatic, as well as many other metabelian groups.

math.GR↗

arXiv2008

Unary Automatic Graphs: An Algorithmic Perspective

This paper studies infinite graphs produced from a natural unfolding operation applied to finite graphs. Graphs produced via such operations are of finite degree and automatic over the unary alphabet (that is, they can be described by finite automata over unary alphabet). We investigate algorithmic properties of such unfolded graphs given their finite presentations. In particular, we ask whether a given node belongs to an infinite component, whether two given nodes in the graph are reachable from one another, and whether the graph is connected. We give polynomial-time algorithms for each of these questions. For a fixed input graph, the algorithm for the first question is in constant time and the second question is decided using an automaton that recognizes the reachability relation in a uniform way. Hence, we improve on previous work, in which non-elementary or non-uniform algorithms were found.

math.LO↗

arXiv2008

Model Theoretic Complexity of Automatic Structures

We study the complexity of automatic structures via well-established concepts from both logic and model theory, including ordinal heights (of well-founded relations), Scott ranks of structures, and Cantor-Bendixson ranks (of trees). We prove the following results: 1) The ordinal height of any automatic well- founded partial order is bounded by ω^ω; 2) The ordinal heights of automatic well-founded relations are unbounded below the first non-computable ordinal; 3) For any computable ordinal there is an automatic structure of Scott rank at least that ordinal. Moreover, there are automatic structures of Scott rank the first non-computable ordinal and its successor; 4) For any computable ordinal, there is an automatic successor tree of Cantor-Bendixson rank that ordinal.

math.LO↗

arXiv2008

Three Lectures on Automatic Structures

This paper grew out of three tutorial lectures on automatic structures given by the first author at the Logic Colloquium 2007. We discuss variants of automatic structures related to several models of computation: word automata, tree automata, Buchi automata, and Rabin automata. Word automata process finite strings, tree automata process finite labeled trees, Buchi automata process infinite strings, and Rabin automata process infinite binary labeled trees. Automatic structures are mathematical objects which can be represented by (word, tree, Buchi, or Rabin) automata. The study of properties of automatic structures is a relatively new and very active area of research.

math.LO↗

arXiv2007

Automatic Structures: Richness and Limitations

We study the existence of automatic presentations for various algebraic structures. An automatic presentation of a structure is a description of the universe of the structure by a regular set of words, and the interpretation of the relations by synchronised automata. Our first topic concerns characterising classes of automatic structures. We supply a characterisation of the automatic Boolean algebras, and it is proven that the free Abelian group of infinite rank, as well as certain Fraisse limits, do not have automatic presentations. In particular, the countably infinite random graph and the random partial order do not have automatic presentations. Furthermore, no infinite integral domain is automatic. Our second topic is the isomorphism problem. We prove that the complexity of the isomorphism problem for the class of all automatic structures is Σ_1^1-complete.

cs.DM↗